# 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:

1. coloryellow = The height of subjects are lower when they like yellow, and are higher if they like white.
2. rat:colorred = The effect of rat preference enhances the preference of red, and these two promote height of subjects.
3. 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?

• Welcome to the site.Interpreting interactions in this sort of model is quite similar to interpreting them in "regular" regression. This has been discussed here often. If you search on the term "interaction" you will find a bunch of posts. – Peter Flom - Reinstate Monica Feb 21 '14 at 12:29

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.

• Thanks very much Sven. Your reply enlighten my knowledge towards interaction output. ^^ – user3288202 Feb 21 '14 at 12:59
• Sven, is your answer 100% correct? I wanted to comment, but I do not have enough rep. Shouldn't the interpretation be, for example for rat:sexmale, that the difference between females and males is smaller for rats? – user139190 Mar 18 '17 at 18:32
• @MichaelR I agree with your comment. Here, I mean directed difference. Hence, higher is related to a numerically higher abolute value (less negative or more positive). – Sven Hohenstein Mar 18 '17 at 20:07
• Seven, I am wonder your interpretation regarding the main effect. For example, the rat, is not this mean that the difference between rat and lion and across other factors such as gender and color? In another word, generally there is no difference between rat and lion. – Ping Tang May 28 '17 at 2:07
• @PingTang Since rat is also part of interactions, the main effect hold for the reference levels of the factors that are in interaction with rat. – Sven Hohenstein May 28 '17 at 2:19