How to interpret 2-way and 3-way interaction in lmer? 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?
 A: 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.


*

*The intercept is the mean of the dependent variable in the three base levels.

*rat is the difference between rat and lion (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 of rat is estimated for data where color = white and sex = female.

*sexmale is the difference between males and females (where animal = lion and color = white).

*colorred is the difference between red and white (where animal = lion and sex = female).

*coloryellow is the difference between yellow and white (where animal = lion and sex = female).

*rat:sexmale: The difference between lions and rats is higher for males than for females (where color = white).

*rat:colorred: The difference between lions and rats is higher for red than for white (where sex = female).  

*rat:coloryellow: The difference between lions and rats is higher for yellow than for white (where sex = female).  

*sexmale:colorred: The difference between males and females is higher for red than for white (where animal = lion).               

*sexmale:coloryellow: The difference between males and females is higher for yellow than for white (where animal = lion).                    

*rat:sexmale:colorred: Three-factor interaction. The effect rat:sexmale is different for red compared to white.           

*rat:sexmale:coloryellow: Three-factor interaction. The effect rat:sexmale is different for yellow compared to white.        


To test further contrasts, you have to run another analysis.
