Skip to main content
MedCalc
Mail a PDF copy of this page to:
(Your email address will not be added to a mailing list)
working
Show menu

Probit regression (Dose-Response analysis)

Command:Statistics
Next selectRegression
Next selectProbit 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:

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

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:

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

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

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

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:

probit data entry - binary format

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:

probit data entry - grouped format

Required input

Probit regression. Dialog box for grouped data.

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

Variables in case of grouped data

Filter

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

Options

Results

Probit regression. Results tables.

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:

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

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:

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

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:

probit = −2.61 + 6.36 × 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.

Dose-response plot

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

Dose-response plot

Literature

See also

External links

Recommended book

Probit Analysis
David Finney

Buy from Amazon US - CA - UK - DE - FR - ES - IT

Originally published in 1947, this classic study by D. J. Finney was the first to examine and explain a branch of statistical method widely used in connection with the biological assay of insecticides, fungicides, drugs, vitamins, etc. It standardized the computations and terminology and made its use easier for a biologist without statistical expertise, whilst also outlining the underlying mathematical theory. Finney had made several important contributions to the method in the past, and his own results are also included.