Common-language effect size (probabilistic index)

A threshold-free responder-type measure: the probability that a randomly chosen treated patient has a better change score than a randomly chosen control. Under a Normal model this is exact, \(\mathrm{CLES} = \Phi(\delta)\) with \(\delta = (\mu_e - \mu_c) / \sqrt{\sigma_e^2 + \sigma_c^2}\) (the sign is flipped when direction = "lower"). Per-study \(\delta\) values are pooled by fixed- or random-effect inverse variance and back-transformed, so no minimal important difference is required.

Usage

responder_cles(
  data,
  direction = c("higher", "lower"),
  pooling = c("fixed", "random"),
  tau_method = c("DL", "REML"),
  ci_method = c("wald", "hksj"),
  conf_level = 0.95
)

Arguments

data

A data frame with columns study, change_e, sd_e, n_e, change_c, sd_c, n_c. See sample_responder_data.

direction

"higher" (a larger change is better) or "lower".

pooling

"fixed" (default) or "random" effects.

tau_method

Between-study variance estimator for random effects: "DL" (default) or "REML".

ci_method

Random-effects interval method: "wald" (default) or "hksj" (Hartung-Knapp-Sidik-Jonkman, better for small numbers of studies).

conf_level

Confidence level (default 0.95).

Value

A list with:

studies

Per-study data frame: study, delta, cles, cles_lb, cles_ub.

cles, cles_lb, cles_ub

Pooled CLES and its interval.

delta, se_delta

Pooled standardized difference and its SE.

tau2, i2, q, q_p, pi_lb, pi_ub

Heterogeneity statistics; the prediction interval is back-transformed to the CLES scale.

pooling, k

Settings echoed back.

References

McGraw KO, Wong SP (1992). A common language effect size statistic. Psychological Bulletin, 111(2), 361 to 365.

Examples

cles <- responder_cles(sample_responder_data)
cles$cles
#> [1] 0.6899041