causal inferance page

pages/altman_bland_analysis_of_methods.md

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+# Stats Analysis of methods
+
+created: 16 Sept 2023
 
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+---
+
 Altman, D. G., and J. M. Bland. "Measurement in Medicine: The Analysis of Method Comparison Studies." Journal of the Royal Statistical Society. Series D (The Statistician), vol. 32, no. 3, 1983, pp. 307–17. JSTOR, <https://doi.org/10.2307/2987937>.
 
 The paper is a pivotal guide discussing the analysis of method comparison studies, particularly in the field of medicine.

pages/altman_bland_ci_from_p.md

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+title: Stats CI from P
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+---
+
+# Stats CI from P
+
+* TOC
+{:toc}
 
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+---
 
+{: .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: <https://doi.org/10.1136/bmj.d2090> (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)?
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  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

pages/altman_bland_correlation.md

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+title: Stats Correlation, regression and repeated data
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-title: Stats Correlation, regression and repeated data
-created: 29 June 2021
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+# Stats Correlation, regression and repeated data
+
+created: 29 June 2021
 
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+---
+
 ## Introduction
 
 This topic is introduced as the first paper 

pages/altman_bland_odds_ratios.md

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-created: 4 July 2021
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+created: 4 July 2021
+# Stats Odds ratios, SE & CI
 
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+---
+
 ## Introduction
 
 Altman & Bland review the use of odds ratio (OR), standard error (SE), and confidence interval (CI) with some examples in 

pages/altman_bland_roc_curve.md

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+created: 2021-07-16
+# Stats Receiver operating characteristic plots
 
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+---
+
 ## 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) <a href="http://www.bmj.com/cgi/content/full/309/6948/188"> Diagnostic tests 3: receiver operating characteristic plots.</a> <b>309</b>, 188,

pages/altman_bland_sensitivity_specificity.md

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+title: Stats Sensitivity and specificity
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-title: Stats Sensitivity and specificity
-created: 19 July 2021
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+created: 19 July 2021
+
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