Supported outcome families

MCPower fits four outcome families, chosen with the family= argument to the constructor (the app exposes the same choice as an Outcome type dropdown: Linear, Logit, Probit, Poisson). Each family has its own link function and its own way of setting the reference-level baseline; everything else — the formula, effect sizes, find_power/find_sample_size — works the same way across all four.

Outcome type family= Link Baseline setter Estimator Typical use
Linear "ols" (default) identity none — effects are direct shifts on the outcome's SD scale OLS (clustered: linear mixed model, family="lme") continuous scores, indices, reaction times
Logit "logit" logit (log-odds) `set_baseline_probability(p)` GLM (clustered: GLMM) yes/no outcomes where effects read naturally as odds ratios
Probit "probit" probit (Φ⁻¹) `set_baseline_probability(p)` GLM (clustered: GLMM) yes/no outcomes where a normal-latent-variable model is preferred
Poisson "poisson" log `set_baseline_rate(λ)` GLM (clustered: GLMM) counts and event rates

"lme" is not a fifth row here — it is the clustered form of the continuous (linear) family, selected the same way clustered logit/probit/Poisson are: add a (1|group) term to the formula and call set_cluster. See mixed-effects models for the clustered path in general.

Baseline probability (logit and probit)

Both binary families need one extra number the linear family doesn't: the outcome probability when every predictor sits at its reference level. Set it with set_baseline_probability(p), p strictly between 0 and 1 — required for family="logit" and family="probit", since without it the engine has no reference point to generate realistic binary outcomes from.

The two families turn that same p into different intercepts: logit maps it through the log-odds transform (log(p / (1 - p))), probit through the normal inverse CDF (Φ⁻¹(p)). Effect sizes are standardised on each family's own link scale, so a given standardised effect is not numerically identical between logit and probit — the two links have different scale factors — but the two families answer the same question (does this predictor move a binary outcome) and typically agree closely on power for the same design. Both start from the same small/medium/large 0.20 / 0.50 / 0.80 Cohen benchmarks (see Cohen's benchmarks), but only logit additionally offers the odds-ratio preset set (beta = log(OR)) — a probit beta is already Cohen's d on the latent scale, not a log-odds, so exp(beta) there is not an odds ratio. See logistic effect sizes for the odds-ratio benchmarks.

Baseline rate (Poisson)

A count outcome needs its own reference point: the expected count when every predictor is at its reference level. Set it with set_baseline_rate(lambda), lambda > 0 — required for family="poisson" for the same reason set_baseline_probability is required for the binary families: without a baseline the engine cannot generate realistic counts. Internally lambda becomes the log-link intercept, ln(lambda).

Poisson effects are shifts on the log-rate scale — a predictor's coefficient is a log-rate-ratio, and exp(beta) is the multiplicative change in the expected count per unit (or per level, for binary/factor predictors). MCPower reuses the logit family's odds-ratio preset numbers (small 0.41 / medium 0.92 / large 1.39, Chen et al. 2010) as rate-ratio anchors for Poisson — same beta values, read as RR = exp(beta) instead of an odds ratio. See logistic effect sizes for where those numbers come from.

Clustered counts and probits

Clustering works for all four families, but the random-intercept is sized differently depending on the link:

  • Logit / probit (clustered): set_cluster(grouping, ICC=..., n_clusters=...) — the ICC is read on the latent scale, using each link's own latent-residual variance (logit: π²/3; probit: 1). You still just pass an ICC; the conversion is internal.
  • Poisson (clustered): Poisson has no natural latent-scale ICC, so the random intercept is sized directly by its raw variance: set_cluster(grouping, tau_squared=..., n_clusters=...). Passing ICC= for family="poisson" is an error, and tau_squared= is only accepted for Poisson.

See Clustered logistic (GLMM) for the worked binary case; the Poisson case follows the same pattern with tau_squared in place of ICC.

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