Purpose

This workflow analyzes the same repeated outcome with three frameworks: repeated-measures ANOVA, a multivariate test of change contrasts, and a linear mixed-effects growth model. The comparison clarifies how assumptions, missing data, time coding, and the research question affect the analysis.

The included longitudinal data are entirely synthetic. The methods are related but not interchangeable. Select a framework from the design, time structure, estimand, and assumptions rather than from which produces the smallest p-value.

How the frameworks differ

Framework View of time Typical strength Important limitation
Repeated-measures ANOVA Categorical occasions Direct omnibus mean comparison Complete cases and covariance assumptions
Multivariate approach Vector of repeated outcomes or contrasts Avoids sphericity Complete cases; more parameters
Growth model Numeric or flexible time function Individual trajectories and incomplete schedules Requires random-effects and covariance decisions

User settings

Simulate and import the example

Rows: 960
Columns: 5
$ student_id <chr> "R001", "R001", "R001", "R001", "R002", "R002", "R002", "R0~
$ program    <fct> Standard, Standard, Standard, Standard, Standard, Standard,~
$ month      <dbl> 0, 6, 12, 18, 0, 6, 12, 18, 0, 6, 12, 18, 0, 6, 12, 18, 0, ~
$ score      <dbl> 50.8, 45.5, 48.4, 46.1, 56.4, 60.4, 53.1, 67.5, 45.5, 54.2,~
$ time       <dbl> 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3,~

Data audit

stage n
Students imported 240
Complete repeated-measures profiles 177
Students contributing to growth model 240
month records observed missing missing_percent
0 240 221 19 7.9
6 240 225 15 6.2
12 240 222 18 7.5
18 240 222 18 7.5

The ANOVA and multivariate demonstrations use complete profiles. The mixed model uses every observed outcome under its likelihood assumptions. Neither choice repairs data that are missing not at random.

Describe repeated outcomes

Means and covariance

month n mean sd se conf_low conf_high
0 221 50.61 8.06 0.54 49.54 51.67
6 225 53.99 8.22 0.55 52.91 55.07
12 222 56.21 8.35 0.56 55.11 57.32
18 222 57.89 9.21 0.62 56.68 59.11
score_m0 score_m6 score_m12 score_m18
score_m0 64.97 51.48 49.49 54.33
score_m6 51.48 67.52 51.44 56.78
score_m12 49.49 51.44 69.79 61.45
score_m18 54.33 56.78 61.45 84.81

Individual and mean trajectories

Repeated-measures ANOVA

df1 df2 f_value p_value gg_epsilon gg_df1 gg_df2 gg_p_value
3 528 109.277 0 0.689 2.066 363.648 0

This omnibus test asks whether occasion means are equal among complete profiles. The Greenhouse-Geisser epsilon summarizes departure from sphericity and is used here to correct the numerator and denominator degrees of freedom. Report the corrected p-value when epsilon is meaningfully below 1. The multivariate and mixed-model analyses provide alternatives with different assumptions and estimands.

Multivariate test of change

hotelling_t2 approx_f df1 df2 p_value
258.476 85.18 3 174 0

The one-sample Hotelling test is a repeated-measures multivariate test of whether the vector of successive mean changes equals zero. It avoids sphericity but still needs complete profiles, multivariate assumptions, and adequate sample size relative to the number of contrasts.

Quadratic growth model

term estimate std_error df t_value p_value conf_low conf_high
(Intercept) 49.404 0.697 647 70.914 0.000 48.036 50.772
time 3.764 0.453 647 8.318 0.000 2.876 4.653
I(time^2) -0.521 0.135 647 -3.854 0.000 -0.786 -0.255
programEnhanced 2.361 0.983 238 2.401 0.017 0.424 4.299
time:programEnhanced 0.489 0.293 647 1.671 0.095 -0.086 1.064 
student_id = pdLogChol(time) 
            Variance  StdDev   Corr  
(Intercept) 45.445895 6.741357 (Intr)
time         1.589124 1.260605 0.088 
Residual    15.537890 3.941813

Time is centered at baseline and one unit represents 6 months. The time coefficient is therefore the instantaneous linear component at baseline; the quadratic term allows the rate of change to bend over follow-up.

Model-predicted trajectories

Growth-model diagnostics

Also review the random-effects distribution, within-person covariance, influential students, nonlinear time alternatives, and missing-data assumptions. More flexible covariance structures require enough repeated observations to estimate reliably.

Dynamic results template

Repeated outcomes were available for 240 synthetic students across 4 scheduled occasions. The repeated-measures ANOVA included 177 complete profiles and, after the Greenhouse-Geisser correction (epsilon = 0.69), provided evidence that at least one occasion mean differed, F(2.07, 363.65) = 109.28, p < .001. A multivariate Hotelling test of the successive change vector produced T-squared = 258.48, approximate F(3, 174) = 85.18, p < .001.

The mixed-effects growth model used 890 observed scores from 240 students. At baseline, the estimated 6-month linear component was 3.76 points, 95% CI [2.88, 4.65], p < .001. The quadratic coefficient was -0.52, 95% CI [-0.79, -0.26], p < .001, supporting curvature over the observed period. The enhanced-program linear component differed from the standard-program component by 0.49 points per 6 months, 95% CI [-0.09, 1.06], p = .095. These frameworks use different estimation samples and target different summaries, so their results should be interpreted within their own assumptions rather than treated as interchangeable tests.

Reporting checklist

  • State the number, timing, and coding of repeated occasions.
  • Report the analysis sample used by each framework and the missing-data approach.
  • Describe covariance or sphericity assumptions and any corrections.
  • Define the growth model’s time origin, time unit, fixed effects, and random effects.
  • Present estimated means or trajectories with uncertainty.
  • Explain why the chosen framework answers the research question.

Exports

file
synthetic_repeated_measures_wide.csv
synthetic_repeated_measures_long.csv
wave_descriptives.csv
repeated_measures_anova.csv
multivariate_change_test.csv
growth_model_fixed_effects.csv
growth_model_predictions.csv

References

  • Gueorguieva, R., & Krystal, J. H. (2004). Move over ANOVA: Progress in analyzing repeated-measures data. Archives of General Psychiatry, 61(3), 310–317.
  • Singer, J. D., & Willett, J. B. (2003). Applied Longitudinal Data Analysis. Oxford University Press.
Paid version: The download includes all editable R code, the synthetic-data generator, documented outputs, and complete software-version details.
Dissertation Stats Helper · Reproducible statistical guidance for doctoral research · Choose a longitudinal framework from the estimand, schedule, missingness, and covariance structure.