Interpreting interactions when main effect is not significant

Question

I just started fooling around with R and I am quite struggling with mixed models: I have the following experimental layout:

Seven different viruses, including ctrl virus;

27 mice, randomly infected with one virus, resulting in 7 groups of 3 to 5 mice;

followed daily over 21 days, resulting in cumulative clinical score

Now I want to analyse the data and see how the virus has an impact on the cumulative clinical score, when compared to ctrl virus. Initially I thought using a linear mixed model with Virus (Vir_) and time (Day) as fixed effects, but then V_Vir1_Vir15 is not significantly different from Ctrl virus, whereas NV_Vir8 is, which I cannot understand in respect to the graph.

  > lNull3 <- lme(score ~  Day * Vir_,data=LinMixOK3, random= ~ 1 | Mouse, method='REML')
> summary(lNull3)
Linear mixed-effects model fit by REML
 Data: LinMixOK3 
       AIC      BIC    logLik
  3822.383 3892.192 -1895.192

Random effects:
 Formula: ~1 | Mouse
        (Intercept) Residual
StdDev:    2.030927 5.778948

Fixed effects: score ~ Day * Vir_ 
                          Value Std.Error  DF   t-value p-value
(Intercept)           -0.048221 1.4000504 564 -0.034443  0.9725
Day                    0.118012 0.0868499 564  1.358809  0.1748
Vir_NV_Vir1_Vir15     -5.636561 2.1000756 564 -2.683980  0.0075
Vir_NV_Vir8           -5.386561 2.1000756 564 -2.564937  0.0106
Vir_V_Vir1_Vir15      -2.113834 2.1000756 564 -1.006551  0.3146
Vir_V_Vir8             0.077866 2.1000756 564  0.037078  0.9704
Vir_Vir1               0.909881 2.2862728  16  0.397976  0.6959
Vir_Vir15              1.442161 2.2862728  16  0.630791  0.5371
Day:Vir_NV_Vir1_Vir15  2.200452 0.1302749 564 16.890835  0.0000
Day:Vir_NV_Vir8        1.463439 0.1302749 564 11.233467  0.0000
Day:Vir_V_Vir1_Vir15   1.700452 0.1302749 564 13.052797  0.0000
Day:Vir_V_Vir8        -0.007199 0.1302749 564 -0.055263  0.9559
Day:Vir_Vir1           0.267457 0.1418253 564  1.885821  0.0598
Day:Vir_Vir15          0.111425 0.1418253 564  0.785648  0.4324

However in the Day:Virus interaction results Day:Vir_NV_Vir1_Vir15, Day:Vir_NV_Vir8 and Day:Vir_V_Vir1_Vir15 are all significant.

On the other hand, when only considering the virus effect, results are consistent with the graph:

> lNull3b<- lme(score ~  Vir_,data=LinMixOK3, random= ~ 1 | Mouse, method='REML')
> summary(lNull3b)
Linear mixed-effects model fit by REML
 Data: LinMixOK3 
       AIC      BIC    logLik
  4443.458 4482.834 -2212.729

Random effects:
 Formula: ~1 | Mouse
        (Intercept) Residual
StdDev:    1.654594 10.14644

Fixed effects: score ~ Vir_ 
                      Value Std.Error  DF  t-value p-value
(Intercept)        1.190909  1.217969 571 0.977783  0.3286
Vir_NV_Vir1_Vir15 17.468182  1.826953 571 9.561376  0.0000
Vir_NV_Vir8        9.979545  1.826953 571 5.462399  0.0000
Vir_V_Vir1_Vir15  15.740909  1.826953 571 8.615937  0.0000
Vir_V_Vir8         0.002273  1.826953 571 0.001244  0.9990
Vir_Vir1           3.718182  1.988934  16 1.869434  0.0800
Vir_Vir15          2.612121  1.988934  16 1.313327  0.2076

Where does this come from ? How should I interpret this ? Is my approach right ?

I use R, so an example in R would be nice.

Thanks !

Answer 1

In general, you need to control for lower level main effects so that the "ratio" interpretation of the interaction term holds. However, the interpretation of the main effect differs substantially after adjusting for an interaction. Usually, unless appropriate transformations have been applied, the main effect is not even worth looking at (much like an intercept).

Suppose, for instance, you're looking at CVD incidence versus age and sex (model 1: no interaction), then the interpretation of the "sex" effect is: the risk ratio for CVD comparing men to women of the same age. Now consider model 2 adjusting for main effects as well as their interaction, the interpretation of "sex" is the risk ratio for CVD comparing men to women at age 0. Needless to say, the extrapolation is so great, it's not even worth considering.

However--and this is a subtle point--if you are interested in a global test for whether or not the age curves for CVD risk vary between men and women you can use model 2 to calculate the two degree of freedom test for the significance of sex as well as its interaction in the age adjusted CVD risk model. In such a case, you might be surprised to see that neither age nor the age-sex interaction is significant, but jointly, there is substantial statistical evidence to conclude that these risk curves are different for men and women.