Pools correlations across independent studies on the Fisher's Z
scale and reports the results back in the correlation metric. Optionally,
each study's correlation is first corrected for attenuation due to
measurement error in either or both variables (the Spearman correction of
correction_for_attenuation, the basic artifact correction of
Hunter and Schmidt's psychometric meta-analysis), using reliabilities you
supply, for example from the reliability family. That
combination, synthesis connected to an actual reliability toolkit, is the
measurement-aware path: the pooled quantity is then the construct-level
correlation rather than the attenuated observed one.
meta_r(
r,
n,
reliability_x = NULL,
reliability_y = NULL,
method = c("reml", "pm", "dl", "fe"),
hartung_knapp = TRUE,
conf_level = 0.95
)A data.frame (class dmar_tbl) with the same
rows as meta_es: the estimate,
lower_limit / upper_limit, and prediction interval rows
in the correlation metric; the se, test statistic, and
heterogeneity rows on the Fisher's Z scale where the model lives.
Numeric vector of observed correlations, one per study, each in (-1, 1).
Per-study sample sizes (integer, at least 4).
Optional per-study reliabilities in
(0, 1] for the two measured variables; a single value is recycled
across studies. When either is supplied, each correlation is
disattenuated by \(r_i / \sqrt{\rho_{xx,i}\, \rho_{yy,i}}\) before
pooling (a reliability left NULL is treated as 1). The
reliabilities are treated as known.
Passed to meta_es.
Ken Kelley [email protected]
Pooling uses \(z_i = \mathrm{atanh}(r_i)\) with sampling variance \(1 / (n_i - 3)\); the pooled estimate, its confidence limits, and the prediction interval are transformed back through \(\tanh\). The heterogeneity quantities (tau, tau-squared, I-squared, H-squared, Q) remain on the Fisher's Z scale, where the model lives; tau is therefore the between-study standard deviation of the z-scale correlations.
When corrections are applied, the corrected correlation's variance is
computed from its own \(n_i\) on the z scale, the conventional
simple treatment when reliabilities are taken as known constants; the
more elaborate artifact-distribution machinery of Hunter and Schmidt
(2004) is deliberately out of scope here. A corrected correlation that
exceeds 1 in magnitude (possible when an observed \(r\) outruns the
supplied reliabilities) is an error at the pooling stage, unlike the
single-study correction_for_attenuation, which reports it
with a warning: \(\mathrm{atanh}\) is undefined there.
Hunter, J. E., & Schmidt, F. L. (2004). Methods of meta-analysis: Correcting error and bias in research findings (2nd ed.). Sage.
meta_es for the engine;
correction_for_attenuation for the single-study
correction and its connection to latent variable modeling;
reliability for estimating the reliabilities;
convert_r_Z / convert_Z_r for the
transformation used.
Other meta-analysis:
combine_p(),
meta_contrast(),
meta_es(),
meta_smd(),
plot_forest()
# Five validity studies of the same selection instrument.
r <- c(.28, .35, .22, .40, .31)
n <- c(120, 85, 200, 60, 150)
meta_r(r, n)
# The same studies corrected for criterion unreliability (reliability
# 0.80 in every study): the construct-level validity.
meta_r(r, n, reliability_y = 0.80)
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