Using Logit Coefficients to Obtain the Effects of Independent Variables on Changes in Probabilities

Unnecessarily awkward procedures for interpreting logit coefficients in terms of odds-ratios or ratios of odds-ratios appear in sociological research. Because odds-ratios are not linearly related to probabilities, statements about relative odds-ratios do not identify the effects of group membership as measured by dummy independent variables on a dichotomous dependent variable. Procedures from econometrics allow calculating the probabilities of being in the category of interest of a dichotomous dependent variable. The effects of dummy independent variables equal the differences in probabilities among the different groups. For continuous independent variables, the maximum possible effect on changes in probability is always one-quarter of the value of the logit coefficient. Interpreting logit coefficients for dichotomous dependent variables remains a problem in sociological research as illustrated by the variation in the ways that logit coefficients are used. Smith and Jarjoura's (1989) criminological research focuses only on the direction and statistical significance of the coefficients. Kelley's (1990) study in the sociology of work uses strategies derived from econometrics, although no source is cited. Baer, Grabb, and Johnston's (1990) work in political sociology apparently adapts methods from log-linear analysis in which the dependent and independent variables are all categorical. Given the absence in the sociological methodological literature of a full treatment on interpreting coefficients in logit models with dichotomous dependent variables, this lack of consistency is not surprising. The only attention to this problem is found in Long (1987:421-5). The proposed strategy of exponentiating logit coefficients, however, does not permit one to relate logit coefficients directly to probabilities and, thus, can be misleading. My purpose is to show how to interpret logit coefficients in terms of probabilities. I explain i) The Uriversity of North Carolina Press Social Forces, December 1991, 70(2):509-518 * I am grateful to Harold L Orbach, William C. Rau, and David B. Rottman for detailed comments on earlier drafts. Colleagues in the Department of Economics and the Department of Statistics also provided valuable consultations. The Department of Mathematics at Kansas State University generously provided the resources to produce this article. Any remainingfaults are solely the author's responsibility. Direct correspondence to Dennis W. Roncek, Department of Sociology, Kansas State University, Manhattan, KS 66506. This content downloaded from 157.55.39.177 on Tue, 15 Nov 2016 03:52:55 UTC All use subject to http://about.jstor.org/terms 510 / Social Forces 70:2, December 1991 how to find the effects of dummy independent variables that represent group membership, as well as those of continuous variables, using simple strategies from econometrics. Additionally, I show how to estimate the feasibility of change for models with at least one continuous independent variable. Interpreting Logit Coefficients for Dummy Independent Variables THE PROBLEM WiTH A COMMON PROCEDURE Baer, Grabb and Johnston (1990) illustrates an alluring pitfall in interpreting logit coefficients for dummy independent variables. Although they do not cite Long (1987), they used the method of exponentiating logit coefficients to analyze the effects of being Canadian on two dichotomous dependent variables, whether an individual responds "yes" to feeling (1) that they have no say in government and (2) that the government does not care. They examine how English-Canadians and Quebeckers each differ from Americans. Two dummy independent variables reflect the three ethnicities. Americans are the omitted group. Baer, Grabb, and Johnston interpret the exponentiated logit coefficients for being English or being a Quebecker as the "times as likely" that members of each group respond "yes" to each of the two questions relative to Americans. Their procedure implies that there is a special advantage to using dichotomous independent variables because it is possible to interpret coefficients in terms of "times as likely" since such variables only have two categories. Understanding why their procedure is misleading requires examining the form of the logit model in equation 1 which resembles that of an ordinary regression. A In( ) =a +b1 *Xi +b2 *X2+''' (1) A In equation 1,ln(P/1 P) is the natural logarithm of the predicted value of the A odds-ratio (P/1 P) of being in the category of interest, i.e., having a score of 1 on the dichotomous dependent variable. P is the probability of having a score of 1. The a is a constant analogous to the intercept in ordinary least squares (OLS). The b's are logistic regression coefficients. Aldrich and Nelson (1984) provide a concise review of the advantages of logistic regression over multiple regression (OLS) for dichotomous dependent variables. Equation l's resemblance to OLS is limited. Its b-coefficients, at first glance, are the change in the natural logarithm of the predicted odds-ratio for a one unit change in a specific independent variable (Studenmund & Cassidy 1987).1 Yet, the logarithm of the odds-ratio is not easily related to probabilities because the logarithm is a nonlinear function of the odds-ratio, which itself is a nonlinear function of the probability of being in the category of interest. A simple example shows the complex relationship of odds-ratios to probabilities. An odds-ratio of .4 obtained by exponentiating the sum of the This content downloaded from 157.55.39.177 on Tue, 15 Nov 2016 03:52:55 UTC All use subject to http://about.jstor.org/terms Logit Coefficients and Independent Variables / 511 constant of the logit equation and the products of selected values of the independent variables times their coefficients as given in the right side of equation 1 is twice an odds-ratio of .2 for a case with other values of the independent variables. To find the probability associated with a particular odds-ratio, one solves for the probability implied by the odds-ratio, P4l P. For an odds-ratio of .4, P/i P a .4, thus P .4 .4P and 1.4P .4, hence, P-.286. For an odds ratio of .2, Pa.167. Thus, the ratio of these odds-ratios should not be interpreted as being, two times as likely. Because Baer, Grabb, and Johnston's exponentiating of logit coefficients gives the ratio of odds-ratios, their conclusions about how many more "times likely" English-Canadians or Quebeckers respond "yes" to either of the dependent variables than Americans should not be interpreted as probabilities. For a dichotomous independent variable, the exponentiated logit coefficient eb is a ratio of odds-ratios.2 The ratio of odds-ratios, however, has a less direct relationship to the probability of responding "yes" than odds-ratios have. The results from the bottom of their Table 5 for the "no say" variable allow estimating how the three ethnic groups differ in the approximate probabilities of feeling that an individual has 'no say" and further expanding the information obtainable from a logit analysis.3 PROCEDURE FOR INTERPRETING THE EFFECTS OF DUNMMY VARIABLES First, the nonlinear nature of the logit model requires choosing substantively interesting cases. Consider "middle-status" individuals who, for Baer, Grabb, and Johnston, would be employed, earning between $9,000 and $30,000, between age 30 and 55, and college-educated. Second, compute the natural logarithm of the predicted odds-ratio from equation 1 for English-Canadians, Quebeckers, and Americans. Their logit coefficients are in the "Coefficients" column in Table 1. Using these coefficients, the values of their ten dummy variables that indicate "middle status" English-Canadians, and treating the constant term as 0 yields the logarithm of the predicted odds-ratio from equation 1 for this group.