1、Are you looking for the right interactions? Part 2: Statistically testing for interaction effects with dichotomous outcome variables Melanie M. Wall Departments of Psychiatry and Biostatistics New York State Psychiatric Institute and Mailman School of Public Health Columbia University mmwallcolumbia
2、.edu Joint Presentation with Sharon Schwartz (Part 1) Department of Epidemiology Mailman School of Public Health Columbia University,1,Data from Brown and Harris (1978) 2X2X2 Table,OR = Odds Ratio (95% Confidence Interval) -compare to 1 RR = Risk Ratio (95% Confidence Interval) -compare to 1 RD = Ri
3、sk Difference (95% Confidence Interval) -compare to 0,2,Does Vulnerability Modify the Effect of Stress on Depression?,On the multiplicative Odds Ratio scale, is 10.9 sig different from 13.8? Test whether the ratio of the odds ratios (i.e. 13.8/10.9 = 1.27) is significantly different from 1.On the mu
4、ltiplicative Risk Ratio scale, is 9.9 sig different from 9.8? Test whether the ratio of the risk ratios (i.e. 9.8/9.9 = 0.99) is significantly different from 1.On the additive Risk Difference scale, is 0.092 sig different from 0.284? Test whether the difference in the risk differences (i.e. 0.28-0.0
5、9 = 0.19) is significantly different from 0.Rothman calls this difference in the risk differences the “interaction contrast (IC)” IC = (P11 - P10) (P01 - P00),3,95% confidence intervals for Odds Ratios overlap - no statistically significant multiplicative interaction OR scale 95% confidence interval
6、s for Risk Ratios overlap - no statistically significant multiplicative interaction RR scale95% confidence intervals for Risk Differences do not overlap - statistically significant additive interaction,Comparing stress effects across vulnerability groups Different conclusions on multiplicative vs ad
7、ditive scale,4,In general, it is possible to reach different conclusions on the two different multiplicative scales “distributional interaction” (Campbell, Gatto, Schwartz 2005),Modeling Probabilities Binomial modeling with logit, log, or linear link,5,Test for multiplicative interaction on the OR s
8、cale- Logistic Regression with a cross-product,IN SAS: proc logistic data = brownharris descending; model depressn = stressevent lack_intimacy stressevent*lack_intimacy; oddsratio stressevent / at(lack_intimacy = 0 1); oddsratio lack_intimacy / at(stressevent = 0 1); run;Analysis of Maximum Likeliho
9、od EstimatesStandard Wald Parameter DF Estimate Error Chi-Square Pr ChiSq Intercept 1 -4.5591 0.7108 41.1409 .0001 stressevent 1 2.3869 0.7931 9.0576 0.0026 lack_intimacy 1 1.1579 1.0109 1.3120 0.2520 stresseve*lack_intim 1 0.2411 1.0984 0.0482 0.8262Wald Confidence Interval for Odds Ratios Label Es
10、timate 95% Confidence Limits stressevent at lack_intimacy=0 10.880 2.299 51.486 stressevent at lack_intimacy=1 13.846 3.122 61.408 lack_intimacy at stressevent=0 3.183 0.439 23.086 lack_intimacy at stressevent=1 4.051 1.745 9.405,exp(.2411) = 1.27 = Ratio of Odds ratios =13.846/10.880 Not significan
11、tly different from 1,“multiplicative interaction” on OR scale is not significant,6,Test for interaction: Are the lines Parallel?,Log Odds scale,Probability scale,Cross product term in logistic regression is magnitude of deviation of these lines from being parallel p-value = 0.8262 - cannot reject th
12、at lines on logit scale are parallel Thus, no statistically significant multiplicative interaction on OR scale,Test for whether lines are parallel on probability scale is same as H0: IC = 0. Need to construct a statistical test for IC = P11-P10-P01+P00,7,P10,P00,P01,P11,Dont fall into the trap of co
13、ncluding there must be effect modification because one association was statistically significant while the other one was not. In other words, just because a significant effect is found in one group and not in the other, does NOT mean the effects are necessarily different in the two groups (regardles
14、s of whether we use OR, RR, or RD). Remember, statistical significance is not only a function of the effect (OR, RR, or RD) but also the sample size and the baseline risk. Both of these can differ across groups. McKee and Vilhjalmsson (1986) point out that Brown and Harris (1978) wrongfully applied
15、this logic to conclude there was statistical evidence of effect modification (fortunately there conclusion was correct despite an incorrect statistical test ),The Problem with Comparing Statistical Significance of Effects Across Groups,8,Risk = b0 + b1 * STRESS + b2 * LACKINT + b3*STRESS*LACKINT NOT
16、E: b3 = ICIN SAS: proc genmod data = individual descending; model depressn = stressevent lack_intimacy stressevent*lack_intimacy/ link = identity dist = binomial lrci; estimate RD of stressevent when intimacy = 0 stressevent 1; estimate RD of stressevent when intimacy = 1 stressevent 1 stressevent*l
17、ack_intimacy 1; run;Analysis Of Maximum Likelihood Parameter Estimates Likelihood RatioStandard 95% Confidence WaldParameter DF Estimate Error Limits Chi-Square PrChiSqIntercept 1 0.0104 0.0073 0.0017 0.0317 2.02 0.1551stressevent 1 0.0919 0.0331 0.0368 0.1675 7.70 0.0055lack_intimacy 1 0.0219 0.023
18、6 -0.0139 0.0870 0.86 0.3534stresseve*lack_intim 1 0.1916 0.0667 0.0588 0.3219 8.26 0.0040Contrast Estimate ResultsMean Mean StandardLabel Estimate Confidence Limits ErrorRD of stressevent when intimacy = 0 0.0919 0.0270 0.1568 0.0331RD of stressevent when intimacy = 1 0.2835 0.1701 0.3969 0.0578,9,
19、Testing for additive interaction on the probability scale Strategy #1: Use linear binomial regression with a cross-product,Interaction is statistically significant “additive interaction”. Reject H0: IC = 0, i.e. Reject parallel lines on probability scale,link=identity dist=binomial tells SAS to do l
20、inear binomial regression. lrci outputs likelihood ratio (profile likelihood) confidence intervals.,Different strategies for statistically testing additive interactions on the probability scale,The IC is the Difference of Risk Differences. IC = (P11 - P10) (P01 - P00) = P11-P10-P01+P00 From Cheung (
21、2007) “Now that many commercially available statistical packages have the capacity to fit log binomial and linear binomial regression models, there is no longer any good justification for fitting logistic regression models and estimating odds ratios when the odds ratio is not of scientific interest”
22、 Inside quote from Spiegelman and Herzmark (2005).Fit a linear binomial regression Risk = b0 + b1 * EXPO + b2 * VULN + b3*EXPO*VULN. The b3 = IC and so a test for coefficient b3 is a test for IC. Can be implemented directly in PROC GENMOD. PROS: Contrast of interest is directly estimated and tested
23、and covariates easily included CONS: Linear model for probabilities can be greater than 1 and less than 0 and thus maximum likelihood estimation can be a problem. Wald-type confidence intervals can have poor coverage (Storer et al 1983), better to use profile likelihood confidence intervals. Fit a l
24、ogistic regression log(Risk/(1-Risk) = b0 + b1 * EXPO + b2 * VULN + b3*EXPO*VULN, then back-transform parameters to the probability scale to calculate IC. Can be implemented directly in PROC NLMIXED. PROS: logistic model more computationally stable since smooth decrease/increase to 0 and 1. CONS: ba
25、ck-transforming can be tricky for estimator and standard errors particularly in presence of covariates. Covariate adjusted probabilities are obtained from average marginal predictions in the fitted logistic regression model (Greenland 2004). Homogeneity of covariate effects on odds ratio scale is no
26、t the same as homogeneity on risk difference scale and this may imply misspecification (Kalilani and Atashili 2006; Skrondal 2003). Instead of IC, use IC ratio. Divide the IC by P00 and get a contrast of risk ratios:IC Ratio = P11/P00 -P10/P00 -P01/P00+P00/P00 = RR(11) RR(10) RR(01) + 1 called the R
27、elative Excess Risk due to Interaction (RERI). Many papers on inference for RERI,10,Test for additive interaction on the probability scale Strategy #2: Use logistic regression and back-transform estimates to form contrasts on the probability scale,PROC NLMIXED DATA=individual; *logistic regression m
28、odel is; odds = exp(b0 +b1*stressevent + b2*lack_intimacy + b3*stressevent*lack_intimacy); pi = odds/(1+odds); MODEL depressnBINARY(pi);estimate p00 exp(b0)/(1+exp(b0); estimate p10 exp(b0+b1)/(1+exp(b0+b1); estimate p01 exp(b0+b2)/(1+exp(b0+b2); estimate p11 exp(b0+b1+b2+b3)/(1+exp(b0+b1+ b2+b3);es
29、timate p11-p10 exp(b0+b1+b2+b3)/(1+exp(b0+b1+ b2+b3)- exp(b0+b1)/(1+exp(b0+b1); estimate p01-p00 exp(b0+b2)/(1+exp(b0+b2) - exp(b0)/(1+exp(b0);estimate IC= interaction contrast = p11-p10 - p01 + p00 exp(b0+b1+b2+b3)/(1+exp(b0+b1+ b2+b3) - exp(b0+b1)/(1+exp(b0+b1) - exp(b0+b2)/(1+exp(b0+b2) + exp(b0)
30、/(1+exp(b0);estimate ICR= RERI using RR = p11/p00 - p10/p00 - p01/p00 + 1exp(b0+b1+b2+b3)/(1+exp(b0+b1+ b2+b3)/ (exp(b0)/(1+exp(b0)- exp(b0+b1)/(1+exp(b0+b1)/ (exp(b0)/(1+exp(b0)- exp(b0+b2)/(1+exp(b0+b2) / (exp(b0)/(1+exp(b0) + 1; estimate ICR= RERI using OR exp(b1+b2+b3) - exp(b1) - exp(b2) +1; RU
31、N;,11,Strategy #2 Output from NLMIXED,Parameter EstimatesStandard Parameter Estimate Error DF t Value Pr |t| Alpha Lower Upper Gradientb0 -4.5591 0.7108 419 -6.41 |t| Lower Upper p00 0.01036 0.00728 419 1.42 0.1559 -0.00397 0.0246 p10 0.1023 0.03230 419 3.17 0.0017 0.03878 0.1658 p01 0.03226 0.02244
32、 419 1.44 0.1513 -0.01185 0.0763 p11 0.3158 0.05332 419 5.92 .0001 0.2110 0.4206 p11-p10 0.2135 0.06234 419 3.43 0.0007 0.09098 0.3361 p01-p00 0.02190 0.02359 419 0.93 0.3539 -0.02448 0.0682 IC =p11-p10-p01+p00 0.1916 0.06666 419 2.87 0.0042 0.06060 0.3226RERI using RR 18.4915 13.8661 419 1.33 0.183
33、1 -8.7644 45.7473 RERI using OR 31.0138 24.3583 419 1.27 0.2036 -16.8659 78.8936,12,The IC estimator is same as before (slide 9) but slightly different s.e., p-value and 95% confidence interval still conclude there is a significant additive interaction.Results for RERI (using RR and OR) indicate tha
34、t there is NOT a significant additive interaction. This conflicts with the conclusion that the IC is highly significant. The cause of the discrepancy is related to estimation of standard errors and confidence intervals. Literature indicates Wald-type confidence intervals perform poorly for RERI (Hos
35、mer and Lemeshow 1992; Assman et al 1996). Proc NLMIXED uses Delta method to obtain standard errors of back-transformed parameters and Wald-type confidence intervals, i.e. (estimate) +- 1.96*(standard error) . Possible to obtain profile likelihood confidence intervals using a separate macro (Richard
36、son and Kaufman 2009) or PROC NLP (nonlinear programming) (Kuss et al 2010). Also possible to bootstrap (Assman et al 1996 and Nie et al 2010) or incorporate prior information (Chu et al 2011),IC estimator same as strategy #1, but slightly different s.e., p-value, 95% conf interval,Conclusion,The ap
37、propriate scale on which to assess interaction effects with dichotomous outcomes has been a controversial topic in epidemiology for years, but awareness of this controversy is not yet wide spread enough. This would not be a problem if the status quo for examining effect modification (i.e. testing in
38、teraction effects in logistic regression) was actually the “RIGHT” thing to do, but, persuasive arguments have been made from the sufficient cause framework that the additive probability scale (not the multiplicative odds ratio scale) should be used to assess the presence of synergistic effects (Dar
39、roch 1997, Rothman and Greenland 1998, Schwartz 2006, Vanderwheel and Robins 2007,2008) There are now straightforward ways within existing software to estimate and test the statistical significance of additive interaction effects. Additional work is needed getting the word out that effect modificati
40、on should not (just) be looked at using Odds Ratios.,13,Appendix Material,14,Reading in the Brown and Harris data into SAS,data a; input stressevent lack_intimacy depressn count; *When entering the counts, it is necessary to subtract the cases from the denominator to get the non-cases, e.g. 9/88 bec
41、omes 9 cases and 79 (88-9) noncases; cards;0 0 0 1910 0 1 20 1 0 600 1 1 21 0 0 791 0 1 91 1 0 521 1 1 24 ; *This data step creates 419 records corresponding to the counts above, basically one record for each of individuals in the study; data individual; set a; do i = 1 to count; output; end; drop i
42、; run;,15,Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix,SAS (Proc GENMOD) code for fitting logit, log, and linear binomial models,*logistic binomial regression; proc genmod data = individual descending; model depressn = stressevent lack_intimacy stressevent*lack_intimacy/ l
43、ink = logit dist = binomial lrci type3; estimate OR of stressevent when intimacy = 0 stressevent 1/exp; estimate OR of stressevent when intimacy = 1 stressevent 1 stressevent*lack_intimacy 1/exp; estimate OR interaction contrast stressevent*lack_intimacy 1/exp; run;*log binomial regression; proc gen
44、mod data = individual descending; model depressn = stressevent lack_intimacy stressevent*lack_intimacy/ link = log dist = binomial lrci type3; estimate RR of stressevent when intimacy = 0 stressevent 1/exp; estimate RR of stressevent when intimacy = 1 stressevent 1 stressevent*lack_intimacy 1/exp; e
45、stimate RR interaction contrast stressevent*lack_intimacy 1/exp; run;*linear binomial regression; proc genmod data = individual descending; model depressn = stressevent lack_intimacy stressevent*lack_intimacy/ link = identity dist = binomial lrci type3; estimate RD of stressevent when intimacy = 0 s
46、tressevent 1; estimate RD of stressevent when intimacy = 1 stressevent 1 stressevent*lack_intimacy 1; estimate RD interaction contrast stressevent*lack_intimacy 1; run;,16,Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix,Partial output from SAS Proc GENMOD code on previous sli
47、de,Logistic binomial regression Analysis Of Maximum Likelihood Parameter EstimatesLikelihood RatioStandard 95% Confidence WaldParameter DF Estimate Error Limits Chi-Square Pr ChiSqIntercept 1 -4.5591 0.7108 -6.3576 -3.4207 41.14 ChiSqIntercept 1 -4.5695 0.7034 -6.3593 -3.4529 42.20 ChiSqIntercept 1 0.0104 0.0073 0.0017 0.0317 2.02 0.1551stressevent 1 0.0919 0.0331 0.0368 0.1675 7.70 0.0055lack_intimacy 1 0.0219 0.0236 -0.0139 0.0870 0.86 0.3534stresseve*lack_intim 1 0.1916 0.0667 0.0588 0.3219 8.26 0.0040,17,Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix,
