From b1000e016028f90d433f33e378f0de918e137a8f Mon Sep 17 00:00:00 2001 From: DylanLawless Date: Tue, 31 Dec 2024 18:41:58 +0100 Subject: [PATCH] causal inferance page --- assets/images/file_count_plot-1.png | Bin 49266 -> 47324 bytes assets/images/word-count-plots-1.png | Bin 31067 -> 31152 bytes pages/altman_bland_analysis_of_methods.md | 13 +- pages/altman_bland_ci_from_p.md | 88 +- pages/altman_bland_correlation.md | 11 +- pages/altman_bland_odds_ratios.md | 12 +- pages/altman_bland_roc_curve.md | 12 +- pages/altman_bland_sensitivity_specificity.md | 5 +- pages/causal_inference_whole_game.md | 893 +++++++++++++++ .../figure-gfm/fig-ate-density-net-data-1.png | Bin 0 -> 22275 bytes .../fig-bootstrap-estimates-net-data-1.png | Bin 0 -> 22105 bytes .../figure-gfm/fig-love-plot-net-data-1.png | Bin 0 -> 34720 bytes .../figure-gfm/fig-malaria-risk-density-1.png | Bin 0 -> 36618 bytes ...mirror-histogram-net-data-unweighted-1.png | Bin 0 -> 22828 bytes 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The paper is a pivotal guide discussing the analysis of method comparison studies, particularly in the field of medicine. diff --git a/pages/altman_bland_ci_from_p.md b/pages/altman_bland_ci_from_p.md index b538f24..459f417 100644 --- a/pages/altman_bland_ci_from_p.md +++ b/pages/altman_bland_ci_from_p.md @@ -1,39 +1,49 @@ - -nav_order: 5 +--- +layout: default math: mathjax - - - - - - +title: Stats CI from P +nav_order: 5 +--- + +# Stats CI from P + +* TOC +{:toc} - - +--- +{: .warning } +This text is largely copied directly from the following source while we build an example closer to our needs. Please see the original source as ollows. + +This topic is from a series of BMJ statistical notes by Altman & Bland. +BMJ 2011; 343 doi: (Published 08 August 2011) +Cite this as: BMJ 2011;343:d2090. Douglas G Altman, professor of statistics in medicine, J Martin Bland, professor of health statistics. + +## How to obtain the confidence interval from a P value Confidence intervals (CIs) are widely used in reporting statistical analyses of research data, and are usually considered to be more informative than P values from significance tests.1 2 Some published articles, however, report estimated effects and P values, but do not give CIs (a practice BMJ now strongly discourages). Here we show how to obtain the confidence interval when only the observed effect and the P value were reported. The method is outlined in the box below in which we have distinguished two cases. -Steps to obtain the confidence interval (CI) for an estimate of effect from the P value and the estimate (Est) -(a) CI for a difference - 1 calculate the test statistic for a normal distribution test, z, from P3: z = −0.862 + √[0.743 − 2.404×log(P)] - 2 calculate the standard error: SE = Est/z (ignoring minus signs) - 3 calculate the 95% CI: Est –1.96×SE to Est + 1.96×SE. -(b) CI for a ratio - For a ratio measure, such as a risk ratio, the above formulas should be used with the estimate Est on the log scale (eg, the log risk ratio). Step 3 gives a CI on the log scale; to derive the CI on the natural scale we need to exponentiate (antilog) Est and its CI.4 -Notes +**Steps to obtain the confidence interval (CI) for an estimate of effect from the P value and the estimate (Est)** -All P values are two sided. +* (a) CI for a difference + 1. calculate the test statistic for a normal distribution test, z, from P3: z = −0.862 + √[0.743 − 2.404×log(P)] + 2. calculate the standard error: SE = Est/z (ignoring minus signs) + 3. calculate the 95% CI: Est –1.96×SE to Est + 1.96×SE. +* (b) CI for a ratio + - For a ratio measure, such as a risk ratio, the above formulas should be used with the estimate Est on the log scale (eg, the log risk ratio). Step 3 gives a CI on the log scale; to derive the CI on the natural scale we need to exponentiate (antilog) Est and its CI.4 -All logarithms are natural (ie, to base e). 4 +**Notes** -For a 90% CI, we replace 1.96 by 1.65; for a 99% CI we use 2.57. +* All P values are two sided. +* All logarithms are natural (ie, to base e). 4 +* For a 90% CI, we replace 1.96 by 1.65; for a 99% CI we use 2.57. + +## (a) Calculating the confidence interval for a difference -(a) Calculating the confidence interval for a difference We consider first the analysis comparing two proportions or two means, such as in a randomised trial with a binary outcome or a measurement such as blood pressure. For example, the abstract of a report of a randomised trial included the statement that “more patients in the zinc group than in the control group recovered by two days (49% v 32%, P=0.032).”5 The difference in proportions was Est = 17 percentage points, but what is the 95% confidence interval (CI)? @@ -43,23 +53,39 @@ Following the steps in the box we calculate the CI as follows: z = –0.862+ √[0.743 – 2.404×log(0.032)] = 2.141; SE = 17/2.141 = 7.940, so that 1.96×SE = 15.56 percentage points; 95% CI is 17.0 – 15.56 to 17.0 + 15.56, or 1.4 to 32.6 percentage points. -(b) Calculating the confidence interval for a ratio (log transformation needed) + +## (b) Calculating the confidence interval for a ratio (log transformation needed) + The calculation is trickier for ratio measures, such as risk ratio, odds ratio, and hazard ratio. We need to log transform the estimate and then reverse the procedure, as described in a previous Statistics Note.6 For example, the abstract of a report of a cohort study includes the statement that “In those with a [diastolic blood pressure] reading of 95-99 mm Hg the relative risk was 0.30 (P=0.034).”7 What is the confidence interval around 0.30? Following the steps in the box we calculate the CI as follows: - z = –0.862+ √[0.743 – 2.404×log(0.034)] = 2.117; - Est = log (0.30) = −1.204; - SE = −1.204/2.117 = −0.569 but we ignore the minus sign, so SE = 0.569, and 1.96×SE = 1.115; - 95% CI on log scale = −1.204 − 1.115 to −1.204 + 1.115 = −2.319 to −0.089; - 95% CI on natural scale = exp (−2.319) = 0.10 to exp (−0.089) = 0.91. - Hence the relative risk is estimated to be 0.30 with 95% CI 0.10 to 0.91. -Limitations of the method +* z = $$–0.862+ √[0.743 – 2.404×log(0.034)] = 2.117$$; +* Est = $$log (0.30) = −1.204$$; +* SE = −1.204/2.117 = −0.569 but we ignore the minus sign, so SE = 0.569, and 1.96×SE = 1.115; +* 95% CI on log scale = −1.204 − 1.115 to −1.204 + 1.115 = −2.319 to −0.089; +* 95% CI on natural scale = exp (−2.319) = 0.10 to exp (−0.089) = 0.91. +* Hence the relative risk is estimated to be 0.30 with 95% CI 0.10 to 0.91. + +## Limitations of the method + The methods described can be applied in a wide range of settings, including the results from meta-analysis and regression analyses. The main context where they are not correct is in small samples where the outcome is continuous and the analysis has been done by a t test or analysis of variance, or the outcome is dichotomous and an exact method has been used for the confidence interval. However, even here the methods will be approximately correct in larger studies with, say, 60 patients or more. -P values presented as inequalities +## P values presented as inequalities + Sometimes P values are very small and so are presented as P<0.0001 or something similar. The above method can be applied for small P values, setting P equal to the value it is less than, but the z statistic will be too small, hence the standard error will be too large and the resulting CI will be too wide. This is not a problem so long as we remember that the estimate is better than the interval suggests. When we are told that P>0.05 or the difference is not significant, things are more difficult. If we apply the method described here, using P=0.05, the confidence interval will be too narrow. We must remember that the estimate is even poorer than the confidence interval calculated would suggest. + + +## References + +1. Gardner MJ, Altman DG. Confidence intervals rather than P values: estimation rather than hypothesis testing. BMJ1986;292:746-50.Abstract/FREE Full TextGoogle Scholar +1. Moher D, Hopewell S, Schulz KF, Montori V, Gøtzsche PC, Devereaux PJ, et al. CONSORT 2010. Explanation and Elaboration: updated guidelines for reporting parallel group randomised trials. BMJ2010;340:c869.FREE Full TextGoogle Scholar +1. Lin J-T. Approximating the normal tail probability and its inverse for use on a pocket calculator. Appl Stat1989;38:69-70.CrossRefGoogle Scholar +1. Bland JM, Altman DG. Statistics Notes. Logarithms. BMJ1996;312:700.FREE Full TextGoogle Scholar +1. Roy SK, Hossain MJ, Khatun W, Chakraborty B, Chowdhury S, Begum A, et al. Zinc supplementation in children with cholera in Bangladesh: randomised controlled trial. BMJ2008;336:266-8.Abstract/FREE Full TextGoogle Scholar +1. Altman DG, Bland JM. Interaction revisited: the difference between two estimates. BMJ2003;326:219.FREE Full TextGoogle Scholar +1. Lindblad U, Råstam L, Rydén L, Ranstam J, Isacsson S-O, Berglund G. Control of blood pressure and risk of first acute myocardial infarction: Skaraborg hypertension project. BMJ1994;308:681. Abstract/FREE Full TextGoogle Scholar diff --git a/pages/altman_bland_correlation.md b/pages/altman_bland_correlation.md index 505ae5b..8c08ef8 100644 --- a/pages/altman_bland_correlation.md +++ b/pages/altman_bland_correlation.md @@ -1,16 +1,19 @@ --- +title: Stats Correlation, regression and repeated data nav_order: 5 math: mathjax layout: default -title: Stats Correlation, regression and repeated data -created: 29 June 2021 --- -{{ page.title }} -================ + +# Stats Correlation, regression and repeated data + +created: 29 June 2021 * TOC {:toc} +--- + ## Introduction This topic is introduced as the first paper diff --git a/pages/altman_bland_odds_ratios.md b/pages/altman_bland_odds_ratios.md index 2bba7de..6ceae04 100644 --- a/pages/altman_bland_odds_ratios.md +++ b/pages/altman_bland_odds_ratios.md @@ -1,16 +1,18 @@ --- -nav_order: 5 -math: mathjax layout: default +math: mathjax title: Stats Odds ratios, SE & CI -created: 4 July 2021 +nav_order: 5 --- -{{ page.title }} -================ + +created: 4 July 2021 +# Stats Odds ratios, SE & CI * TOC {:toc} +--- + ## Introduction Altman & Bland review the use of odds ratio (OR), standard error (SE), and confidence interval (CI) with some examples in diff --git a/pages/altman_bland_roc_curve.md b/pages/altman_bland_roc_curve.md index 3d52977..68959fa 100644 --- a/pages/altman_bland_roc_curve.md +++ b/pages/altman_bland_roc_curve.md @@ -1,16 +1,18 @@ --- -nav_order: 5 -math: mathjax layout: default +math: mathjax title: Stats Receiver operating characteristic plots -created: 2021-07-16 +nav_order: 5 --- -{{ page.title }} -================ + +created: 2021-07-16 +# Stats Receiver operating characteristic plots * TOC {:toc} +--- + ## Receiver operating characteristic plots This article covers the fifth paper in the series of statistics notes altman1994diagnostic ([ lit-altman_bland.md ]( https://github.com/DylanLawless/notes/blob/main/202106291417-lit-altman_bland.md )): 5. Altman DG, Bland JM. (1994) Diagnostic tests 3: receiver operating characteristic plots. 309, 188, diff --git a/pages/altman_bland_sensitivity_specificity.md b/pages/altman_bland_sensitivity_specificity.md index 148f009..c388435 100644 --- a/pages/altman_bland_sensitivity_specificity.md +++ b/pages/altman_bland_sensitivity_specificity.md @@ -1,13 +1,14 @@ --- +title: Stats Sensitivity and specificity nav_order: 5 math: mathjax layout: default -title: Stats Sensitivity and specificity -created: 19 July 2021 --- {{ page.title }} ================ +created: 19 July 2021 + * TOC {:toc} diff --git a/pages/causal_inference_whole_game.md b/pages/causal_inference_whole_game.md new file mode 100644 index 0000000..7567db2 --- /dev/null +++ b/pages/causal_inference_whole_game.md @@ -0,0 +1,893 @@ +--- +title: "Causal inference mosquito nets" +author: "Ref, Dylan Lawless" +output: + # md_document: # faster but missing features + bookdown::markdown_document2: # slower but supports features like cross-referencing + variant: gfm + preserve_yaml: true +layout: default +nav_order: 5 +math: mathjax +--- + +# 1 Causal inference - mosquito nets and malaria + +Last update: + + ## [1] "2024-12-31" + +This doc was built with: +`rmarkdown::render("causal_inference_whole_game.Rmd", output_file = "../pages/causal_inference_whole_game.md")` + +{: .warning } + +This text is largely copied directly from the following source while we +build an example closer to our needs. Please see the original source as +follows. + +This example is described in the textbook: Causal Inference in R, by +Malcolm Barrett, Lucy D’Agostino McGowan, and Travis Gerke. +, chapter 2. + + + +We will use simulated data to answer a more specific question: Does +using insecticide-treated bed nets compared to no nets decrease the risk +of contracting malaria after 1 year? Data [simulated by Dr. Andrew +Heiss](https://evalsp21.classes.andrewheiss.com/example/matching-ipw/#program-background): + +> …researchers are interested in whether using mosquito nets decreases +> an individual’s risk of contracting malaria. They have collected data +> from 1,752 households in an unnamed country and have variables related +> to environmental factors, individual health, and household +> characteristics. The data is **not experimental**—researchers have no +> control over who uses mosquito nets, and individual households make +> their own choices over whether to apply for free nets or buy their own +> nets, as well as whether they use the nets if they have them. + +This data includes a variable that measures the likelihood of +contracting malaria, something we wouldn’t likely have in real life. +This let’s us know the actual effect size to understand the methods +better. The simulated data is in `net_data` from the +{[causalworkshop](https://github.com/r-causal/causalworkshop)} package, +which includes ten variables: + + + +- `id` : an ID variable +- `net` and `net_num` : a binary variable indicating if the participant + used a net (1) or didn’t use a net (0) +- `malaria_risk` : risk of malaria scale ranging from 0-100 +- `income` : weekly income, measured in dollars +- `health` : a health score scale ranging from 0–100 +- `household` : number of people living in the household +- `eligible` : a binary variable indicating if the household is eligible + for the free net program. +- `temperature` : the average temperature at night, in Celsius +- `resistance` : Insecticide resistance of local mosquitoes. A scale of + 0–100, with higher values indicating higher resistance. + +The distribution of malaria risk appears to be quite different by net +usage. + +``` r +# devtools::install_github("r-causal/causalworkshop") +``` + +``` r +library(tidyverse) +library(causalworkshop) +net_data |> + ggplot(aes(malaria_risk, fill = net)) + + geom_density(color = NA, alpha = .8) +``` + +

+ +A density plot of malaria risk for those who did and did not use nets. The risk of malaria is lower for those who use nets. +

+ +Figure 1.1: A density +plot of malaria risk for those who did and did not use nets. The risk of +malaria is lower for those who use nets. +

+ +
+ +In figure 1.1, the density +of those who used nets is to the left of those who did not use nets. The +mean difference in malaria risk is about 16.4, suggesting net use might +be protective against malaria. + +``` r +net_data |> + group_by(net) |> + summarize(malaria_risk = mean(malaria_risk)) +``` + + ## # A tibble: 2 × 2 + ## net malaria_risk + ## + ## 1 FALSE 43.9 + ## 2 TRUE 27.5 + +And that’s what we see with simple linear regression, as well, as we +would expect. + +``` r +library(broom) +net_data |> + lm(malaria_risk ~ net, data = _) |> + tidy() +``` + + ## # A tibble: 2 × 5 + ## term estimate std.error statistic p.value + ## + ## 1 (Intercept) 43.9 0.377 116. 0 + ## 2 netTRUE -16.4 0.741 -22.1 1.10e-95 + +## 1.1 Draw our assumptions using a causal diagram + +The problem that we face is that other factors may be responsible for +the effect we’re seeing. In this example, we’ll focus on confounding: a +common cause of net usage and malaria will bias the effect we see unless +we account for it somehow. One of the best ways to determine which +variables we need to account for is to use a causal diagram. These +diagrams, also called **causal directed acyclic graphs (DAGs)**, +visualize the assumptions that we’re making about the causal +relationships between the exposure, outcome, and other variables we +think might be related. + +Here’s the DAG that we’re proposing for this question. + +
+ +A proposed causal diagram of the effect of bed net use on malaria. This directed acyclic graph (DAG) states our assumption that bed net use causes a reduction in malaria risk. It also says that we assume: malaria risk is impacted by net usage, income, health, temperature, and insecticide resistance; net usage is impacted by income, health, temperature, eligibility for the free net program, and the number of people in a household; eligibility for the free net programs is impacted by income and the number of people in a household; and health is impacted by income. +

+ +Figure 1.2: A proposed causal +diagram of the effect of bed net use on malaria. This directed acyclic +graph (DAG) states our assumption that bed net use causes a reduction in +malaria risk. It also says that we assume: malaria risk is impacted by +net usage, income, health, temperature, and insecticide resistance; net +usage is impacted by income, health, temperature, eligibility for the +free net program, and the number of people in a household; eligibility +for the free net programs is impacted by income and the number of people +in a household; and health is impacted by income. +

+ +
+ +We’ll explore how to create and analyze DAGs in @sec-dags. + +In DAGs, each point represents a variable, and each arrow represents a +cause. In other words, this diagram declares what we think the causal +relationships are between these variables. In figure +1.2, we’re saying that we believe: + +- Malaria risk is causally impacted by net usage, income, health, + temperature, and insecticide resistance. +- Net usage is causally impacted by income, health, temperature, + eligibility for the free net program, and the number of people in a + household. +- Eligibility for the free net programs is determined by income and the + number of people in a household. +- Health is causally impacted by income. + +You may agree or disagree with some of these assertions. That’s a good +thing! Laying bare our assumptions allows us to consider the scientific +credibility of our analysis. Another benefit of using DAGs is that, +thanks to their mathematics, we can determine precisely the subset of +variables we need to account for if we assume this DAG is correct. + +{: .note } + +**Assembling DAGs** In this exercise, we’re providing you with a +reasonable DAG based on knowledge of how the data were generated. In +real life, setting up a DAG is a challenge requiring deep thought, +domain expertise, and (often) collaboration between several experts. + +The chief problem we’re dealing with is that, when we analyze the data +we’re working with, we see the impact of net usage on malaria risk *and +of all these other relationships*. In DAG terminology, we have more than +one open causal pathway. If this DAG is correct, we have *eight* causal +pathways: the path between net usage and malaria risk and seven other +*confounding* pathways. + +
+ +In the proposed DAG, there are eight open pathways that contribute to the causal effect seen in the naive regression: the true effect (in green) of net usage on malaria risk and seven other confounding pathways (in orange). The naive estimate is wrong because it is a composite of all these effects. +

+ +Figure 1.3: In the +proposed DAG, there are eight open pathways that contribute to the +causal effect seen in the naive regression: the true effect (in green) +of net usage on malaria risk and seven other confounding pathways (in +orange). The naive estimate is wrong because it is a composite of all +these effects. +

+ +
+ +When we calculate a naive linear regression that only includes net usage +and malaria risk, the effect we see is incorrect because the seven other +confounding pathways in figure +1.3 distort it. In DAG +terminology, we need to *block* these open pathways that distort the +causal estimate we’re after. (We can block paths through several +techniques, including stratification, matching, weighting, and more. +We’ll see several methods throughout the book.) Luckily, by specifying a +DAG, we can precisely determine the variables we need to control for. +For this DAG, we need to control for three variables: health, income, +and temperature. These three variables are a *minimal adjustment set*, +the minimum set (or sets) of variables you need to block all confounding +pathways. We’ll discuss adjustment sets further in @sec-dags. + +## 1.2 Model our assumptions + +We’ll use a technique called **inverse probability weighting (IPW)** to +control for these variables, which we’ll discuss in detail in +@sec-using-ps. We’ll use logistic regression to predict the probability +of treatment—the propensity score. Then, we’ll calculate inverse +probability weights to apply to the linear regression model we fit +above. The propensity score model includes the exposure—net use—as the +dependent variable and the minimal adjustment set as the independent +variables. + +{: .note } + +**Modeling the functional form** Generally speaking, we want to lean on +domain expertise and good modeling practices to fit the propensity score +model. For instance, we may want to allow continuous confounders to be +non-linear using splines, or we may want to add essential interactions +between confounders. Because these are simulated data, we know we don’t +need these extra parameters (so we’ll skip them), but in practice, you +often do. We’ll discuss this more in @sec-using-ps. + +The propensity score model is a logistic regression model with the +formula `net ~ income + health + temperature`, which predicts the +probability of bed net usage based on the confounders income, health, +and temperature. + +``` r +propensity_model <- glm( + net ~ income + health + temperature, + data = net_data, + family = binomial() +) + +# the first six propensity scores +head(predict(propensity_model, type = "response")) +``` + + ## 1 2 3 4 5 6 + ## 0.2464 0.2178 0.3230 0.2307 0.2789 0.3060 + +We can use propensity scores to control for confounding in various ways. +In this example, we’ll focus on weighting. In particular, we’ll compute +the inverse probability weight for the **average treatment effect +(ATE)**. The ATE represents a particular causal question: what if +*everyone* in the study used bed nets vs. what if *no one* in the study +used bed nets? + +To calculate the ATE, we’ll use the broom and propensity packages. +broom’s `augment()` function extracts prediction-related information +from the model and joins it to the data. propensity’s `wt_ate()` +function calculates the inverse probability weight given the propensity +score and exposure. + +For inverse probability weighting, the ATE weight is the inverse of +probability of receiving the treatment you actually received. In other +words, if you used a bed net, the ATE weight is the inverse of the +probability that you used a net, and if you did *not* use a net, it is +the the inverse of the probability that you did *not* use a net. + +``` r +library(broom) +library(propensity) +net_data_wts <- propensity_model |> + augment(data = net_data, type.predict = "response") |> + # .fitted is the value predicted by the model + # for a given observation + mutate(wts = wt_ate(.fitted, net)) + +net_data_wts |> + select(net, .fitted, wts) |> + head() +``` + + ## # A tibble: 6 × 3 + ## net .fitted wts + ## + ## 1 FALSE 0.246 1.33 + ## 2 FALSE 0.218 1.28 + ## 3 FALSE 0.323 1.48 + ## 4 FALSE 0.231 1.30 + ## 5 FALSE 0.279 1.39 + ## 6 FALSE 0.306 1.44 + +`wts` represents the amount each observation will be up-weighted or +down-weighted in the outcome model we will soon fit. For instance, the +16th household used a bed net and had a predicted probability of 0.41. +That’s a pretty low probability considering they did, in fact, use a +net, so their weight is higher at 2.42. In other words, this household +will be up-weighted compared to the naive linear model we fit above. The +first household did *not* use a bed net; they’re predicted probability +of net use was 0.25 (or put differently, a predicted probability of +*not* using a net of 0.75). That’s more in line with their observed +value of `net`, but there’s still some predicted probability of using a +net, so their weight is 1.28. + +## 1.3 Diagnose our models + +The goal of propensity score weighting is to weight the population of +observations such that the distribution of confounders is balanced +between the exposure groups. Put another way, we are, in principle, +removing the arrows between the confounders and exposure in the DAG, so +that the confounding paths no longer distort our estimates. Here’s the +distribution of the propensity score by group, created by +`geom_mirror_histogram()` from the halfmoon package for assessing +balance in propensity score models: + +``` r +library(halfmoon) +ggplot(net_data_wts, aes(.fitted)) + + geom_mirror_histogram( + aes(fill = net), + bins = 50 + ) + + scale_y_continuous(labels = abs) + + labs(x = "propensity score") +``` + +
+ +A mirrored histogram of the propensity scores of those who used nets (top, blue) versus those who did not use nets (bottom, orange). The range of propensity scores is similar between groups, with those who used nets slightly to the left of those who didn't, but the shapes of the distribution are different. +

+ +Figure +1.4: A mirrored histogram of the propensity scores of those who used +nets (top, blue) versus those who did not use nets (bottom, orange). The +range of propensity scores is similar between groups, with those who +used nets slightly to the left of those who didn’t, but the shapes of +the distribution are different. +

+ +
+ +The weighted propensity score creates a pseudo-population where the +distributions are much more similar: + +``` r +ggplot(net_data_wts, aes(.fitted)) + + geom_mirror_histogram( + aes(group = net), + bins = 50 + ) + + geom_mirror_histogram( + aes(fill = net, weight = wts), + bins = 50, + alpha = .5 + ) + + scale_y_continuous(labels = abs) + + labs(x = "propensity score") +``` + +
+ +A mirrored histogram of the propensity scores of those who used nets (top, blue) versus those who did not use nets (bottom, orange). The shaded region represents the unweighted distribution, and the colored region represents the weighted distributions. The ATE weights up-weight the groups to be similar in range and shape of the distribution of propensity scores. +

+ +Figure 1.5: +A mirrored histogram of the propensity scores of those who used nets +(top, blue) versus those who did not use nets (bottom, orange). The +shaded region represents the unweighted distribution, and the colored +region represents the weighted distributions. The ATE weights up-weight +the groups to be similar in range and shape of the distribution of +propensity scores. +

+ +
+ +In this example, the unweighted distributions are not awful—the shapes +are somewhat similar here, and they overlap quite a bit—but the weighted +distributions in figure +1.4 are much +more similar. + +{: .note } + +**Unmeasured confounding** Propensity score weighting and most other +causal inference techniques only help with *observed* confounders—ones +that we model correctly, at that. Unfortunately, we still may have +unmeasured confounding, which we’ll discuss below. Randomization is one +causal inference technique that *does* deal with unmeasured confounding, +one of the reasons it is so powerful. + +We might also want to know how well-balanced the groups are by each +confounder. One way to do this is to calculate the **standardized mean +differences (SMDs)** for each confounder with and without weights. We’ll +calculate the SMDs with `tidy_smd()` then plot them with `geom_love()`. + +``` r +plot_df <- tidy_smd( + net_data_wts, + c(income, health, temperature), + .group = net, + .wts = wts +) + +ggplot( + plot_df, + aes( + x = abs(smd), + y = variable, + group = method, + color = method + ) +) + + geom_love() +``` + +
+ +A love plot representing the standardized mean differences (SMD) between exposure groups of three confounders: temperature, income, and health. Before weighting, there are considerable differences in the groups. After weighting, the confounders are much more balanced between groups. +

+ +Figure 1.6: A love plot +representing the standardized mean differences (SMD) between exposure +groups of three confounders: temperature, income, and health. Before +weighting, there are considerable differences in the groups. After +weighting, the confounders are much more balanced between groups. +

+ +
+ +A standard guideline is that balanced confounders should have an SMD of +less than 0.1 on the absolute scale. 0.1 is just a rule of thumb, but if +we follow it, the variables in figure +1.6 are well-balanced after +weighting (and unbalanced before weighting). + +Before we apply the weights to the outcome model, let’s check their +overall distribution for extreme weights. Extreme weights can +destabilize the estimate and variance in the outcome model, so we want +to be aware of it. We’ll also discuss several other types of weights +that are less prone to this issue in @sec-estimands. + +``` r +net_data_wts |> + ggplot(aes(wts)) + + geom_density(fill = "#CC79A7", color = NA, alpha = 0.8) +``` + +
+ +A density plot of the average treatment effect (ATE) weights. The plot is skewed, with higher values towards 8. This may indicate a problem with the model, but the weights aren't so extreme to destabilize the variance of the estimate. +

+ +Figure 1.7: A density +plot of the average treatment effect (ATE) weights. The plot is skewed, +with higher values towards 8. This may indicate a problem with the +model, but the weights aren’t so extreme to destabilize the variance of +the estimate. +

+ +
+ +The weights in figure 1.7 +are skewed, but there are no outrageous values. If we saw extreme +weights, we might try trimming or stabilizing them, or consider +calculating an effect for a different estimand, which we’ll discuss in +@sec-estimands. It doesn’t look like we need to do that here, however. + +## 1.4 Estimate the causal effect + +We’re now ready to use the ATE weights to (attempt to) account for +confounding in the naive linear regression model. Fitting such a model +is pleasantly simple in this case: we fit the same model as before but +with `weights = wts`, which will incorporate the inverse probability +weights. + +``` r +net_data_wts |> + lm(malaria_risk ~ net, data = _, weights = wts) |> + tidy(conf.int = TRUE) +``` + + ## # A tibble: 2 × 7 + ## term estimate std.error statistic p.value conf.low + ## + ## 1 (Inte… 42.7 0.442 96.7 0 41.9 + ## 2 netTR… -12.5 0.624 -20.1 5.50e-81 -13.8 + ## # ℹ 1 more variable: conf.high + +The estimate for the average treatment effect is -12.5 (95% CI -13.8, +-11.3). Unfortunately, the confidence intervals we’re using are wrong +because they don’t account for the uncertainty in estimating the +weights. Generally, confidence intervals for propensity score weighted +models will be too narrow unless we account for this uncertainty. The +nominal coverage of the confidence intervals will thus be wrong (they +aren’t 95% CIs because their coverage is much lower than 95%) and may +lead to misinterpretation. + +We’ve got several ways to address this problem, which we’ll discuss in +detail in @sec-outcome-model, including the bootstrap, robust standard +errors, and manually accounting for the estimation procedure with +empirical sandwich estimators. For this example, we’ll use the +bootstrap, a flexible tool that calculates distributions of parameters +using re-sampling. The bootstrap is a useful tool for many causal models +where closed-form solutions to problems (particularly standard errors) +don’t exist or when we want to avoid parametric assumptions inherent to +many such solutions; see @sec-appendix-bootstrap for a description of +what the bootstrap is and how it works. We’ll use the rsample package +from the tidymodels ecosystem to work with bootstrap samples. + +Because the bootstrap is so flexible, we need to think carefully about +the sources of uncertainty in the statistic we’re calculating. It might +be tempting to write a function like this to fit the statistic we’re +interested in (the point estimate for `netTRUE`): + +``` r +library(rsample) + +fit_ipw_not_quite_rightly <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} +``` + +However, this function won’t give us the correct confidence intervals +because it treats the inverse probability weights as fixed values. +They’re not, of course; we just estimated them using logistic +regression! We need to account for this uncertainty by bootstrapping the +*entire modeling process*. For every bootstrap sample, we need to fit +the propensity score model, calculate the inverse probability weights, +then fit the weighted outcome model. + +``` r +library(rsample) + +fit_ipw <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit propensity score model + propensity_model <- glm( + net ~ income + health + temperature, + data = .df, + family = binomial() + ) + + # calculate inverse probability weights + .df <- propensity_model |> + augment(type.predict = "response", data = .df) |> + mutate(wts = wt_ate(.fitted, net)) + + # fit correctly bootstrapped ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} +``` + +Now that we know precisely how to calculate the estimate for each +iteration let’s create the bootstrapped dataset with rsample’s +`bootstraps()` function. The `times` argument determines how many +bootstrapped datasets to create; we’ll do 1,000. + +``` r +bootstrapped_net_data <- bootstraps( + net_data, + times = 1000, + # required to calculate CIs later + apparent = TRUE +) + +bootstrapped_net_data +``` + + ## # Bootstrap sampling with apparent sample + ## # A tibble: 1,001 × 2 + ## splits id + ## + ## 1 Bootstrap0001 + ## 2 Bootstrap0002 + ## 3 Bootstrap0003 + ## 4 Bootstrap0004 + ## 5 Bootstrap0005 + ## 6 Bootstrap0006 + ## 7 Bootstrap0007 + ## 8 Bootstrap0008 + ## 9 Bootstrap0009 + ## 10 Bootstrap0010 + ## # ℹ 991 more rows + +The result is a nested data frame: each `splits` object contains +metadata that rsample uses to subset the bootstrap samples for each of +the 1,000 samples. We actually have 1,001 rows because `apparent = TRUE` +keeps a copy of the original data frame, as well, which is needed for +some times of confidence interval calculations. Next, we’ll run +`fit_ipw()` 1,001 times to create a distribution for `estimate`. At its +heart, the calculation we’re doing is + +``` r +fit_ipw(bootstrapped_net_data$splits[[n]]) +``` + +Where *n* is one of 1,001 indices. We’ll use purrr’s `map()` function to +iterate across each `split` object. + +``` r +ipw_results <- bootstrapped_net_data |> + mutate(boot_fits = map(splits, fit_ipw)) + +ipw_results +``` + + ## # Bootstrap sampling with apparent sample + ## # A tibble: 1,001 × 3 + ## splits id boot_fits + ## + ## 1 Bootstrap0001 + ## 2 Bootstrap0002 + ## 3 Bootstrap0003 + ## 4 Bootstrap0004 + ## 5 Bootstrap0005 + ## 6 Bootstrap0006 + ## 7 Bootstrap0007 + ## 8 Bootstrap0008 + ## 9 Bootstrap0009 + ## 10 Bootstrap0010 + ## # ℹ 991 more rows + +The result is another nested data frame with a new column, `boot_fits`. +Each element of `boot_fits` is the result of the IPW for the +bootstrapped dataset. For example, in the first bootstrapped data set, +the IPW results were: + +``` r +ipw_results$boot_fits[[1]] +``` + + ## # A tibble: 2 × 5 + ## term estimate std.error statistic p.value + ## + ## 1 (Intercept) 42.7 0.465 91.9 0 + ## 2 netTRUE -11.8 0.657 -18.0 1.04e-66 + +Now we have a distribution of estimates: + +``` r +ipw_results |> + # remove original data set results + filter(id != "Apparent") |> + mutate( + estimate = map_dbl( + boot_fits, + # pull the `estimate` for `netTRUE` for each fit + \(.fit) .fit |> + filter(term == "netTRUE") |> + pull(estimate) + ) + ) |> + ggplot(aes(estimate)) + + geom_histogram(fill = "#D55E00FF", color = "white", alpha = 0.8) +``` + +
+ +"A histogram of 1,000 bootstrapped estimates of the effect of net use on malaria risk. The spread of these estimates accounts for the dependency and uncertainty in the use of IPW weights." +

+ +Figure 1.8: “A +histogram of 1,000 bootstrapped estimates of the effect of net use on +malaria risk. The spread of these estimates accounts for the dependency +and uncertainty in the use of IPW weights.” +

+ +
+ +Figure figure 1.8 +gives a sense of the variation in `estimate`, but let’s calculate 95% +confidence intervals from the bootstrapped distribution using rsample’s +`int_t()` : + +``` r +boot_estimate <- ipw_results |> + # calculate T-statistic-based CIs + int_t(boot_fits) |> + filter(term == "netTRUE") + +boot_estimate +``` + + ## # A tibble: 1 × 6 + ## term .lower .estimate .upper .alpha .method + ## + ## 1 netTRUE -13.4 -12.5 -11.7 0.05 student-t + +Now we have a confounder-adjusted estimate with correct standard errors. +The estimate of the effect of *all* households using bed nets versus +*no* households using bed nets on malaria risk is -12.5 (95% CI -13.4, +-11.7). Bed nets do indeed seem to reduce malaria risk in this study. + +## 1.5 Conduct sensitivity analysis on the effect estimate + +We’ve laid out a roadmap for taking observational data, thinking +critically about the causal question we want to ask, identifying the +assumptions we need to get there, then applying those assumptions to a +statistical model. Getting the correct answer to the causal question +relies on getting our assumptions more or less right. But what if we’re +more on the less correct side? + +Spoiler alert: the answer we just calculated is *wrong*. After all that +effort! + +When conducting a causal analysis, it’s a good idea to use sensitivity +analyses to test your assumptions. There are many potential sources of +bias in any study and many sensitivity analyses to go along with them +(@sec-sensitivity); here, we’ll focus on the assumption of no +confounding. + +Let’s start with a broad sensitivity analysis; then, we’ll ask questions +about specific unmeasured confounders. When we have less information +about unmeasured confounders, we can use tipping point analysis to ask +how much confounding it would take to tip my estimate to the null. In +other words, what would the strength of the unmeasured confounder have +to be to explain our results away? The tipr package is a toolkit for +conducting sensitivity analyses. Let’s examine the tipping point for an +unknown, normally-distributed confounder. The `tip_coef()` function +takes an estimate (a beta coefficient from a regression model, or the +upper or lower bound of the coefficient). It further requires either +the 1) scaled differences in means of the confounder between exposure +groups or 2) effect of the confounder on the outcome. For the estimate, +we’ll use `conf.high`, which is closer to 0 (the null), and ask: how +much would the confounder have to affect malaria risk to have an +unbiased upper confidence interval of 0? We’ll use tipr to calculate +this answer for 5 scenarios, where the mean difference in the confounder +between exposure groups is 1, 2, 3, 4, or 5. + +``` r +library(tipr) +tipping_points <- tip_coef(boot_estimate$.upper, exposure_confounder_effect = 1:5) + +tipping_points |> + ggplot(aes(confounder_outcome_effect, exposure_confounder_effect)) + + geom_line(color = "#009E73", linewidth = 1.1) + + geom_point(fill = "#009E73", color = "white", size = 2.5, shape = 21) + + labs( + x = "Confounder-Outcome Effect", + y = "Scaled mean differences in\n confounder between exposure groups" + ) +``` + +
+ +A tipping point analysis under several confounding scenarios where the unmeasured confounder is a normally-distributed continuous variable. The line represents the strength of confounding necessary to tip the upper confidence interval of the causal effect estimate to 0. The x-axis represents the coefficient of the confounder-outcome relationship adjusted for the exposure and the set of measured confounders. The y-axis represents the scaled mean difference of the confounder between exposure groups. +

+ +Figure 1.9: A tipping point +analysis under several confounding scenarios where the unmeasured +confounder is a normally-distributed continuous variable. The line +represents the strength of confounding necessary to tip the upper +confidence interval of the causal effect estimate to 0. The x-axis +represents the coefficient of the confounder-outcome relationship +adjusted for the exposure and the set of measured confounders. The +y-axis represents the scaled mean difference of the confounder between +exposure groups. +

+ +
+ +If we had an unmeasured confounder where the standardized mean +difference between exposure groups was 1, the confounder would need to +decrease malaria risk by about -11.7. That’s pretty strong relative to +other effects, but it may be feasible if we have an idea of something we +might have missed. Conversely, suppose the relationship between net use +and the unmeasured confounder is very strong, with a mean scaled +difference of 5. In that case, the confounder-malaria relationship only +needs to be -2.3. Now we have to consider: which of these scenarios are +plausible given our domain knowledge and the effects we see in this +analysis? + +Now let’s consider a much more specific sensitivity analysis. Some +ethnic groups, such as the Fulani, have a genetic resistance to malaria +\[@arama2015\]. Let’s say that in our simulated data, an unnamed ethnic +group in the unnamed country shares this genetic resistance to malaria. +For historical reasons, bed net use in this group is also very high. We +don’t have this variable in `net_data`, but let’s say we know from the +literature that in this sample, we can estimate at: + +1. People with this genetic resistance have, on average, a lower + malaria risk by about 10. +2. About 26% of people who use nets in our study have this genetic + resistance. +3. About 5% of people who don’t use nets have this genetic resistance. + +With this amount of information, we can use tipr to adjust the estimates +we calculated for the unmeasured confounder. We’ll use +`adjust_coef_with_binary()` to calculate the adjusted estimates. + +``` r +adjusted_estimates <- boot_estimate |> + select(.estimate, .lower, .upper) |> + unlist() |> + adjust_coef_with_binary( + exposed_confounder_prev = 0.26, + unexposed_confounder_prev = 0.05, + confounder_outcome_effect = -10 + ) + +adjusted_estimates +``` + + ## # A tibble: 3 × 4 + ## effect_adjusted effect_observed + ## + ## 1 -10.4 -12.5 + ## 2 -11.3 -13.4 + ## 3 -9.63 -11.7 + ## # ℹ 2 more variables: + ## # exposure_confounder_effect , + ## # confounder_outcome_effect + +The adjusted estimate for a situation where genetic resistance to +malaria is a confounder is -10.4 (95% CI -11.3, -9.6). + +In fact, these data were simulated with just such a confounder. The true +effect of net use on malaria is about -10, and the true DAG that +generated these data is: + +
+ +The true causal diagram for `net_data`. This DAG is identical to the one we proposed with one addition: genetic resistance to malaria causally reduces the risk of malaria and impacts net use. It's thus a confounder and a part of the minimal adjustment set required to get an unbiased effect estimate. In otherwords, by not including it, we've calculated the wrong effect. +

+ +Figure 1.10: The true causal +diagram for `net_data`. This DAG is identical to the one we proposed +with one addition: genetic resistance to malaria causally reduces the +risk of malaria and impacts net use. It’s thus a confounder and a part +of the minimal adjustment set required to get an unbiased effect +estimate. In otherwords, by not including it, we’ve calculated the wrong +effect. +

+ +
+ +The unmeasured confounder in figure +1.10 is available in the +dataset `net_data_full` as `genetic_resistance`. If we recalculate the +IPW estimate of the average treatment effect of nets on malaria risk, we +get -10.3 (95% CI -11.2, -9.3), much closer to the actual answer of -10. + +What do you think? Is this estimate reliable? Did we do a good job +addressing the assumptions we need to make for a causal effect, mainly +that there is no confounding? How might you criticize this model, and +what would you do differently? Ok, we know that -10 is the correct +answer because the data are simulated, but in practice, we can never be +sure, so we need to continue probing our assumptions until we’re +confident they are robust. We’ll explore these techniques and others in +@sec-sensitivity. + + +To calculate this effect, we: + +1. Specified a causal question (for the average treatment effect) +2. Drew our assumptions using a causal diagram (using DAGs) +3. Modeled our assumptions (using propensity score weighting) +4. Diagnosed our models (by checking confounder balance after + weighting) +5. Estimated the causal effect (using inverse probability weighting) +6. Conducted sensitivity analysis on the effect estimate (using tipping + point analysis) + +We can dive more deeply into propensity score techniques, explore other +methods for estimating causal effects, and, most importantly, make sure +that the assumptions we’re making are reasonable, even if we’ll never +know for sure. diff --git a/pages/causal_inference_whole_game_files/figure-gfm/fig-ate-density-net-data-1.png b/pages/causal_inference_whole_game_files/figure-gfm/fig-ate-density-net-data-1.png new file mode 100644 index 0000000000000000000000000000000000000000..fb190e93b32dadc53a7a551f726b536aa2d49c11 GIT binary patch literal 22275 zcmb@uby!s07dAX}C;}n|tq2GRC?O><0@5X^v`DAY9f}GlUD88$cY}g-cO%^(-Mo8b zhUfQQ@Adun{qgwdIdk^ed+inXTI=4&TTVt450?ZN0)gO(iwVm^AgC}11Z5f<6I^lp zKE)1!px-bO5|T3!5*4yAwXjrp|Ms1psGhl=rO`WiQ9%fV%hy*)!;nnr7JpQEA>+f& zPyHz=K{w7hLW)e~dpt*%&h-k-#vccp7QF~)S(}J4qGg*sO){eAzx9HkK5aU9`a-Rq z?!Iy6`ZuP8BDn?33?dbX@)$hrn{nCRAgVh{4Q53I< z&%;?a>Arx?31j2;S-lzQ)p%T 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Error t value Pr(>|t|) + ## (Intercept) -1.71e+03 4.98e+02 -3.43 0.019 * + ## DateNumeric 8.74e-02 2.50e-02 3.49 0.017 * ## --- ## Signif. codes: - ## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' - ## 0.1 ' ' 1 + ## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## - ## Residual standard error: 6.361 on 4 degrees of freedom - ## Multiple R-squared: 0.8884, Adjusted R-squared: 0.8605 - ## F-statistic: 31.84 on 1 and 4 DF, p-value: 0.004857 + ## Residual standard error: 15 on 5 degrees of freedom + ## Multiple R-squared: 0.709, Adjusted R-squared: 0.651 + ## F-statistic: 12.2 on 1 and 5 DF, p-value: 0.0175 - ## Estimated Annual Growth Rate (pages/year): 23.82 + ## Estimated Annual Growth Rate (pages/year): 31.91 - ## Current count: 47 + ## Current count: 77 ![](../assets/images/word-count-plots-1.png) diff --git a/src/causal_inference_whole_game.Rmd b/src/causal_inference_whole_game.Rmd new file mode 100644 index 0000000..96f634e --- /dev/null +++ b/src/causal_inference_whole_game.Rmd @@ -0,0 +1,1002 @@ +--- +title: "Causal inference mosquito nets" +author: "Ref, Dylan Lawless" +output: + # md_document: # faster but missing features + bookdown::markdown_document2: # slower but supports features like cross-referencing + variant: gfm + preserve_yaml: true +layout: default +nav_order: 5 +math: mathjax +--- + +# Causal inference - mosquito nets and malaria {#sec-whole-game} + +Last update: +```{r, date, echo = FALSE} +print(Sys.Date()) +``` + + +This doc was built with: `rmarkdown::render("causal_inference_whole_game.Rmd", output_file = "../pages/causal_inference_whole_game.md")` + +{: .warning } + +This text is largely copied directly from the following source while we build an example closer to our needs. Please see the original source as follows. + +This example is described in the textbook: +Causal Inference in R, by +Malcolm Barrett, +Lucy D’Agostino McGowan, and +Travis Gerke. +, chapter 2. + + + +```{r, echo=FALSE, warning=FALSE, message=FALSE} +library(tidyverse) + +# source(here::here("R/ggdag-mask.R")) + +# this is all a hack to make work with quick plotting +# TODO: when `geom_dag_label_repel2` exists, add to namespace as 1 then delete this first bit +# copied from source to avoid recursion issue in overriding in ggdag namsespace +ggdag_geom_dag_label_repel <- function( + mapping = NULL, + data = NULL, + parse = FALSE, + ..., + box.padding = grid::unit(0.35, "lines"), + label.padding = grid::unit(0.25, "lines"), + point.padding = grid::unit(1.5, "lines"), + label.r = grid::unit(0.15, "lines"), + label.size = 0.25, + segment.color = "grey50", + segment.size = 0.5, + arrow = NULL, + force = 1, + max.iter = 2000, + nudge_x = 0, + nudge_y = 0, + na.rm = FALSE, + show.legend = NA, + inherit.aes = TRUE +) { + ggplot2::layer( + data = data, + mapping = mapping, + stat = ggdag:::StatNodesRepel, + geom = ggrepel::GeomLabelRepel, + position = "identity", + show.legend = show.legend, + inherit.aes = inherit.aes, + params = list( + parse = parse, + box.padding = box.padding, + label.padding = label.padding, + point.padding = point.padding, + label.r = label.r, + label.size = label.size, + segment.colour = segment.color %||% + segment.colour, + segment.size = segment.size, + arrow = arrow, + na.rm = na.rm, + force = force, + max.iter = max.iter, + nudge_x = nudge_x, + nudge_y = nudge_y, + segment.alpha = 1, + ... + ) + ) +} + +geom_dag_label_repel_internal <- function(..., seed = 10) { + ggdag_geom_dag_label_repel( + mapping = aes(x, y, label = label), + # TODO: make sure this looks ok. slightly different from above + box.padding = 2, + max.overlaps = Inf, + inherit.aes = FALSE, + family = getOption("book.base_family"), + seed = seed, + label.size = NA, + label.padding = 0.01 + ) +} + +# apply to quick functions as well +assignInNamespace("geom_dag_label_repel", geom_dag_label_repel_internal, ns = "ggdag") + +# override some other clumsy internals in ggdag until addressed + +assignInNamespace("scale_color_hue", ggplot2::scale_color_discrete, ns = "ggplot2") +assignInNamespace("scale_edge_colour_hue", \(...) ggraph::scale_edge_colour_manual(..., values = ggokabeito::palette_okabe_ito()), ns = "ggraph") + +# force ggraph to respect palette +assignInNamespace("scale_edge_color_discrete", ggokabeito::scale_edge_color_okabe_ito, ns = "ggraph") + +``` + +```{r, echo=FALSE, warning=FALSE, message=FALSE} +# source(here::here("R/setup.R")) +options( + tidyverse.quiet = TRUE, + propensity.quiet = TRUE, + tipr.verbose = FALSE, + htmltools.dir.version = FALSE, + width = 55, + digits = 4, + ggplot2.discrete.colour = ggokabeito::palette_okabe_ito(), + ggplot2.discrete.fill = ggokabeito::palette_okabe_ito(), + ggplot2.continuous.colour = "viridis", + ggplot2.continuous.fill = "viridis", + book.base_family = "sans", + book.base_size = 14 +) + +library(ggplot2) + +theme_set( + theme_bw( + # theme_minimal( + base_size = getOption("book.base_size"), + base_family = getOption("book.base_family") + ) %+replace% + theme( + # panel.grid.minor = element_blank(), + legend.position = "bottom" + ) +) + +theme_dag <- function() { + ggdag::theme_dag(base_family = getOption("book.base_family")) +} + +geom_dag_label_repel <- function(..., seed = 10) { + ggdag_geom_dag_label_repel( + aes(x, y, label = label), + box.padding = 3.5, + inherit.aes = FALSE, + max.overlaps = Inf, + family = getOption("book.base_family"), + seed = seed, + label.size = NA, + label.padding = 0.1, + size = getOption("book.base_size") / 3, + ... + ) +} + +est_ci <- function(.df, rsample = FALSE) { + if (!is.data.frame(.df) && is.numeric(.df)) { + return( + glue::glue("{round(.df[[1]], digits = 1)} (95% CI {round(.df[[2]], digits = 1)}, {round(.df[[3]], digits = 1)})") + ) + } + + if (rsample) { + glue::glue("{round(.df$.estimate, digits = 1)} (95% CI {round(.df$.lower, digits = 1)}, {round(.df$.upper, digits = 1)})") + } else { + glue::glue("{.df$estimate} (95% CI {.df$conf.low}, {.df$conf.high})") + } +} + +# based on https://github.com/hadley/r-pkgs/blob/main/common.R +status <- function(type) { + status <- switch( + type, + unstarted = "is unstarted, but don't worry, it's on our roadmap", + polishing = "has its foundations written but is still undergoing changes", + wip = "is actively undergoing work and may be restructured or changed. It may also be incomplete", + complete = "is mostly complete, but we might make small tweaks or copyedits", + stop("Invalid `type`", call. = FALSE) + ) + + class <- switch( + type, + complete = , + polishing = "callout-note", + wip = "callout-warning", + unstarted = "callout-warning" + ) + + knitr::asis_output(paste0( + "::: ", + class, + "\n", + "## Work-in-progress 🚧\n", + "You are reading the work-in-progress first edition of *Causal Inference in R*. ", + "This chapter ", + status, + ". \n", + ":::\n" + )) +} + +``` + + + + + +We will use simulated data to answer a more specific question: Does using insecticide-treated bed nets compared to no nets decrease the risk of contracting malaria after 1 year? +Data [simulated by Dr. Andrew Heiss](https://evalsp21.classes.andrewheiss.com/example/matching-ipw/#program-background): + +> ...researchers are interested in whether using mosquito nets decreases an individual's risk of contracting malaria. +> They have collected data from 1,752 households in an unnamed country and have variables related to environmental factors, individual health, and household characteristics. +> The data is **not experimental**---researchers have no control over who uses mosquito nets, and individual households make their own choices over whether to apply for free nets or buy their own nets, as well as whether they use the nets if they have them. + +This data includes a variable that measures the likelihood of contracting malaria, something we wouldn't likely have in real life. +This let's us know the actual effect size to understand the methods better. +The simulated data is in `net_data` from the {[causalworkshop](https://github.com/r-causal/causalworkshop)} package, which includes ten variables: + + + +* `id` : an ID variable +* `net` and `net_num` : a binary variable indicating if the participant used a net (1) or didn't use a net (0) +* `malaria_risk` : risk of malaria scale ranging from 0-100 +* `income` : weekly income, measured in dollars +* `health` : a health score scale ranging from 0--100 +* `household` : number of people living in the household +* `eligible` : a binary variable indicating if the household is eligible for the free net program. +* `temperature` : the average temperature at night, in Celsius +* `resistance` : Insecticide resistance of local mosquitoes. A scale of 0--100, with higher values indicating higher resistance. + +The distribution of malaria risk appears to be quite different by net usage. + +```{r} +# devtools::install_github("r-causal/causalworkshop") +``` + +```{r fig-malaria-risk-density} +#| label: fig-malaria-risk-density +#| fig.cap: > +#| A density plot of malaria risk for those who did and did not use nets. +#| The risk of malaria is lower for those who use nets. +library(tidyverse) +library(causalworkshop) +net_data |> + ggplot(aes(malaria_risk, fill = net)) + + geom_density(color = NA, alpha = .8) +``` + +```{r} +#| echo = FALSE +means <- net_data |> + group_by(net) |> + summarize(malaria_risk = mean(malaria_risk)) |> + pull(malaria_risk) +``` + +In figure \@ref(fig:fig-malaria-risk-density), the density of those who used nets is to the left of those who did not use nets. +The mean difference in malaria risk is about `r round(means[[1]] - means[[2]], digits = 1)`, suggesting net use might be protective against malaria. + +```{r} +net_data |> + group_by(net) |> + summarize(malaria_risk = mean(malaria_risk)) +``` + +And that's what we see with simple linear regression, as well, as we would expect. + +```{r} +library(broom) +net_data |> + lm(malaria_risk ~ net, data = _) |> + tidy() +``` + +## Draw our assumptions using a causal diagram + +The problem that we face is that other factors may be responsible for the effect we're seeing. +In this example, we'll focus on confounding: a common cause of net usage and malaria will bias the effect we see unless we account for it somehow. +One of the best ways to determine which variables we need to account for is to use a causal diagram. +These diagrams, also called **causal directed acyclic graphs (DAGs)**, visualize the assumptions that we're making about the causal relationships between the exposure, outcome, and other variables we think might be related. + +Here's the DAG that we're proposing for this question. + +```{r fig-net-data-dag} +#| label: fig-net-data-dag +#| echo: false +#| fig.width: 7 +#| fig.cap: > +#| A proposed causal diagram of the effect of bed net use on malaria. +#| This directed acyclic graph (DAG) states our assumption that bed net +#| use causes a reduction in malaria risk. It also says that we assume: +#| malaria risk is impacted by net usage, income, health, temperature, +#| and insecticide resistance; net usage is impacted by income, health, +#| temperature, eligibility for the free net program, and the number of +#| people in a household; eligibility for the free net programs is impacted +#| by income and the number of people in a household; and health is impacted +#| by income. +library(ggdag, warn.conflicts = FALSE) +library(ggokabeito) +mosquito_dag <- dagify( + malaria_risk ~ net + income + health + temperature + resistance, + net ~ income + health + temperature + eligible + household, + eligible ~ income + household, + health ~ income, + exposure = "net", + outcome = "malaria_risk", + coords = list( + x = c( + malaria_risk = 7, + net = 3, + income = 4, + health = 5, + temperature = 6, + resistance = 8.5, + eligible = 2, + household = 1 + ), + y = c( + malaria_risk = 2, + net = 2, + income = 3, + health = 1, + temperature = 3, + resistance = 2, + eligible = 3, + household = 2 + ) + ), + labels = c( + malaria_risk = "Risk of malaria", + net = "Mosquito net", + income = "Income", + health = "Health", + temperature = "Nighttime temperatures", + resistance = "Insecticide resistance", + eligible = "Eligible for program", + household = "Number in the household" + ) +) + +mosquito_dag |> + tidy_dagitty() |> + node_status() |> + ggplot( + aes(x, y, xend = xend, yend = yend, color = status) + ) + + geom_dag_edges() + + geom_dag_point() + + geom_dag_label_repel() + + scale_color_okabe_ito(na.value = "grey90") + + theme_dag() + + theme(legend.position = "none") + + coord_cartesian(clip = "off") +``` + +We'll explore how to create and analyze DAGs in @sec-dags. + +In DAGs, each point represents a variable, and each arrow represents a cause. +In other words, this diagram declares what we think the causal relationships are between these variables. +In figure \@ref(fig:fig-net-data-dag), we're saying that we believe: + +- Malaria risk is causally impacted by net usage, income, health, temperature, and insecticide resistance. +- Net usage is causally impacted by income, health, temperature, eligibility for the free net program, and the number of people in a household. +- Eligibility for the free net programs is determined by income and the number of people in a household. +- Health is causally impacted by income. + +You may agree or disagree with some of these assertions. +That's a good thing! +Laying bare our assumptions allows us to consider the scientific credibility of our analysis. +Another benefit of using DAGs is that, thanks to their mathematics, we can determine precisely the subset of variables we need to account for if we assume this DAG is correct. + +{: .note } + +**Assembling DAGs** +In this exercise, we're providing you with a reasonable DAG based on knowledge of how the data were generated. +In real life, setting up a DAG is a challenge requiring deep thought, domain expertise, and (often) collaboration between several experts. + +The chief problem we're dealing with is that, when we analyze the data we're working with, we see the impact of net usage on malaria risk *and of all these other relationships*. +In DAG terminology, we have more than one open causal pathway. +If this DAG is correct, we have *eight* causal pathways: the path between net usage and malaria risk and seven other *confounding* pathways. + +```{r fig-net-data-confounding} +#| label: fig-net-data-confounding +#| echo: false +#| fig.width: 14 +#| fig.height: 10 +#| fig.cap: > +#| In the proposed DAG, there are eight open pathways that contribute to the +#| causal effect seen in the naive regression: the true effect (in green) of +#| net usage on malaria risk and seven other confounding pathways (in orange). +#| The naive estimate is wrong because it is a composite of all these effects. +glyph <- function(data, params, size) { + data$shape <- 15 + data$size <- 12 + ggplot2::draw_key_point(data, params, size) +} + +mosquito_dag |> + dag_paths() |> + mutate( + effects = case_when( + set == "1" & path == "open path" ~ "true effect", + path == "open path" ~ "confounding effect", + TRUE ~ NA_character_ + ), + effects = factor(effects, c("true effect", "confounding effect")) + ) |> + ggplot(aes(x = x, y = y, xend = xend, yend = yend, color = effects, alpha = path)) + + geom_dag_edges(aes(edge_alpha = path, edge_colour = effects), show.legend = FALSE) + + geom_dag_point( + data = function(.x) dplyr::filter(.x, is.na(path)), + key_glyph = glyph + ) + + geom_dag_point( + data = function(.x) dplyr::filter(.x, !is.na(path)), + key_glyph = glyph + ) + + facet_wrap(vars(fct_inorder(factor(set)))) + + expand_plot( + expand_x = expansion(c(0.25, 0.25)), + expand_y = expansion(c(0.1, 0.1)) + ) + + theme_dag() + + theme( + legend.position = "top", + legend.spacing.x = unit(8, "mm"), + legend.text = element_text(size = rel(2.5)), + legend.box.margin = margin(b = 20), + strip.text = element_blank() + ) + + coord_cartesian(clip = "off") + + scale_alpha_manual( + drop = FALSE, + values = c("open path" = 1), + na.value = .5, + breaks = "open path" + ) + + ggraph::scale_edge_alpha_manual( + drop = FALSE, + values = c("open path" = 1), + na.value = .5, + breaks = "open path" + ) + + scale_color_okabe_ito( + name = NULL, + na.value = "grey90", + order = c(3, 6), + breaks = c("true effect", "confounding effect") + ) + + scale_edge_color_okabe_ito( + name = NULL, + na.value = "grey90", + order = c(3, 6), + breaks = c("true effect", "confounding effect") + ) + + guides(alpha = "none", edge_alpha = "none") +``` + +When we calculate a naive linear regression that only includes net usage and malaria risk, the effect we see is incorrect because the seven other confounding pathways in figure \@ref(fig:fig-net-data-confounding) distort it. +In DAG terminology, we need to *block* these open pathways that distort the causal estimate we're after. +(We can block paths through several techniques, including stratification, matching, weighting, and more. We'll see several methods throughout the book.) Luckily, by specifying a DAG, we can precisely determine the variables we need to control for. +For this DAG, we need to control for three variables: `r glue::glue_collapse(as.list(dagitty::adjustmentSets(mosquito_dag))[[1]], sep = ", ", last = ", and ")`. +These three variables are a *minimal adjustment set*, the minimum set (or sets) of variables you need to block all confounding pathways. +We'll discuss adjustment sets further in @sec-dags. + +## Model our assumptions + +We'll use a technique called **inverse probability weighting (IPW)** to control for these variables, which we'll discuss in detail in @sec-using-ps. +We'll use logistic regression to predict the probability of treatment---the propensity score. +Then, we'll calculate inverse probability weights to apply to the linear regression model we fit above. +The propensity score model includes the exposure---net use---as the dependent variable and the minimal adjustment set as the independent variables. + +{: .note } + +**Modeling the functional form** +Generally speaking, we want to lean on domain expertise and good modeling practices to fit the propensity score model. +For instance, we may want to allow continuous confounders to be non-linear using splines, or we may want to add essential interactions between confounders. +Because these are simulated data, we know we don't need these extra parameters (so we'll skip them), but in practice, you often do. +We'll discuss this more in @sec-using-ps. + +The propensity score model is a logistic regression model with the formula `net ~ income + health + temperature`, which predicts the probability of bed net usage based on the confounders income, health, and temperature. + +```{r} +propensity_model <- glm( + net ~ income + health + temperature, + data = net_data, + family = binomial() +) + +# the first six propensity scores +head(predict(propensity_model, type = "response")) +``` + +We can use propensity scores to control for confounding in various ways. +In this example, we'll focus on weighting. +In particular, we'll compute the inverse probability weight for the **average treatment effect (ATE)**. +The ATE represents a particular causal question: what if *everyone* in the study used bed nets vs. what if *no one* in the study used bed nets? + +To calculate the ATE, we'll use the broom and propensity packages. +broom's `augment()` function extracts prediction-related information from the model and joins it to the data. +propensity's `wt_ate()` function calculates the inverse probability weight given the propensity score and exposure. + +For inverse probability weighting, the ATE weight is the inverse of probability of receiving the treatment you actually received. +In other words, if you used a bed net, the ATE weight is the inverse of the probability that you used a net, and if you did *not* use a net, it is the the inverse of the probability that you did *not* use a net. + +```{r} +library(broom) +library(propensity) +net_data_wts <- propensity_model |> + augment(data = net_data, type.predict = "response") |> + # .fitted is the value predicted by the model + # for a given observation + mutate(wts = wt_ate(.fitted, net)) + +net_data_wts |> + select(net, .fitted, wts) |> + head() +``` + +`wts` represents the amount each observation will be up-weighted or down-weighted in the outcome model we will soon fit. +For instance, the 16th household used a bed net and had a predicted probability of `r round(net_data_wts$.fitted[[16]], digits = 2)`. +That's a pretty low probability considering they did, in fact, use a net, so their weight is higher at `r round(net_data_wts$wts[[16]], digits = 2)`. +In other words, this household will be up-weighted compared to the naive linear model we fit above. +The first household did *not* use a bed net; they're predicted probability of net use was `r round(net_data_wts$.fitted[[1]], digits = 2)` (or put differently, a predicted probability of *not* using a net of `r 1 - round(net_data_wts$.fitted[[1]], digits = 2)`). +That's more in line with their observed value of `net`, but there's still some predicted probability of using a net, so their weight is `r round(net_data_wts$wts[[2]], digits = 2)`. + +## Diagnose our models + +The goal of propensity score weighting is to weight the population of observations such that the distribution of confounders is balanced between the exposure groups. +Put another way, we are, in principle, removing the arrows between the confounders and exposure in the DAG, so that the confounding paths no longer distort our estimates. +Here's the distribution of the propensity score by group, created by `geom_mirror_histogram()` from the halfmoon package for assessing balance in propensity score models: + +```{r fig-mirror-histogram-net-data-unweighted} +#| label: fig-mirror-histogram-net-data-unweighted +#| fig.cap: > +#| A mirrored histogram of the propensity scores of those who used nets (top, blue) versus those who did not use nets (bottom, orange). The range of propensity scores is similar between groups, with those who used nets slightly to the left of those who didn't, but the shapes of the distribution are different. +library(halfmoon) +ggplot(net_data_wts, aes(.fitted)) + + geom_mirror_histogram( + aes(fill = net), + bins = 50 + ) + + scale_y_continuous(labels = abs) + + labs(x = "propensity score") +``` + +The weighted propensity score creates a pseudo-population where the distributions are much more similar: + +```{r fig-mirror-histogram-net-data-weighted} +#| label: fig-mirror-histogram-net-data-weighted +#| fig.cap: > +#| A mirrored histogram of the propensity scores of those who used nets (top, blue) versus those who did not use nets (bottom, orange). The shaded region represents the unweighted distribution, and the colored region represents the weighted distributions. The ATE weights up-weight the groups to be similar in range and shape of the distribution of propensity scores. +ggplot(net_data_wts, aes(.fitted)) + + geom_mirror_histogram( + aes(group = net), + bins = 50 + ) + + geom_mirror_histogram( + aes(fill = net, weight = wts), + bins = 50, + alpha = .5 + ) + + scale_y_continuous(labels = abs) + + labs(x = "propensity score") +``` + +In this example, the unweighted distributions are not awful---the shapes are somewhat similar here, and they overlap quite a bit---but the weighted distributions in figure \@ref(fig:fig-mirror-histogram-net-data-unweighted) are much more similar. + +{: .note } + +**Unmeasured confounding** +Propensity score weighting and most other causal inference techniques only help with *observed* confounders---ones that we model correctly, at that. +Unfortunately, we still may have unmeasured confounding, which we'll discuss below. +Randomization is one causal inference technique that *does* deal with unmeasured confounding, one of the reasons it is so powerful. + +We might also want to know how well-balanced the groups are by each confounder. +One way to do this is to calculate the **standardized mean differences (SMDs)** for each confounder with and without weights. +We'll calculate the SMDs with `tidy_smd()` then plot them with `geom_love()`. + +```{r fig-love-plot-net-data} +#| label: fig-love-plot-net-data +#| fig.cap: > +#| A love plot representing the standardized mean differences (SMD) between exposure groups of three confounders: temperature, income, and health. Before weighting, there are considerable differences in the groups. After weighting, the confounders are much more balanced between groups. +plot_df <- tidy_smd( + net_data_wts, + c(income, health, temperature), + .group = net, + .wts = wts +) + +ggplot( + plot_df, + aes( + x = abs(smd), + y = variable, + group = method, + color = method + ) +) + + geom_love() +``` + +A standard guideline is that balanced confounders should have an SMD of less than 0.1 on the absolute scale. +0.1 is just a rule of thumb, but if we follow it, the variables in figure \@ref(fig:fig-love-plot-net-data) are well-balanced after weighting (and unbalanced before weighting). + +Before we apply the weights to the outcome model, let's check their overall distribution for extreme weights. +Extreme weights can destabilize the estimate and variance in the outcome model, so we want to be aware of it. +We'll also discuss several other types of weights that are less prone to this issue in @sec-estimands. + +```{r fig-ate-density-net-data} +#| label: fig-ate-density-net-data +#| fig.cap: > +#| A density plot of the average treatment effect (ATE) weights. The plot is skewed, with higher values towards 8. This may indicate a problem with the model, but the weights aren't so extreme to destabilize the variance of the estimate. +net_data_wts |> + ggplot(aes(wts)) + + geom_density(fill = "#CC79A7", color = NA, alpha = 0.8) +``` + +The weights in figure \@ref(fig:fig-ate-density-net-data) are skewed, but there are no outrageous values. +If we saw extreme weights, we might try trimming or stabilizing them, or consider calculating an effect for a different estimand, which we'll discuss in @sec-estimands. +It doesn't look like we need to do that here, however. + +## Estimate the causal effect + +We're now ready to use the ATE weights to (attempt to) account for confounding in the naive linear regression model. +Fitting such a model is pleasantly simple in this case: we fit the same model as before but with `weights = wts`, which will incorporate the inverse probability weights. + +```{r} +net_data_wts |> + lm(malaria_risk ~ net, data = _, weights = wts) |> + tidy(conf.int = TRUE) +``` + +```{r} +#| include = FALSE +estimates <- net_data_wts |> + lm(malaria_risk ~ net, data = _, weights = wts) |> + tidy(conf.int = TRUE) |> + filter(term == "netTRUE") |> + select(estimate, starts_with("conf")) |> + mutate(across(everything(), round, digits = 1)) +``` + +The estimate for the average treatment effect is `r est_ci(estimates)`. +Unfortunately, the confidence intervals we're using are wrong because they don't account for the uncertainty in estimating the weights. +Generally, confidence intervals for propensity score weighted models will be too narrow unless we account for this uncertainty. +The nominal coverage of the confidence intervals will thus be wrong (they aren't 95% CIs because their coverage is much lower than 95%) and may lead to misinterpretation. + +We've got several ways to address this problem, which we'll discuss in detail in @sec-outcome-model, including the bootstrap, robust standard errors, and manually accounting for the estimation procedure with empirical sandwich estimators. +For this example, we'll use the bootstrap, a flexible tool that calculates distributions of parameters using re-sampling. +The bootstrap is a useful tool for many causal models where closed-form solutions to problems (particularly standard errors) don't exist or when we want to avoid parametric assumptions inherent to many such solutions; see @sec-appendix-bootstrap for a description of what the bootstrap is and how it works. +We'll use the rsample package from the tidymodels ecosystem to work with bootstrap samples. + +Because the bootstrap is so flexible, we need to think carefully about the sources of uncertainty in the statistic we're calculating. +It might be tempting to write a function like this to fit the statistic we're interested in (the point estimate for `netTRUE`): + +```{r} +#| eval = FALSE +library(rsample) + +fit_ipw_not_quite_rightly <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} +``` + +However, this function won't give us the correct confidence intervals because it treats the inverse probability weights as fixed values. +They're not, of course; we just estimated them using logistic regression! +We need to account for this uncertainty by bootstrapping the *entire modeling process*. +For every bootstrap sample, we need to fit the propensity score model, calculate the inverse probability weights, then fit the weighted outcome model. + +```{r} +library(rsample) + +fit_ipw <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit propensity score model + propensity_model <- glm( + net ~ income + health + temperature, + data = .df, + family = binomial() + ) + + # calculate inverse probability weights + .df <- propensity_model |> + augment(type.predict = "response", data = .df) |> + mutate(wts = wt_ate(.fitted, net)) + + # fit correctly bootstrapped ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} +``` + +Now that we know precisely how to calculate the estimate for each iteration let's create the bootstrapped dataset with rsample's `bootstraps()` function. +The `times` argument determines how many bootstrapped datasets to create; we'll do 1,000. + +```{r} +bootstrapped_net_data <- bootstraps( + net_data, + times = 1000, + # required to calculate CIs later + apparent = TRUE +) + +bootstrapped_net_data +``` + +The result is a nested data frame: each `splits` object contains metadata that rsample uses to subset the bootstrap samples for each of the 1,000 samples. +We actually have 1,001 rows because `apparent = TRUE` keeps a copy of the original data frame, as well, which is needed for some times of confidence interval calculations. +Next, we'll run `fit_ipw()` 1,001 times to create a distribution for `estimate`. +At its heart, the calculation we're doing is + +``` r +fit_ipw(bootstrapped_net_data$splits[[n]]) +``` + +Where *n* is one of 1,001 indices. +We'll use purrr's `map()` function to iterate across each `split` object. + +```{r} +ipw_results <- bootstrapped_net_data |> + mutate(boot_fits = map(splits, fit_ipw)) + +ipw_results +``` + +The result is another nested data frame with a new column, `boot_fits`. +Each element of `boot_fits` is the result of the IPW for the bootstrapped dataset. +For example, in the first bootstrapped data set, the IPW results were: + +```{r} +ipw_results$boot_fits[[1]] +``` + +Now we have a distribution of estimates: + +```{r fig-bootstrap-estimates-net-data} +#| label: fig-bootstrap-estimates-net-data +#| message: false +#| warning: false +#| fig.cap: > +#| "A histogram of 1,000 bootstrapped estimates of the effect of net use on malaria risk. The spread of these estimates accounts for the dependency and uncertainty in the use of IPW weights." +ipw_results |> + # remove original data set results + filter(id != "Apparent") |> + mutate( + estimate = map_dbl( + boot_fits, + # pull the `estimate` for `netTRUE` for each fit + \(.fit) .fit |> + filter(term == "netTRUE") |> + pull(estimate) + ) + ) |> + ggplot(aes(estimate)) + + geom_histogram(fill = "#D55E00FF", color = "white", alpha = 0.8) +``` + +Figure figure \@ref(fig:fig-bootstrap-estimates-net-data) gives a sense of the variation in `estimate`, but let's calculate 95% confidence intervals from the bootstrapped distribution using rsample's `int_t()` : + +```{r} +boot_estimate <- ipw_results |> + # calculate T-statistic-based CIs + int_t(boot_fits) |> + filter(term == "netTRUE") + +boot_estimate +``` + +Now we have a confounder-adjusted estimate with correct standard errors. +The estimate of the effect of *all* households using bed nets versus *no* households using bed nets on malaria risk is `r est_ci(boot_estimate, rsample = TRUE)`. +Bed nets do indeed seem to reduce malaria risk in this study. + +## Conduct sensitivity analysis on the effect estimate + +We've laid out a roadmap for taking observational data, thinking critically about the causal question we want to ask, identifying the assumptions we need to get there, then applying those assumptions to a statistical model. +Getting the correct answer to the causal question relies on getting our assumptions more or less right. +But what if we're more on the less correct side? + +Spoiler alert: the answer we just calculated is *wrong*. +After all that effort! + +When conducting a causal analysis, it's a good idea to use sensitivity analyses to test your assumptions. +There are many potential sources of bias in any study and many sensitivity analyses to go along with them (@sec-sensitivity); here, we'll focus on the assumption of no confounding. + +Let's start with a broad sensitivity analysis; then, we'll ask questions about specific unmeasured confounders. +When we have less information about unmeasured confounders, we can use tipping point analysis to ask how much confounding it would take to tip my estimate to the null. +In other words, what would the strength of the unmeasured confounder have to be to explain our results away? +The tipr package is a toolkit for conducting sensitivity analyses. +Let's examine the tipping point for an unknown, normally-distributed confounder. +The `tip_coef()` function takes an estimate (a beta coefficient from a regression model, or the upper or lower bound of the coefficient). +It further requires either the 1) scaled differences in means of the confounder between exposure groups or 2) effect of the confounder on the outcome. +For the estimate, we'll use `conf.high`, which is closer to 0 (the null), and ask: how much would the confounder have to affect malaria risk to have an unbiased upper confidence interval of 0? +We'll use tipr to calculate this answer for 5 scenarios, where the mean difference in the confounder between exposure groups is 1, 2, 3, 4, or 5. + +```{r} +#| echo: false +options(tipr.verbose = FALSE) +``` + +```{r} +#| label: fig-tip-coef-net +#| fig.cap: > +#| A tipping point analysis under several confounding scenarios where the unmeasured confounder is a normally-distributed continuous variable. The line represents the strength of confounding necessary to tip the upper confidence interval of the causal effect estimate to 0. The x-axis represents the coefficient of the confounder-outcome relationship adjusted for the exposure and the set of measured confounders. The y-axis represents the scaled mean difference of the confounder between exposure groups. +library(tipr) +tipping_points <- tip_coef(boot_estimate$.upper, exposure_confounder_effect = 1:5) + +tipping_points |> + ggplot(aes(confounder_outcome_effect, exposure_confounder_effect)) + + geom_line(color = "#009E73", linewidth = 1.1) + + geom_point(fill = "#009E73", color = "white", size = 2.5, shape = 21) + + labs( + x = "Confounder-Outcome Effect", + y = "Scaled mean differences in\n confounder between exposure groups" + ) +``` + +If we had an unmeasured confounder where the standardized mean difference between exposure groups was 1, the confounder would need to decrease malaria risk by about `r round(tipping_points$confounder_outcome_effect[[1]], digits = 1)`. +That's pretty strong relative to other effects, but it may be feasible if we have an idea of something we might have missed. +Conversely, suppose the relationship between net use and the unmeasured confounder is very strong, with a mean scaled difference of 5. +In that case, the confounder-malaria relationship only needs to be `r round(tipping_points$confounder_outcome_effect[[5]], digits = 1)`. +Now we have to consider: which of these scenarios are plausible given our domain knowledge and the effects we see in this analysis? + +Now let's consider a much more specific sensitivity analysis. +Some ethnic groups, such as the Fulani, have a genetic resistance to malaria [@arama2015]. +Let's say that in our simulated data, an unnamed ethnic group in the unnamed country shares this genetic resistance to malaria. +For historical reasons, bed net use in this group is also very high. +We don't have this variable in `net_data`, but let's say we know from the literature that in this sample, we can estimate at: + +1. People with this genetic resistance have, on average, a lower malaria risk by about 10. +2. About 26% of people who use nets in our study have this genetic resistance. +3. About 5% of people who don't use nets have this genetic resistance. + +With this amount of information, we can use tipr to adjust the estimates we calculated for the unmeasured confounder. +We'll use `adjust_coef_with_binary()` to calculate the adjusted estimates. + +```{r} +adjusted_estimates <- boot_estimate |> + select(.estimate, .lower, .upper) |> + unlist() |> + adjust_coef_with_binary( + exposed_confounder_prev = 0.26, + unexposed_confounder_prev = 0.05, + confounder_outcome_effect = -10 + ) + +adjusted_estimates +``` + +The adjusted estimate for a situation where genetic resistance to malaria is a confounder is `r est_ci(adjusted_estimates$effect_adjusted)`. + +In fact, these data were simulated with just such a confounder. +The true effect of net use on malaria is about -10, and the true DAG that generated these data is: + +```{r fig-net-data-true-dag} +#| label: fig-net-data-true-dag +#| echo: false +#| fig.cap: > +#| The true causal diagram for `net_data`. This DAG is identical to the one we proposed with one addition: genetic resistance to malaria causally reduces the risk of malaria and impacts net use. It's thus a confounder and a part of the minimal adjustment set required to get an unbiased effect estimate. In otherwords, by not including it, we've calculated the wrong effect. +mosquito_dag_full <- dagify( + malaria_risk ~ net + income + health + temperature + insecticide_resistance + genetic_resistance, + net ~ income + health + temperature + eligible + household + genetic_resistance, + eligible ~ income + household, + health ~ income, + exposure = "net", + outcome = "malaria_risk", + coords = list( + x = c( + malaria_risk = 7, + net = 3, + income = 4, + health = 5, + temperature = 6, + insecticide_resistance = 8.5, + eligible = 2, + household = 1, + genetic_resistance = 8.5 + ), + y = c( + malaria_risk = 2, + net = 2, + income = 3, + health = 1, + temperature = 3, + insecticide_resistance = 2, + eligible = 3, + household = 2, + genetic_resistance = 1 + ) + ), + labels = c( + malaria_risk = "Risk of malaria", + net = "Mosquito net", + income = "Income", + health = "Health", + temperature = "Nighttime temperatures", + insecticide_resistance = "Insecticide resistance", + eligible = "Eligible for program", + household = "Number in household", + genetic_resistance = "Malaria resistance" + ) +) + +mosquito_dag_full |> + tidy_dagitty() |> + node_status() |> + ggplot( + aes(x, y, xend = xend, yend = yend, color = status) + ) + + geom_dag_edges() + + geom_dag_point() + + geom_dag_label_repel() + + scale_color_okabe_ito(na.value = "grey90") + + theme_dag() + + theme(legend.position = "none") + + coord_cartesian(clip = "off") +``` + +```{r} +#| include: false +fit_ipw_full <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit propensity score model + propensity_model <- glm( + net ~ income + health + temperature + genetic_resistance, + data = .df, + family = binomial() + ) + + # calculate inverse probability weights + .df <- propensity_model |> + augment(type.predict = "response", data = .df) |> + mutate(wts = wt_ate(.fitted, net)) + + # fit correctly bootstrapped ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} + +bootstrapped_net_data_full <- bootstraps( + net_data_full, + times = 1000, + # required to calculate CIs later + apparent = TRUE +) + +ipw_results_full <- bootstrapped_net_data_full |> + mutate(boot_fits = map(splits, fit_ipw_full)) + +boot_estimate_full <- ipw_results_full |> + # calculate T-statistic-based CIs + int_t(boot_fits) |> + filter(term == "netTRUE") +``` + +The unmeasured confounder in figure \@ref(fig:fig-net-data-true-dag) is available in the dataset `net_data_full` as `genetic_resistance`. +If we recalculate the IPW estimate of the average treatment effect of nets on malaria risk, we get `r est_ci(boot_estimate_full, rsample = TRUE)`, much closer to the actual answer of -10. + +What do you think? +Is this estimate reliable? +Did we do a good job addressing the assumptions we need to make for a causal effect, mainly that there is no confounding? +How might you criticize this model, and what would you do differently? +Ok, we know that -10 is the correct answer because the data are simulated, but in practice, we can never be sure, so we need to continue probing our assumptions until we're confident they are robust. +We'll explore these techniques and others in @sec-sensitivity. + + +To calculate this effect, we: + +1. Specified a causal question (for the average treatment effect) +2. Drew our assumptions using a causal diagram (using DAGs) +3. Modeled our assumptions (using propensity score weighting) +4. Diagnosed our models (by checking confounder balance after weighting) +5. Estimated the causal effect (using inverse probability weighting) +6. Conducted sensitivity analysis on the effect estimate (using tipping point analysis) + +We can dive more deeply into propensity score techniques, explore other methods for estimating causal effects, and, most importantly, make sure that the assumptions we're making are reasonable, even if we'll never know for sure. diff --git a/src/causal_inference_whole_game_raw.Rmd b/src/causal_inference_whole_game_raw.Rmd new file mode 100644 index 0000000..b40f16a --- /dev/null +++ b/src/causal_inference_whole_game_raw.Rmd @@ -0,0 +1,902 @@ +# The whole game: mosquito nets and malaria {#sec-whole-game} + +{{< include 00-setup.qmd >}} + +```{r} +#| echo: false +# TODO: remove when first edition complete +status("complete") +``` + +In this chapter, we'll analyze data using techniques we learn in this book. +We'll play the [whole game](https://www.gse.harvard.edu/news/uk/09/01/education-bat-seven-principles-educators) of causal analysis using a few key steps: + +1. Specify a causal question +2. Draw our assumptions using a causal diagram +3. Model our assumptions +4. Diagnose our models +5. Estimate the causal effect +6. Conduct sensitivity analysis on the effect estimate + +We'll focus on the broader ideas behind each step and what they look like all together; however, we don't expect you to fully digest each idea. +We'll spend the rest of the book taking up each step in detail. + +## Specify a causal question + +In this guided exercise, we'll attempt to answer a causal question: does using a bed net reduce the risk of malaria? + +Malaria remains a serious public health issue. +While malaria incidence has decreased since 2000, 2020 and the COVID-19 pandemic saw an increase in cases and deaths due primarily to service interruption [@worldma]. +About 86% of malaria deaths occurred in 29 countries. +Nearly half of all malaria deaths occurred in just six of those countries: Nigeria (27%), the Democratic Republic of the Congo (12%), Uganda (5%), Mozambique (4%), Angola (3%), and Burkina Faso (3%). +Most of these deaths occurred in children under 5 [@Fink2022]. +Malaria also poses severe health risks to pregnant women and worsens birth outcomes, including early delivery and low birth weight. + +Bed nets prevent morbidity and mortality due to malaria by providing a barrier against infective bites by the chief host of malaria parasites, the mosquito. +Humans have used bed nets since ancient times. +Herodotus, the 5th century BC Greek author of *The Histories*, observed Egyptians using their fishing nets as bed nets: + +> Against the gnats, which are very abundant, they have contrived as follows:---those who dwell above the fen-land are helped by the towers, to which they ascend when they go to rest; for the gnats by reason of the winds are not able to fly up high: but those who dwell in the fen-land have contrived another way instead of the towers, and this is it:---every man of them has got a casting net, with which by day he catches fish, but in the night he uses it for this purpose, that is to say he puts the casting-net round about the bed in which he sleeps, and then creeps in under it and goes to sleep: and the gnats, if he sleeps rolled up in a garment or a linen sheet, bite through these, but through the net they do not even attempt to bite [@thehist]. + +Many modern nets are also treated with insecticide, dating back to Russian soldiers in World War II [@nevill1996], although some people still use them as fishing nets [@gettleman2015]. + +It's easy to imagine a randomized trial that deals with this question: participants in a study are randomly assigned to use a bed net, and we follow them over time to see if there is a difference in malaria risk between groups. +Randomization is often the best way to estimate a causal effect of an intervention because it reduces the number of assumptions we need to make for that estimate to be valid (we will discuss these assumptions in @sec-assump). +In particular, randomization addresses confounding very well, accounting for confounders about which we may not even know. + +Several landmark trials have studied the effects of bed net use on malaria risk, with several essential studies in the 1990s. +A 2004 meta-analysis found that insecticide-treated nets reduced childhood mortality by 17%, malarial parasite prevalence by 13%, and cases of uncomplicated and severe malaria by about 50% (compared to no nets) [@lengeler2004]. +Since the World Health Organization began recommending insecticide-treated nets, insecticide resistance has been a big concern. +However, a follow-up analysis of trials found that it has yet to impact the public health benefits of bed nets [@pryce2018]. + +Trials have also been influential in determining the economics of bed net programs. +For instance, one trial compared free net distribution versus a cost-share program (where participants pay a subsidized fee for nets). +The study's authors found that net uptake was similar between the groups and that free net distribution --- because it was easier to access --- saved more lives, and was cheaper per life saved than the cost-sharing program [@cohen2010]. + +There are several reasons we might not be able to conduct a randomized trial, including ethics, cost, and time. +We have substantial, robust evidence in favor of bed net use, but let's consider some conditions where observational causal inference could help. + +- Imagine we are at a time before trials on this subject, and let's say people have started to use bed nets for this purpose on their own. + Our goal may still be to conduct a randomized trial, but we can answer questions more quickly with observed data. + In addition, this study's results might guide trials' design or intermediary policy suggestions. + +- Sometimes, it is also not ethical to conduct a trial. + An example of this in malaria research is a question that arose in the study of bed net effectiveness: does malaria control in early childhood result in delayed immunity to the disease, resulting in severe malaria or death later in life? + Since we now know bed net use is very effective, *withholding* nets would be unethical. + A recent observational study found that the benefits of bed net use in childhood on all-cause mortality persist into adulthood [@Fink2022]. + +- We may also want to estimate a different effect or the effect for another population than in previous trials. + For example, both randomized and observational studies helped us better understand that insecticide-based nets improve malaria resistance in the entire community, not just among those who use nets, so long as net usage is high enough [@howard2000; @hawley2003]. + +As we'll see in @sec-strat-outcome and @sec-g-comp, the causal inference techniques that we'll discuss in this book are often beneficial even when we're able to randomize. + +When we conduct an observational study, it's still helpful to think through the randomized trial we would run were it possible. +The trial we're trying to emulate in this causal analysis is the **target trial**. Considering the target trial helps us make our causal question more accurate. +We'll use this framework more explicitly in @sec-designs, but for now, let's consider the causal question posed earlier: does using a bed net (a mosquito net) reduce the risk of malaria? +This question is relatively straightforward, but it is still vague. +As we saw in @sec-causal-question, we need to clarify some key areas: + +- **What do we mean by "bed net"?** + There are several types of nets: untreated bed nets, insecticide-treated bed nets, and newer long-lasting insecticide-treated bed nets. + +- **Risk compared to what?** + Are we, for instance, comparing insecticide-treated bed nets to *no* net? + Untreated nets? + Or are we comparing a new type of net, like long-lasting insecticide-treated bed nets, to nets that are already in use? + +- **Risk as defined by what?** + Whether or not a person contracted malaria? + Whether a person died of malaria? + +- **Risk among whom?** + What is the population to which we're trying to apply this knowledge? + Who is it practical to include in our study? + Who might we need to exclude? + +We will use simulated data to answer a more specific question: Does using insecticide-treated bed nets compared to no nets decrease the risk of contracting malaria after 1 year? +In this particular data, [simulated by Dr. Andrew Heiss](https://evalsp21.classes.andrewheiss.com/example/matching-ipw/#program-background): + +> ...researchers are interested in whether using mosquito nets decreases an individual's risk of contracting malaria. +> They have collected data from 1,752 households in an unnamed country and have variables related to environmental factors, individual health, and household characteristics. +> The data is **not experimental**---researchers have no control over who uses mosquito nets, and individual households make their own choices over whether to apply for free nets or buy their own nets, as well as whether they use the nets if they have them. + +Because we're using simulated data, we'll have direct access to a variable that measures the likelihood of contracting malaria, something we wouldn't likely have in real life. +We'll stick with this measure because we know the actual effect size. +We can also safely assume that the population in our dataset represents the population we want to make inferences about (the unnamed country) because the data are simulated as such. +We can find the simulated data in `net_data` from the {[causalworkshop](https://github.com/r-causal/causalworkshop)} package, which includes ten variables: + + + +`id` + +: an ID variable + +`net` and `net_num` + +: a binary variable indicating if the participant used a net (1) or didn't use a net (0) + +`malaria_risk` + +: risk of malaria scale ranging from 0-100 + +`income` + +: weekly income, measured in dollars + +`health` + +: a health score scale ranging from 0--100 + +`household` + +: number of people living in the household + +`eligible` + +: a binary variable indicating if the household is eligible for the free net program. + +`temperature` + +: the average temperature at night, in Celsius + +`resistance` + +: Insecticide resistance of local mosquitoes. + A scale of 0--100, with higher values indicating higher resistance. + +The distribution of malaria risk appears to be quite different by net usage. + +```{r} +#| label: fig-malaria-risk-density +#| fig.cap: > +#| A density plot of malaria risk for those who did and did not use nets. +#| The risk of malaria is lower for those who use nets. +library(tidyverse) +library(causalworkshop) +net_data |> + ggplot(aes(malaria_risk, fill = net)) + + geom_density(color = NA, alpha = .8) +``` + +```{r} +#| echo = FALSE +means <- net_data |> + group_by(net) |> + summarize(malaria_risk = mean(malaria_risk)) |> + pull(malaria_risk) +``` + +In @fig-malaria-risk-density, the density of those who used nets is to the left of those who did not use nets. +The mean difference in malaria risk is about `r round(means[[1]] - means[[2]], digits = 1)`, suggesting net use might be protective against malaria. + +```{r} +net_data |> + group_by(net) |> + summarize(malaria_risk = mean(malaria_risk)) +``` + +And that's what we see with simple linear regression, as well, as we would expect. + +```{r} +library(broom) +net_data |> + lm(malaria_risk ~ net, data = _) |> + tidy() +``` + +## Draw our assumptions using a causal diagram + +The problem that we face is that other factors may be responsible for the effect we're seeing. +In this example, we'll focus on confounding: a common cause of net usage and malaria will bias the effect we see unless we account for it somehow. +One of the best ways to determine which variables we need to account for is to use a causal diagram. +These diagrams, also called **causal directed acyclic graphs (DAGs)**, visualize the assumptions that we're making about the causal relationships between the exposure, outcome, and other variables we think might be related. + +Here's the DAG that we're proposing for this question. + +```{r} +#| label: fig-net-data-dag +#| echo: false +#| fig.width: 7 +#| fig.cap: > +#| A proposed causal diagram of the effect of bed net use on malaria. +#| This directed acyclic graph (DAG) states our assumption that bed net +#| use causes a reduction in malaria risk. It also says that we assume: +#| malaria risk is impacted by net usage, income, health, temperature, +#| and insecticide resistance; net usage is impacted by income, health, +#| temperature, eligibility for the free net program, and the number of +#| people in a household; eligibility for the free net programs is impacted +#| by income and the number of people in a household; and health is impacted +#| by income. +library(ggdag, warn.conflicts = FALSE) +library(ggokabeito) +mosquito_dag <- dagify( + malaria_risk ~ net + income + health + temperature + resistance, + net ~ income + health + temperature + eligible + household, + eligible ~ income + household, + health ~ income, + exposure = "net", + outcome = "malaria_risk", + coords = list( + x = c( + malaria_risk = 7, + net = 3, + income = 4, + health = 5, + temperature = 6, + resistance = 8.5, + eligible = 2, + household = 1 + ), + y = c( + malaria_risk = 2, + net = 2, + income = 3, + health = 1, + temperature = 3, + resistance = 2, + eligible = 3, + household = 2 + ) + ), + labels = c( + malaria_risk = "Risk of malaria", + net = "Mosquito net", + income = "Income", + health = "Health", + temperature = "Nighttime temperatures", + resistance = "Insecticide resistance", + eligible = "Eligible for program", + household = "Number in the household" + ) +) + +mosquito_dag |> + tidy_dagitty() |> + node_status() |> + ggplot( + aes(x, y, xend = xend, yend = yend, color = status) + ) + + geom_dag_edges() + + geom_dag_point() + + geom_dag_label_repel() + + scale_color_okabe_ito(na.value = "grey90") + + theme_dag() + + theme(legend.position = "none") + + coord_cartesian(clip = "off") +``` + +We'll explore how to create and analyze DAGs in @sec-dags. + +In DAGs, each point represents a variable, and each arrow represents a cause. +In other words, this diagram declares what we think the causal relationships are between these variables. +In @fig-net-data-dag, we're saying that we believe: + +- Malaria risk is causally impacted by net usage, income, health, temperature, and insecticide resistance. +- Net usage is causally impacted by income, health, temperature, eligibility for the free net program, and the number of people in a household. +- Eligibility for the free net programs is determined by income and the number of people in a household. +- Health is causally impacted by income. + +You may agree or disagree with some of these assertions. +That's a good thing! +Laying bare our assumptions allows us to consider the scientific credibility of our analysis. +Another benefit of using DAGs is that, thanks to their mathematics, we can determine precisely the subset of variables we need to account for if we assume this DAG is correct. + +::: callout-tip +## Assembling DAGs + +In this exercise, we're providing you with a reasonable DAG based on knowledge of how the data were generated. +In real life, setting up a DAG is a challenge requiring deep thought, domain expertise, and (often) collaboration between several experts. +::: + +The chief problem we're dealing with is that, when we analyze the data we're working with, we see the impact of net usage on malaria risk *and of all these other relationships*. +In DAG terminology, we have more than one open causal pathway. +If this DAG is correct, we have *eight* causal pathways: the path between net usage and malaria risk and seven other *confounding* pathways. + +```{r} +#| label: fig-net-data-confounding +#| echo: false +#| fig.width: 14 +#| fig.height: 10 +#| fig.cap: > +#| In the proposed DAG, there are eight open pathways that contribute to the +#| causal effect seen in the naive regression: the true effect (in green) of +#| net usage on malaria risk and seven other confounding pathways (in orange). +#| The naive estimate is wrong because it is a composite of all these effects. +glyph <- function(data, params, size) { + data$shape <- 15 + data$size <- 12 + ggplot2::draw_key_point(data, params, size) +} + +mosquito_dag |> + dag_paths() |> + mutate( + effects = case_when( + set == "1" & path == "open path" ~ "true effect", + path == "open path" ~ "confounding effect", + TRUE ~ NA_character_ + ), + effects = factor(effects, c("true effect", "confounding effect")) + ) |> + ggplot(aes(x = x, y = y, xend = xend, yend = yend, color = effects, alpha = path)) + + geom_dag_edges(aes(edge_alpha = path, edge_colour = effects), show.legend = FALSE) + + geom_dag_point( + data = function(.x) dplyr::filter(.x, is.na(path)), + key_glyph = glyph + ) + + geom_dag_point( + data = function(.x) dplyr::filter(.x, !is.na(path)), + key_glyph = glyph + ) + + facet_wrap(vars(fct_inorder(factor(set)))) + + expand_plot( + expand_x = expansion(c(0.25, 0.25)), + expand_y = expansion(c(0.1, 0.1)) + ) + + theme_dag() + + theme( + legend.position = "top", + legend.spacing.x = unit(8, "mm"), + legend.text = element_text(size = rel(2.5)), + legend.box.margin = margin(b = 20), + strip.text = element_blank() + ) + + coord_cartesian(clip = "off") + + scale_alpha_manual( + drop = FALSE, + values = c("open path" = 1), + na.value = .5, + breaks = "open path" + ) + + ggraph::scale_edge_alpha_manual( + drop = FALSE, + values = c("open path" = 1), + na.value = .5, + breaks = "open path" + ) + + scale_color_okabe_ito( + name = NULL, + na.value = "grey90", + order = c(3, 6), + breaks = c("true effect", "confounding effect") + ) + + scale_edge_color_okabe_ito( + name = NULL, + na.value = "grey90", + order = c(3, 6), + breaks = c("true effect", "confounding effect") + ) + + guides(alpha = "none", edge_alpha = "none") +``` + +When we calculate a naive linear regression that only includes net usage and malaria risk, the effect we see is incorrect because the seven other confounding pathways in @fig-net-data-confounding distort it. +In DAG terminology, we need to *block* these open pathways that distort the causal estimate we're after. +(We can block paths through several techniques, including stratification, matching, weighting, and more. We'll see several methods throughout the book.) Luckily, by specifying a DAG, we can precisely determine the variables we need to control for. +For this DAG, we need to control for three variables: `r glue::glue_collapse(as.list(dagitty::adjustmentSets(mosquito_dag))[[1]], sep = ", ", last = ", and ")`. +These three variables are a *minimal adjustment set*, the minimum set (or sets) of variables you need to block all confounding pathways. +We'll discuss adjustment sets further in @sec-dags. + +## Model our assumptions + +We'll use a technique called **inverse probability weighting (IPW)** to control for these variables, which we'll discuss in detail in @sec-using-ps. +We'll use logistic regression to predict the probability of treatment---the propensity score. +Then, we'll calculate inverse probability weights to apply to the linear regression model we fit above. +The propensity score model includes the exposure---net use---as the dependent variable and the minimal adjustment set as the independent variables. + +::: callout-tip +## Modeling the functional form + +Generally speaking, we want to lean on domain expertise and good modeling practices to fit the propensity score model. +For instance, we may want to allow continuous confounders to be non-linear using splines, or we may want to add essential interactions between confounders. +Because these are simulated data, we know we don't need these extra parameters (so we'll skip them), but in practice, you often do. +We'll discuss this more in @sec-using-ps. +::: + +The propensity score model is a logistic regression model with the formula `net ~ income + health + temperature`, which predicts the probability of bed net usage based on the confounders income, health, and temperature. + +```{r} +propensity_model <- glm( + net ~ income + health + temperature, + data = net_data, + family = binomial() +) + +# the first six propensity scores +head(predict(propensity_model, type = "response")) +``` + +We can use propensity scores to control for confounding in various ways. +In this example, we'll focus on weighting. +In particular, we'll compute the inverse probability weight for the **average treatment effect (ATE)**. +The ATE represents a particular causal question: what if *everyone* in the study used bed nets vs. what if *no one* in the study used bed nets? + +To calculate the ATE, we'll use the broom and propensity packages. +broom's `augment()` function extracts prediction-related information from the model and joins it to the data. +propensity's `wt_ate()` function calculates the inverse probability weight given the propensity score and exposure. + +For inverse probability weighting, the ATE weight is the inverse of probability of receiving the treatment you actually received. +In other words, if you used a bed net, the ATE weight is the inverse of the probability that you used a net, and if you did *not* use a net, it is the the inverse of the probability that you did *not* use a net. + +```{r} +library(broom) +library(propensity) +net_data_wts <- propensity_model |> + augment(data = net_data, type.predict = "response") |> + # .fitted is the value predicted by the model + # for a given observation + mutate(wts = wt_ate(.fitted, net)) + +net_data_wts |> + select(net, .fitted, wts) |> + head() +``` + +`wts` represents the amount each observation will be up-weighted or down-weighted in the outcome model we will soon fit. +For instance, the 16th household used a bed net and had a predicted probability of `r round(net_data_wts$.fitted[[16]], digits = 2)`. +That's a pretty low probability considering they did, in fact, use a net, so their weight is higher at `r round(net_data_wts$wts[[16]], digits = 2)`. +In other words, this household will be up-weighted compared to the naive linear model we fit above. +The first household did *not* use a bed net; they're predicted probability of net use was `r round(net_data_wts$.fitted[[1]], digits = 2)` (or put differently, a predicted probability of *not* using a net of `r 1 - round(net_data_wts$.fitted[[1]], digits = 2)`). +That's more in line with their observed value of `net`, but there's still some predicted probability of using a net, so their weight is `r round(net_data_wts$wts[[2]], digits = 2)`. + +## Diagnose our models + +The goal of propensity score weighting is to weight the population of observations such that the distribution of confounders is balanced between the exposure groups. +Put another way, we are, in principle, removing the arrows between the confounders and exposure in the DAG, so that the confounding paths no longer distort our estimates. +Here's the distribution of the propensity score by group, created by `geom_mirror_histogram()` from the halfmoon package for assessing balance in propensity score models: + +```{r} +#| label: fig-mirror-histogram-net-data-unweighted +#| fig.cap: > +#| A mirrored histogram of the propensity scores of those who used nets (top, blue) versus those who did not use nets (bottom, orange). The range of propensity scores is similar between groups, with those who used nets slightly to the left of those who didn't, but the shapes of the distribution are different. +library(halfmoon) +ggplot(net_data_wts, aes(.fitted)) + + geom_mirror_histogram( + aes(fill = net), + bins = 50 + ) + + scale_y_continuous(labels = abs) + + labs(x = "propensity score") +``` + +The weighted propensity score creates a pseudo-population where the distributions are much more similar: + +```{r} +#| label: fig-mirror-histogram-net-data-weighted +#| fig.cap: > +#| A mirrored histogram of the propensity scores of those who used nets (top, blue) versus those who did not use nets (bottom, orange). The shaded region represents the unweighted distribution, and the colored region represents the weighted distributions. The ATE weights up-weight the groups to be similar in range and shape of the distribution of propensity scores. +ggplot(net_data_wts, aes(.fitted)) + + geom_mirror_histogram( + aes(group = net), + bins = 50 + ) + + geom_mirror_histogram( + aes(fill = net, weight = wts), + bins = 50, + alpha = .5 + ) + + scale_y_continuous(labels = abs) + + labs(x = "propensity score") +``` + +In this example, the unweighted distributions are not awful---the shapes are somewhat similar here, and they overlap quite a bit---but the weighted distributions in @fig-mirror-histogram-net-data-weighted are much more similar. + +::: callout-caution +## Unmeasured confounding + +Propensity score weighting and most other causal inference techniques only help with *observed* confounders---ones that we model correctly, at that. +Unfortunately, we still may have unmeasured confounding, which we'll discuss below. + +Randomization is one causal inference technique that *does* deal with unmeasured confounding, one of the reasons it is so powerful. +::: + +We might also want to know how well-balanced the groups are by each confounder. +One way to do this is to calculate the **standardized mean differences (SMDs)** for each confounder with and without weights. +We'll calculate the SMDs with `tidy_smd()` then plot them with `geom_love()`. + +```{r} +#| label: fig-love-plot-net-data +#| fig.cap: > +#| A love plot representing the standardized mean differences (SMD) between exposure groups of three confounders: temperature, income, and health. Before weighting, there are considerable differences in the groups. After weighting, the confounders are much more balanced between groups. +plot_df <- tidy_smd( + net_data_wts, + c(income, health, temperature), + .group = net, + .wts = wts +) + +ggplot( + plot_df, + aes( + x = abs(smd), + y = variable, + group = method, + color = method + ) +) + + geom_love() +``` + +A standard guideline is that balanced confounders should have an SMD of less than 0.1 on the absolute scale. +0.1 is just a rule of thumb, but if we follow it, the variables in @fig-love-plot-net-data are well-balanced after weighting (and unbalanced before weighting). + +Before we apply the weights to the outcome model, let's check their overall distribution for extreme weights. +Extreme weights can destabilize the estimate and variance in the outcome model, so we want to be aware of it. +We'll also discuss several other types of weights that are less prone to this issue in @sec-estimands. + +```{r} +#| label: fig-ate-density-net-data +#| fig.cap: > +#| A density plot of the average treatment effect (ATE) weights. The plot is skewed, with higher values towards 8. This may indicate a problem with the model, but the weights aren't so extreme to destabilize the variance of the estimate. +net_data_wts |> + ggplot(aes(wts)) + + geom_density(fill = "#CC79A7", color = NA, alpha = 0.8) +``` + +The weights in @fig-ate-density-net-data are skewed, but there are no outrageous values. +If we saw extreme weights, we might try trimming or stabilizing them, or consider calculating an effect for a different estimand, which we'll discuss in @sec-estimands. +It doesn't look like we need to do that here, however. + +## Estimate the causal effect + +We're now ready to use the ATE weights to (attempt to) account for confounding in the naive linear regression model. +Fitting such a model is pleasantly simple in this case: we fit the same model as before but with `weights = wts`, which will incorporate the inverse probability weights. + +```{r} +net_data_wts |> + lm(malaria_risk ~ net, data = _, weights = wts) |> + tidy(conf.int = TRUE) +``` + +```{r} +#| include = FALSE +estimates <- net_data_wts |> + lm(malaria_risk ~ net, data = _, weights = wts) |> + tidy(conf.int = TRUE) |> + filter(term == "netTRUE") |> + select(estimate, starts_with("conf")) |> + mutate(across(everything(), round, digits = 1)) +``` + +The estimate for the average treatment effect is `r est_ci(estimates)`. +Unfortunately, the confidence intervals we're using are wrong because they don't account for the uncertainty in estimating the weights. +Generally, confidence intervals for propensity score weighted models will be too narrow unless we account for this uncertainty. +The nominal coverage of the confidence intervals will thus be wrong (they aren't 95% CIs because their coverage is much lower than 95%) and may lead to misinterpretation. + +We've got several ways to address this problem, which we'll discuss in detail in @sec-outcome-model, including the bootstrap, robust standard errors, and manually accounting for the estimation procedure with empirical sandwich estimators. +For this example, we'll use the bootstrap, a flexible tool that calculates distributions of parameters using re-sampling. +The bootstrap is a useful tool for many causal models where closed-form solutions to problems (particularly standard errors) don't exist or when we want to avoid parametric assumptions inherent to many such solutions; see @sec-appendix-bootstrap for a description of what the bootstrap is and how it works. +We'll use the rsample package from the tidymodels ecosystem to work with bootstrap samples. + +Because the bootstrap is so flexible, we need to think carefully about the sources of uncertainty in the statistic we're calculating. +It might be tempting to write a function like this to fit the statistic we're interested in (the point estimate for `netTRUE`): + +```{r} +#| eval = FALSE +library(rsample) + +fit_ipw_not_quite_rightly <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} +``` + +However, this function won't give us the correct confidence intervals because it treats the inverse probability weights as fixed values. +They're not, of course; we just estimated them using logistic regression! +We need to account for this uncertainty by bootstrapping the *entire modeling process*. +For every bootstrap sample, we need to fit the propensity score model, calculate the inverse probability weights, then fit the weighted outcome model. + +```{r} +library(rsample) + +fit_ipw <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit propensity score model + propensity_model <- glm( + net ~ income + health + temperature, + data = .df, + family = binomial() + ) + + # calculate inverse probability weights + .df <- propensity_model |> + augment(type.predict = "response", data = .df) |> + mutate(wts = wt_ate(.fitted, net)) + + # fit correctly bootstrapped ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} +``` + +Now that we know precisely how to calculate the estimate for each iteration let's create the bootstrapped dataset with rsample's `bootstraps()` function. +The `times` argument determines how many bootstrapped datasets to create; we'll do 1,000. + +```{r} +bootstrapped_net_data <- bootstraps( + net_data, + times = 1000, + # required to calculate CIs later + apparent = TRUE +) + +bootstrapped_net_data +``` + +The result is a nested data frame: each `splits` object contains metadata that rsample uses to subset the bootstrap samples for each of the 1,000 samples. +We actually have 1,001 rows because `apparent = TRUE` keeps a copy of the original data frame, as well, which is needed for some times of confidence interval calculations. +Next, we'll run `fit_ipw()` 1,001 times to create a distribution for `estimate`. +At its heart, the calculation we're doing is + +``` r +fit_ipw(bootstrapped_net_data$splits[[n]]) +``` + +Where *n* is one of 1,001 indices. +We'll use purrr's `map()` function to iterate across each `split` object. + +```{r} +ipw_results <- bootstrapped_net_data |> + mutate(boot_fits = map(splits, fit_ipw)) + +ipw_results +``` + +The result is another nested data frame with a new column, `boot_fits`. +Each element of `boot_fits` is the result of the IPW for the bootstrapped dataset. +For example, in the first bootstrapped data set, the IPW results were: + +```{r} +ipw_results$boot_fits[[1]] +``` + +Now we have a distribution of estimates: + +```{r} +#| label: fig-bootstrap-estimates-net-data +#| message: false +#| warning: false +#| fig.cap: > +#| "A histogram of 1,000 bootstrapped estimates of the effect of net use on malaria risk. The spread of these estimates accounts for the dependency and uncertainty in the use of IPW weights." +ipw_results |> + # remove original data set results + filter(id != "Apparent") |> + mutate( + estimate = map_dbl( + boot_fits, + # pull the `estimate` for `netTRUE` for each fit + \(.fit) .fit |> + filter(term == "netTRUE") |> + pull(estimate) + ) + ) |> + ggplot(aes(estimate)) + + geom_histogram(fill = "#D55E00FF", color = "white", alpha = 0.8) +``` + +@fig-bootstrap-estimates-net-data gives a sense of the variation in `estimate`, but let's calculate 95% confidence intervals from the bootstrapped distribution using rsample's `int_t()` : + +```{r} +boot_estimate <- ipw_results |> + # calculate T-statistic-based CIs + int_t(boot_fits) |> + filter(term == "netTRUE") + +boot_estimate +``` + +Now we have a confounder-adjusted estimate with correct standard errors. +The estimate of the effect of *all* households using bed nets versus *no* households using bed nets on malaria risk is `r est_ci(boot_estimate, rsample = TRUE)`. +Bed nets do indeed seem to reduce malaria risk in this study. + +## Conduct sensitivity analysis on the effect estimate + +We've laid out a roadmap for taking observational data, thinking critically about the causal question we want to ask, identifying the assumptions we need to get there, then applying those assumptions to a statistical model. +Getting the correct answer to the causal question relies on getting our assumptions more or less right. +But what if we're more on the less correct side? + +Spoiler alert: the answer we just calculated is *wrong*. +After all that effort! + +When conducting a causal analysis, it's a good idea to use sensitivity analyses to test your assumptions. +There are many potential sources of bias in any study and many sensitivity analyses to go along with them (@sec-sensitivity); here, we'll focus on the assumption of no confounding. + +Let's start with a broad sensitivity analysis; then, we'll ask questions about specific unmeasured confounders. +When we have less information about unmeasured confounders, we can use tipping point analysis to ask how much confounding it would take to tip my estimate to the null. +In other words, what would the strength of the unmeasured confounder have to be to explain our results away? +The tipr package is a toolkit for conducting sensitivity analyses. +Let's examine the tipping point for an unknown, normally-distributed confounder. +The `tip_coef()` function takes an estimate (a beta coefficient from a regression model, or the upper or lower bound of the coefficient). +It further requires either the 1) scaled differences in means of the confounder between exposure groups or 2) effect of the confounder on the outcome. +For the estimate, we'll use `conf.high`, which is closer to 0 (the null), and ask: how much would the confounder have to affect malaria risk to have an unbiased upper confidence interval of 0? +We'll use tipr to calculate this answer for 5 scenarios, where the mean difference in the confounder between exposure groups is 1, 2, 3, 4, or 5. + +```{r} +#| echo: false +options(tipr.verbose = FALSE) +``` + +```{r} +#| label: fig-tip-coef-net +#| fig.cap: > +#| A tipping point analysis under several confounding scenarios where the unmeasured confounder is a normally-distributed continuous variable. The line represents the strength of confounding necessary to tip the upper confidence interval of the causal effect estimate to 0. The x-axis represents the coefficient of the confounder-outcome relationship adjusted for the exposure and the set of measured confounders. The y-axis represents the scaled mean difference of the confounder between exposure groups. +library(tipr) +tipping_points <- tip_coef(boot_estimate$.upper, exposure_confounder_effect = 1:5) + +tipping_points |> + ggplot(aes(confounder_outcome_effect, exposure_confounder_effect)) + + geom_line(color = "#009E73", linewidth = 1.1) + + geom_point(fill = "#009E73", color = "white", size = 2.5, shape = 21) + + labs( + x = "Confounder-Outcome Effect", + y = "Scaled mean differences in\n confounder between exposure groups" + ) +``` + +If we had an unmeasured confounder where the standardized mean difference between exposure groups was 1, the confounder would need to decrease malaria risk by about `r round(tipping_points$confounder_outcome_effect[[1]], digits = 1)`. +That's pretty strong relative to other effects, but it may be feasible if we have an idea of something we might have missed. +Conversely, suppose the relationship between net use and the unmeasured confounder is very strong, with a mean scaled difference of 5. +In that case, the confounder-malaria relationship only needs to be `r round(tipping_points$confounder_outcome_effect[[5]], digits = 1)`. +Now we have to consider: which of these scenarios are plausible given our domain knowledge and the effects we see in this analysis? + +Now let's consider a much more specific sensitivity analysis. +Some ethnic groups, such as the Fulani, have a genetic resistance to malaria [@arama2015]. +Let's say that in our simulated data, an unnamed ethnic group in the unnamed country shares this genetic resistance to malaria. +For historical reasons, bed net use in this group is also very high. +We don't have this variable in `net_data`, but let's say we know from the literature that in this sample, we can estimate at: + +1. People with this genetic resistance have, on average, a lower malaria risk by about 10. +2. About 26% of people who use nets in our study have this genetic resistance. +3. About 5% of people who don't use nets have this genetic resistance. + +With this amount of information, we can use tipr to adjust the estimates we calculated for the unmeasured confounder. +We'll use `adjust_coef_with_binary()` to calculate the adjusted estimates. + +```{r} +adjusted_estimates <- boot_estimate |> + select(.estimate, .lower, .upper) |> + unlist() |> + adjust_coef_with_binary( + exposed_confounder_prev = 0.26, + unexposed_confounder_prev = 0.05, + confounder_outcome_effect = -10 + ) + +adjusted_estimates +``` + +The adjusted estimate for a situation where genetic resistance to malaria is a confounder is `r est_ci(adjusted_estimates$effect_adjusted)`. + +In fact, these data were simulated with just such a confounder. +The true effect of net use on malaria is about -10, and the true DAG that generated these data is: + +```{r} +#| label: fig-net-data-true-dag +#| echo: false +#| fig.cap: > +#| The true causal diagram for `net_data`. This DAG is identical to the one we proposed with one addition: genetic resistance to malaria causally reduces the risk of malaria and impacts net use. It's thus a confounder and a part of the minimal adjustment set required to get an unbiased effect estimate. In otherwords, by not including it, we've calculated the wrong effect. +mosquito_dag_full <- dagify( + malaria_risk ~ net + income + health + temperature + insecticide_resistance + genetic_resistance, + net ~ income + health + temperature + eligible + household + genetic_resistance, + eligible ~ income + household, + health ~ income, + exposure = "net", + outcome = "malaria_risk", + coords = list( + x = c( + malaria_risk = 7, + net = 3, + income = 4, + health = 5, + temperature = 6, + insecticide_resistance = 8.5, + eligible = 2, + household = 1, + genetic_resistance = 8.5 + ), + y = c( + malaria_risk = 2, + net = 2, + income = 3, + health = 1, + temperature = 3, + insecticide_resistance = 2, + eligible = 3, + household = 2, + genetic_resistance = 1 + ) + ), + labels = c( + malaria_risk = "Risk of malaria", + net = "Mosquito net", + income = "Income", + health = "Health", + temperature = "Nighttime temperatures", + insecticide_resistance = "Insecticide resistance", + eligible = "Eligible for program", + household = "Number in household", + genetic_resistance = "Malaria resistance" + ) +) + +mosquito_dag_full |> + tidy_dagitty() |> + node_status() |> + ggplot( + aes(x, y, xend = xend, yend = yend, color = status) + ) + + geom_dag_edges() + + geom_dag_point() + + geom_dag_label_repel() + + scale_color_okabe_ito(na.value = "grey90") + + theme_dag() + + theme(legend.position = "none") + + coord_cartesian(clip = "off") +``` + +```{r} +#| include: false +fit_ipw_full <- function(.split, ...) { + # get bootstrapped data frame + .df <- as.data.frame(.split) + + # fit propensity score model + propensity_model <- glm( + net ~ income + health + temperature + genetic_resistance, + data = .df, + family = binomial() + ) + + # calculate inverse probability weights + .df <- propensity_model |> + augment(type.predict = "response", data = .df) |> + mutate(wts = wt_ate(.fitted, net)) + + # fit correctly bootstrapped ipw model + lm(malaria_risk ~ net, data = .df, weights = wts) |> + tidy() +} + +bootstrapped_net_data_full <- bootstraps( + net_data_full, + times = 1000, + # required to calculate CIs later + apparent = TRUE +) + +ipw_results_full <- bootstrapped_net_data_full |> + mutate(boot_fits = map(splits, fit_ipw_full)) + +boot_estimate_full <- ipw_results_full |> + # calculate T-statistic-based CIs + int_t(boot_fits) |> + filter(term == "netTRUE") +``` + +The unmeasured confounder in @fig-net-data-true-dag is available in the dataset `net_data_full` as `genetic_resistance`. +If we recalculate the IPW estimate of the average treatment effect of nets on malaria risk, we get `r est_ci(boot_estimate_full, rsample = TRUE)`, much closer to the actual answer of -10. + +What do you think? +Is this estimate reliable? +Did we do a good job addressing the assumptions we need to make for a causal effect, mainly that there is no confounding? +How might you criticize this model, and what would you do differently? +Ok, we know that -10 is the correct answer because the data are simulated, but in practice, we can never be sure, so we need to continue probing our assumptions until we're confident they are robust. +We'll explore these techniques and others in @sec-sensitivity. + + +To calculate this effect, we: + +1. Specified a causal question (for the average treatment effect) +2. Drew our assumptions using a causal diagram (using DAGs) +3. Modeled our assumptions (using propensity score weighting) +4. Diagnosed our models (by checking confounder balance after weighting) +5. Estimated the causal effect (using inverse probability weighting) +6. Conducted sensitivity analysis on the effect estimate (using tipping point analysis) + +Throughout the rest of the book, we'll follow these broad steps in examples from many domains. +We'll dive more deeply into propensity score techniques, explore other methods for estimating causal effects, and, most importantly, make sure, over and over again, that the assumptions we're making are reasonable---even if we'll never know for sure. diff --git a/src/file_counts/file_count_log.md b/src/file_counts/file_count_log.md index d12cecb..b4ec454 100644 --- a/src/file_counts/file_count_log.md +++ b/src/file_counts/file_count_log.md @@ -4,3 +4,4 @@ 20240826 30 20241120 36 20241129 47 +20241231 77