Tukey-Kramer Simultaneous Confidence Intervals for Pairwise Contrasts
Source:R/ci_tukey_kramer.R
ci_tukey_kramer.RdComputes the Tukey-Kramer simultaneous confidence intervals for all
\(a (a - 1) / 2\) pairwise contrasts among \(a\) group means in
a one-way design with possibly unequal group sample sizes (Tukey,
1953; Kramer, 1956; Hayter, 1984), and returns the result in tidy
long form. Each interval has individual coverage at least the
specified conf_level and the family-wise coverage is at
least conf_level.
Arguments
- x
Either (a) a fitted
lmoraovobject with a single one-way factor predictor, or (b) a numeric vector of observations, in which casegroupmust also be supplied.- group
When
xis a vector, a factor (or coercible to factor) of group labels, same length asx.- conf_level
Family-wise confidence level. Default
0.95.
Value
A data.frame with one row per pairwise
contrast. Columns: contrast (e.g., "B - A"),
mean_difference, se, q_statistic
(the studentized-range \(q\)), lower_limit,
upper_limit, p_adjusted.
Details
Formula. For groups \(i, j\) with means \(\bar y_i,
\bar y_j\) and sample sizes \(n_i, n_j\), the Tukey-Kramer
simultaneous CI is
$$\bar y_i - \bar y_j \;\pm\; q_{\alpha, a, \nu}
\sqrt{\frac{\mathit{MS}_E}{2} \left(\frac{1}{n_i} + \frac{1}{n_j}\right)},$$
where \(q_{\alpha, a, \nu}\) is the upper \(\alpha\) quantile of
the studentized-range distribution with \(a\) groups and \(\nu\)
error degrees of freedom (stats::qtukey()).
Why Tukey-Kramer. Hayter (1984) proved that the
Tukey-Kramer procedure is conservative for unbalanced designs (the
coverage probability is at least conf_level). For balanced
designs it reduces to Tukey's HSD and the coverage is exactly
conf_level.
Adjusted p-values. Each pairwise \(p\)-value is computed from the studentized-range distribution: \(p = 1 - \mathrm{ptukey}(|q|, a, \nu)\).
References
Hayter, A. J. (1984). A proof of the conjecture that the Tukey-Kramer multiple comparisons procedure is conservative. Annals of Statistics, 12(1), 61–75.
Kramer, C. Y. (1956). Extension of multiple range tests to group means with unequal numbers of replications. Biometrics, 12(3), 307–310.
Tukey, J. W. (1953). The problem of multiple comparisons. Unpublished manuscript, Princeton University.
See also
cv_tukey_hsd, ci_dunnett,
ci_scheffe, TukeyHSD
Other hypothesis tests:
adjusted_means(),
ancova(),
anova_within(),
ci_dunnett(),
ci_scheffe(),
compare_cov_structures(),
contrast_test(),
correlations_test(),
equivalence_r(),
equivalence_smd(),
factorial_anova(),
manova_split_plot(),
mauchly_test(),
mixed_anova(),
obrien_test(),
pairwise_within(),
randomization_test(),
randomization_test_paired(),
regions_of_significance(),
simple_effects_AB(),
summary_t_test(),
welch_t()
Author
Ken Kelley kkelley@nd.edu
Examples
# 1. Balanced one-way: the six marketing panels of the test_market
# data, four outlets per panel, so the procedure is exactly Tukey's
# HSD. Panels 5 and 6 separate from panel 1.
fit <- lm(brand_movement ~ panel, data = test_market)
ci_tukey_kramer(fit)
#> contrast mean_difference se q_statistic lower_limit upper_limit p_adjusted
#> 2 - 1 0.11 0.269 0.409 -1.1 1.32 0.9997
#> 3 - 1 0.685 0.269 2.55 -0.522 1.89 0.4882
#> 4 - 1 0.87 0.269 3.24 -0.337 2.08 0.2481
#> 5 - 1 1.25 0.269 4.63 0.0375 2.45 0.0411
#> 6 - 1 1.3 0.269 4.82 0.0875 2.5 0.0315
#> 3 - 2 0.575 0.269 2.14 -0.632 1.78 0.6607
#> 4 - 2 0.76 0.269 2.83 -0.447 1.97 0.3797
#> 5 - 2 1.14 0.269 4.22 -0.0725 2.34 0.0725
#> 6 - 2 1.19 0.269 4.41 -0.0225 2.39 0.0562
#> 4 - 3 0.185 0.269 0.689 -1.02 1.39 0.9961
#> 5 - 3 0.56 0.269 2.08 -0.647 1.77 0.6840
#> 6 - 3 0.61 0.269 2.27 -0.597 1.82 0.6055
#> 5 - 4 0.375 0.269 1.4 -0.832 1.58 0.9162
#> 6 - 4 0.425 0.269 1.58 -0.782 1.63 0.8674
#> 6 - 5 0.05 0.269 0.186 -1.16 1.26 1.0000
#>
#> Confidence level: 95%
# 2. Same data via vector / group interface:
ci_tukey_kramer(test_market$brand_movement, group = test_market$panel)
#> contrast mean_difference se q_statistic lower_limit upper_limit p_adjusted
#> 2 - 1 0.11 0.269 0.409 -1.1 1.32 0.9997
#> 3 - 1 0.685 0.269 2.55 -0.522 1.89 0.4882
#> 4 - 1 0.87 0.269 3.24 -0.337 2.08 0.2481
#> 5 - 1 1.25 0.269 4.63 0.0375 2.45 0.0411
#> 6 - 1 1.3 0.269 4.82 0.0875 2.5 0.0315
#> 3 - 2 0.575 0.269 2.14 -0.632 1.78 0.6607
#> 4 - 2 0.76 0.269 2.83 -0.447 1.97 0.3797
#> 5 - 2 1.14 0.269 4.22 -0.0725 2.34 0.0725
#> 6 - 2 1.19 0.269 4.41 -0.0225 2.39 0.0562
#> 4 - 3 0.185 0.269 0.689 -1.02 1.39 0.9961
#> 5 - 3 0.56 0.269 2.08 -0.647 1.77 0.6840
#> 6 - 3 0.61 0.269 2.27 -0.597 1.82 0.6055
#> 5 - 4 0.375 0.269 1.4 -0.832 1.58 0.9162
#> 6 - 4 0.425 0.269 1.58 -0.782 1.63 0.8674
#> 6 - 5 0.05 0.269 0.186 -1.16 1.26 1.0000
#>
#> Confidence level: 95%