Confidence Interval for an (Unstandardized) Contrast in ANCOVA With One Covariate

Calculates the confidence interval for an unstandardized contrast in the one-covariate ANCOVA. Two procedures are available through the procedure argument. The default ("t") returns a single per-comparison interval based on the t distribution: it gives the correct \((1 -\) conf_level\()\) coverage for one contrast chosen in advance, and its standard error includes the \((\sum c_i \bar X_i)^2 / SS_{\mathrm{within}(x)}\) term that accounts for the covariate separation between the groups in that one contrast. The "bryant_paulson" procedure instead returns Bryant–Paulson simultaneous (familywise) intervals over a whole family of contrasts of adjusted means; when selected, ci_c_ancova simply forwards its arguments to ci_c_ancova_bp and returns that result. See Details for which to use when.

Usage

ci_c_ancova(
  psi = NULL,
  adj_means = NULL,
  s_ancova = NULL,
  c_weights,
  n,
  cov_means,
  SSwithin_x,
  conf_level = 0.95,
  procedure = c("t", "bryant_paulson"),
  ...
)

Arguments

psi

The unstandardized contrast of adjusted means

adj_means

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

s_ancova

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

c_weights

The contrast weights

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)

procedure

The interval procedure, one of "t" (the default, a single per-comparison interval based on the t distribution) or "bryant_paulson" (Bryant–Paulson simultaneous intervals for a family of contrasts of adjusted means). When "bryant_paulson" is chosen the call is forwarded to ci_c_ancova_bp; see Details.

...

Allows one to potentially include parameter values for inner functions. When procedure = "bryant_paulson", these are passed on to ci_c_ancova_bp (for example num_covariates, df, or contrast_type).

Value

A 3-row data.frame with columns term and value (numeric). The term values are "lower_limit" (the lower confidence limit on the unstandardized ANCOVA contrast), "psi" (the unstandardized contrast point estimate), and "upper_limit" (the upper limit).

When procedure = "bryant_paulson", the return value is whatever ci_c_ancova_bp returns (a table with one row per contrast and columns contrast, estimate, lower_limit, and upper_limit).

Details

Per-comparison versus simultaneous. The two procedures answer different questions and are not interchangeable. Use the default procedure = "t" when a single contrast was planned in advance: the interval has exact per-comparison coverage and its width reflects the covariate adjustment for that specific contrast through the \((\sum c_i \bar X_i)^2 / SS_{\mathrm{within}(x)}\) term. Use procedure = "bryant_paulson" when several contrasts (for example all pairwise comparisons of adjusted means) are examined together and the coverage statement must hold simultaneously across the family: the Bryant–Paulson generalized studentized range supplies a larger critical value that controls the familywise error rate and, because the covariates are random, correctly absorbs the extra sampling uncertainty from estimating the covariate adjustment (which holds on average over the covariate distribution, so the per-contrast separation term is not added again). A per-comparison interval used as if it were simultaneous understates the family error rate; a simultaneous interval used for one planned contrast is wider than necessary. See ci_c_ancova_bp for the full description of the simultaneous procedure and its arguments.

Note

Be sure to use the standard deviation and not the error variance for s_ancova, not the square of this value which would come from the source table (i.e., do not use the variance of the error but rather use the square root).

If n receives a single number, that number is considered as the sample size per group. If n receives a vector, the vector is considered as the sample size of each group.

Be sure to use fractions not the integers to specify c_weights. For example, in an ANCOVA of four groups, if the user wants to compare the mean of group 1 and 2 with the mean of group 3 and 4, c_weights should be specified as c(0.5, 0.5, -0.5, -0.5) rather than c(1, 1, -1, -1). Make sure the sum of the contrast weights are zero.

References

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

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. 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.)

See also

ci_c,ci_sc_ancova, ci_c_ancova_bp for the Bryant–Paulson simultaneous procedure

Other confidence intervals for effect sizes: ci_R2(), ci_c(), 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(), ci_smd_c(), ci_snr(), ci_src(), ci_srsnr(), contrast_adjusted(), plot_smd()

Author

Ken Kelley kkelley@nd.edu

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 to 3 are
# 17, 17.7, and 17.4, respectively, and the adjusted means of groups 1 to 3
# are 7.5, 12, and 14, respectively. The error variance in ANCOVA is 29,
# and the sum of squares within groups from ANOVA on the covariate is 752.5.

# To obtain the confidence interval for adjusted mean of group 1 versus group 2:
ci_c_ancova(adj_means = c(7.5, 12, 14), s_ancova = sqrt(29),
            c_weights = c(1, -1, 0), n = 10,
            cov_means = c(17, 17.7, 17.4), SSwithin_x = 752.5)
#>  term        value
#>  lower_limit -9.46
#>  psi         -4.5 
#>  upper_limit 0.458
#> 
#> Confidence level: 95%

# That interval is the right one for a single contrast planned in advance.
# For the family of all three pairwise comparisons of the adjusted means,
# with coverage that holds simultaneously across the family, select
# procedure = "bryant_paulson"; the call is forwarded to ci_c_ancova_bp().
# That route inverts the Bryant-Paulson distribution numerically to get its
# multiplier, which takes about half a second, so it is shown here rather
# than run.
# ci_c_ancova(adj_means = c(7.5, 12, 14), s_ancova = sqrt(29), n = 10,
#             procedure = "bryant_paulson")
# The simultaneous limits are wider, which is the price of the family
# statement: group 1 against group 2 runs from -10.61 to 1.61, where the
# single planned contrast above ran from -9.47 to 0.47.