Provides the Critical Value for the Tukey Honestly Significant Difference (HSD) Test

Usage

cv_tukey_hsd(alpha_level, df, groups, verbose = TRUE)

Arguments

alpha_level: Type I error rate (i.e., the false positive rate). For the Tukey HSD test, the full alpha_level applies to the upper tail of the Studentized range distribution (see Details).
df: The error degrees of freedom from the ANOVA (a positive number; typically N - groups).
groups: The number of groups whose means are being compared (an integer of at least 2).
verbose: Provides extra information about areas under the curve.

Value

Returns the critical value in a output style (a data.frame with one row per critical value, following the format used by cv_t and cv_z).

Details

The Tukey HSD test compares all pairs of group means using the Studentized range distribution (qtukey). Because the Studentized range is the absolute difference between the largest and smallest sample means (scaled by a standard error), it is non-negative and the associated distribution has support on \([0, \infty)\). As a consequence, the Type I error rate alpha_level is not split between two tails in the way it is for the (symmetric) t- and z-distributions in cv_t and cv_z; rather, the full alpha_level applies to the upper tail.

The reported critical value is on the scale used for pairwise comparisons of group means, i.e., \(q_{1-\alpha, k, df} / \sqrt{2}\), where \(k\) is the number of groups. A pair of means is declared significantly different when the absolute standardized difference between them exceeds this critical value.

The Tukey HSD critical value is a quantile of the Studentized range distribution, which base R supplies through qtukey, so unlike cv_dunnett and cv_smm this function needs no multivariate distribution machinery and does not require the mvtnorm package. Maxwell, Delaney, and Kelley (2027, Chapter 5) develop the Tukey method as the procedure for all-pairwise comparisons within the multiple-comparisons problem.

References

Tukey, J. W. (1953). The problem of multiple comparisons. Unpublished manuscript, Princeton University.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 5 on the multiple-comparisons problem, where the Tukey method for all-pairwise comparisons is developed.)

Author

Ken Kelley kkelley@nd.edu

Examples

# Following the all-pairwise comparisons setting of Maxwell, Delaney, and
# Kelley (2027, Chapter 5): every pair of k = 3 group means is compared,
# with 27 error degrees of freedom, holding the family-wise error rate at
# .05.
cv_tukey_hsd(alpha_level = .05, df = 27, groups = 3)
#>  term     value area_less area_greater
#>  upper_cv 2.48  0.95      0.05        

# Using DMAR's test_market data (6 marketing panels, N = 24).
fit <- aov(brand_movement ~ panel, data = test_market)
cv_tukey_hsd(
  alpha_level  = .05,
  df     = df.residual(fit),
  groups = nlevels(test_market$panel)
)
#>  term     value area_less area_greater
#>  upper_cv 3.18  0.95      0.05        

# A more stringent alpha with a simple (non-verbose) result.
cv_tukey_hsd(alpha_level = .01, df = 27, groups = 3, verbose = FALSE)
#>  term     value
#>  upper_cv 3.18