Sample Size for Two-Group Linear Mixed Model: Difference in Slopes

Example clinical scenario

A chronic kidney disease study measures eGFR at baseline and every six months for two years. The primary hypothesis is that the treatment group has a slower mean annual decline than the reference group, while participant-specific baseline eGFR and rates of change are represented by random effects.

Formula and methodological note

The model includes fixed group, linear time, and group-by-time terms. For visit design X and participant covariance V=ZGZᵀ+R, the per-participant slope-information variance is the slope diagonal of (XᵀV⁻¹X)⁻¹. The variance of the two-group slope difference is this value multiplied by (1/n reference + 1/n comparison). Exact whole-number allocation multiples are searched until normal-theory Wald power reaches the target.

The method assumes a continuous outcome, equally spaced common visits, linear mean trajectories, independent participants, common correctly specified covariance across groups, and credible population variance components. Dropout is recruitment inflation only; partially observed or informative missingness is not modelled. It is not for categorical-time contrasts, nonlinear trajectories, target-visit MMRM effects, more than two groups, cluster randomization, crossed effects, or non-Gaussian outcomes. This tool supports research planning and does not replace statistical consultation.

Technical results

Method references

1. Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2nd ed. Oxford University Press; 2002.
2. Donohue MC, Edland SD. Longitudinal power calculations in the R package longpower. R package documentation.