Estimate standardized mean difference (Cohen's d) for an independent groups contrast
\loadmathjax CI_smd_ind_contrast returns the point estimate and confidence interval for a standardized mean difference (smd aka Cohen's d aka Hedges g). A standardized mean difference is a difference in means standardized to a standard deviation: \mjdeqn d = \frac \psisd = psi/s
CI_smd_ind_contrast( means, sds, ns, contrast, conf_level = 0.95, assume_equal_variance = FALSE, correct_bias = TRUE )
means: A vector of 2 or more meanssds: A vector of standard deviations, same length as meansns: A vector of sample sizes, same length as meanscontrast: A vector of group weights, same length as meansconf_level: The confidence level for the confidence interval, in decimal form. Defaults to 0.95.assume_equal_variance: Defaults to FALSEcorrect_bias: Defaults to TRUE; attempts to correct the slight upward bias in d derived from a sample. As of 8/9/2023 - Bias correction has been added for more than 2 groups when equal variance is not assumed, based on recent updates to statpsychReturns a list with these named elements:
effect_size - the point estimate from the sample
lower - lower bound of the CI
upper - upper bound of the CI
numerator - the numerator for Cohen's d_biased; the mean difference in the contrast
denominator - the denominator for Cohen's d_biased; if equal variance is assumed this is sd_pooled, otherwise sd_avg
df - the degrees of freedom used for correction and CI calculation
se - the standard error of the estimate; warning not totally sure about this yet
moe - margin of error; 1/2 length of the CI
d_biased - Cohen's d without correction applied
properties - a list of properties for the result
Properties
effect_size_name - if equal variance assumed d_s, otherwise d_avg
effect_size_name_html - html representation of d_name
denominator_name - if equal variance assumed sd_pooled otherwise sd_avg
denominator_name_html - html representation of denominator name
bias_corrected - TRUE/FALSE if bias correction was applied
message - a message explaining denominator and correction status
message_html - html representation of message
A standardized mean difference turns out to be complicated.
First, it has many names:
Second, the choice of the standardizer requires thought:
The choice of standardizer is important, so it's noted in the subscript:
A third complication is the issue of bias: d estimated from a sample has a slight upward bias at smaller sample sizes. With total sample size > 30, this slight bias becomes fairly negligible (kind of like the small upward bias in a sample standard deviation).
This bias can be corrected when equal variance is assumed or when the design of the study is simple (2 groups). For complex designs (>2 groups) without the assumption of equal variance, there is now also an approximate approach to correcting bias from Bonett.
Corrections for bias produce a long-run reduction in average bias. Corrections for bias are approximate.
When equal variance is assumed, the standardized mean difference is d_s, defined in Kline, p. 196:
\mjdeqn d_s = \frac \psisd_pooledd_s = psi / sd_s
where psi is defined in Kline, equation 7.8:
\mjdeqn \psi =\sum _i=1^ac_iM_ipsi = sum(contrasts*means)
and where sd_pooled is defined in Kline, equation 3.11 \mjdeqn sd_pooled = \frac \sum _i=1^a (n_i -1) s_i^2 \sum _i=1^a (n_i-1)sqrt(sum(variances*dfs) / sum(dfs))
The CI for d_s is derived from lambda-prime transformation from Lecoutre, 2007 with code adapted from Cousineau & Goulet-Pelletier, 2020. Kelley, 2007 explains the general approach for linear contrasts.
This approach to generating the CI is 'exact', meaning coverage should be as desired if all assumptions are met (ha!).
Correction of upward bias can be applied.
When equal variance is not assumed, the standardized mean difference is d_avg, defined in Bonett, equation 6:
\mjdeqn d_avg = \frac \psisd_avgd_avg = psi / sd_avg
Where sd_avg is the square root of the average of the group variances, as given in Bonett, explanation of equation 6:
\mjdeqn sd_avg = \sqrt \frac \sum _i=1^a s_i^2asqrt(mean(variances))
# Example from Kline, 2013 # Data in Table 3.4 # Worked out in Chapter 7 # See p. 202, non-central approach # With equal variance assumed and no correction, should give: # d_s = -0.8528028 [-2.121155, 0.4482578] res <- esci::CI_smd_ind_contrast( means = c(13, 11, 15), sds = c(2.738613, 2.236068, 2.000000), ns = c(5, 5, 5), contrast = contrast <- c(1, 0, -1), conf_level = 0.95, assume_equal_variance = TRUE, correct_bias = FALSE ) # Example from [statpsych::ci.lc.stdmean.bs()] should give: # Estimate SE LL UL # Unweighted standardizer: -1.273964 0.3692800 -2.025039 -0.5774878 # Weighted standardizer: -1.273964 0.3514511 -1.990095 -0.6124317 # Group 1 standardizer: -1.273810 0.4849842 -2.343781 -0.4426775 res <- esci::CI_smd_ind_contrast( means = c(33.5, 37.9, 38.0, 44.1), sds = c(3.84, 3.84, 3.65, 4.98), ns = c(10,10,10,10), contrast = c(.5, .5, -.5, -.5), conf_level = 0.95, assume_equal_variance = FALSE, correct_bias = TRUE )
estimate_mdiff_ind_contrast for friendly version that also returns raw score effect sizes for this designUseful links