Confidence Limits for the Standardized Mean Difference Using the Control Group Standard Deviation as the Divisor

Computes the exact noncentral t-based confidence limits for the standardized mean difference that uses the control group standard deviation as the divisor (Glass's g). Standardizing by the control group alone keeps the scale of the effect anchored in the untreated population, which matters when the treatment may alter variability as well as the mean.

Usage

ci_smd_c(
  ncp = NULL,
  smd_c = NULL,
  n_C = NULL,
  n_E = NULL,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL,
  tol = 1e-09,
  ...
)

Arguments

ncp: The estimated noncentrality parameter, this is generally the observed t-statistic from comparing the control and experimental group (assuming homogeneity of variance)
smd_c: The standardized mean difference (using the control group standard deviation in the denominator)
n_C: The sample size for the control group
n_E: The sample size for experimental group
conf_level: The confidence level (1-Type I error rate)
alpha_lower: The Type I error rate for the lower tail
alpha_upper: The Type I error rate for the upper tail
tol: The tolerance of the iterative method for determining the critical values
...: Potentially include parameter for inner functions

Value

A 3-row data.frame with columns term and value. The term values are "lower_limit" (the lower bound of the confidence interval), "smd_c" (the standardized mean difference standardized by the control group standard deviation), and "upper_limit" (the upper bound).

Warning

This function uses conf_limits_nct, 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 conf_limits_nct 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.

References

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. doi:10.1177/0013164401614002

Glass, G. V. (1976). Primary, secondary, and meta-analysis of research. Educational Researcher, 5, 3–8.

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

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. 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. doi:10.1037/1082-989X.11.4.363

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.)

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.

Author

Ken Kelley kkelley@nd.edu

Examples

ci_smd_c(smd_c = .5, n_C = 100, n_E = 100, conf_level = .95)
#>  term        value
#>  lower_limit 0.213
#>  smd_c       0.5  
#>  upper_limit 0.785
#> 
#> Confidence level: 95%