Understanding covariance in MMRM

🔑 1. Within-Subject Covariance (Repeated Measures Covariance)

This is the main type of covariance modeled in MMRM.

➤ What it is:

• The correlation of repeated measurements within the same subject over time.

• It captures how a subject’s outcome at Week 12, Week 24, Week 48, etc., are related to each other.

➤ How it’s handled:

• This is modeled using the REPEATED statement in SAS (or nlme::lme in R).

• You specify a covariance structure, such as:

Covariance Structure	Description
UN (Unstructured)	Every timepoint has its own variance; all pairwise covariances are estimated. Very flexible but high parameter count.
CS (Compound Symmetry)	Constant variance and constant correlation between any two visits.
AR(1)	Correlation declines as timepoints are further apart (auto-regressive).
TOEP (Toeplitz)	Fixed correlation pattern depending on lag between visits.
VC (Variance Components)	No correlation, but separate variances per visit.
FA (Factor Analytic)	Parsimonious alternative to UN; good for many timepoints.

🔢 Number of parameters for common structures (assuming k timepoints):

Structure	# of parameters
UN	k(k+1)/2
CS	2 (variance + correlation)
AR(1)	2 (variance + ρ)
TOEP(k−1)	k variances + (k−1) covariances

---

🔑 2. Between-Subject Covariance

• Not explicitly modeled in MMRM.

• Why? Because MMRM assumes that each subject is independent of the others.

• The focus is entirely on within-subject repeated measures.

However:

• The random errors across different subjects are assumed independent, unless you’re using random effects models (e.g., linear mixed models with subject-level random intercepts).

MMRM does not include random effects like subject-level intercepts. Instead, it uses a marginal model with a fully specified within-subject covariance.

---

🔍 Summary Table

Covariance Type	Modeled in MMRM?	Example Structures	# of Parameters (k=5 visits)
Within-Subject	✅ Yes	UN, CS, AR(1), TOEP, FA	Varies (e.g., 15 for UN)
Between-Subject	❌ No	(Assumed independent)	0
Subject-Level Random Effects	❌ No in MMRM	(Only in LME models)	N/A

---

🧪 Bonus: What Happens If You Use Random Effects?

If you use PROC MIXED with a RANDOM intercept / subject=ID; statement instead of REPEATED, you’re fitting a linear mixed-effects model (LME), not a marginal MMRM. That has different assumptions:

• Conditional model (subject-specific)

• Assumes random intercepts (or slopes)

• Simpler within-subject residual structure (often i.i.d.)

MMRM, by contrast:

• Is marginal

• No subject-specific random effects

• Models entire within-subject covariance matrix directly

---

Final Notes

• Always choose your within-subject covariance structure based on:

  • Number of visits

  • Sample size

  • Missing data pattern

  • Model fit (AIC/BIC/Likelihood Ratio Test)