Random Effects Meta-Analysis of Standardized Mean Differences

Pools two-group standardized mean differences across independent studies. Each study contributes its standardized mean difference and per-group sample sizes; the function computes the within-study sampling variances, applies the Hedges small-sample bias correction by default (the same \(J\) factor as expected_smd and smd), and fits the random effects model of meta_es, returning the pooled effect with its confidence interval, tau and tau-squared with intervals, I-squared, Cochran's Q, and a prediction interval for the effect in a new study.

Usage

meta_smd(
  smd,
  n_1,
  n_2,
  unbiased = TRUE,
  method = c("reml", "pm", "dl", "fe"),
  hartung_knapp = TRUE,
  conf_level = 0.95
)

Arguments

smd

Numeric vector of standardized mean differences (Cohen's d), one per study, positive in the direction of the common hypothesis.

n_1, n_2

Per-group sample sizes for each study.

unbiased

Logical: convert each d to Hedges g (the small-sample unbiased estimator) before pooling? Default TRUE. Set FALSE to pool the raw d values, for example when reproducing a historical analysis such as Raudenbush (1984) that predates routine use of the correction.

method, hartung_knapp, conf_level

Passed to meta_es: the tau-squared estimator ("reml" default), the Hartung-Knapp small-sample adjustment (default TRUE), and the confidence level.

Value

A data.frame (class dmar_tbl) with the same rows as meta_es.

Details

The within-study variance is the standard large-sample form $$v_i = \frac{n_{1i} + n_{2i}}{n_{1i} n_{2i}} + \frac{g_i^2}{2 (n_{1i} + n_{2i})},$$ computed from the bias-corrected \(g_i\) when unbiased = TRUE (Hedges, 1981; Borenstein, Hedges, Higgins, & Rothstein, 2009). All reported quantities are in the standardized mean difference metric.

References

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Wiley.

Hedges, L. V. (1981). Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128.

Raudenbush, S. W. (1984). Magnitude of teacher expectancy effects on pupil IQ as a function of the credibility of expectancy induction: A synthesis of findings from 18 experiments. Journal of Educational Psychology, 76(1), 85–97.

See also

meta_es for the engine and the reported rows; smd and ci_smd for the single-study quantities; plot_forest for the picture; teacher_expectancy for the example data.

Other meta-analysis: combine_p(), meta_contrast(), meta_es(), meta_r(), plot_forest()

Author

Ken Kelley kkelley@nd.edu

Examples

# Pool the teacher expectancy studies (Raudenbush, 1984). Hedges g and
# the Hartung-Knapp adjustment are on by default; the prediction
# interval shows where a new expectancy study would be expected to land.
data(teacher_expectancy)
meta_smd(smd = teacher_expectancy$d,
         n_1 = teacher_expectancy$n_experimental,
         n_2 = teacher_expectancy$n_control)
#>  term             value  
#>  estimate         0.0544 
#>  se               0.0352 
#>  t                1.55   
#>  p_value          0.1392 
#>  lower_limit      -0.0195
#>  upper_limit      0.128  
#>  prediction_lower -0.0198
#>  prediction_upper 0.129  
#>  tau2             0      
#>  tau2_lower       0      
#>  tau2_upper       0.048  
#>  tau              0      
#>  I2               0      
#>  I2_lower         0      
#>  I2_upper         64.6   
#>  H2               1      
#>  Q                16.7   
#>  Q_df             18     
#>  Q_p              0.5468 
#>  k                19     
#> 
#> Confidence level: 95%