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As an example I use Pinheiro, J. C. & Bates, D. M. 2000. Mixed-effects models in S and S-PLUS. Springer, New York. page 225. Rats whose body mass has been measured are fed by 3 different diets over time.

Response: Body mass, fixed effects Time*Diet, random effect ~Time|Rat. The main question with this test was if the interaction term is significant (i.e. difference in growth rate between diets). However, my question is could I also look at the p-values of the main effects to say that body mass increased significantly with time for Diet1 (which is the "dummy variable")?

From Pinheiro, J. C. & Bates, D. M. (2000)

 Fixed effects: weight ~Time * Diet
                  Value        St.error         DF        t-value         p-value 
 Intercept       251.60         13.068         157         19.254          <.0001
 Time              0.36          0.088          13          4.084          0.0001
 Diet2           200.78         22.657          13          8.862          <.0001
 Diet3           252.17         22.662         157         11.127          <.0001
 Time*Diet2        0.60          0.155         157          3.871          0.0002
 Time*Diet3        0.30          0.156         157          1.893          0.0602 

As stated by Pinheiro & Bates, the growth rate of diet 2 (TimeDiet2) differs significantly from diet 1. Although could I state like this for the effect of time on Diet1: f(x) = 251.60 (+/-13.068) + 0.36 x (+/- 0.088), t = 4.084, p = 0.0001? I have seen that people have claimed that it is wrong to interpret p-values for the main effects when the interaction is significant. And is it more proper to split the data and run the test (weight ~Time) for each diet separately, when only looking at the effect of time on body mass?

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If there is interaction in the model, the interpretations for main effects would change. For example, in your model, if there is no interaction between Time and Diet, Diet2 means the difference between Diet2 and Diet1 regardless the value of Time; however, if you add the interaction Time*Diet, Diet2 means the difference between Diet2 and Diet1 when Time equals 0, i.e. the difference of intercepts. This depends on model formula, but not significance.

You can say that "body mass increased significant with Time for Diet1" just based on the significance of Time; but for Diet2 and Diet3, to test the slopes, you may need to test the linear combinations of parameters, say for Diet2, to test the significance of Time+Time*Diet2.

  1. for Diet1: weight = 251.60 + 0.36*Time;
  2. for Diet2: weight = (251.60 + 200.78) + (0.36 + 0.60)*Time;
  3. for Diet3: weight = (251.60 + 252.17) + (0.36 + 0.30)*Time.

You can split the data and run the test separately, but it is formal to integrate the three regressions into one by using the interaction.

By the way, if you need to test the significance of Time*Diet, you may use anova() since it is actually a factor.

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