Linear Mixed Model for a
Longitudinal Classroom Study

RESEARCH QUESTION

What the Study Set Out to Test

This linear mixed model example follows an education research team comparing a new reading-support program with standard instruction. Reading fluency was assessed in fall, winter and spring, and the team wants to know whether fluency grew differently in the two groups across the school year.

The question is about a difference in growth, so the group-by-time interaction is the primary test, specified before spring scores were examined. A gain in one group and no gain in the other would not by itself show that growth differed. Classrooms were not randomized to the program, so the result is framed as a difference in growth, not a program effect.

STUDY DESIGN AND DATA

Repeated Measures Nested in Classrooms

Around 400 students in 20 classrooms, 10 per program group, each assessed at three waves. The dataset holds a fluency score per wave, program group, classroom, and baseline variables such as grade and prior reading level. Some students left before spring, so not every student has all three scores.

METHOD AND ASSUMPTIONS

Why a Mixed Model, and What Must Hold

Scores from the same student are correlated, and students in the same classroom share a teacher and setting, so a repeated measures ANOVA on complete cases would ignore the clustering and drop students who left early. The mixed model keeps every available score, includes time, program group and their interaction as fixed effects, and uses random effects for students and classrooms. Because program was allocated by classroom, the group comparison rests on 20 clusters, and the plan treats that number as modest when judging precision.

The plan names a random slope for time as the preferred structure. When it is fitted in SPSS MIXED, the slope variance at the classroom level is close to zero and the output warns that a covariance parameter is redundant, which signals that the data cannot support that term. Following the plan's stated fallback, the classroom-level random slope is dropped, residual covariance structures are compared by AIC under REML, and the interaction is tested with ML. Missing spring scores are judged plausibly missing at random given earlier scores, and the team is told what that judgment assumes.

OUTPUT AND INTERPRETATION

What the Results Showed

The output has the group-by-time interaction with its 95% confidence interval, the estimated means at each wave by group with confidence intervals, and the variance components with the intraclass correlation for classrooms. A profile plot shows the two growth patterns, and the interval, not the p-value alone, carries the message about the size and precision of the difference in growth.

A sensitivity analysis reports the complete-case count and refits the model on students with all three waves, to show whether the conclusion depends on how early leavers are handled. The write-up states what the estimates support for a design in which classrooms were not randomized, and what the simplification of the random effects does and does not change.

REPORTING

What the Researcher Received

A written report formatted to APA 7, with statistical wording for the methods, matched to the output, for the team to check and incorporate, including the explanation of the simplified random-effects structure. The research question, substantive interpretation and conclusions remain with the research team.