Constructs an exact-coverage confidence interval for the population standardized mean difference \(\delta = (\mu_1 - \mu_2)/\sigma\) (Cohen's d when expressed as a sample quantity) for two independent groups under bivariate normality with equal variances. The interval is obtained by inverting the noncentral t sampling distribution of the rescaled statistic \(t = \hat d \sqrt{n_1 n_2 / (n_1 + n_2)}\), which under the independent groups equal variances model is exactly noncentral t with \(n_1 + n_2 - 2\) degrees of freedom and noncentrality parameter \(\lambda = \delta \sqrt{n_1 n_2 / (n_1 + n_2)}\) (Hedges, 1981). The confidence limits are then rescaled back to the \(\delta\) metric. This is the same construction Steiger and Fouladi (1997) and Kelley (2007) describe for noncentral effect size CIs.
ci_smd(
ncp = NULL,
smd = NULL,
n_1 = NULL,
n_2 = NULL,
conf_level = 0.95,
alpha_lower = NULL,
alpha_upper = NULL,
tol = 1e-09,
...
)A 3-row data.frame with columns term and value. The
term values are "lower_limit" (the lower bound of the
confidence interval on the standardized mean difference), "smd" (the
point estimate), and "upper_limit" (the upper bound).
The estimated noncentrality parameter, this is generally the observed t-statistic from comparing the two groups and assumes homogeneity of variance
The standardized mean difference (using the pooled standard deviation in the denominator)
The sample size for Group 1
The sample size for Group 2
The confidence level (1-Type I error rate)
The Type I error rate for the lower tail
The Type I error rate for the upper tail
The tolerance of the iterative method for determining the critical values
Allows one to potentially include parameter values for inner functions
This function uses ci_nc_t, which has as one of its arguments tol (and can be modified with tol of the present function).
If the present function fails to converge (i.e., if it runs but does not report a solution), it is likely that the tol value is too restrictive and should be increased by a factor of 10, but probably by no more than 100.
Running the function ci_nc_t directly will report the actual probability values of the limits found. This should be done if any modification to tol is necessary in order to ensure acceptable confidence limits for the noncentral t parameter have been achieved.
Ken Kelley [email protected]
ncp-input vs. smd-input paths. The function accepts the effect
size in either of two equivalent metrics: the observed
t-statistic (via ncp) or the sample standardized mean
difference (via smd). The two paths are mathematically
equivalent under the equal variances assumption (since
\(t = \hat d \sqrt{n_1 n_2 / (n_1 + n_2)}\)); pick whichever is
easier to obtain. Supply exactly one. Both paths internally call
ci_nc_t to invert the noncentral t
distribution at the specified two-tailed (or asymmetric, via
alpha_lower / alpha_upper) confidence level.
Independent vs.\ paired comparison. ci_smd assumes two
independent groups with a common variance. DMAR does not
currently provide a confidence interval for the standardized mean
difference in a paired or within-subject design, whose sampling
distribution depends on the correlation between the paired
measurements; applying the independent groups interval to paired data
gives the wrong coverage. (ci_smd_c is not a paired
interval either; it is the interval for Glass's estimator, which
standardizes the difference between two independent groups by the
control group standard deviation.)
Bias correction (Hedges' g). ci_smd reports the CI on
d; if the bias-corrected g is desired, multiply the
bounds by the Hedges and Olkin (1985) correction factor
\(J(\nu) = 1 - 3/(4 \nu - 1)\) (with \(\nu = n_1 + n_2 - 2\)).
Because \(J(\nu)\) is a constant, the rescaling preserves coverage.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
Cumming, G., & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61(4), 532--574. tools:::Rd_expr_doi("10.1177/0013164401614002")
Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107--128.
Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Academic Press.
Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals. Educational and Psychological Measurement, 65(1), 51--69. tools:::Rd_expr_doi("10.1177/0013164404264850")
Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1--24. tools:::Rd_expr_doi("10.18637/jss.v020.i08")
Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in parameter estimation via narrow confidence intervals. Psychological Methods, 11(4), 363--385. tools:::Rd_expr_doi("10.1037/1082-989X.11.4.363")
Kelley, K., Maxwell, S. E., & Rausch, J. R. (2003). Obtaining power or obtaining precision: Delineating methods of sample size planning. Evaluation and the Health Professions, 26(3), 258--287. tools:::Rd_expr_doi("10.1177/0163278703255242")
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 4 on individual comparisons and Chapter 3 on one-way ANOVA.)
Maxwell, S. E., Kelley, K., & Rausch, J. R. (2008). Sample size planning for statistical power and accuracy in parameter estimation. Annual Review of Psychology, 59, 537--563. tools:::Rd_expr_doi("10.1146/annurev.psych.59.103006.093735")
Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221--257). Mahwah, NJ: Lawrence Erlbaum.
smd, smd_c, ci_smd_c,
ss_aipe_smd, ss_power_smd,
plot_smd, ci_nc_t
Other confidence intervals for effect sizes:
ci_R2(),
ci_c(),
ci_c_ancova(),
ci_c_ancova_bp(),
ci_correlation,
ci_cv(),
ci_eta_squared(),
ci_eta_squared_generalized(),
ci_eta_squared_partial(),
ci_mahalanobis(),
ci_omega_squared(),
ci_pvaf(),
ci_rc(),
ci_reg_coef(),
ci_rmsea(),
ci_sc(),
ci_sc_ancova(),
ci_sm(),
ci_smd_c(),
ci_snr(),
ci_src(),
ci_srsnr(),
contrast_adjusted(),
plot_smd()
# Steiger and Fouladi (1997) example values.
ci_smd(ncp = 2.6, n_1 = 10, n_2 = 10, conf_level = 1 - .05)
ci_smd(ncp = 2.4, n_1 = 300, n_2 = 300, conf_level = 1 - .05)
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