Provides the Critical Value of the Studentized Maximum Modulus Distribution

Usage

cv_smm(alpha_level, df, n_comparisons, verbose = TRUE)

Arguments

alpha_level: Type I error rate (i.e., the family-wise false-positive rate).
df: Error degrees of freedom (typically $N - k$ in a one-way ANOVA). May be Inf, in which case the known-variance (normal) limit is used.
n_comparisons: The number of simultaneous comparisons ($m$) for which a family-wise critical value is desired.
verbose: Provides extra information about areas under the curve.

Value

Returns the critical value as a data.frame, following the format used by cv_t.

Details

The Studentized maximum modulus (SMM) distribution is the distribution of $$\max_{i=1,\ldots,m}\, |Z_i| \,/\, S,$$ where $Z_1, \ldots, Z_m$ are independent standard normal variates and $S$ is independent of the $Z_i$ and equal to $\sqrt{\chi^2_{df} / df}$.

What the SMM is used for. The SMM critical value is the multiplier that turns a set of $m$ individual estimates into a family of simultaneous confidence intervals, or equivalently a family of tests, that jointly control the family-wise error rate at level alpha_level. Because the modulus is the largest of $m$ standardized statistics in absolute value, requiring that maximum to clear the critical value bounds the chance of any one of the $m$ intervals failing to cover (or any one of the $m$ tests producing a false positive). Maxwell, Delaney, and Kelley (2027, Chapter 5) describe this use in the context of the multiple-comparisons problem: when a researcher forms several means or contrasts and wants the stated coverage to hold across the whole set rather than one interval at a time, the SMM supplies the simultaneous critical value. A pair-by-pair construction, applied to $m$ comparisons, is $\hat\psi_i \pm c_{\alpha;m,df}\,\mathit{SE}_{\hat\psi_i}$ with $c_{\alpha;m,df}$ the SMM critical value returned here. When $m = 1$ it reduces to the ordinary two-sided t critical value.

How it is computed. The $m$ statistics are independent given the common scale estimate $S$, so the joint distribution factorizes after conditioning on $S$ and the $m$-dimensional integral that defines the maximum modulus collapses to a single one-dimensional integral, $$P\!\left(\max_i |Z_i|/S \le c\right) = \int_0^\infty \bigl[\,2\,\Phi(c\,s) - 1\,\bigr]^{m}\, f_S(s)\; ds,$$ where $f_S$ is the density of $S = \sqrt{\chi^2_{df}/df}$. This package evaluates that integral with integrate and inverts it with uniroot, so the returned value is deterministic and accurate to the solver tolerance (there is no Monte Carlo simulation and no random seed). When df = Inf the scale is degenerate at 1 and the closed form $c = \Phi^{-1}\!\bigl((1 + (1-\alpha)^{1/m})/2\bigr)$ is returned; when $m = 1$ the value is exactly $t_{1-\alpha/2,\,df}$.

References

Stoline, M. R., & Ury, H. K. (1979). Tables of the Studentized maximum modulus distribution and an application to multiple comparisons among means. Technometrics, 21(1), 87–93.

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 simultaneous confidence intervals for several means or contrasts are developed; Appendix Table A.5 reports SMM critical values.)

Author

Ken Kelley kkelley@nd.edu

Examples

# Following the simultaneous-intervals setting of Maxwell, Delaney, and
# Kelley (2027, Chapter 5): a researcher forms m = 5 comparisons and wants
# all five confidence intervals to hold simultaneously at the .05 level,
# with 36 error degrees of freedom.
cv_smm(alpha_level = .05, df = 36, n_comparisons = 5)
#>  term     value area_less area_greater
#>  upper_cv 2.7   0.95      0.05        

# When m = 1, the SMM critical value reduces to the two-sided
# t critical value:
cv_smm(alpha_level = .05, df = 36, n_comparisons = 1)$value
#> [1] 2.028094
cv_t(alpha_level = .05, df = 36)$value[2]   # upper_cv from cv_t
#> [1] 2.028094