Power for a random-slope growth curve model

Every seedling is measured at several weekly time points, and the question is whether height moves with week on average. Because seedlings start at different heights and grow at different rates, each seedling gets its own intercept and its own slope of week. As an MCPower formula: seedling_height ~ week + (week | seedling), fit as a linear mixed model (the default MLE estimator). The test of interest is the average week slope, judged against how widely the individual growth rates scatter.

Variations

Drop the random slope. If individual growth rates are not your concern, use seedling_height ~ week + (1 | seedling) — random intercepts only. Every seedling shares one average slope, which usually raises power for week because the slope no longer competes with between-seedling slope variance.
More seedlings vs. more measurements. Power for a fixed total sample_size shifts with the seedling/measurement split; raise n_clusters (more seedlings, fewer measurements each) or lower it (fewer seedlings, denser follow-up) to see which your design can afford.
Tighter or looser individual trajectories. Raise slope_variance to model seedlings whose growth rates differ a lot (harder to detect the average slope), or lower it toward zero to approach a common-slope design.
Swap the trend strength. Replace week=0.25 with week=0.10 (small) or week=0.40 (large) to bracket a plausible average growth rate.
Same design, other fields:
- systolic_bp ~ phase + (phase|patient) — one patient measured at several clinical phases; individual BP trajectories vary in slope (clinical).
- life_satisfaction ~ wave + (wave|individual) — one person surveyed repeatedly; individual well-being slopes vary (social science).

Not this setup?

If you'd rather have…

Random intercept only — same week-trend design but the simpler form: random intercept only, no random slope of week.
Conditional growth with a fertilizer treatment — adds a treatment that moderates the individual slopes, layering conditional growth on top of this same random-slope structure.
Treatment x week, random intercept only — a treatment x week interaction but with random intercept only — a between-arm trajectory difference without random slopes.
Random slope over sites — another random-slope design, but the slope varies over clusters (sites) for a treatment effect rather than over time within seedlings.
Random slope on a binary outcome — the same random-slope idea on a binary outcome: a continuous predictor with a random slope per subject (logistic GLMM).

Copy-paste setup

from mcpower import MCPower

# Repeated-measures growth curve: each seedling is measured at several weekly
# time points and we ask whether height changes with `week`. `(week | seedling)`
# gives every seedling its own random intercept (baseline height) AND its own
# random slope of week (personal growth rate), so the test of the average
# `week` slope is judged against how much those individual growth rates scatter.
# family="lme" with the default MLE estimator fits the mixed model.
model = MCPower("seedling_height ~ week + (week | seedling)", family="lme")

# `week` is a continuous (normally distributed) within-seedling predictor.
model.set_variable_type("week=normal")

# Average weekly growth slope on the continuous benchmark scale:
#   week=0.25 -> a medium average rate of change across seedlings.
model.set_effects("week=0.25")

# Clustering by seedling: 40 seedlings, conditional ICC 0.3 (baseline
# heights correlate within a seedling after `week` is accounted for), and a
# random slope of `week` whose variance 0.05 sets how much individual growth
# rates differ; slope_intercept_corr ties taller seedlings to faster growth.
model.set_cluster(
    "seedling",
    ICC=0.3,
    n_clusters=40,
    random_slopes=["week"],
    slope_variance=0.05,
    slope_intercept_corr=0.2,
)

model.set_seed(2137)

# Power for the average `week` slope. sample_size is the total number of
# observations (seedlings x measurements per seedling).
model.find_power(sample_size=200, target_test="week")

suppressMessages(library(mcpower))

# Repeated-measures growth curve: each seedling is measured at several weekly
# time points and we ask whether height changes with `week`. `(week | seedling)`
# gives every seedling its own random intercept (baseline height) AND its own
# random slope of week (personal growth rate), so the test of the average
# `week` slope is judged against how much those individual growth rates scatter.
# family="lme" with the default MLE estimator fits the mixed model.
model <- MCPower$new("seedling_height ~ week + (week | seedling)", family = "lme")

# `week` is a continuous (normally distributed) within-seedling predictor.
model$set_variable_type("week=normal")

# Average weekly growth slope on the continuous benchmark scale:
#   week=0.25 -> a medium average rate of change across seedlings.
model$set_effects("week=0.25")

# Clustering by seedling: 40 seedlings, conditional ICC 0.3 (baseline
# heights correlate within a seedling after `week` is accounted for), and a
# random slope of `week` whose variance 0.05 sets how much individual growth
# rates differ; corr_with_intercept ties taller seedlings to faster growth.
model$set_cluster(
  "seedling",
  ICC = 0.3,
  n_clusters = 40L,
  random_slopes = list(
    list(predictor = "week", variance = 0.05, corr_with_intercept = 0.2)
  )
)

model$set_seed(2137)

# Power for the average `week` slope. sample_size is the total number of
# observations (seedlings x measurements per seedling).
invisible(model$find_power(sample_size = 200, target_test = "week"))