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I have recently been asked to check the sphericity of my data to confirm it meets the assumptions of the linear mixed models I have generated (using lmerTest in R). I have read conflicting information about whether sphericity is indeed an assumption of LMMs. (1) Is sphericity an assumption of LMMs, and (2) if so, how can I address non-sphericity?

My models include continuous and categorical variables, with subject ID as a random intercept to account for multiple observations per subject.

Something like: y ~ age + sex + (1 | id)

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No, they don't. That's one of the reasons to use them in place of repeated measures ANOVA, which does assume this. There are many references for this, it's in several books on mixed models (aka multilevel models) but I don't have those books any more. But I am pretty sure they are listed in e.g. Gelman and Hill Data Analysis Using Regression and Hierarchical/Multilevel Models and many other texts on the subject.

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Magezi (2015). Linear mixed-effects models for within-participant psychology experiments: an introductory tutorial and free, graphical user interface (LMMgui). Frontiers in Psychology.

Which looks pretty good and non-technical.

Chapter 12 of Introduction to Multilevel Models by Shaw and Flake, lists these assumptions of MLMs:

In brief, the assumptions underlying MLMs are as follows:

  • The model is correctly specified (i.e., all the predictors associated with the outcome and relevant random effects are included);
  • The functional form is correct (e.g., the relationship between the predictors and outcome is linear if using a linear model);
  • Level-1 residuals are independent and normally distributed;
  • Level-2 residuals are independent and multivariate normally distributed;
  • Residuals at level-1 and level-2 are unrelated;
  • Predictors at one level are not related to errors at another level (homoscedasticity).
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  • $\begingroup$ Isn't the assumption of sphericity the same as the assumption of homoscedasticity? Although I don't understand the phrase of the last bullet point, that's also part of LMMs, not? $\endgroup$
    – Sextus Empiricus
    Commented Dec 1, 2023 at 14:03
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    $\begingroup$ As far as I remember, sphericity corresponds to random intercepts, i.e., the same correlation among all measurements within the same level of the grouping variable (here id) and constant variance. Hence, you relax the sphericity assumption when you specify a more complex random-effects structure. $\endgroup$
    – Dimitris Rizopoulos
    Commented Dec 1, 2023 at 14:55
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    $\begingroup$ Spherecity is broader than homoscedasticity. Sphericity means that all the covariances are the same. This would mean that correlation between time 1 and time 2 is the same as time 1 and time 3. In LMMs, you can specify covariance patterns that violate this. $\endgroup$
    – Peter Flom
    Commented Dec 1, 2023 at 14:55
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    $\begingroup$ This is where I find it confusing, because I have read the linked resource from Magezi, and indeed there is mention of sphericty for repeated measure ANOVAs, and not for LMM. However, the comment "relax the sphericity assumption when you specify a more complex random-effects structure" seems to indicate that ther is a sphericity assumption? I ask because I have a paper under revision and I got a reviewer comment asking me to address sphericity in LMMs. I have checked the other assumptions, including homoscedasticity, for my models, but the sphericity comment has me stumped. $\endgroup$
    – HarD
    Commented Dec 1, 2023 at 15:05
  • $\begingroup$ That wording is not ideal. I think what they mean is "apply a random effects structure that is more complex than compound symmetry". (If you look into how SAS handles this in PROC MIXED, you can get some good descriptions, even if you don't use SAS; I find that documentation much clearer than what R has). $\endgroup$
    – Peter Flom
    Commented Dec 1, 2023 at 15:14

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