Correlated Predictors & Power Analysis - Python

So far the model has assumed that stress, exercise, and sleep are drawn independently. Real predictors rarely are — and ignoring their correlations means your power estimate could be off. This rung shows how to tell MCPower about predictor correlations and why it matters.

Why correlation erodes power

When two predictors are correlated they carry overlapping information about the outcome. The regression must disentangle those overlapping signals, and that disentanglement costs statistical precision. The more correlated the predictors, the harder each one is to detect — see [[concepts/correlations|why correlation matters]] for the derivation.

The effect is not always symmetric: a predictor with a modest correlation to one strong predictor may lose only a little power, while one caught between two strongly correlated companions loses substantially more.

The study

We model self-reported wellbeing as a function of three lifestyle predictors: stress (negative), exercise, and sleep. All three have plausible medium effects. Because they are lifestyle variables they are correlated in real populations — stressed people sleep less, and they also exercise less.

Three ways to describe the same structure

MCPower accepts predictor correlations in three equivalent forms. Pick whichever fits your workflow.

1. String form — good for a sparse set of pairs

Name each pair explicitly with corr(a, b)=value:

from mcpower import MCPower

model = MCPower("wellbeing = stress + exercise + sleep")
model.set_effects("stress=-0.3, exercise=0.25, sleep=0.3")
model.set_correlations("corr(stress, exercise)=-0.3, corr(stress, sleep)=0.4, corr(exercise, sleep)=0.2")

result = model.find_power(sample_size=150, target_test="all", verbose=False)
print(result.summary())

==================================================
  MCPower · Power Analysis
==================================================
formula: wellbeing = stress + exercise + sleep
estimator: OLS  N=150  sims=1600  α=0.05  target=80%
effects: stress=-0.30, exercise=0.25, sleep=0.30

Per-test power
───────────────────────────────────
Test                 Power   Target
───────────────────────────────────
Overall F            99.9%      80%
stress               85.8%      80%
exercise             74.8%      80%
sleep                87.1%      80%
───────────────────────────────────

Power & 95% CI
───────────────────────────────────────────
Test                 Power           CI 95%
───────────────────────────────────────────
Overall F            99.9%   [99.6%,  100%]
stress               85.8%   [84.0%, 87.4%]
exercise             74.8%   [72.6%, 76.8%]
sleep                87.1%   [85.3%, 88.6%]
───────────────────────────────────────────
95% CIs are Monte-Carlo (Wilson), n_sims=1600.

Joint significance distribution
────────────────────────
k     Exactly   At least
────────────────────────
0        0.4%       100%
1        6.8%      99.6%
2       37.8%      92.9%
3       55.1%      55.1%
────────────────────────

Plots: result.plot() to view, result.plot('chart.png') to save.

stress reaches 85.8% power, sleep 87.1%, but exercise falls just short at 74.8% — exercise is the most "squeezed" predictor, sandwiched between two variables it correlates with in opposite directions.

2. Matrix form — good for a complete correlation structure

Pass a matrix — a plain list of lists (or a numpy array, if you have numpy installed) — whose rows and columns follow the predictor order in the formula (stress, exercise, sleep):

from mcpower import MCPower

model = MCPower("wellbeing = stress + exercise + sleep")
model.set_effects("stress=-0.3, exercise=0.25, sleep=0.3")

# rows/cols follow the predictors in formula order: stress, exercise, sleep
# (a list of lists; a numpy array works too, if you have numpy installed)
correlation_matrix = [
    [ 1.0, -0.3, 0.4],
    [-0.3,  1.0, 0.2],
    [ 0.4,  0.2, 1.0],
]
model.set_correlations(correlation_matrix)

result = model.find_power(sample_size=150, target_test="all", verbose=False)
print(result.summary())

==================================================
  MCPower · Power Analysis
==================================================
formula: wellbeing = stress + exercise + sleep
estimator: OLS  N=150  sims=1600  α=0.05  target=80%
effects: stress=-0.30, exercise=0.25, sleep=0.30

Per-test power
───────────────────────────────────
Test                 Power   Target
───────────────────────────────────
Overall F            99.9%      80%
stress               85.8%      80%
exercise             74.8%      80%
sleep                87.1%      80%
───────────────────────────────────

Power & 95% CI
───────────────────────────────────────────
Test                 Power           CI 95%
───────────────────────────────────────────
Overall F            99.9%   [99.6%,  100%]
stress               85.8%   [84.0%, 87.4%]
exercise             74.8%   [72.6%, 76.8%]
sleep                87.1%   [85.3%, 88.6%]
───────────────────────────────────────────
95% CIs are Monte-Carlo (Wilson), n_sims=1600.

Joint significance distribution
────────────────────────
k     Exactly   At least
────────────────────────
0        0.4%       100%
1        6.8%      99.6%
2       37.8%      92.9%
3       55.1%      55.1%
────────────────────────

Plots: result.plot() to view, result.plot('chart.png') to save.

The per-test powers are identical to the string form: 85.8% / 74.8% / 87.1%. The two inputs describe exactly the same correlation structure, so you can use whichever is more convenient — a sparse string for a handful of pairs, a full matrix when you have a complete correlation table from a pilot study.

3. Dict form — a Python-native shorthand

If you would rather keep correlations as Python data, pass a dict keyed by variable-name pairs. It is exactly equivalent to the string form, so it produces the same result:

model.set_correlations({
    ("stress", "exercise"): -0.3,
    ("stress", "sleep"): 0.4,
    ("exercise", "sleep"): 0.2,
})

Reading the output

The joint significance distribution shows that all three effects are simultaneously detected in 55.1% of simulations at N = 150. That is the number to budget for when you want to claim that every predictor is significant — not just any one of them.

next → Logistic regression