[R-sig-ME] Interpreting interaction terms

Thu Jul 15 23:09:06 CEST 2010

In situations like this it helps to think about *simple* examples.  You can do this
just in terms of ordinary garden-variety fixed effects only anova; no need to get
into mixed models and lmer, etc.

Suppose that you have a 2 x 2 design with cell means

	2 3
	3 2

If you were to fit a linear model with no interaction to a y-vector exactly equal to the cell means
(no noise at all) you would find that the A and B effect estimates were exactly 0 (since the row
means and column means are all equal to the grand mean).  However there ***is*** of course interaction
here, the values of the (alpha beta)_ij being +/- 0.5.

If you fit a model *with* interaction you get an error sum of squares equal to 0 and get a non-zero
interaction sum of squares.  The ``main effect'' sums of squares will still of course be 0.

So how can we concoct an example where the main effects appear to be 0 when an additive model is fitted,
but all effects (main and interaction) appear *not* to be 0 when a model with interaction is fitted?

Perturb the basic example a little bit so that the row and column means are not *exactly* equal to
the grand mean, but are close enough so that they will show up as being ``non-significant'' when
their sums of squares are compared with the interaction sum of squares (which becomes the error
sum of squares when you fit an additive model).

When you fit a model with interaction, the error sum of squares gets reduced since the interaction
sum of squares is no longer absorbed into it, and the ``little differences'' between the row and
column means and the grand mean show up as significant.

To do a numerical example in R, replicate the design twice and add a tiny bit of noise so that the
error sum of squares for the model with interaction is not exactly zero (which makes anova() complain
a bit):

mu <- rep(c(2.1,3.2,3,2.1),2)
a  <- rep(factor(c(1,1,2,2)),2)
b  <- rep(factor(c(1,2,1,2)),2)
set.seed(42)
y  <- mu + rnorm(8,0,0.0001)

anova(lm(y ~ a + b)) # Neither a nor b significant.
anova(lm(y ~ a * b)) # All effects significant.

Notice that ``significance'' --- or meaning --- of main effects when there is interaction
in the model is of dubious import.  A significant main effect when there is interaction can be
interpreted as saying that there are differences between the levels of this effect when these
levels are *averaged* over the levels of the *other* factor.  In other words there are differences
between the row means, or column means, in the cell means model. There are those who will fulminate
vehemently about the folly of giving *any* consideration at all to such an interpretation.  And
indeed it is not likely to be of much interest in any practical application.

HTH.

	cheers,

		Rolf Turner

On 16/07/2010, at 6:50 AM, Chris Mcowen wrote:

> Dear List -
> 
> I am fitting some models and have a question regarding the interpretation.
> 
> When i fit the model with no interaction term   only one significant term - pollen dispersal - is significant
> 
> However when i introduce an interaction term ( based on a priori knowledge) not only do the interaction terms become significant ( A*B) but also the terms A + B become significant on there own. Furthermore, new factors become slightly significant.
> 
> Please see below for models ( sorry for formatting).
> 
> My question is, how do i interpret this? Is factor A or B significant on their own? Furthermore, what about the factor that appears significant when the interaction term is added?
> 
> Thanks for your help
> 
> Chris
> 
> 
> NO INTERACTION TERMS
> 
> Starting model - model31 <- lmer(netf~1+(1|order/family) + endosperm + geophyte + breedingsystem + woodyness + seasonality + geophyte + fruit + pollendispersal, family=binomial)
> 
> MAM -  (by removing the factor with the highest pvalue) - model31 <- lmer(netf~1+(1|order/family) + pollendispersal, family=binomial)
> 
> Generalized linear mixed model fit by the Laplace approximation 
> Formula: netf ~ 1 + (1 | order/family) + pollendispersal 
>  AIC  BIC logLik deviance
> 1245 1266 -618.6     1237
> Random effects:
> Groups       Name        Variance Std.Dev.
> family:order (Intercept) 5.3978   2.3233  
> order        (Intercept) 0.0000   0.0000  
> Number of obs: 1242, groups: family:order, 43; order, 9
> 
> Fixed effects:
>                Estimate Std. Error z value Pr(>|z|)  
> (Intercept)       0.9065     0.9071   0.999   0.3177  
> pollendispersal  -1.7225     0.6782  -2.540   0.0111 *
> 
> INTERACTION TERM INCLUDED
> 
> Starting model - model32 <- lmer(netf~1+(1|order/family) + endosperm + geophyte + breedingsystem* woodyness + seasonality + geophyte + fruit + pollendispersal, family=binomial)
> 
> MAM- model33 <- lmer(netf~1+(1|order/family) + breedingsystem*fruit + seasonality + fruit + pollendispersal, family=binomial)
> Generalized linear mixed model fit by the Laplace approximation 
> Formula: netf ~ 1 + (1 | order/family) + breedingsystem * fruit + seasonality +      fruit + pollendispersal 
>  AIC  BIC logLik deviance
> 1246 1287 -614.8     1230
> Random effects:
> Groups       Name        Variance   Std.Dev.  
> family:order (Intercept) 4.8431e+00 2.2007e+00
> order        (Intercept) 4.2899e-11 6.5497e-06
> Number of obs: 1242, groups: family:order, 43; order, 9
> 
> Fixed effects:
>                     Estimate Std. Error z value Pr(>|z|)  
> (Intercept)            2.9283     2.0712   1.414   0.1574  
> breedingsystem        -1.4739     0.7695  -1.915   0.0554 .
> fruit                 -2.4174     1.1130  -2.172   0.0299 *
> seasonality            0.7199     0.4367   1.648   0.0993 .
> pollendispersal       -1.6574     0.6686  -2.479   0.0132 *
> breedingsystem:fruit   1.0401     0.5029   2.068   0.0386 *
> 
> 
> Chris Mcowen
> PhD Student
> 
> Room 15
> Sir Harold Mitchell Building
> University of St Andrews
> St Andrews
> Fife
> KY16 9TH
> UK
> Phone 01334 463381 
> 
> 
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> 
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