I am trying to inspect the data from a 2 x 2 factorial design. The experiment was run by other researchers and the design was settled upon before.

Participants were tested 3 times using 3 different texts. These texts can be defined by two factors with two levels each. I will give you a comparable example:

Factor 1: Lexical complexity (average vs. high) Factor 2: Readability (average vs. high)

Three different texts were created based on the two factors and data was collected from each participant for each of these three texts.

Text 1: high lexical complexity + average readability Text 2: average lexical complexity + high readability Text 3: average lexical complexity + average readability

I am trying to fit a linear-mixed effects models with intercepts-only random effects for participant. I am trying to estimate a model that includes an interaction between Lexical Complexity and Readability, but to no success. The model is rank deficient and the interaction is not estimated.

Any idea what could be wrong here?

  • $\begingroup$ Greetings and welcome to CV! A few things...first, can you plot what the variation between/within subjects is? One can for example plot the by-subject variation in each association (as shown here). Second, what is the model syntax? It could just be an error in your code. $\endgroup$ Feb 14 at 11:31
  • $\begingroup$ Hello Shawn and thank you for your reply! The model syntax is this one: lmer(Mean_F0 ~ LexicalComplexitySumC + ReadabilitySumC + (1|participant_number), data = session2, REML = F, control=lmerControl(calc.derivs=FALSE, optimizer = "bobyqa")). The factors are sum coded, that is where the SumC ending comes from (-1 for average lexical density, 1 for high density; -1 for average readability , 1 for high readability) $\endgroup$
    – xcvfg
    Feb 14 at 11:34
  • $\begingroup$ Thanks for sharing. But the second question I asked remains unanswered. Given there are only two levels of each factor and three data points per cluster, I suspect there isn't enough variation to calculate in a mixed model, where if so a mixed model becomes somewhat redundant and difficult to fit. $\endgroup$ Feb 14 at 11:53
  • $\begingroup$ Dear Shawn, I generated the variances, I only have 10 DPs for the 10 participants. $participant_number (Intercept) 1 -66.35224 2 13.77819 3 23.56233 4 -35.26209 5 25.54489 6 -41.15692 7 51.23483 8 16.49912 9 -50.78433 10 62.89639. I also tried to just fit a simple linear model instead, but it did not work in terms of the interaction. I just got NA values for the interaction, but this time no warning that it was rank deficient. What could lead to this? $\endgroup$
    – xcvfg
    Feb 14 at 12:23

1 Answer 1


The problem is in the design; you don't really have 2x2, you have something like 2x1.5. Whenever there is high lexical complexity, there is average readability, so the interaction can't be estimated. This is why the model is rank deficient.

And, even in the main effects model, there is surely collinearity.

  • $\begingroup$ Dear Peter, thank you for your reply. I am not sure how to tackle this data. As mentioned this is not my design and I am just in charge of making sense of the data after it has already been collected. I am not sure how I could possibly approach this. Do you have any suggestions? $\endgroup$
    – xcvfg
    Feb 14 at 12:38
  • $\begingroup$ Not really. This is a prime example of Ronald Fisher's maxim "To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of." $\endgroup$
    – Peter Flom
    Feb 14 at 13:11
  • 1
    $\begingroup$ Ah good point @PeterFlom I didnt even notice that til you pointed it out. $\endgroup$ Feb 14 at 13:38

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