Provides the Critical Value for Dunnett's Many-to-One Comparisons Procedure

Usage

cv_dunnett(
  alpha_level,
  df,
  n_comparisons,
  alternative = "not_equal",
  verbose = TRUE
)

Arguments

alpha_level: Type I error rate (i.e., the family-wise false-positive rate).
df: Error degrees of freedom (typically $N - k$, where $k$ is the total number of groups including the control). May be Inf, in which case the known-variance (normal) limit is used.
n_comparisons: The number of treatment-versus-control comparisons (i.e., $k - 1$ where $k$ is the total number of groups).
alternative: The form of the alternative hypothesis: one of "not_equal" (two-sided; default), "greater" (treatments greater than control), or "less" (treatments less than control).
verbose: Provides extra information about areas under the curve.

Value

Returns the critical value in a output style (a data.frame following the format used by cv_t).

Details

Dunnett's procedure controls the family-wise error rate for the special case of comparing each of $k - 1$ treatments to a single control (the many-to-one comparisons setting), using a multivariate t reference with constant pairwise correlation $1/2$. That correlation is exact for a balanced design: each comparison is $(\bar Y_i - \bar Y_0)$, all sharing the one control mean $\bar Y_0$ of variance $\sigma^2/n$, while each comparison has variance $2\sigma^2/n$, so any two comparisons correlate $(\sigma^2/n)/(2\sigma^2/n) = 1/2$. Maxwell, Delaney, and Kelley (2027, Chapter 5) develop the many-to-one comparisons problem and tabulate these critical values.

How it is computed. The critical value is a quantile of a $(k-1)$-dimensional multivariate t distribution whose correlation matrix has a unit diagonal and off-diagonal entries of 1/2. Because that correlation is a single common value, the comparisons have the one-factor representation $Z_i = \sqrt{1/2}\,W + \sqrt{1/2}\,U_i$ with a shared factor $W$ and independent $U_i$, all standard normal. Conditioning on $W$ and on the common scale estimate $S = \sqrt{\chi^2_{df}/df}$ makes the comparisons independent, so the $(k-1)$-dimensional integral collapses to two nested one-dimensional integrals: $$P\!\left(\max_i T_i \le d\right) = \int_0^\infty\!\!\int_{-\infty}^{\infty} \Bigl[\Phi\bigl((d\,s - \sqrt{1/2}\,w)/\sqrt{1/2}\bigr)\Bigr]^{k-1} \phi(w)\,dw\; f_S(s)\,ds$$ for the one-sided value (the two-sided value replaces the bracket with the probability that $|T_i| \le d$). This package evaluates those integrals with integrate and inverts with uniroot, so the returned value is deterministic and accurate to the solver tolerance, with no Monte Carlo simulation and no random seed. When $k - 1 = 1$ the value is the ordinary one- or two-sided t critical value.

Note

The constant-correlation assumption holds for balanced designs (equal n per group); for severely unbalanced designs use glht instead.

References

Dunnett, C. W. (1955). A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association, 50(272), 1096–1121.

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; Dunnett's many-to-one comparisons procedure is developed there, and Appendix Tables A.6 and A.7 report its critical values for two- and one-tailed tests.)

Author

Ken Kelley kkelley@nd.edu

Examples

# Following the many-to-one comparisons setting of Maxwell, Delaney, and
# Kelley (2027, Chapter 5): three treatment groups each compared to a single
# control (k = 4 groups, so 3 comparisons) with 36 error degrees of freedom.
# The two-sided critical value controls the family-wise error rate at .05.
cv_dunnett(alpha_level = .05, df = 36, n_comparisons = 3)
#>  term     value area_less area_greater
#>  upper_cv 2.45  0.95      0.05        

# When the treatments are expected only to exceed the control, a one-sided
# ("treatments greater than control") critical value is smaller.
cv_dunnett(alpha_level = .05, df = 36, n_comparisons = 3, alternative = "greater")
#>  term     value area_less area_greater
#>  upper_cv 2.13  0.95      0.05