MedCalc

# Probit regression (Dose-Response analysis)

 Command: StatisticsRegressionProbit regression (Dose-Response analysis)

## Description

The probit regression procedure fits a probit sigmoid dose-response curve and calculates values (with 95% CI) of the dose variable that correspond to a series of probabilities. For example the ED50 (median effective dose) or (LD50 median lethal dose) are the values corresponding to a probability of 0.50, the Limit-of-Detection (CLSI, 2012) is the value corresponding to a probability of 0.95.

The probit regression equation has the form:

$$\operatorname{probit}(p) = a + b \times X$$

Where X is the (possibly log-transformed) dose variable and probit(p) is the value of the inverse standard normal cumulative distribution function Φ-1 corresponding with a probability p:

$$\operatorname{probit}(p) = \Phi^{-1}(p)$$

Probit(p) can be transformed to a probability p using the standard normal cumulative distribution function Φ:

$$p = \Phi(\operatorname{probit}(p))$$

MedCalc fits the regression coefficients a and b using the method of maximum likelihood.

## How to enter data

You can enter the data in binary format or in grouped format.

### Binary

In the binary format, you have 2 variables, one variable for the dose (concentration) and one for the binary response.

For each single measurement, there is a row with the dose and the response, which is coded 0 (no response) and 1 (response).

For example:

### Grouped

In the grouped format, you have 3 variables, one variable for the dose, one for the total number of measurements, and one for the number of measurements with a response.

For example:

## Required input

### Data type

Select the option corresponding to the way you have entered the data: binary or grouped (see above).

### Dose variable

Select the dose variable.

### Variables in case of binary data

• Response variable: the response variable must be binary or dichotomous, and should only contain data coded as 0 (no response) or 1 (response). If your data are coded differently, you can use the Define status tool to recode your data.

### Variables in case of grouped data

• Total number of cases: select the variable that contains the number of measurements for each dose.
• Number of responses: select the variable that contains the number of responses for each dose.

### Filter

(Optionally) enter a data filter in order to include only a selected subgroup of cases in the analysis.

### Options

• Log transformation: select this option if the dose variable requires a logarithmic transformation. When the dose variable contains 0 values, MedCalc will automatically add a small number to the data in order to make the logarithmic transformation possible. This small number will be subtracted when the results are backtransformed for presentation.
• Dose-response plot: select this option to obtain a dose-response plot.

Markers: click this option to have the data represented in the graph as markers. Note that when you have selected logarithmic transformation and the dose variable contains 0 values, these values cannot be represented in the graph as markers.

## Results

### Sample size and cases with negative and positive outcome

First the program gives sample size and the number and proportion of cases with and without response.

### Overall model fit

The null model −2 Log Likelihood is given by −2 * ln(L0) where L0 is the likelihood of obtaining the observations in the "null" model, a model without the dose variable.

The full model −2 Log Likelihood is given by −2 * ln(L) where L is the likelihood of obtaining the observations with the dose variable incorporated in the model.

The difference of these two yields a Chi-Squared statistic which is a measure of how well the dose variable affects the response variable.

Cox & Snell R2 and Nagelkerke R2 are other goodness of fit measures known as pseudo R-squareds. Note that Cox & Snell's pseudo R-squared has a maximum value that is not 1. Nagelkerke R2 adjusts Cox & Snell's so that the range of possible values extends to 1.

### Regression coefficients

The regression coefficients are the coefficients a (constant) and b (slope) of the regression equation:

$$\operatorname{probit}(p) = a + b \times X$$

The Wald statistic is the regression coefficient divided by its standard error squared: (b/SE)2.

### Log transformation

When you have selected logarithmic transformation of the dose variable, then a and b are in fact the coefficients of the regression equation:

$$\operatorname{probit}(p) = a + b \times \operatorname{Log}(X)$$

### Use of the fitted equation

The predicted probability of a positive response can be calculated using the regression equation.

When the regression equation is for example:

$$\operatorname{probit} = -2.61 + 6.36 \times Dose$$

then for a Dose of 0.500 probit(p) equals 0.57. Probit(p) can be transformed to p by the MedCalc spreadsheet function NORMSDIST(z) or the equivalent Excel function.

Alternatively, you can use the following table.

Probit(p)p
2.3260.99
1.6450.95
1.2820.90
0.8420.80
0.0000.50
-0.8420.20
-1.2820.10
-1.6450.05
-2.3260.01

In the example, with probit(p) equal to 0.57, p = 0.72.

A probability p can be transformed to Probit(p) using the table above or using the MedCalc spreadsheet function NORMSINV(p) or the equivalent Excel function. For a probability p=0.5 you find in the table that probit(p)=0. When the regression equation is

probit = −2.61 + 6.36 × Dose

then

Dose = (probit+2.61)/6.36

and therefore dose = 2.61/6.36 = 0.41.

## Dose-Response table

This table lists a series of Probabilities with corresponding Dose, with a 95% confidence interval for the dose (Finney, 1947).

Values in light gray text color are dose values that fall outside the observed range of the dose variable.

### Log transformation

When you have selected logarithmic transformation of the dose variable, MedCalc will backtransform the results and display the dose variable on its original scale in the Dose-Response table.

## Graph

This graph shows the probabilities, ranging from 0 to 1, and the corresponding dose. Two additional curves represent the 95% confidence interval for the dose.

The dose and 95% confidence interval, corresponding with a particular probability, are taken from a horizontal line at that probability level.

## Literature

• CLSI (2012) Evaluation of detection capability for clinical laboratory measurement procedures; Approved guideline - 2nd edition. CLSI document EP17-A2. Wayne, PA: Clinical and Laboratory Standards Institute.
• Finney DJ (1947) Probit Analysis. A statistical treatment of the sigmoid response curve. Cambridge: Cambridge University Press.