# Probit regression

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:

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

SciStat.com 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).

### 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, SciStat.com 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(L_{0}) where L_{0} 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 R^{2} and Nagelkerke R^{2} 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 R^{2} 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:

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:

### 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 x Dose

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

Alternatively, you can use the following table

.Probit(p) | p |
---|---|

2.326 | 0.99 |

1.645 | 0.95 |

1.282 | 0.90 |

0.842 | 0.80 |

0.000 | 0.50 |

-0.842 | 0.20 |

-1.282 | 0.10 |

-1.645 | 0.05 |

-2.326 | 0.01 |

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 SciStat.com 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 x 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, SciStat.com 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 - 2
^{nd}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.

## See also

## Link

Go to Probit regression.