Lars K. Madson                                                                                            Annotated Bibliography

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Roncek, Dennis W. 1991.  Using logit coefficients to obtain the effects of independent variables

on change in probabilities.  Social Forces.  70(2): 509-518.

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Introduction

            Roncek argues that a variety of hypotheses can be posed in which predictive probability models are tests.  The impacts on the probability of dichotomous dependent variable can be tested through the use of a logit model or logistic regression.  The purpose of the logit equation is to estimate the predicted probability of the dichotomous dependent variable.  In this case, the intent is to determine the change in predicted probability for a dichotomous dependent variable regressed on a series of independent variables.  In order to simplify the calculation of the change in predicted probability, dichotomous indepe3ndent variables, measured as 0 or 1, and continuous independent variables, evaluated at the mean, can be used to calculate the change in predicted probability between the absence or presence of a particular independent variable, controlling for specified independent variable.  The advantage to this procedure is that the change in predicted probability can be interpreted in terms of “times as likely” for the dependent variable between units of analysis with the absence of a chosen independent variable and the same units of analysis with the independent variable present.  A similar procedure may be to examine the change in predicted probability of a dichotomous dependent variable between male and female, white and not white, or some other dichotomous independent variable.

Model

Logit models are estimated in the form:

1n(P_Y/1-P_Y) = a_i + B₁(X₁) + B₂(X₂) + B₃(X₃) + B_i(X_i) + e_i.

in which the equation resembles an ordinary least square regression.  In the equation, 1n(P_Y/1-P_Y) is the natural logarithm of the predicted value of the odds-ratio 1n(P_Y/1-P_Y) of the dichotomous dependent variable (i.e., having a score of 1).  P is the probability of having a score of 1 (Pindyck and Rubinfeld 1976). The B’s are the logistic regression coefficients for the independent variables, which will be used to calculate the change in predicted probabilities of the dichotomous dependent variable between varying independent variables and specific units of analysis (i.e., assigning a score of 1 for the dichotomous independent variable of interest, and the mean for the continuously measured independent variable of interest, and the mean for the continuously measured independent variables), as previously discussed.

Technique

            The techniques for calculating the change in predicted probabilities consists of the following:  (1) selecting the type of case to calculate the effect of the absence or presence of a selected independent variable on the dependent variable; (2) computing the natural logarithm of the predicted odds-ratios by setting the constant to 0, multiplying the dichotomous independent variable logit coefficients by 1 for the desired case and the continuous independent variable logit coefficients by their mean, and summing for the case with and without the selected independent variable coefficient, respectively; (3) exponentiating the natural logarithm of the predicted odds-ratios for the case with and without the selected independent variable, respectively; (4) calculating the predicted probability of dependent variable for the case with and without the selected independent variable by dividing the predicted odds-ratio by 1 plus the predicted odds-ratio, respectively; and (5) subtracting the presence model’s predicted probability from the absences model’s predicted probability, rendering the change in predicted probability for the presence of the independent variable (Roncek 1991).

Example

            An example of the results, as hypothesized, for 25 year-old burglary convicts, males will increase the predicted probability of re-offending by 35 percent relative to females of the same age and conviction.  In other words, 25 year-old males convicted of burglary are 35 percent more likely to re-offend (i.e., recidivism is the dichotomous dependent variable) than 25 year-old females convicted of burglary.  Furthermore, the change in predicted probability for recidivism will render the change in predicted probability for not offending again.  For the above example, if the change in predicted probability for re-offending is a 35 percent increase for males, then the change in predicted probability for not re-offending would be 35 percent decrease for females.

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References

Pindyck, R.S. and D.L. Rubinfeld.  1976.  Econometrics Models and Econometric Forecasts.

New York, NY:  McGraw Hill Book Co.

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