Provides the Critical Value for the Bryant–Paulson ANCOVA Multiple-Comparison Procedure

Computes the critical value of the Bryant–Paulson generalized studentized range, the reference distribution for multiple comparisons of adjusted means in an analysis of covariance with random covariates. The single-step value is the multiplier for simultaneous confidence intervals on pairwise differences of adjusted means; the "duncan" option instead returns the significant range of the stepwise Duncan multiple-range procedure.

Usage

cv_bryant_paulson(
  alpha_level,
  df,
  groups,
  covariates = 1,
  procedure = c("tukey", "duncan"),
  verbose = TRUE
)

Arguments

alpha_level: Type I error rate (i.e., the false positive rate). As with cv_tukey_hsd, the full alpha_level applies to the upper tail of the (non-negative) generalized studentized range distribution; it is not split between two tails (see Details).
df: The ANCOVA error degrees of freedom (a positive number; in a one-way ANCOVA, df = N - groups - covariates).
groups: The number of groups whose adjusted means are being compared (an integer of at least 2).
covariates: The number of random covariates in the ANCOVA, the parameter $p$ of the Bryant–Paulson distribution (a non-negative integer). With covariates = 0 the critical value reduces to the ordinary studentized range and the result equals $\sqrt2$ times cv_tukey_hsd (see Details).
procedure: One of "tukey" (default) or "duncan". "tukey" returns the single-step simultaneous critical value $q_{\alpha;p,k,\nu}$ used for familywise (Tukey–Kramer-type) confidence intervals; "duncan" returns the stepwise Duncan multiple-range significant range $r_{\alpha;p,k,\nu}$ tabled by Bryant and Bruvold (1980, Table 2) (see Details).
verbose: Provides extra information (the tail areas) about the critical value.

Value

Returns the critical value in a output style (a data.frame with class dmar_tbl and one row per critical value, following the format used by cv_tukey_hsd and cv_t). The value is on the studentized-range scale (the scale on which Bryant and Paulson tabulate their critical values and on which ci_c_ancova_bp uses them). When verbose = TRUE and procedure = "tukey", the upper- and lower-tail areas of the Bryant–Paulson distribution at the critical value are also returned; for procedure = "duncan" the tail areas are NA because the significant range is a stepwise quantity rather than a single quantile.

Details

The Bryant–Paulson procedure is the analysis-of-covariance generalization of Tukey's method (cv_tukey_hsd) for comparing adjusted means when the covariate is random. Because the covariate adjustment must be estimated, the studentized range of adjusted means is stochastically larger than the ordinary studentized range, so the Bryant–Paulson critical value exceeds Tukey's; using the latter would give intervals that are too narrow and a familywise error rate above alpha_level. The single-step (procedure = "tukey") value is the multiplier for a family of simultaneous confidence intervals on the pairwise differences of adjusted means that jointly hold at level $1 - \alpha$. Maxwell, Delaney, and Kelley (2027, Chapter 9) develop multiple comparisons of adjusted means in the analysis of covariance, the setting this critical value serves.

The reference distribution is the Bryant–Paulson generalized studentized range, implemented in qbryant_paulson. Its quantiles are not a standard base-R distribution and are not the multivariate t quantiles that cv_dunnett and cv_smm obtain from mvtnorm; they are computed directly by qbryant_paulson, so this function depends on neither base-R nor mvtnorm multiple-mean machinery.

Scale. The returned value is on the studentized-range scale, $q_{\alpha;p,k,\nu}$, the scale of Bryant and Paulson's (1976) and Bryant and Bruvold's (1980) tables and of Eq. (2.4) of the latter. A pair of adjusted means is declared different when $|\hat\theta_i - \hat\theta_j| > q_{\alpha;p,k,\nu}\,\hat\sigma_{y\mid x}\sqrt{1/n}$. This differs from cv_tukey_hsd, which divides its value by $\sqrt2$ to report on the pairwise mean-difference scale; divide the value here by $\sqrt2$ to obtain that scale. With covariates = 0, cv_bryant_paulson returns exactly $\sqrt2 \times$ cv_tukey_hsd.

Duncan multiple-range. For procedure = "duncan" the value is the “significant range” of Duncan's stepwise test as extended to ANCOVA by Bryant and Bruvold (1980, Section 4). With variable protection levels $\alpha_k = 1 - (1-\alpha)^{k-1}$, $$r_{\alpha;p,2,\nu} = q_{\alpha;p,2,\nu}, \qquad r_{\alpha;p,k,\nu} = \max\{\, r_{\alpha;p,k-1,\nu},\; q_{\alpha_k;p,k,\nu} \,\}, \quad k > 2.$$ These are the values in Bryant and Bruvold's Table 2, reproduced by this function to the tabled two-decimal precision (see the package tests).

References

Bryant, J. L., & Paulson, A. S. (1976). An extension of Tukey's method of multiple comparisons to experimental designs with random concomitant variables. Biometrika, 63, 631–638.

Bryant, J. L., & Bruvold, N. T. (1980). Multiple comparison procedures in the analysis of covariance. Journal of the American Statistical Association, 75(372), 874–880. doi:10.2307/2287175

Duncan, D. B. (1955). Multiple range and multiple F tests. Biometrics, 11, 1–42.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 9, where multiple comparisons of adjusted means in the analysis of covariance are developed; Appendix Table A.8 reports these critical values.)

Author

Ken Kelley kkelley@nd.edu

Examples

# Multiple comparisons of adjusted means in ANCOVA (the setting of Maxwell,
# Delaney, and Kelley, 2027, Chapter 9), using the worked example of Bryant
# and Bruvold (1980): 6 panels, 1 random covariate, 14 error df,
# alpha_level = .05. The single-step value is the multiplier for simultaneous
# confidence intervals on the pairwise differences of adjusted means. With a
# covariate present there is no closed form for it: the Bryant-Paulson
# distribution function is integrated numerically and then inverted, which
# takes about half a second, so the call is shown here rather than run.
# cv_bryant_paulson(alpha_level = .05, df = 14, groups = 6, covariates = 1)
# It returns 4.83, the entry in Table 1 of Bryant and Paulson (1976) and the
# multiplier behind the simultaneous intervals of the 1980 worked example.

# With no covariates the reference distribution is the ordinary studentized
# range, which base R supplies directly, so these calls are quick and do run
# here. The critical value is sqrt(2) times the Tukey HSD critical value.
cv_bryant_paulson(alpha_level = .05, df = 14, groups = 6, covariates = 0)$value
#> [1] 4.638538
sqrt(2) * cv_tukey_hsd(alpha_level = .05, df = 14, groups = 6)$value
#> [1] 4.638538

# The stepwise Duncan multiple-range significant range comes from
# procedure = "duncan". With covariates = 0 it reduces to Duncan's (1955)
# own significant studentized range, 3.37 for a stretch of 6 groups on 14
# error degrees of freedom.
cv_bryant_paulson(alpha_level = .05, df = 14, groups = 6, covariates = 0,
                  procedure = "duncan")
#>  term     value area_less area_greater
#>  upper_cv 3.37  <NA>      <NA>        
# One random covariate raises that range to 3.50, the entry in Table 2 of
# Bryant and Bruvold (1980). Being stepwise, it inverts a separate
# Bryant-Paulson quantile for every stretch from 2 to 6 groups, so it costs
# several seconds and is not run here either. The package tests and the
# tests check it against the paper's Section 4 example.
# cv_bryant_paulson(alpha_level = .05, df = 14, groups = 6, covariates = 1,
#                   procedure = "duncan")