Provides the Critical Value for the Scheffé Procedure
Arguments
- alpha_level
Type I error rate (i.e., the family-wise false-positive rate).
- df_numerator
The numerator degrees of freedom (typically the number of groups minus 1, \(k - 1\), in a one-way ANOVA).
- df_denominator
The denominator (error) degrees of freedom (typically \(N - k\) in a one-way ANOVA).
- 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_tukey_hsd).
Details
The Scheffé critical value protects the family-wise error rate for the simultaneous test of any contrast (or family of contrasts) in a fixed-effects ANOVA, including data-driven contrasts selected after looking at the data. It is therefore the most conservative of the standard procedures.
The critical value, on the scale of a t-statistic, is $$t_{\mathrm{crit}}^{\mathrm{Scheffe}} = \sqrt{(k-1)\, F_{1-\alpha,\,k-1,\,df_{\mathrm{denominator}}}},$$ so that a contrast \(\hat\psi\) with standard error \(\mathit{SE}_{\hat\psi}\) is declared significant when \(|\hat\psi/\mathit{SE}_{\hat\psi}| > t_{\mathrm{crit}}^{\mathrm{Scheffe}}\). The corresponding simultaneous confidence interval is \(\hat\psi \pm t_{\mathrm{crit}}^{\mathrm{Scheffe}} \cdot \mathit{SE}_{\hat\psi}\).
Like the Studentized range distribution underlying Tukey HSD, the
Scheffé reference is one-sided (the underlying F statistic is
non-negative), so alpha_level is not split between two tails.
The Scheffé critical value is a function of a univariate F
quantile, which base R supplies through qf, so unlike
cv_dunnett and cv_smm this function needs no
multivariate distribution machinery. Scheffé's procedure earns its
simultaneous protection over the infinite family of all possible
contrasts by projecting onto the overall F test rather than by
integrating a multivariate t density; that is why a single
univariate quantile suffices and the mvtnorm package is not
required here. Maxwell, Delaney, and Kelley (2027, Chapter 5) develop the
Scheffé method as the procedure for arbitrary contrasts within the
multiple-comparisons problem.
References
Scheffe, H. (1953). A method for judging all contrasts in the analysis of variance. Biometrika, 40, 87–104.
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 Scheffé method for arbitrary contrasts is developed.)
See also
cv_t, cv_tukey_hsd,
contrast_test
Other critical values:
cv_bonferroni_f(),
cv_bryant_paulson(),
cv_chisq(),
cv_dunnett(),
cv_f(),
cv_smm(),
cv_t(),
cv_tukey_hsd(),
cv_z()
Author
Ken Kelley kkelley@nd.edu
Examples
# Following the arbitrary-contrasts setting of Maxwell, Delaney, and Kelley
# (2027, Chapter 5): a one-way ANOVA with k = 4 groups and 36 error degrees
# of freedom (e.g., n = 10 per group). The Scheffé critical value protects
# the family-wise error rate over any contrast, including contrasts chosen
# after looking at the data.
cv_scheffe(alpha_level = .05, df_numerator = 3, df_denominator = 36)
#> term value area_less area_greater
#> upper_cv 2.93 0.95 0.05
# The price of that protection is a larger multiplier than the unadjusted
# t critical value at the same error degrees of freedom:
cv_t(alpha_level = .05, df = 36)
#> term value area_less area_greater
#> lower_cv -2.03 0.025 0.975
#> upper_cv 2.03 0.975 0.025