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I have the following two mixed models

Full model (with the three-way interaction)

m_Kunkle_pacc3_n <- lmer(pacc3_old ~ PRS_Kunkle*AgeAtVisit +
                     PRS_Kunkle*I(AgeAtVisit^2) +
                     APOE_score*AgeAtVisit + APOE_score*I(AgeAtVisit^2) + PRS_Kunkle*APOE_score + famhist +
                     + gender + EdYears_Coded_Max20 +  VisNo + X1 + X2 + X3 + X4 + X5 +
                     (1 |family/DBID),
                   data = WRAP_all, REML = F)

Nested model (exclude the three-way interaction (two variables are excluded: three-way interaction with linear age and quadratic age))

m_Kunkle_pacc3 <- lmer(pacc3_old ~ PRS_Kunkle*AgeAtVisit*APOE_score + 
                     PRS_Kunkle*I(AgeAtVisit^2)*APOE_score +
                     + gender + EdYears_Coded_Max20 +  VisNo + famhist + X1 + X2 + X3 + X4 + X5 +
                     (1 |family/DBID),
                   data = WRAP_all, REML = F)

I used the likelihood ratio test to test the difference between full model and nested model, am I correct in testing the significance of this three-way interaction?

pacc3_LRT_Kunkle <- anova(m_Kunkle_pacc3, m_Kunkle_pacc3_n, test = "chisq")

Many thanks

zjppdozen
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1 Answers1

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If you are interested in testing the significance of the three-way interaction, I think in general you should do that within the context of a single model. You first select a model based on theoretical considerations and sometimes certain indices, and then look at the parameters of the model you pick. For example, the BIC is related to a model's negative log-likelihood penalized by its complexity (it also depends on your sample size), and you can use the BIC to select a model among competing choices. Once you pick a model that has a certain interaction term within it, you evaluate its coefficient. I should warn you that interpreting three-way interactions can be very challenging, so you should consider that in the context of your problem as well.

TLDR; comparing a model that has a term with one that does not have it (whether you look at their R^2, compare likelihoods or penalized likelihoods, etc.) will tell you something about the whole model, not the parameter itself.

Arshiya
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