Odds versus Probabilities in Logit Equations: A Reply to Roncek

This article takes issue with propositions tendered in Roncek (1991) regarding definitions of and ratios, interpretation of partial slope, and an emphasis on using predicted probabilities to convey impact of categorical predictors in logit analysis. This article correctly defines and ratios, provides correct interpretation of partial slope, and argues that interpretation of results in terms of and is at preferable to a focus on probabilities. Although logit modeling is widely used in sociological research, there is still considerable confusion about interpretation of logistic regression results. This confusion centers on an intractable problem in logit model: As long as probability of an event is response of interest, there is measure that exactly summarizes impact on response of a unit increase in a given explanatory variable, net of other predictors in model. This poses an interpretive dilemma for researchers accustomed to regression models in which 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 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 is an exact summary measure of net multiplicative impact on of an event for each unit increase in a given predictor, and is therefore a multiplicative analog of partial slope in linear regression; and (2) and have an intuitively appealing interpretation and are therefore worthy of consideration in place of probabilities as 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 to most social scientists, they are not inherently easier to interpret. Indeed, are readily understood by laypersons, particularly in context of games of chance. The is mathematically defined (in a population) as n/(1 i), of probability that an event occurs (X) to probability that it does not occur (1 i). Or, equivalently, it is of number of that an event occurs, to number of that it does not occur, in n trials. When we speak of having 2 to 1 odds, therefore, we mean that event occurs twice as often as not, or that event is twice as to occur as not. If our odds, on other hand, were 1 to 10, then event occurs only a tenth as often as not, or it is only one-tenth as to occur as not. Therefore, have to use Roncek's phrase a times as likely interpretation with respect to probabilities of interest. I shall let P/(1 P) denote sample estimate of odds. Roncek incorrectly refers to this as predicted odds ratio. The is indeed a ratio, but it is a of probabilities rather than odds. On other hand, what Roncek refers to as a ratio of ratios is, instead, simply odds ratio. This label suggests, quite aptly, a of odds. Although this correction in terminology may appear trivial, it makes quite a difference in interpretability of results. Saying (incorrectly) that the of for English Canadians versus Americans is sounds much more awkward than saying (correctly) that the for English Canadians versus Americans is 1.345. Given former phrasing, it would be wonder that researchers would eschew interpreting their results in terms of odds. The interpretation of is strictly in terms of and not probabilities, and Roncek is absolutely correct in pointing this out. His criticism of Baer, Grabb, and Johnston's (1990) interpretation of is well taken. However (continuing with Baer, Grabb, & Johnston [1990] example), if (for of having no in government) for English Canadians versus Americans is 1.345, then it is correct to say that of having no in government for English Canadians are 1.345 higher than they are for Americans. Provided that one is comfortable with concept of an odds, are readily understood.' The Slope of a Nonlinear Function More relevant to my discussion is interpretation of partial slope in logit model. Contrary to Roncek's assertion (see Roncek 1991:516, note 1), 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 function for a unit increase in Xi net of other predictors. Rather it represents change along a line tangent to function for a unit increase in Xj. In particular, partial derivative with respect to Xj of a function, say, of XI, X2, .. . , XK, is slope of line tangent to function at a specific value of Xi, at constant values of all other X's (Anton 1984). It is only for linear functions that partial derivative indicates change along function for a unit increase in Xi, and 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 tangent line at a given point is synonymous with 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 partial slope in logit modeling (e.g., Aldrich & Nelson 1984; Hanushek & Jackson 1977; Judge et al., 1985), incorrect interpretation apparently persists (see also Cleary & Angel 1984, for another instance of researchers incorrectly interpreting 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 partial slope as it applies to logistic distribution function, which, in logistic regression, relates probability of being in category of interest on Y to independent variables. For simplicity, I have depicted 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, curve is simply a reflection across P axis of curve shown, such that 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 figure, function is decidedly nonlinear, taking on shape of an elongated S (and appropriately referred to as a 'sigmoid curve). As I have also shown, entire curve lies between values of 0 and 1 on P (vertical) axis. The rate of change of P with increase in X varies all along curve, reaching a maximum at a P-value of .5, which occurs when X = -a/b. This value is also curve's inflection point X-value for which curve changes from concave up to concave down (from left to right along X-axis). Of special interest in figure is tangent line to curve at point x. (I shall adopt convention of letting an upper case X represent independent variable in general, and letting a lower case x represent a specific value of independent variable.) This is line that touches curve only at P(x). The slope of this line, P'(x), is first derivative (or first partial derivative if more than one X were involved in function) of function with respect to X, evaluated at x. In terms of function, P'(x) indicates instantaneous rate at which P changes with X at point x.2 This value depends on value of X at which it is evaluated, as is evident both in graph and from formula for 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 tangent line at x, but increment to P is considerably smaller along 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 tangent line P increases from P(x) to [P(x) + P'(x)], while along function, P increases from P(x) to P(x + 1). Consequently, b(P)(1 P) never represents 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 Function and along Tangent Line at a Point x, for a Unit Increase in X