# Direct and indirect effects in a logit model

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1 Department of Social Research Methodology Vrije Universiteit Amsterdam

2 Outline The aim

3 The Total Effect X Y

4 The Total Effect parental class attend college

5 The Indirect Effect Z a b X Y

6 The Indirect Effect during high school a b parental class attend college

7 The Direct Effect Z a b X c Y

8 The Direct Effect during high school a b parental class c attend college

9 The aim Z The aim is to find the size of the indirect effect relative to the total effect. a b X c Y

10 Outline The aim

11 Estimation When using regress: 1. college = class + 2. college = class

12 Estimation When using regress: 1. college = class + 2. college = class The direct effect is the effect of class in model 1.

13 Estimation When using regress: 1. college = class + 2. college = class The direct effect is the effect of class in model 1. The total effect is the effect of class in model 2.

14 Estimation When using regress: 1. college = class + 2. college = class The direct effect is the effect of class in model 1. The total effect is the effect of class in model 2. The indirect effect is the total effect - direct effect.

15 Estimation When using regress: 1. college = class + 2. college = class The direct effect is the effect of class in model 1. The total effect is the effect of class in model 2. The indirect effect is the total effect - direct effect. This won t work when using logit

16 Why the naive method doesn t work Easiest explained when there is no indirect effect.

17 Why the naive method doesn t work Easiest explained when there is no indirect effect. The total effect = the direct effect + the indirect effect.

18 Why the naive method doesn t work Easiest explained when there is no indirect effect. The total effect = the direct effect + the indirect effect. So, the total effect should be the same as the direct effect when there is no indirect effect.

19 Why the naive method doesn t work Easiest explained when there is no indirect effect. The total effect = the direct effect + the indirect effect. So, the total effect should be the same as the direct effect when there is no indirect effect. So, the effect of class in a model that controls for (the direct effect ) should be the same as the effect of class in a model that does not control for (the total effect ).

20 Effect while controlling for log odds proportion high medium low 1.5 transformation controlled 3 proportion high status log odds log odds low status proportion effect controlled

21 Averaging the proportions 3 log odds proportion high medium low not controlled 1.5 transformation controlled 3 proportion high status log odds log odds low status proportion effect controlled

22 Effect while not controlling for 3 log odds proportion high status log odds log odds low status proportion proportion high medium low not controlled transformation controlled not constrolled effect controlled not constrolled

23 Outline The aim

24 Indirect effect present log odds prop. high status log odds log odds low status prop proportion high medium low not controlled transformation controlled not constrolled effect controlled not constrolled

25 Indirect effect 3 log odds prop. high status log odds indirect effect log odds low status prop proportion factual high medium low not controlled counterfactual high low not controlled

26 Direct effect 3 log odds prop. high status log odds direct effect log odds low status prop proportion factual high medium low not controlled counterfactual high low not controlled

27 Direct and indirect effects in logit 3 log odds proportion high status log odds indirect effect direct effect total effect log odds low status proportion proportion factual high medium low not controlled counterfactual high low not controlled

28 The logic can be reversed 3 log odds total effect direct effect proportion factual high medium low not controlled prop. high status log odds indirect effect log odds low status prop. counterfactual high low not constrolled

29 Extension Erikson et al. (2005) propose to compute the average proportions given the observed and counterfactual distribution of by assuming that is normally distributed, and then integrate over this normal distribution.

30 Extension Erikson et al. (2005) propose to compute the average proportions given the observed and counterfactual distribution of by assuming that is normally distributed, and then integrate over this normal distribution. Alternatively, these averages can be computed by predicting the observed and counterfactual proportions, add them up and divide by the number of respondents in that group.

31 Extension Erikson et al. (2005) propose to compute the average proportions given the observed and counterfactual distribution of by assuming that is normally distributed, and then integrate over this normal distribution. Alternatively, these averages can be computed by predicting the observed and counterfactual proportions, add them up and divide by the number of respondents in that group. The latter method has the advantage of making less assumptions about the distribution of, as it integrates over the empirical distribution of instead of over a normal distribution.

32 Outline The aim

33 Descriptives. table ocf57 if!missing(hsrankq, college), /// > contents(mean college mean hsrankq freq) /// > format(%9.3g) stubwidth(15) occupation of r father in 1957 mean(college) mean(hsrankq) Freq. lower ,218 middle higher ,837

34 The ldecomp package ldecomp depvar [ if ] [ in ] [ weight ], direct(varname) indirect(varlist) [ obspr predpr predodds or rindirect normal range(##) nip(#) interactions nolegend nodecomp nobootstrap bootstrap_options ]

35 Decomposition of log odds ratios. ldecomp college, direct(ocf57) indirect(hsrankq) rind nolegend (running _ldecomp on estimation sample) Bootstrap replications (50) Bootstrap results Number of obs = 8923 Replications = 50 Observed Bootstrap Normal-based Coef. Std. Err. z P> z [95% Conf. Interval] 2/1 total indirect direct indirect direct /1 total indirect direct indirect direct /2 total indirect direct indirect direct

36 Relative effects 2/1r 3/1r 3/2r method method average method method average method method average

37 Decomposition of odds ratios. ldecomp college, direct(ocf57) indirect(hsrankq) or nolegend (running _ldecomp on estimation sample) Bootstrap replications (50) Bootstrap results Number of obs = 8923 Replications = 50 Observed Bootstrap Normal-based Odds Ratio Std. Err. z P> z [95% Conf. Interval] 2/1 total indirect direct indirect direct /1 total indirect direct indirect direct /2 total indirect direct indirect direct

38 Does it matter? Table: Comparing different estimates of the size of indirect effect relative to the size of the total effect generalization (Erikson et al. 2005) naive middle v. low method method average high v. low method method average high v. middle method method average

39 Discussion This is an area of active research

40 Discussion There are unanswered questions:

41 Discussion There are unanswered questions: The need to take the average indirect effect is less than elegant.

42 Discussion There are unanswered questions: The need to take the average indirect effect is less than elegant. How does it relate to the alternative method proposed by Fairlie (2005) and implemented by Ben Jann as the fairlie package?

43 References Buis, M. L.. Erikson, R., J. H. Goldthorpe, M. Jackson, M. Yaish, and D. R. Cox. On class differentials in educational attainment. Proceedings of the National Academy of Science, 102: , Fairlie, R. W. An extension of the Blinder-Oaxaca decomposition technique to logit and probit models. Journal of Economic and Social Measurement, 30: , 2005.

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