# R lmer model: degree of freedom and chi square values are zero

I have built the following models:

full <- lmer(DV~ A*B + (1|speaker), data, REML=FALSE)

A <- lmer(DV~ A+ A:B + (1|speaker), data, REML=FALSE)
B <- lmer(DV~ B+ A:B + (1|speaker), data, REML=FALSE)
interaction <- lmer(DV~ A + B + (1|speaker), data, REML=FALSE)


I use anova to compare the first full model to the other ones:

anova(full, A)
anova(full, B)
anova(full, interaction)


The first two comparisons generated results with both df and chi square values being zeros, as shown below:

However, I have also tried to compare the null model with another model only include A or B:

null <- lmer(DV~ 1 + (1|speaker), data, REML=FALSE)
AA <- lmer(DV~ A + (1|speaker), data, REML=FALSE)
BB <- lmer(DV~ B + (1|speaker), data, REML=FALSE)
AB <- lmer(DV~ A:B + (1|speaker), data, REML=FALSE)


all the comparisons generated reasonable results (i.e. not 0 df and all comparisons are significant)

I have looked online and found this post: https://www.researchgate.net/post/What_is_a_Likelihood_ratio_test_with_0_degree_of_freedom

And my guess is that maybe for my full model, the interaction might be able to predict everything without the main effects (A and B).

I have a few questions:

1. Is my guess possibly true?
2. If it is true, why did the comparison with the null model show a significant effect?
3. On a more general scale, when I build linear mixed effect models, can I start from the Null model and add a factor at a time, then compare with the previous models? Or do I have to reduce from the full model?
4. If I use A+B as the base model:
base <- lmer(DV~ A+B + (1|speaker), data, REML=FALSE)

A <- lmer(DV~ A + (1|speaker), data, REML=FALSE)
B <- lmer(DV~ B + (1|speaker), data, REML=FALSE)
interaction <- lmer(DV~ A*B + (1|speaker), data, REML=FALSE)


Is it ok to report the comparison between the base model and A, B, interaction respectively?

Please find the data file and the R markdown document here: dropbox.com/sh/88m8h6blow2xbn5/AABiNccsUlu3AlfPyamQP4n_a?dl=0 I also asked a question about the procedures I used in the R script in this post R lmer model: add factors or reduce factors

I'd be most grateful if you could help me please. Thank you!

• What is C? I see models full, A, B and interaction. But then you do anova(full, C) – Jeremy Miles Aug 21 '20 at 23:55
• It's better to copy and paste output with commands, for that reason. – Jeremy Miles Aug 21 '20 at 23:56
• sorry, C was supposed to be 'interaction'. I have changed it now. Thanks! – RoroMario Aug 22 '20 at 0:00
• Also, it can be overwhelming to people who want to answer if you ask multiple questions. I would consider asking one question at a time. First, to understand what's happening with the models. Then to discuss what comparisons should be done. (Someone might only want to answer one question, but feel like they shouldn't answer only half). – Jeremy Miles Aug 22 '20 at 0:14
• @StatsStudent Thanks for your answer! I think you must be right, because I also got a warning after I run the interaction only model [AB <- lmer(DV~ A:B + (1|speaker), data, REML=FALSE)] saying "fixed-effect model matrix is rank deficient so dropping 1 column / coefficient". But I don't understand why since I have 576 observations; A has 4 levels; B has 2 levels. What should I do if your explanation is true? – RoroMario Aug 22 '20 at 22:13

This happens because models full, A and B are in fact the same. They are just parameterised differently. To see this, inspect the estimates for the full model:

(Intercept)  6.03977    0.34949  17.282
AT2         -0.55051    0.07597  -7.246
AT3         -1.16472    0.07597 -15.331
AT4          0.48228    0.07597   6.348
BS          -0.64024    0.07597  -8.427
AT2:BS       0.35379    0.10744   3.293
AT3:BS       0.47244    0.10824   4.365
AT4:BS       0.05247    0.10744   0.488


In model A, we have removed the main effect for the variable B and then obtain:

            Estimate Std. Error t value
(Intercept)  6.03977    0.34949  17.282
AT2         -0.55051    0.07597  -7.246
AT3         -1.16472    0.07597 -15.331
AT4          0.48228    0.07597   6.348
AT1:BS      -0.64024    0.07597  -8.427
AT2:BS      -0.28645    0.07597  -3.770
AT3:BS      -0.16781    0.07710  -2.177
AT4:BS      -0.58777    0.07597  -7.737


We immediately see that the estimates for the intercept AT2- AT4 are the same. The estimate for AT1:BS in the second model is identical to the estimate for the main effect for B in the full model (because the second model does not include the main effect for B). Then, for the same reason, the remaining interaction terms in the second model will be the sum of the main effect for B in the full model, and the equivalent interaction terms:

> -0.64024 + 0.35379
[1] -0.28645
> -0.64024 + 0.47244
[1] -0.1678
> -0.64024 + 0.05247
[1] -0.58777


I think it is good general advice to always include both main effects in a model which includes their interaction. This type of problem will then not occur.