Pools independent effect sizes given their sampling variances: the general
engine behind meta_smd and meta_r, exposed for
any effect metric whose estimates are approximately normal with known
variances. The random effects model is the default and the fit reports the
full uncertainty picture in one table: the pooled estimate with its
confidence interval, the between-study standard deviation tau, the
between-study variance tau-squared with its Q-profile confidence
interval, I-squared with an interval mapped from the tau-squared limits,
H-squared, Cochran's Q test, and, always, a prediction interval for the
effect in a new study. Reporting the prediction interval by default is
deliberate: when heterogeneity is real, the confidence interval for the
average effect understates what the next study will show, and the package
treats “where will the next study land” as part of the answer, not
an option.
meta_es(
yi,
vi,
method = c("reml", "pm", "dl", "fe"),
hartung_knapp = TRUE,
conf_level = 0.95
)A data.frame (class dmar_tbl) with rows
estimate, se, the test statistic (t under
Hartung-Knapp, z otherwise), p_value,
lower_limit / upper_limit, prediction_lower /
prediction_upper, tau2 with tau2_lower /
tau2_upper, tau, I2 with limits, H2,
Q / Q_df / Q_p, and k. The estimator and
adjustment are recorded in the "method" and
"hartung_knapp" attributes.
Numeric vector of effect sizes, one per independent study.
Sampling variances of yi, one per study.
Between-study variance estimator: "reml" (restricted
maximum likelihood, the default), "pm" (Paule-Mandel),
"dl" (DerSimonian-Laird), or "fe" (a fixed effect /
common effect analysis, which assumes tau-squared is zero and reports
no prediction interval).
Logical: apply the Hartung-Knapp-Sidik-Jonkman
small-sample adjustment (the pooled standard error rescaled from the
weighted residuals, with a t reference on \(k - 1\) degrees of
freedom)? Default TRUE: with the small numbers of studies
typical in psychology and education it keeps the confidence interval
near its nominal coverage, where the conventional normal interval is
anticonservative. Ignored for method = "fe".
Confidence level for all intervals. Defaults to 0.95.
Ken Kelley [email protected]
The model is \(y_i = \mu + u_i + e_i\) with \(u_i \sim N(0, \tau^2)\) and \(e_i \sim N(0, v_i)\), \(v_i\) treated as known. The \(\tau^2\) confidence interval inverts the generalized Q statistic (Viechtbauer, 2007); the I-squared interval maps the \(\tau^2\) interval through the typical within-study variance of Higgins and Thompson (2002). The prediction interval follows Higgins, Thompson, and Spiegelhalter (2009), using t with \(k - 2\) degrees of freedom, and requires at least three studies.
I-squared is reported because readers expect it, but note its
well-known limitation: it is a proportion of variability, not an
amount, so the same tau matched with larger studies yields a larger
I-squared. The quantity with direct scientific meaning is tau (the
between-study standard deviation, in the metric of yi) together
with the prediction interval.
DerSimonian, R., & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7(3), 177--188.
Hartung, J., & Knapp, G. (2001). On tests of the overall treatment effect in meta-analysis with normally distributed responses. Statistics in Medicine, 20(12), 1771--1782. tools:::Rd_expr_doi("10.1002/sim.791")
Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. Statistics in Medicine, 21(11), 1539--1558. tools:::Rd_expr_doi("10.1002/sim.1186")
Higgins, J. P. T., Thompson, S. G., & Spiegelhalter, D. J. (2009). A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A, 172(1), 137--159. tools:::Rd_expr_doi("10.1111/j.1467-985X.2008.00552.x")
Viechtbauer, W. (2007). Confidence intervals for the amount of heterogeneity in meta-analysis. Statistics in Medicine, 26(1), 37--52. tools:::Rd_expr_doi("10.1002/sim.2514")
meta_smd and meta_r for the metric-
specific front ends; meta_contrast for focused moderator
contrasts; combine_p for combined significance tests;
plot_forest to see the studies and the pool together.
Other meta-analysis:
combine_p(),
meta_contrast(),
meta_r(),
meta_smd(),
plot_forest()
# The teacher expectancy studies (Raudenbush, 1984), pooled in the d
# metric with variances from the standard large-sample formula.
data(teacher_expectancy)
d <- teacher_expectancy$d
n_e <- teacher_expectancy$n_experimental
n_c <- teacher_expectancy$n_control
v <- (n_e + n_c) / (n_e * n_c) + d^2 / (2 * (n_e + n_c))
meta_es(d, v)
# A fixed effect (common effect) analysis of the same studies.
meta_es(d, v, method = "fe")
Run the code above in your browser using DataLab