DMAR (version 1.0.0)

ci_sc_ancova: Confidence Interval for a Standardized Contrast in ANCOVA With One Covariate

Description

Calculate the confidence interval for a standardized contrast in ANCOVA with one covariate. The standardizer (i.e., the divisor) can be either the error standard deviation of the ANOVA model (i.e., the model excluding the covariate) or of the ANCOVA model.

Usage

ci_sc_ancova(
  psi = NULL,
  adj_means = NULL,
  s_anova = NULL,
  s_ancova = NULL,
  standardizer = "s_ancova",
  c_weights,
  n,
  cov_means,
  SSwithin_x,
  conf_level = 0.95
)

Value

A 3-row data.frame with columns term and value

(numeric). The term values are "lower_limit" (the lower confidence limit on the standardized ANCOVA contrast), "psi"

(the standardized contrast), and "upper_limit" (the upper limit). The divisor used in standardization (either "s_anova"

or "s_ancova") is attached as the "standardizer"

attribute of the returned data.frame.

Arguments

psi

Unstandardized contrast of adjusted means

adj_means

The vector that contains the adjusted mean of each group on the dependent variable

s_anova

The standard deviation of the errors from the ANOVA model (i.e., the square root of the mean square error from ANOVA)

s_ancova

The standard deviation of the errors from the ANCOVA model (i.e., the square root of the mean square error from ANCOVA)

standardizer

Which error standard deviation the user wants to use, the value of which can be either "s_ancova" or "s_anova"

c_weights

The contrast weights (chose weights so that the positive c-weights sum to 1 and the negative c-weights sum to -1; i.e., use fractional values not integers).

n

Either a single number that indicates the sample size per group, or a vector that contains the sample size of each group

cov_means

A vector that contains the group means of the covariate

SSwithin_x

The sum of squares within groups obtained from the summary table for ANOVA on the covariate

conf_level

The desired confidence interval coverage, (i.e., 1 - Type I error rate)

Author

Ken Kelley [email protected]

Details

The argument SSwithin_x is the sum of squares within groups for the covariate, taken from the ANOVA source table in which the covariate (not the outcome) is the dependent variable. Published reports do not always print this quantity directly. When a report gives the covariate group means, the group sample sizes, and the F statistic from the one-way ANOVA on the covariate, SSwithin_x can be recovered algebraically. The worked example below follows Lai and Kelley (2012): three groups of sizes 19, 18, and 19 (so \(N = 56\)) have covariate means 60.08, 57.08, and 57.97, and the covariate ANOVA reports \(F = 0.756\) with 2 and 53 degrees of freedom. The sum of squares between groups for the covariate, computed from the group means and sample sizes, is approximately 88.5, so the mean square between groups is approximately \(88.5 / 2 = 44.3\). Because F is the ratio of the mean square between groups to the mean square within groups, the mean square within groups is approximately \(44.3 / 0.756 = 58.6\), and the sum of squares within groups is that mean square times its degrees of freedom, approximately \(58.6 \times 53 = 3103\). That recovered value is what you would pass to SSwithin_x. The “Examples” section reproduces this computation in code.

References

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, 363--385. tools:::Rd_expr_doi("10.1037/1082-989X.11.4.363")

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 350--370. tools:::Rd_expr_doi("10.1111/j.2044-8317.2011.02029.x")

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 9.)

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.

See Also

ci_c_ancova, ci_sc

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_sm(), ci_smd(), ci_smd_c(), ci_snr(), ci_src(), ci_srsnr(), contrast_adjusted(), plot_smd()

Examples

# Maxwell, Delaney, & Kelley (2027) offer an example that 30 depressive
# individuals are randomly assigned to three groups, 10 in each, and ANCOVA
# is performed on the posttest scores using the participants' pretest
# scores as the covariate. The means of pretest scores of group 1, 2, and 3 are
# 17, 17.7, and 17.4, respectively, whereas the adjusted means of groups 1, 2, and 3
# are 7.5, 12, and 14, respectively. The error variance in ANCOVA is 29 and thus
# 5.385165 is the error standard deviation, with the sum of squares within groups
# from an ANOVA on the covariate is 752.5.

# To obtained the confidence interval for the standardized adjusted mean difference
# between group 1 and 2, using the ANCOVA error standard deviation:
ci_sc_ancova(adj_means = c(7.5, 12, 14), s_ancova = 5.385165, c_weights = c(1, -1, 0),
             n = 10, cov_means = c(17, 17.7, 17.4), SSwithin_x = 752.5)

# Or, with less error in rounding:
ci_sc_ancova(adj_means = c(7.54, 11.98, 13.98), s_ancova = 5.393, c_weights = c(-1, 0, 1),
             n = 10, cov_means = c(17, 17.7, 17.4), SSwithin_x = 752.5)

# Now, using the standard deviation from ANOVA (and not ANCOVA as above), we have:
ci_sc_ancova(adj_means = c(7.54, 11.98, 13.98), s_anova = 6.294, s_ancova = 5.393,
             c_weights = c(-1, 0, 1),n = 10, cov_means = c(17, 17.7, 17.4),
             SSwithin_x = 752.5, standardizer = "s_anova", conf_level = .95)

# Recovering SSwithin_x from a covariate ANOVA F when a report does not print
# it directly (see the Details section). This example follows Lai and Kelley
# (2012): three groups of sizes 19, 18, and 19 have covariate means 60.08,
# 57.08, and 57.97, and the one-way ANOVA on the covariate reports F = 0.756.
cov_means_ex  <- c(60.08, 57.08, 57.97)
n_ex          <- c(19, 18, 19)
grand_x       <- sum(n_ex * cov_means_ex) / sum(n_ex)
ss_between_x  <- sum(n_ex * (cov_means_ex - grand_x)^2)
ms_between_x  <- ss_between_x / (length(n_ex) - 1)
ms_within_x   <- ms_between_x / 0.756
SSwithin_x_ex <- ms_within_x * (sum(n_ex) - length(n_ex))

ci_sc_ancova(adj_means = c(63.88, 62.39, 56.48), s_ancova = 20.267,
             c_weights = c(0.5, 0.5, -1), n = n_ex, cov_means = cov_means_ex,
             SSwithin_x = SSwithin_x_ex)