3.3.1. Dose‐responscurve

The logit function is used to calculate a sigmoid curve (S-shaped curve) as close as possible, minimized in the Y-axis, by calculating a dose-response curve between the blank and maxRespons (Bmax) (see figure A). With the calculated parameters of the logit formula, the dose is calculated from a measured response. The dose-response curve has the following relationship:

eLog (dose) = eLog ((response - blank) / (Bmax - response))

eLog(x) = eLog ((y-C) / (D-y)).

All responses are transformed with the logit function (eLog ((response - blank) / (Bmax - response)) and a e-based logarithm is taken from the dose (see Figure B).

Left: Figure A Dose-responsecurve .......................... Right: Figure B Dose-responsecurve after transformation

A linear regression is applied to these transformed values and the correlation R is calculated. The calculation of the logit regression parameters is iterative, that is to say in this case that a Bmax is searched for where the correlation R is maximal. A response slightly larger than the response of the highest dose is used as the initial Bmax parameter and with this Bmax the responses are transformed and the correlation R is calculated. The Bmax is then increased by 0.001% and compared to see if the correlation increases. This process is repeated until the correlation drops. The Bmax is then lowered by 10 times smaller steps until the correlation drops again. The Bmax is then increased again by 10 times smaller. This process is repeated until the correlation is equal to the previous correlation or a limit in iteration steps is reached. The latter is the case if a perfect straight line is calculated. Calculating the dose is done with the following formula:

All values to be entered in the formula are placed in the worksheet.

• The blank is a mandatory measurement point in the calibration line. The blank has concentration 0 (zero). • It is incorrect to subtract the blank from the measurements and then use it in the logit regression. In the algorithm, the blank in the numerator is already subtracted from the numerator and not in the denominator. Sometimes the measuring equipment corrects this on its own and <= 0 responses can occur. It is better to turn off that correction and to work with the most raw measurement data. Beside this the blank is masked from the results. • It is wrong to first averaging responses and then use them in the logit regression. By averaging, the spread of the test is masked and an incorrect measuring point cannot be removed because it is already average. • Five dose calibration lines are preferred, pipetted in duplicate or triplicate. More than seven points is discouraged. This is unnecessary, and often causes too long calibration lines that flatten. • Preferably invalidate no more than two measurements with a five dose calibration pipetted in duplicate or no more than three points with triplicate measurements. • Do not invalidate all measuring points of one dose. • Use logarithm spaced dilutions: e.g. • 1:2, 1:4, 1:16,1:32 or 1:3, 1:9,1:27,1:81