# Power calculations for `emmeans` analyses

#### Frederik Aust

#### 16 May, 2022

    library("Superpower")
    library("emmeans")

When conducting exact ANOVA power analyses with `Superpower` it is possible to calculate the power for both the omnibus \(F\)-tests and planned contrasts or post-hoc comparisons. It is possible to use this approach to calculate power for standard contrasts, such as pairwise contrasts between cells. Here I provide a brief overview of how the output of `ANOVA_exact()` can be used to perform power analyses for tailored planned contrasts and follow-up tests.

# A note of caution

All power analyses for `emmeans`-objects are based on the \(F\)- and \(t\)-values from the analyses of the dataset simulated by `ANOVA_exact()` assuming two-sided testing. Thus, the `emmeans_power()` does not honor adjustments of the testing procedure due to either one-sided testing (including two one-sided tests) or corrections for multiple comparisons via the `adjust` option in `emmeans`. As noted below, for the Bonferroni-adjustment this limitation can be overcome by manually adjusting `alpha_level`.

# Post hoc pairwise comparisons

First, we will set up a 2 \(\times\) 3 repeated measures design. When calling `ANOVA_exact()` pairwise comparisons of expected marginal means are added by setting `emm = TRUE` and `contrast_type = "pairwise"` (default).

    # Set up a within design with 2 factors, each with 2 and 3 levels
    design_result <- ANOVA_design(
      design = "2w*3w",
      n = 40,
      mu = c(0.3, 0, 0.5, 0.3, 0, 0),
      sd = 2,
      r = 0.8,
      label_list = list("condition" = c("cheerful", "sad"),
                        "voice" = c("human", "robot", "cartoon"))
    )

    exact_result <- ANOVA_exact(
      design_result,
      alpha_level = 0.05,
      verbose = FALSE,
      emm = TRUE,
      contrast_type = "pairwise"
    )

The result contains the power calculations for both the omnibus \(F\)-tests and pairwise post-hoc comparisons.

    exact_result$main_results

| power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | ---
condition | 29.08380 | 0.0507099 | 0.2311251 | 2.083333
voice | 50.12584 | 0.0621242 | 0.2573700 | 5.166667
condition:voice | 41.62968 | 0.0507099 | 0.2311251 | 4.166667

    head(exact_result$emm_results)

contrast | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | ---
condition_cheerful voice_cartoon - condition_sad voice_cartoon | 68.37160 | 0.1381215 | 0.4003204 | 6.25
condition_cheerful voice_cartoon - condition_cheerful voice_human | 16.41134 | 0.0250000 | 0.1601282 | 1.00
condition_cheerful voice_cartoon - condition_sad voice_human | 16.41134 | 0.0250000 | 0.1601282 | 1.00
condition_cheerful voice_cartoon - condition_cheerful voice_robot | 68.37160 | 0.1381215 | 0.4003204 | 6.25
condition_cheerful voice_cartoon - condition_sad voice_robot | 68.37160 | 0.1381215 | 0.4003204 | 6.25
condition_sad voice_cartoon - condition_cheerful voice_human | 30.99652 | 0.0545455 | 0.2401922 | 2.25

The output also contains the `emmeans`-object on which these power calculations are based. By manipulating this object it is possible to tailor the power analyses to the contrasts desired for the planned study. That is, based on the dataset simulated with `ANOVA_exact()` we can write out the analysis code for `emmeans`-contrasts, just as we would if we were to analyze the empirical data, and use the output to perform the corresponding power analysis.

# Customized `emmeans` contrasts

The `emmeans` reference grid and contrasts are included in the output of `ANOVA_exact()`.

    knitr::kable(exact_result$emmeans$emmeans)

condition | voice | emmean | SE | df | lower.CL | upper.CL
--- | --- | --- | --- | --- | --- | ---
condition_cheerful | voice_cartoon | 0.5 | 0.3162278 | 39 | -0.139631 | 1.139631
condition_sad | voice_cartoon | 0.0 | 0.3162278 | 39 | -0.639631 | 0.639631
condition_cheerful | voice_human | 0.3 | 0.3162278 | 39 | -0.339631 | 0.939631
condition_sad | voice_human | 0.3 | 0.3162278 | 39 | -0.339631 | 0.939631
condition_cheerful | voice_robot | 0.0 | 0.3162278 | 39 | -0.639631 | 0.639631
condition_sad | voice_robot | 0.0 | 0.3162278 | 39 | -0.639631 | 0.639631

    knitr::kable(exact_result$emmeans$contrasts)

contrast | estimate | SE | df | t.ratio | p.value
--- | --- | --- | --- | --- | ---
condition_cheerful voice_cartoon - condition_sad voice_cartoon | 0.5 | 0.2 | 39 | 2.5 | 0.0167338
condition_cheerful voice_cartoon - condition_cheerful voice_human | 0.2 | 0.2 | 39 | 1.0 | 0.3234749
condition_cheerful voice_cartoon - condition_sad voice_human | 0.2 | 0.2 | 39 | 1.0 | 0.3234749
condition_cheerful voice_cartoon - condition_cheerful voice_robot | 0.5 | 0.2 | 39 | 2.5 | 0.0167338
condition_cheerful voice_cartoon - condition_sad voice_robot | 0.5 | 0.2 | 39 | 2.5 | 0.0167338
condition_sad voice_cartoon - condition_cheerful voice_human | -0.3 | 0.2 | 39 | -1.5 | 0.1416670
condition_sad voice_cartoon - condition_sad voice_human | -0.3 | 0.2 | 39 | -1.5 | 0.1416670
condition_sad voice_cartoon - condition_cheerful voice_robot | 0.0 | 0.2 | 39 | 0.0 | 1.0000000
condition_sad voice_cartoon - condition_sad voice_robot | 0.0 | 0.2 | 39 | 0.0 | 1.0000000
condition_cheerful voice_human - condition_sad voice_human | 0.0 | 0.2 | 39 | 0.0 | 1.0000000
condition_cheerful voice_human - condition_cheerful voice_robot | 0.3 | 0.2 | 39 | 1.5 | 0.1416670
condition_cheerful voice_human - condition_sad voice_robot | 0.3 | 0.2 | 39 | 1.5 | 0.1416670
condition_sad voice_human - condition_cheerful voice_robot | 0.3 | 0.2 | 39 | 1.5 | 0.1416670
condition_sad voice_human - condition_sad voice_robot | 0.3 | 0.2 | 39 | 1.5 | 0.1416670
condition_cheerful voice_robot - condition_sad voice_robot | 0.0 | 0.2 | 39 | 0.0 | 1.0000000

By using `emmeans_power()` on the contrasts, we can reproduce the results of the previous power analysis for the pairwise comparisons.

    head(emmeans_power(exact_result$emmeans$contrasts))

contrast | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | ---
condition_cheerful voice_cartoon - condition_sad voice_cartoon | 68.37160 | 0.1381215 | 0.4003204 | 6.25
condition_cheerful voice_cartoon - condition_cheerful voice_human | 16.41134 | 0.0250000 | 0.1601282 | 1.00
condition_cheerful voice_cartoon - condition_sad voice_human | 16.41134 | 0.0250000 | 0.1601282 | 1.00
condition_cheerful voice_cartoon - condition_cheerful voice_robot | 68.37160 | 0.1381215 | 0.4003204 | 6.25
condition_cheerful voice_cartoon - condition_sad voice_robot | 68.37160 | 0.1381215 | 0.4003204 | 6.25
condition_sad voice_cartoon - condition_cheerful voice_human | 30.99652 | 0.0545455 | 0.2401922 | 2.25

Now, we can manipulate the `emmeans` reference grid to perform additional power analyses. In the following example, we calculate the power for the contrasts between sad and cheerful condition for each voice.

    simple_condition_effects <- emmeans(
      exact_result$emmeans$emmeans,
      specs = ~ condition | voice
    )

    emmeans_power(pairs(simple_condition_effects))

contrast | voice | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | --- | ---
condition_cheerful - condition_sad | voice_cartoon | 68.3716 | 0.1381215 | 0.4003204 | 6.25
condition_cheerful - condition_sad | voice_human | 5.0000 | 0.0000000 | 0.0000000 | 0.00
condition_cheerful - condition_sad | voice_robot | 5.0000 | 0.0000000 | 0.0000000 | 0.00

We may also calculate the power for testing all condition means against an arbitrary constant.

    emmeans_power(test(simple_condition_effects, null = 0.5))

condition | voice | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | --- | ---
condition_cheerful | voice_cartoon | 5.000000 | 0.0000000 | 0.0000000 | 0.0
condition_sad | voice_cartoon | 33.831141 | 0.0602410 | 0.2531848 | 2.5
condition_cheerful | voice_human | 9.462673 | 0.0101523 | 0.1012739 | 0.4
condition_sad | voice_human | 9.462673 | 0.0101523 | 0.1012739 | 0.4
condition_cheerful | voice_robot | 33.831141 | 0.0602410 | 0.2531848 | 2.5
condition_sad | voice_robot | 33.831141 | 0.0602410 | 0.2531848 | 2.5

Finally, we can calculate the power for custom contrasts between any linear combination of conditions.

    custom_contrast <- contrast(
      exact_result$emmeans$emmeans,
      list(robot_vs_sad_human = c(0, 0, 0, 1, -0.5, -0.5))
    )

    emmeans_power(custom_contrast)

contrast | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | ---
robot_vs_sad_human | 39.34971 | 0.0714286 | 0.2773501 | 3

Although `emmeans_power()` currently ignores adjustments for multiple comparisons, it is possible to calculate the power for Bonferroni-corrected tests by adjusting `alpha_level`.

    n_contrasts <- nrow(as.data.frame(simple_condition_effects))

    emmeans_power(
      pairs(simple_condition_effects),
      alpha_level = 0.05 / n_contrasts
    )

contrast | voice | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | --- | ---
condition_cheerful - condition_sad | voice_cartoon | 40.1555499 | 0.1381215 | 0.4003204 | 6.25
condition_cheerful - condition_sad | voice_human | 0.8333333 | 0.0000000 | 0.0000000 | 0.00
condition_cheerful - condition_sad | voice_robot | 0.8333333 | 0.0000000 | 0.0000000 | 0.00

Similarly, if we want to calculate power for a one-sided test, we can doubling `alpha_level`.

    emmeans_power(
      pairs(simple_condition_effects)[1],
      alpha_level = 2 * 0.05
    )

contrast | voice | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | --- | ---
condition_cheerful - condition_sad | voice_cartoon | 79.14507 | 0.1381215 | 0.4003204 | 6.25

Note, that because power is calculated from the squared \(t\)-value, power is only calculated correctly if the alternative hypothesis is true in the simulated dataset. That is, the difference of the condition means is consistent with the tested directional hypothesis.

## Equivalence and non-superiority/-inferiority tests

Because `emmeans` can perform equivalence, non-superiority, and -inferiority tests, `emmeans_power()` can calculate the corresponding power for these tests.

    emmeans_power(
      pairs(simple_condition_effects, side = "equivalence", delta = 0.3)[2]
    )

| contrast | voice | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | --- | --- | ---
2 | condition_cheerful - condition_sad | voice_human | 30.99652 | 0.0545455 | 0.2401922 | 2.25

Note, that because power is calculated from the squared \(t\)-value, power is only calculated correctly if the alternative hypothesis is true in the simulated dataset. That is, the difference between the condition means is consistent with the tested directional hypothesis (smaller than `delta`).

## Joint tests

Another useful application of `emmeans_power()` is to joint tests. Lets assume we plan to test the main effect of voice for each of the two conditions separately using `joint_tests()`. We can then calculate the power for each Bonferroni-corrected \(F\)-test as follows.

    voice_by_condition <- joint_tests(
      exact_result$emmeans$emmeans,
      by = "condition"
    )

    emmeans_power(voice_by_condition, alpha_level = 0.05 / 2)

| model term | condition | power | partial_eta_squared | cohen_f | non_centrality
--- | --- | --- | --- | --- | --- | ---
1 | voice | condition_cheerful | 21.80102 | 0.0751061 | 0.2849651 | 3.167
3 | voice | condition_sad | 10.39583 | 0.0370370 | 0.1961161 | 1.500