Scheffe-Adjusted Simultaneous Confidence Intervals for Contrasts

Computes the Scheffe (1953, 1959) simultaneous confidence intervals on user-specified contrasts among the means of a one-way design. The Scheffe procedure controls the family-wise error rate for any set of contrasts, however many and however post-hoc, which makes it more conservative than Tukey-Kramer or Bonferroni for the specific case of all-pairwise comparisons but optimal for arbitrary post-hoc contrasts.

Usage

ci_scheffe(x, group = NULL, contrasts = NULL, conf_level = 0.95)

Arguments

x

A fitted lm or aov object with a single one-way factor predictor, or a numeric vector of observations with group supplied.

group

Optional factor of group labels when x is a numeric vector.

contrasts

An \(a \times m\) matrix or vector of contrast coefficients (rows = levels, columns = contrasts). Each column must sum to zero. If NULL (default), the function returns intervals for all \(a (a - 1) / 2\) pairwise contrasts.

conf_level

Family-wise confidence level. Default 0.95.

Value

A data.frame with one row per contrast. Columns: contrast (a printed label), contrast_value, se, F_statistic, lower_limit, upper_limit, p_adjusted.

Details

Critical value. For \(a\) groups with \(\nu\) error degrees of freedom, the Scheffe critical value is $$S \;=\; \sqrt{(a - 1) F_{1 - \alpha, a - 1, \nu}},$$ where \(F_{1 - \alpha, a - 1, \nu}\) is the upper \(\alpha\) quantile of the central F distribution. The Scheffe simultaneous CI on a contrast \(\psi = \sum_i c_i \mu_i\) is $$\hat\psi \;\pm\; S \cdot \mathit{SE}(\hat\psi),$$ where \(\mathit{SE}(\hat\psi) = \sqrt{\mathit{MS}_E \sum_i c_i^2 / n_i}\).

Scope. The Scheffe family-wise coverage holds for any number of contrasts, pairwise, complex, or chosen after looking at the data. The trade-off is conservativeness: for all-pairwise comparisons, Tukey-Kramer is uniformly more powerful.

References

Scheffe, H. (1953). A method for judging all contrasts in the analysis of variance. Biometrika, 40(1/2), 87–104.

Scheffe, H. (1959). The analysis of variance. Wiley.

See also

cv_scheffe, ci_tukey_kramer, ci_dunnett

Other hypothesis tests: adjusted_means(), ancova(), anova_within(), ci_dunnett(), ci_tukey_kramer(), 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. All pairwise contrasts among the six marketing panels of the
#    test_market data via the default:
fit <- lm(brand_movement ~ panel, data = test_market)
ci_scheffe(fit)
#>  contrast contrast_value se   F_statistic lower_limit upper_limit p_adjusted
#>  2 - 1    0.11           0.38 0.0168      -1.3        1.52        0.9999    
#>  3 - 1    0.685          0.38 0.65        -0.73       2.1         0.6652    
#>  4 - 1    0.87           0.38 1.05        -0.545      2.28        0.4201    
#>  5 - 1    1.25           0.38 2.15        -0.17       2.66        0.1061    
#>  6 - 1    1.3            0.38 2.32        -0.12       2.71        0.0855    
#>  3 - 2    0.575          0.38 0.458       -0.84       1.99        0.8021    
#>  4 - 2    0.76           0.38 0.8         -0.655      2.17        0.5639    
#>  5 - 2    1.14           0.38 1.78        -0.28       2.55        0.1668    
#>  6 - 2    1.19           0.38 1.95        -0.23       2.6         0.1364    
#>  4 - 3    0.185          0.38 0.0474      -1.23       1.6         0.9984    
#>  5 - 3    0.56           0.38 0.434       -0.855      1.97        0.8186    
#>  6 - 3    0.61           0.38 0.516       -0.805      2.02        0.7611    
#>  5 - 4    0.375          0.38 0.195       -1.04       1.79        0.9605    
#>  6 - 4    0.425          0.38 0.25        -0.99       1.84        0.9342    
#>  6 - 5    0.05           0.38 0.00346     -1.36       1.46        1.0000    
#> 
#> Confidence level: 95%

# 2. A contrast chosen after inspecting the means: the two panels with
#    the highest brand movement (5 and 6) against the two with the
#    lowest (1 and 2). The Scheffe coverage holds for a contrast picked
#    this way, and the interval still excludes zero even though none of
#    the pairwise intervals above does.
cmat <- matrix(c(-0.5, -0.5, 0, 0, 0.5, 0.5), nrow = 6,
               dimnames = list(levels(test_market$panel),
                               "panels 5,6 - panels 1,2"))
ci_scheffe(fit, contrasts = cmat)
#>  contrast                contrast_value se    F_statistic lower_limit
#>  panels 5,6 - panels 1,2 1.22           0.269 4.09        0.215      
#>  upper_limit p_adjusted
#>  2.22        0.0117    
#> 
#> Confidence level: 95%