Standardise by the control group's spread, not the pooled spread

Pooling the two standard deviations assumes the treatment changed the mean and left the variance alone. When the treatment also changes the spread – the usual case for anything that helps some people and not others – the pooled denominator is itself an effect of the treatment, and the standardised difference is no longer comparable across studies. Glass's delta uses the control spread, which the treatment cannot have touched.

Usage

Magsd(m1, m2, s_ctrl, n1, n2)

Arguments

m1, m2

Treatment and control means.

s_ctrl

Control-group standard deviation, strictly positive.

n1, n2

Treatment and control sample sizes; n2 >= 2.

Value

List with delta, var, se, ci_lo, ci_hi, n1, n2.

Details

Formula: Delta = (m1 - m2)/s_ctrl with Var(Delta) = (n1 + n2)/(n1 n2) + Delta^2/(2 (n2 - 1)), n2 being the control group – Glass, McGaw and Smith (1981), Chapter 5; variance from Hedges and Olkin (1985) eq. (5.10).

References

Glass, G. V., McGaw, B. and Smith, M. L. (1981). Meta-Analysis in Social Research. Sage, Chapter 5. Hedges, L. V. and Olkin, I. (1985). Statistical Methods for Meta-Analysis, eq. (5.10).

Examples

Magsd(m1 = 12, m2 = 10, s_ctrl = 4, n1 = 50, n2 = 50)[c("delta", "ci_lo", "ci_hi")]
#> $delta
#> [1] 0.5
#> 
#> $ci_lo
#> [1] 0.09570062
#> 
#> $ci_hi
#> [1] 0.9042994
#>