Solved – Interaction suppresses the main effect? How to interpret it

binomial distributionfixed-effects-modelgeneralized linear modelinteractionlogistic

I have a simple model without interaction and it stated significant effect for all the explanatory variables (continuous variable rok and categorical variables obdobi (levels hn and nehn) and kraj:

Call:
glm(formula = cbind(ml, ad) ~ rok + obdobi + kraj, family = "quasibinomial")

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.8007  -1.1716  -0.5117   1.0864   4.2184  

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept) -107.60761   53.96993  -1.994  0.04674 * 
rok            0.05381    0.02686   2.003  0.04576 * 
obdobinehn    -0.26962    0.11646  -2.315  0.02104 * 
krajJHC        0.68869    0.31009   2.221  0.02683 * 
krajJHM       -0.26607    0.32166  -0.827  0.40855   
krajLBK       -1.11305    0.61942  -1.797  0.07298 . 
krajMSK       -0.61390    0.41828  -1.468  0.14285   
krajOLK       -0.49704    0.36981  -1.344  0.17958   
krajPAK       -1.18444    0.39401  -3.006  0.00279 **
krajPLK       -1.28668    0.49672  -2.590  0.00988 **
krajSTC        0.01872    0.31222   0.060  0.95220   
krajULKV      -0.41950    0.69220  -0.606  0.54478   
krajVYS       -1.17290    0.44614  -2.629  0.00884 **
krajZLK       -0.38170    0.40969  -0.932  0.35198   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for quasibinomial family taken to be 1.645035)

    Null deviance: 1136.22  on 489  degrees of freedom
Residual deviance:  938.02  on 476  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4

Then I added interaction obdobi:kraj:

Call:
glm(formula = cbind(ml, ad) ~ rok + obdobi + kraj + obdobi:kraj, 
    family = "quasibinomial")

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.4635  -1.1706  -0.4597   1.0275   4.6829  

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)   
(Intercept)         -101.49501   54.53576  -1.861  0.06336 . 
rok                    0.05102    0.02715   1.879  0.06086 . 
obdobinehn            -1.11653    0.62058  -1.799  0.07264 . 
krajJHC               -0.16805    0.51957  -0.323  0.74651   
krajJHM               -0.77451    0.53738  -1.441  0.15018   
krajLBK               -3.29567    1.42164  -2.318  0.02087 * 
krajMSK               -0.73640    0.67267  -1.095  0.27420   
krajOLK               -0.41582    0.68758  -0.605  0.54564   
krajPAK               -1.50156    0.63871  -2.351  0.01914 * 
krajPLK               -1.48611    0.75745  -1.962  0.05036 . 
krajSTC               -0.34170    0.52059  -0.656  0.51191   
krajULKV              -1.72550    1.02726  -1.680  0.09369 . 
krajVYS               -1.93603    0.65862  -2.940  0.00345 **
krajZLK               -0.71065    0.65791  -1.080  0.28063   
obdobinehn:krajJHC     1.44638    0.65507   2.208  0.02773 * 
obdobinehn:krajJHM     0.82070    0.67910   1.209  0.22746   
obdobinehn:krajLBK     3.31340    1.61026   2.058  0.04018 * 
obdobinehn:krajMSK     0.12470    0.87281   0.143  0.88645   
obdobinehn:krajOLK     0.04528    0.82529   0.055  0.95627   
obdobinehn:krajPAK     0.48978    0.81921   0.598  0.55022   
obdobinehn:krajPLK     0.23075    1.02316   0.226  0.82167   
obdobinehn:krajSTC     0.50339    0.65976   0.763  0.44585   
obdobinehn:krajULKV    2.49157    1.43679   1.734  0.08356 . 
obdobinehn:krajVYS     1.48201    0.92082   1.609  0.10820   
obdobinehn:krajZLK     0.49357    0.85087   0.580  0.56214   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for quasibinomial family taken to be 1.613648)

    Null deviance: 1136.22  on 489  degrees of freedom
Residual deviance:  899.28  on 465  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4

Strange thing happened – the main effects rok and obdobi are no longer significant! How can this happen? How to interpret this fact? If the interaction obdobi:kraj has significant effect, then the obdobi also has significant effect, right?

Note that the second model differs significantly (tested by anova(..., test = "Chi")).

Thanks in advance!

EDIT: added anova tables of the models (but since this is glm and not simple lm, mean sum of squares and p-values are missing and I don't know how to interpret it…)

> anova(model1)
Analysis of Deviance Table

Model: quasibinomial, link: logit

Response: cbind(ml, ad)

Terms added sequentially (first to last)

        Df Deviance Resid. Df Resid. Dev
NULL                      489    1136.22
rok      1     3.06       488    1133.16
obdobi   1    11.20       487    1121.96
kraj    11   183.94       476     938.02

> anova(model2)
Analysis of Deviance Table

Model: quasibinomial, link: logit

Response: cbind(ml, ad)

Terms added sequentially (first to last)

             Df Deviance Resid. Df Resid. Dev
NULL                           489    1136.22
rok           1     3.06       488    1133.16
obdobi        1    11.20       487    1121.96
kraj         11   183.94       476     938.02
obdobi:kraj  11    38.74       465     899.28

Best Answer

The main effects went from "significant" to "not", but the evidence really didn't change all that much. For example, p=0.047 to p=0.063 for rok isn't, to me, a remarkable change. And a lack of evidence for a coefficient being non-zero isn't the same as saying it is 0.

In considering the coefficient for obdobinehn when the interaction is included, you need to pay careful attention to the factor contrasts that are being used, as the meaning of the coefficient changes and depends on those contrasts.

Note also that if a covariate is involved in an important interaction, then it does have an effect on the outcome, even if it shows no main effect.

I agree with John's comment that it's useful, with factor covariates, to look at an ANOVA table.

Related Solutions

Solved – How to interpret main effects when the interaction effect is not significant

A little niggle

'Now many textbook examples tell me that if there is a significant effect of the interaction, the main effects cannot be interpreted'

I hope that's not true. They should say that if there is an interaction term, say between X and Z called XZ, then the interpretation of the individual coefficients for X and for Z cannot be interpreted in the same way as if XZ were not present. You can definitely interpret it.

Question 2

If the interaction makes theoretical sense then there is no reason not to leave it in, unless concerns for statistical efficiency for some reason override concerns about misspecification and allowing your theory and your model to diverge.

Given that you have left it in, then interpret your model using marginal effects in the same way as if the interaction were significant. For reference, I include a link to Brambor, Clark and Golder (2006) who explain how to interpret interaction models and how to avoid the common pitfalls.

Think of it this way: you often have control variables in a model that turn out not to be significant, but you don't (or shouldn't) go chopping them out at the first sign of missing stars.

Question 1

You ask whether you can 'conclude that the two predictors have an effect on the response?' Apparently you can, but you can also do better. For the model with the interaction term you can report what effect the two predictors actually have on the dependent variable (marginal effects) in a way that is indifferent to whether the interaction is significant, or even present in the model.

The Bottom Line

If you remove the interaction you are re-specifying the model. This may be a reasonable thing to do for many reasons, some theoretical and some statistical, but making it easier to interpret the coefficients is not one of them.

Solved – Accounting for overdispersion in binomial glm using proportions, without quasibinomial

Overdispersion occurs for a number of reasons, but often the case of presence/absence data is because of clustering of observations and correlations between observations.

Taken from Brostrom & Holmberg (2011) Generalised Linear Models with Clustered Data: Fixed and random effects models with glmmML

"Generally speaking, a random effects model is appropriate if the observed clusters may be regarded as a random sample from a (large, possibly infinite) pool of possible clusters. The observed clusters are of no practical interest per se, but the distribution in the pool is. Or this distribution is regarded as a nuisance that needs to be controlled for."

https://cran.r-project.org/web/packages/eha/vignettes/glmmML.pdf

library(lme4) 
library(RVAideMemoire)
Data$obs <- factor(formatC(1:nrow(Data), flag="0", width = 3))
model.glmm <- glmer(cbind(number_pres,number_abs) ~ Var1+Var2+Var3+Var4...+
(1|obs),family = binomial (link = logit),data = Data) 
overdisp.glmer(model.glmm) #Overdispersion for GLMM

Best Answer

Related Solutions

Solved – How to interpret main effects when the interaction effect is not significant

Solved – Accounting for overdispersion in binomial glm using proportions, without quasibinomial

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