Linear Mixed Model Sample Size (Random Slope)

Methodology

This calculator implements the two-level longitudinal mixed-model formulation used by this method for testing the difference between two group slopes when each subject has a random, subject-specific slope. The model includes fixed treatment, time, and treatment-by-time terms together with subject-specific random intercept and random slope terms.

Mixed Model

Here, δ is the difference between the two treatment slopes and is the parameter being tested. The random slope υ_i has variance σ_T², while ρ describes the correlation among repeated measurements on the same subject.

Time Variability

this method assumes equally spaced measurements with time coded as T = 0, 1, …, M−1. The variance of the centered time values is

The random-slope variance is entered through r_T = σ_T² / σ², where σ² represents the variance associated with the fixed-slope model.

Power Formula

Let K₁ and K₂ be the numbers of subjects in Groups 1 and 2, respectively, and let λ = K₁/K₂. Under the normal approximation used by this method, power is calculated as

For sample-size determination, the calculator searches over integer Group 1 sample sizes. For each candidate K₁, Group 2 is set to the smallest integer consistent with the requested allocation ratio, and the exact achieved power from the formula is checked. The reported design is the first one whose achieved power is at least the requested power.

Validation Example

the relevant methodological literature provides a sample-size example with 90% power, α = 0.05, four measurements per subject, a slope difference of 3, standard deviation 9.2, random-slope variance ratio r_T = 0.1, within-subject correlation ρ = 0.5, and equal allocation. This yields 40 subjects in each group, 80 total, with achieved power 0.90335.

The implementation reproduces the this method result: with K₁ = K₂ = 40, calculated power is approximately 0.90335; reducing either group to 39 gives approximately 0.89614, below the 0.90 target. Thus 40 subjects per group is the smallest balanced design meeting the target.

Important Assumptions

• Two treatment groups with subjects randomized to the groups.
• A continuous outcome measured repeatedly within each subject.
• Equally spaced measurement occasions coded 0 through M−1.
• Each subject has a subject-specific random slope.
• The slope-difference estimator is approximated by a normal distribution.
• The calculation is based on the analytical formulation rather than simulation.

References

Ahn, C., Heo, M., & Zhang, S. (2015). Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research. CRC Press, New York. Note that Chapter 5, Section 5.4.2, pages 161–164, as the source of the random-slope two-level longitudinal formulation.

the software, LLC. this method Sample Size Software, Chapter 383: Mixed Models Tests for the Slope Difference in a 2-Level Hierarchical Design with Random Slopes. the relevant methodological literature describes the model, power formula, worked examples, and validation against Ahn, Heo, and Zhang (2015).
View the this method Chapter 383 documentation

The this method procedure documentation explicitly describes this design as a two-level longitudinal study with random assignment of subjects to two groups, repeated measurements within subjects, and subject-specific random slopes. The documentation also reports the validation example used above.