I have a problem with interpreting 2-way and 3-way interactions in lmer. My DV is height which is a continuous variable. All IVs are categorical variables. The first factor is animal, either rat or lion. The second factor is sex, either male or female. The third factor is color: red, white, or yellow. I get confused with interpreting the output:
Fixed effects:
Estimate Std. Error t value
(Intercept) 164.6888 7.8180 21.065
rat -14.1342 8.2889 -1.705
sexmale -16.0883 10.0071 -1.608
colorred 0.5776 6.2473 0.092
coloryellow -14.4048 6.1025 -2.360
rat:sexmale 15.3645 11.8567 1.296
rat:colorred 12.5258 4.4028 2.845
rat:coloryellow 10.3136 4.3196 2.388
sexmale:colorred 2.0272 5.2773 0.384
sexmale:coloryellow 5.7643 5.1669 1.116
rat:sexmale:colorred -5.5144 6.2838 -0.878
rat:sexmale:coloryellow 0.9735 6.1690 2.158
According to Vasishth et al. (2007), the significance of fixed effects can be judged from the absolute t value; if it is higher than 2, then that factor is significant. In interpreting this output, I choose only factors which are significant. Please check if my interpretations are correct:
coloryellow
= The height of subjects are lower when they like yellow, and are higher if they like white.rat:colorred
= The effect of rat preference enhances the preference of red, and these two promote height of subjects.rat:sexmale:coloryellow
= The effect of rat preference, being male, enhances the preference of yellow, and subjects who like rat and yellow and are male have higher height.
From these interpretations, I would like to ask: if I would like to know the effect of lion:sexfemale:colorred
, and rat:sexmale:colorred
compared to rat:sexfemale:coloorred
, do I have to run new statistics?
Best Answer
First of all, the default contrasts for categorial variables in R are treatment contrasts. In treatment contrast, all levels of a factor are compared to the base level (reference category).
The base levels do not appear in the output. In your example, the base levels are:
animal
:lion
color
:white
sex
:female
Note that all effects are estimated with respect to the base levels.
Let's have a look at the effects. You're interpretation is correct.
intercept
is the mean of the dependent variable in the three base levels.rat
is the difference betweenrat
andlion
(with respect to the dependent variable). Note that this is not a global difference, but a difference with respect to the other base levels. The effect ofrat
is estimated for data wherecolor = white
andsex = female
.sexmale
is the difference between males and females (whereanimal = lion
andcolor = white
).colorred
is the difference betweenred
andwhite
(whereanimal = lion
andsex = female
).coloryellow
is the difference betweenyellow
andwhite
(whereanimal = lion
andsex = female
).rat:sexmale
: The difference between lions and rats is higher for males than for females (wherecolor = white
).rat:colorred
: The difference between lions and rats is higher for red than for white (wheresex = female
).rat:coloryellow
: The difference between lions and rats is higher for yellow than for white (wheresex = female
).sexmale:colorred
: The difference between males and females is higher for red than for white (whereanimal = lion
).sexmale:coloryellow
: The difference between males and females is higher for yellow than for white (whereanimal = lion
).rat:sexmale:colorred
: Three-factor interaction. The effectrat:sexmale
is different for red compared to white.rat:sexmale:coloryellow
: Three-factor interaction. The effectrat:sexmale
is different for yellow compared to white.To test further contrasts, you have to run another analysis.