Odds versus Probabilities in Logit Equations: A Reply to Roncek

This article takes issue with propositions tendered in Roncek (1991) regarding the definitions of odds and odds ratios, the interpretation of the partial slope, and an emphasis on using predicted probabilities to convey the impact of categorical predictors in logit analysis. This article correctly defines odds and odds ratios, provides the correct interpretation of the partial slope, and argues that the interpretation of results in terms of odds and odds ratios is at times preferable to a focus on probabilities. Although logit modeling is widely used in sociological research, there is still considerable confusion about the interpretation of logistic regression results. This confusion centers on an intractable problem in the logit model: As long as the probability of an event is the response of interest, there is no measure that exactly summarizes the impact on the response of a unit increase in a given explanatory variable, net of other predictors in the model. This poses an interpretive dilemma for researchers accustomed to regression models in which the partial slope does have that interpretation. (This dilemma of course is not unique to logit models; probit and hazard models are as difficult, if not more, difficult to interpret intuitively.) Dennis W. Roncek's (1991) article in this journal correctly identifies the problem and suggests converting logit coefficients into impacts on probabilities for "particular types of cases." Rather than criticize Roncek's position, this article offers a different perspective, and with it, different advice. Specifically, I wish to make two main points: (1) The odds ratio is an exact summary measure of the net multiplicative impact on the odds of an event for each unit increase in a given predictor, and is therefore a multiplicative analog of the partial slope in linear regression; and (2) odds and odds ratios have an intuitively appealing interpretation and are therefore worthy of consideration in place of probabilities as the substantive "currency" of logit modeling. In addition to establishing these points, I wish to correct some minor errors in Roncek's discussion. * The auithor wishes to thank two anonymous reviewers for constructive comments and suiggestions on earlier drafts of this manuscript. Direct all correspondence to Alfred DeMaris, Department of Sociology, Bowling Green State University, Bowling Green, OH 43403. ? The University of North Carolina Press Social Forces, June 1993, 71(4):1057-1065 This content downloaded from 157.55.39.106 on Sat, 13 May 2017 17:57:45 UTC All use subject to http://about.jstor.org/terms 1058 / Social Forces 71:4, June 1993 Odds and Odds Ratios Although probabilities are more familiar than odds to most social scientists, they are not inherently easier to interpret. Indeed, odds are readily understood by laypersons, particularly in the context of games of chance. The odds is mathematically defined (in a population) as n/(1 i), the ratio of the probability that an event occurs (X) to the probability that it does not occur (1 i). Or, equivalently, it is the ratio of the number of times that an event occurs, to the number of times that it does not occur, in n trials. When we speak of having 2 to 1 odds, therefore, we mean that the event occurs twice as often as not, or that the event is twice as likely to occur as not. If our odds, on the other hand, were 1 to 10, then the event occurs only a tenth as often as not, or it is only one-tenth as likely to occur as not. Therefore, the odds have to use Roncek's phrase a "times as likely" interpretation with respect to the probabilities of interest. I shall let P/(1 P) denote the sample estimate of the odds. Roncek incorrectly refers to this as the predicted "odds ratio." The odds is indeed a ratio, but it is a ratio of probabilities rather than odds. On the other hand, what Roncek refers to as a "ratio of odds ratios" is, instead, simply the "odds ratio." This label suggests, quite aptly, a ratio of odds. Although this correction in terminology may appear trivial, it makes quite a difference in the interpretability of results. Saying (incorrectly) that "the ratio of odds ratios for English Canadians versus Americans is 1.345" sounds much more awkward than saying (correctly) that "the odds ratio for English Canadians versus Americans is 1.345." Given the former phrasing, it would be no wonder that researchers would eschew interpreting their results in terms of odds. The interpretation of the odds ratio is strictly in terms of odds and not probabilities, and Roncek is absolutely correct in pointing this out. His criticism of Baer, Grabb, and Johnston's (1990) interpretation of odds ratios is well taken. However (continuing with the Baer, Grabb, & Johnston [1990] example), if the odds ratio (for the odds of having "no say" in government) for English Canadians versus Americans is 1.345, then it is correct to say that the odds of having "no say" in government for English Canadians are 1.345 times higher than they are for Americans. Provided that one is comfortable with the concept of an odds, odds ratios are readily understood.' The Slope of a Nonlinear Function More relevant to my discussion is the interpretation of the partial slope in the logit model. Contrary to Roncek's assertion (see Roncek 1991:516, note 1), the first-order partial derivative, or slope, of a function with respect to a given independent variable, say Xi, does not in general indicate change in the function for a unit increase in Xi net of other predictors. Rather it represents change along a line tangent to the function for a unit increase in Xj. In particular, the partial derivative with respect to Xj of a function, say, of XI, X2, .. . , XK, is the slope of the line tangent to the function at a specific value of Xi, at constant values of all other X's (Anton 1984). It is only for linear functions that the partial derivative indicates change along the function for a unit increase in Xi, and the This content downloaded from 157.55.39.106 on Sat, 13 May 2017 17:57:45 UTC All use subject to http://about.jstor.org/terms Odds vs. Probabilities in Logit Equations / 1059 reason for this is that the tangent line at a given point is synonymous with the function itself. Although this point has been made before by Petersen (1985), and although other statisticians have been careful to avoid an incorrect interpretation of the partial slope in logit modeling (e.g., Aldrich & Nelson 1984; Hanushek & Jackson 1977; Judge et al., 1985), the incorrect interpretation apparently persists (see also Cleary & Angel 1984, for another instance of researchers incorrectly interpreting the partial slope in logit modeling despite an otherwise excellent discussion of models for binary dependent variables). Figure 1 may aid in this explication by illustrating the partial slope as it applies to the logistic distribution function, which, in logistic regression, relates the probability of being in the category of interest on Y to the independent variables. For simplicity, I have depicted the logistic curve as a function of only one continuous predictor. It is also assumed that a is negative, and b is positive. (When b is negative, the curve is simply a reflection across the P axis of the curve shown, such that the curve then runs from upper left to lower right.) The function itself is ea +bX P(X) = -a(1) 1 +ea +bX(1 As is evident in the figure, the function is decidedly nonlinear, taking on the shape of an elongated S (and appropriately referred to as a 'sigmoid" curve). As I have also shown, the entire curve lies between the values of 0 and 1 on the P (vertical) axis. The rate of change of P with increase in X varies all along the curve, reaching a maximum at a P-value of .5, which occurs when X = -a/b. This value is also the curve's inflection point the X-value for which the curve changes from concave up to concave down (from left to right along the X-axis). Of special interest in the figure is the tangent line to the curve at the point x. (I shall adopt the convention of letting an upper case X represent the independent variable in general, and letting a lower case x represent a specific value of the independent variable.) This is the line that touches the curve only at P(x). The slope of this line, P'(x), is the first derivative (or the first partial derivative if more than one X were involved in the function) of the function with respect to X, evaluated at x. In terms of the function, P'(x) indicates the "instantaneous" rate at which P changes with X at the point x.2 This value depends on the value of X at which it is evaluated, as is evident both in the graph and from the formula for the slope itself: ( ea+bx (____ P'(x)=bI e 1 (2) 1+ea+x) 1 +ea+bx) or, writing this formula in terms of P, P'(x) = b[P(x)][1 P(x)]. Notice, moreover, that when X increases by one unit from x to x+1 there is an increment to P of P'(x) along the tangent line at x, but the increment to P is considerably smaller along the function itself. The increment to P need not necessarily be smaller than P'(x) in absolute value, but it will always be different from P'(x). Hence, along the tangent line P increases from P(x) to [P(x) + P'(x)], while along the function, P increases from P(x) to P(x + 1). Consequently, b(P)(1 P) never represents the exact change in P for a unit increase in X, especially in view of This content downloaded from 157.55.39.106 on Sat, 13 May 2017 17:57:45 UTC All use subject to http://about.jstor.org/terms 1060 / Social Forces 71:4, June 1993 FIGURE 1: The Logistic Distribution Function with One Independent Variable X Showing Change Along the Function and along the Tangent Line at a Point x, for a Unit Increase in X