Mixed model for repeated measures in a longitudinal Solomon design

Analyzes a Solomon four-group design with several posttest occasions by a mixed model for repeated measures (MMRM), and estimates the four Solomon contrasts at each occasion and the change in pretest sensitization from the first occasion to the last. The model is likelihood-based, so it is valid when posttests are missing at random, for example when participants drop out depending on their earlier scores (Fitzmaurice et al., 2011, pp. 497, 505). Per-occasion analyses of the participants still observed, and unweighted generalized estimating equations, require the stronger assumption that the posttests are missing completely at random (Liang & Zeger, 1986; Fitzmaurice et al., 2011, pp. 358, 498).

Usage

fit_solomon_mmrm(
  y_post,
  treat,
  pretested,
  id,
  occasion,
  y_pre = NULL,
  df = c("kenward-roger", "satterthwaite"),
  conf_level = 0.95,
  data = NULL
)

Arguments

y_post: Numeric posttest scores, one per participant and occasion (long format), with NA for missing posttests.
treat: Treatment indicator coded 0/1 (or logical), the same in every row of a participant. Designs with several treatments are not supported; see fit_solomon_glm().
pretested: Pretest indicator coded 0/1 (or logical), the same in every row of a participant.
id: Participant identifier.
occasion: Posttest occasion. A factor keeps its level order; otherwise the sorted unique values are used.
y_pre: Optional pretest score, the same in every row of a participant and missing by design for unpretested participants.
df: Degrees of freedom for tests and intervals: "kenward-roger" (default) or "satterthwaite".
conf_level: Confidence level. Default 0.95.
data: Optional data frame. When supplied, the other data arguments are looked up in it first, as bare column names (y_post = post) or as strings (y_post = "post").

Value

An object of class solomon_mmrm with effects (the Solomon contrasts at each occasion and the change in sensitization), model (the mmrm fit), covariance (the structure used), observed (observed posttests by group and occasion), and the settings used.

Model

The specification follows the example of Mallinckrodt et al. (2008, p. 312), adapted to the Solomon design:

Fixed effects. Occasion x Treatment x Pretested, all categorical.
Pretest adjustment. The pretest enters as in fit_solomon_glm(): the score in the pretested groups and 0 in the unpretested groups, with a separate slope at each occasion ("a full interaction of the covariate with time", p. 312). Pretests absent by design are never imputed.
Covariance. Unstructured within participant, estimated by restricted maximum likelihood (Laird & Ware, 1982). With a pretest, it is estimated separately for pretested and unpretested participants: the pretest adjustment makes the residual covariance conditional on the pretest in the pretested groups only, so a single shared matrix would be misspecified.
Fallback. If the unstructured model does not converge, the heterogeneous Toeplitz, heterogeneous AR(1), and heterogeneous compound symmetry structures are fitted, and the converged one with the smallest AIC is used (Mallinckrodt et al., 2008, p. 312). The output names the structure used.
Inference. t tests and intervals with Kenward-Roger degrees of freedom and adjusted covariance (Kenward & Roger, 1997, as cited in Fitzmaurice et al., 2011, p. 101), as Mallinckrodt et al. (2008, p. 312) specify, or with Satterthwaite (1946) degrees of freedom. The Kenward-Roger adjustment depends on how the covariance is parameterized; it is computed with the matrix's own elements as the parameters, the form that reproduces SAS PROC MIXED (mmrm's "Kenward-Roger-Linear"; Sabanes Bove et al., 2026). Fitzmaurice et al. (2011, p. 101) note that the small-sample properties of both approximations in longitudinal models "have not been extensively studied"; the simulation study posted on issue #57 compares them in Solomon designs.

The model is fitted with the mmrm package (Sabanes Bove et al., 2026), which must be installed.

Validation

A simulation study under a protocol posted on issue #57 before any run (12 scenarios, 2,000 replications each; see the article "Longitudinal Designs: Validating the Repeated-Measures Analysis") generated monotone dropout that depended on the last posttest. With 30 to 120 participants per group:

Coverage and Type I error. With Kenward-Roger degrees of freedom, 95% intervals covered from 0.937 to 0.962 of the time, and Type I error where the true value was zero ranged from 0.041 to 0.0615. Satterthwaite degrees of freedom performed alike.
Bias. The contrasts were unbiased; the largest bias was 0.029 SD.
Normal reference distribution. It gave coverage as low as 0.935 at 30 per group.
Shared covariance. A covariance shared by all four groups misstated the standard errors by up to 12% for the unpretested contrasts and 17% for the pretested ones.
Complete-case analyses. Per-occasion analyses of the participants still observed were biased by up to 0.09 SD at the last occasion.

Lifecycle

Experimental. The study's pre-specified rule for choosing the default degrees of freedom, every tolerance met in 90% of cells and in every cell with 60 or more per group, was not met by either approximation (145 of 156 cells each; 97 and 96 of 104), although Monte Carlo error alone makes the second part unlikely to be met even by a correctly calibrated method. Kenward-Roger remains the default, as Mallinckrodt et al. (2008) specify.

References

Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2011). Applied longitudinal analysis (2nd ed.). Wiley. https://doi.org/10.1002/9781119513469

Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), 963–974. https://doi.org/10.2307/2529876

Liang, K.-Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1), 13–22. https://doi.org/10.1093/biomet/73.1.13

Mallinckrodt, C. H., Lane, P. W., Schnell, D., Peng, Y., & Mancuso, J. P. (2008). Recommendations for the primary analysis of continuous endpoints in longitudinal clinical trials. Drug Information Journal, 42(4), 303–319. https://doi.org/10.1177/009286150804200402

Sabanes Bove, D., Li, L., Dedic, J., Kelkhoff, D., Kunzmann, K., Lang, B. M., Stock, C., Wang, Y., James, D., Sidi, J., Leibovitz, D., Sjoberg, D. D., Krieger, N. I., Panagos, A., & Jones, J. (2026). mmrm: Mixed models for repeated measures (Version 0.3.18) [R package]. https://doi.org/10.32614/CRAN.package.mmrm

Satterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6), 110–114. https://doi.org/10.2307/3002019

Examples

# \donttest{
if (requireNamespace("mmrm", quietly = TRUE)) {
  # Three posttest occasions for the example participants, with dropout.
  set.seed(57)
  d <- solomon_example
  d$id <- seq_len(nrow(d))
  long <- do.call(rbind, lapply(1:3, function(t) {
    x <- d
    x$occasion <- t
    x$y_post <- d$y_post - 2 * (t - 1) + rnorm(nrow(d), 0, 4)
    x
  }))
  long$y_post[long$occasion == 3 & runif(nrow(long)) < 0.3] <- NA
  fit_solomon_mmrm(y_post, treat, pretested, id, occasion, y_pre, data = long)
}
#> Solomon MMRM: mixed model for repeated measures (Mallinckrodt et al., 2008)
#> Occasions: 1, 2, 3
#> Covariance: unstructured, separately for pretested and unpretested participants (REML)
#> Degrees of freedom: Kenward-Roger
#> 
#> Observed posttests by group and occasion:
#>                        1  2  3
#> pretested treatment   30 30 22
#> pretested control     30 30 26
#> unpretested treatment 30 30 23
#> unpretested control   30 30 20
#> 
#>  Occasion Contrast                      Estimate SE    df    t     p    
#>  1        ATE (avg over pretest)         3.027   1.586 114.6  1.91 0.059
#>  1        Pretest x Treatment           -3.301   3.172 114.6 -1.04 0.300
#>  1        Treatment | pretested          1.377   2.298  57.0  0.60 0.552
#>  1        Treatment | unpretested        4.677   2.187  58.0  2.14 0.037
#>  2        ATE (avg over pretest)         3.107   1.730 115.0  1.80 0.075
#>  2        Pretest x Treatment           -3.606   3.460 115.0 -1.04 0.300
#>  2        Treatment | pretested          1.305   2.447  57.0  0.53 0.596
#>  2        Treatment | unpretested        4.910   2.447  58.0  2.01 0.049
#>  3        ATE (avg over pretest)         3.918   1.770 109.1  2.21 0.029
#>  3        Pretest x Treatment           -0.023   3.540 109.1 -0.01 0.995
#>  3        Treatment | pretested          3.906   2.559  56.7  1.53 0.132
#>  3        Treatment | unpretested        3.929   2.446  52.4  1.61 0.114
#>  3 vs 1   Change in Pretest x Treatment  3.277   2.353  89.9  1.39 0.167
#>  95% CI          
#>  [ -0.115, 6.169]
#>  [ -9.584, 2.983]
#>  [ -3.225, 5.978]
#>  [  0.299, 9.055]
#>  [ -0.320, 6.535]
#>  [-10.460, 3.249]
#>  [ -3.596, 6.205]
#>  [  0.013, 9.808]
#>  [  0.409, 7.426]
#>  [ -7.039, 6.993]
#>  [ -1.219, 9.031]
#>  [ -0.979, 8.837]
#>  [ -1.398, 7.953]
# }