18 MMRM in Practice: Linearity, Equivalence, and Fit

Prerequisites: Chapters 8 and 10.

18.1 Learning objectives

By the end of this chapter you should be able to:

State the conditions under which a mixed model reduces exactly to ANCOVA or to a summary-statistic analysis.
Explain the bias-variance trade-off between a random-slopes model and a categorical-time MMRM.
Decide between a slope estimand and a final-visit estimand on scientific grounds.
Report and interpret marginal and conditional \(R^2\) for a mixed model, and know when the estimates are unreliable.

18.2 Orientation

The mixed model for repeated measures is the default analysis for longitudinal trials, and it is often adopted without asking two questions that determine whether it is buying anything.

The first is whether the model reduces to something simpler. With two time points and an unstructured covariance, MMRM is algebraically identical to ANCOVA, not approximately but exactly. Knowing this saves arguments and clarifies what the machinery does.

The second is what the model assumes about the shape of the treatment effect. A random-slopes model constrains the effect to accumulate linearly in time. When that is true, it is more powerful than the unconstrained alternative; when it is false, it is biased. The size of both effects is quantifiable and the choice should be made on biological grounds before the data exist.

18.3 Provenance

Three research compendia support this chapter. 18-mmrm-vs-lm-simple (project mmrmlm) establishes the equivalence conditions under which random-effects models reduce to fixed-effect alternatives. 08-mmrm-linearity-robust (project mmrmrobust) quantifies the bias-variance trade-off between the random-slopes and categorical-time MMRM when the treatment effect departs from linearity, calibrated to CDR-SB in Alzheimer disease trials. 17-mixed-r2 (project mixedr2) compares \(R^2\) measures for mixed models across a 32-scenario simulation.

18.4 The statistician’s contribution

(Judgment 1.) Choosing the estimand, slope or final visit, before choosing the model. These are different scientific claims. A slope estimand asserts that the treatment changes the rate of the disease; a final-visit estimand asserts that patients are better off at a specified time. The estimand determines the model, not the reverse.

(Judgment 2.) Assessing whether linearity is plausible for the treatment effect, not merely for the disease trajectory. These are separate assumptions. A disease may progress linearly while a treatment’s effect on it is delayed, and delayed onset is exactly what breaks the random-slopes model.

(Judgment 3.) Not over-selling model fit. An \(R^2\) for a mixed model is not the familiar quantity, and in small samples the estimate is unstable enough that reporting it without uncertainty is misleading.

18.5 When the mixed model reduces to something simpler

The mmrmlm compendium identifies three conditions.

Condition 1: two time points with unstructured covariance. When MMRM is fitted with an unrestricted \(2 \times 2\) covariance matrix to a baseline and a single follow-up, the treatment effect estimate and its standard error are algebraically identical to those from ANCOVA. This is an exact equivalence, not an approximation.

The practical consequence: for any two-time-point design with complete data, ANCOVA is a sufficient analysis, and the MMRM apparatus adds no information about the treatment contrast. Teams that fit MMRM to pre-post data believing it is more sophisticated are fitting ANCOVA with extra steps. What MMRM does add, and it is not nothing, is graceful handling of missing baselines and a route to including patients missing the follow-up.

Condition 2: small between-subject slope variance. In designs with three or more time points, the random-slopes model becomes equivalent to a summary-statistic analysis, in which each patient’s slope is computed and the slopes are compared, when the ratio \(\sigma_b / \sigma\) is small. Below roughly 0.25, for equally spaced designs with the same total follow-up, the efficiency gain from the mixed model is negligible.

This is the same ratio that governed Chapter 11: where measurement error dominates biological variability in rates, which is common in short trials with noisy endpoints, the elaborate model recovers what a per-patient regression slope would have given.

Condition 3: constrained longitudinal data analysis with complete data. The cLDA model, which constrains the baseline means to be equal across arms because randomization guarantees they are, produces treatment effect estimates effectively identical to ANCOVA when no baseline values are missing. Its advantage appears only with incomplete baselines, where it gains 1 to 4% additional power depending on the missingness rate.

None of this argues against mixed models. They retain genuine advantages when data are incomplete under a plausible MAR assumption, when multiple post-baseline time points carry information about the effect, and when the covariance structure itself is of interest. The point is to know which of those applies to the trial at hand.

Check your understanding: what MMRM buys with two time points

Question. A trial measures the outcome at baseline and week 12 only. Fifteen percent of patients miss the week-12 visit. Is MMRM worth fitting?

Answer.

With complete data, no: it is exactly ANCOVA.

With 15% missing week-12 values, still essentially no, and for a reason worth understanding. A likelihood-based analysis uses each patient’s observed data; a patient with a baseline and no follow-up contributes no information about the treatment effect at week 12, because there is nothing in their record after randomization to inform it. Both MMRM and ANCOVA effectively exclude them. The patients with the missing outcome are simply absent from both analyses.

Where the mixed model earns its place is with intermediate visits. Add a week-6 measurement and a patient who attends week 6 and misses week 12 now contributes: their week-6 value, combined with the estimated correlation between weeks 6 and 12, informs the week-12 contrast under MAR. This is the entire mechanism, and it requires intermediate observations to operate.

The design implication is direct. If dropout is expected and the analysis is MMRM, schedule intermediate visits, because they are what converts dropouts from non-contributors into partial contributors.

18.6 Linearity of the treatment effect

Now the harder question. Two models for a trial with several visits:

Random-slopes model. Treatment modifies the rate of change: \(E[Y_{ij}] = \alpha + \beta t_{ij} + \delta Z_i t_{ij}\). The treatment effect at any time is \(\delta t\), a straight line through the origin. The estimand is the difference in slopes, and the effect at the final visit is an extrapolation along that line.

Categorical-time MMRM. Treatment has a separate effect at each visit: \(E[Y_{ij}] = \mu_j + \delta_j Z_i\). No constraint on shape. The estimand is \(\delta_J\), the difference at the final visit.

The mmrmrobust compendium parameterizes the departure from linearity by an exponent \(\kappa\), with the treatment-effect trajectory proportional to \((t/T)^{\kappa}\), so \(\kappa = 1\) is linear, \(\kappa < 1\) front-loaded (early onset then plateau), and \(\kappa > 1\) back-loaded (delayed onset then acceleration).

The findings:

When \(\kappa = 1\), the random-slopes model has negligible bias and higher power than the categorical-time model, consistent with the general principle that a correctly specified parametric model beats a nonparametric alternative.

As \(\kappa\) departs from 1, the random-slopes model incurs systematic bias in the 18-month treatment contrast. The mechanism: the linear constraint forces the effect to accumulate at a constant rate, so a front-loaded or back-loaded true effect is misallocated across time points, distorting the fitted value at the final visit.

The categorical-time model is approximately unbiased regardless of \(\kappa\), because it places no constraint on the shape.

This is a clean bias-variance trade-off: power when the constraint is right, bias when it is wrong, and no protection from the data, because the trial is rarely large enough to distinguish the shapes reliably.

18.6.1 Which estimand, then

The compendium’s recommendation is that the choice between slope and final-visit estimands should be informed by the expected shape of the treatment effect, decided on biological grounds.

Choose the slope estimand and the random-slopes model when there is a strong biological rationale for a constant rate of slowing, as with a treatment that directly modulates the underlying disease rate.

Choose the final-visit estimand and categorical-time MMRM when the mechanism suggests delayed onset or an early plateau. Amyloid-clearing antibodies are the canonical delayed-onset case: months are required to achieve plaque reduction, so the treatment-placebo difference widens progressively, which is \(\kappa > 1\). Visual inspection of the published CDR-SB trajectories from recent phase III anti-amyloid trials is consistent with this pattern.

This aligns with regulatory positions. The FDA guidance notes that slope analyses may be considered when the course of disease is predictable and approximately linear; the EMA similarly conditions slope-based claims on verifiable linearity; and ICH E9(R1) reinforces that the estimand is chosen on scientific grounds before the model is specified.

# Compare the two models on the same simulated data
library(mmrm); library(lme4)

# categorical time: no shape constraint
m_cat <- mmrm(y ~ trt * factor(visit) + base +
                us(visit | id), data = d)

# random slopes: linear constraint
m_slope <- lmer(y ~ trt:time + time + base + (time | id),
                data = d)

Warning

Fitting both models and reporting the one that gives the smaller \(p\)-value is a serious violation of pre-specification and will produce an inflated type I error. The choice must be made in the SAP. If the team genuinely cannot decide, the safe choice is the categorical-time model, which is unbiased under both shapes and costs power only when linearity happens to hold.

18.7 Reporting fit: \(R^2\) for mixed models

Applied readers ask how much variance the model explains, and the answer for a mixed model is not unique. The standard decomposition gives two quantities:

Marginal \(R^2\), the proportion of total variance explained by the fixed effects alone.

Conditional \(R^2\), the proportion explained by fixed and random effects together.

The mixedr2 compendium compares the Nakagawa-Schielzeth measure and the Johnson extension across 32 scenarios crossing the random-effects structure (intercept-only versus random slopes), the intraclass correlation (0.1 and 0.5), the target marginal \(R^2\) (0.1 and 0.5), and sample size (20 or 50 clusters crossed with 5 or 20 observations per cluster), at 1,000 replicates each.

The findings:

Both methods are approximately unbiased across the simulated conditions, confirming their mathematical equivalence for random-intercept models.
Differences appear only in random-slope scenarios, where the Johnson extension has modestly lower bias.
Small samples (20 clusters of 5) produce substantially wider sampling distributions and lower monotonicity rates than larger samples.
Marginal \(R^2\) is estimated most precisely in high-ICC, large-sample conditions, where the variance decomposition is most identifiable.
Singular fits are common in low-ICC random-slope scenarios, consistent with the known limitations of maximum likelihood for small variance components.

The practical recommendations that follow: use the Nakagawa-Schielzeth measure for random-intercept models and the Johnson extension for random-slope models; report both marginal and conditional values; attach a measure of uncertainty when there are fewer than 30 clusters or fewer than 10 observations per cluster; and report convergence and singularity explicitly, since a singular fit makes the variance decomposition, and therefore \(R^2\), meaningless.

library(performance)
r2_nakagawa(fit)
#> # R2 for Mixed Models
#>   Conditional R2: 0.641
#>      Marginal R2: 0.218

18.8 Worked example: choosing the analysis for an 18-month AD trial

An anti-amyloid antibody, 18-month follow-up, CDR-SB measured at baseline and months 3, 6, 12, 18. Expected dropout 20%.

Estimand. The mechanism requires months of plaque clearance before any clinical effect, so a delayed onset is expected, \(\kappa > 1\). On the compendium’s evidence, the random-slopes model would be biased in the direction of understating the 18-month difference. Choose the final-visit estimand.

Primary model. Categorical-time MMRM, unstructured covariance, fixed effects for treatment, visit, treatment-by-visit, baseline CDR-SB, and stratification factors, Kenward-Roger degrees of freedom, with the month-18 contrast as the primary.

Why intermediate visits are scheduled. They are not merely descriptive. Under 20% dropout, they are the mechanism by which patients who withdraw after month 12 contribute to the month-18 estimate.

Secondary slope analysis. Reported as secondary, with the explicit note that it assumes linear accumulation, and with the estimated trajectory shape plotted so that readers can judge the assumption.

Model fit. Marginal and conditional \(R^2\) reported for the primary model with bootstrap intervals, and convergence and singularity status stated.

What the design would have been if the mechanism were different. For a symptomatic agent with a rapid, sustained effect, the slope estimand with a random-slopes model would have been more powerful, and the visit schedule could have been sparser at the expense of the dropout-handling benefit.

18.9 Collaborating with an LLM on longitudinal models

Prompt 1: ‘Should we use MMRM or ANCOVA?’

What to watch for. Models almost always recommend MMRM, without noting that with two time points and complete data the two are algebraically identical.

Verification. Count the post-baseline time points. If there is one, the answer is that it does not matter for the point estimate.

Prompt 2: ‘Fit a random-slopes model for this trial.’

What to watch for. Code that fits the model without flagging the linearity assumption on the treatment effect, and that reports the slope difference as though it were the effect at the final visit.

Verification. Ask explicitly what the model assumes about the shape of the treatment effect over time, and whether that assumption is defensible for the mechanism.

Prompt 3: ‘Report the R-squared for this mixed model.’

What to watch for. A single number with no indication of whether it is marginal or conditional, and no attention to singularity.

Verification. Check convergence and the variance components. If a variance component is estimated at zero, the decomposition is degenerate.

18.10 Principle in use

Count the time points before choosing the model. Two means ANCOVA; three or more means the covariance structure and the shape assumption start to matter.
Decide the shape assumption from the mechanism, in the SAP. The data will not tell you, and choosing after seeing them invalidates the inference.
Schedule intermediate visits if dropout is expected. They are the machinery that makes MMRM’s MAR handling work.

18.11 Exercises

Simulate a two-time-point trial with complete data. Fit MMRM with unstructured covariance and ANCOVA, and verify the treatment estimates and standard errors agree to numerical precision.
Repeat with 20% of follow-up values missing at random and confirm the two analyses still agree. Then add an intermediate visit and show that they no longer do.
Simulate treatment-effect trajectories with \(\kappa = 0.5, 1.0, 2.0\). Estimate the bias of the random-slopes model and the categorical-time model for the final-visit contrast in each case.
For the same simulations, compute the power of both models at \(\kappa = 1\) and quantify what the categorical-time model gives up when linearity holds.
Reproduce the finding that singular fits are common in low-ICC random-slope scenarios: simulate 500 datasets with 20 clusters of 5 observations and ICC 0.05, and report the singularity rate and the distribution of marginal \(R^2\).

18.12 Further reading

The compendia 18-mmrm-vs-lm-simple, 08-mmrm-linearity-robust, and 17-mixed-r2.
Mallinckrodt et al. (2008), on MMRM as the recommended primary analysis.
Fitzmaurice et al. (2011), Chapters 8 and 15.
Nakagawa & Schielzeth (2013) and Edwards et al. (2008), on \(R^2\) for mixed models.
The mmrm, lme4, lmerTest, and performance R packages.