EMMEANS: Simple-effect analysis and post-hoc multiple comparison.

Description

Easily perform (1) simple-effect (and simple-simple-effect) analyses, including both simple main effects and simple interaction effects, and (2) post-hoc multiple comparisons (e.g., pairwise, sequential, polynomial), with p values adjusted for factors with >= 3 levels.

This function is based on and extends the (1) emmeans::joint_tests(), (2) emmeans::emmeans(), and (3) emmeans::contrast() functions. You only need to specify the model object, to-be-tested effect(s), and moderator(s). Almost all results you need will be displayed in an elegant manner, including effect sizes (partial \(\eta^2\) and Cohen's d) and their confidence intervals (CIs). 90% CIs for partial \(\eta^2\) and 95% CIs for Cohen's d are reported.

To compute Cohen's d and its 95% CI in pairwise comparisons, this function uses the pooled SD: SD_pooled = sqrt(MSE), where MSE is of the effect term extracted from ANOVA table.

Disclaimer: There is substantial disagreement on what is the appropriate pooled SD to use in computing effect sizes. For alternative methods, see emmeans::eff_size() and effectsize::t_to_d(). Users should not take the default output as the only right results and are completely responsible for specifying sd.pooled.

Usage

EMMEANS(
  model,
  effect = NULL,
  by = NULL,
  contrast = "pairwise",
  reverse = TRUE,
  p.adjust = "bonferroni",
  sd.pooled = NULL,
  spss = TRUE,
  digits = 2,
  nsmall = digits
)

Arguments

model

A model object returned by MANOVA.

effect

The effect(s) you want to test. If set to a character string (e.g., "A"), it reports the results of omnibus test or simple main effect. If set to a character vector (e.g., c("A", "B")), it also reports the results of simple interaction effect.

Moderator variable(s). Default is NULL.

contrast

Contrast method for multiple comparisons. Default is "pairwise".

Alternatives can be "pairwise" ("revpairwise"), "seq" ("consec"), "poly", "eff". For details, see ?emmeans::`contrast-methods`.

reverse

The order of levels to be contrasted. Default is TRUE (higher level vs. lower level).

p.adjust

Adjustment method of p values for multiple comparisons. Default is "bonferroni". For polynomial contrasts, default is "none".

Alternatives can be "none", "fdr", "hochberg", "hommel", "holm", "tukey", "mvt", "dunnettx", "sidak", "scheffe", "bonferroni". For details, see stats::p.adjust() and emmeans::summary().

sd.pooled

By default, it uses sqrt(MSE) to compute Cohen's d. Users may also manually set it (e.g., the SD of a reference group, or using effectsize::sd_pooled()).

spss

Return results identical to SPSS. Default is TRUE, which uses the lm (rather than aov) object in model for emmeans::joint_tests() and emmeans::emmeans().

digits, nsmall

Number of decimal places of output. Default is 2.

Value

The same object as returned by MANOVA (for recursive use).

Statistical Details

Some may confuse the statistical terms "simple effects", "post-hoc tests", and "multiple comparisons". Such a confusion is not uncommon. Here, I explain what these terms actually refer to.

1. Simple Effect

When we speak of "simple effect", we are referring to ...

simple main effect
simple interaction effect (only for designs with 3 or more factors)
simple simple effect (only for designs with 3 or more factors)

When the interaction effect in ANOVA is significant, we should then perform a "simple-effect analysis". In ANOVA, we call it "simple-effect analysis"; in regression, we also call it "simple-slope analysis". They are identical in statistical principles. Nonetheless, the situations in ANOVA can be a bit more complex because we sometimes have a three-factors design.

In a regular two-factors design, we only test "simple main effects". That is, on the different levels of a factor "B", the main effects of "A" would be different. However, in a three-factors (or more) design, we may also test "simple interaction effects" and "simple simple effects". That is, on the different combinations of levels of factors "B" and "C", the main effects of "A" would be different.

In SPSS, we usually use the MANOVA and/or the GLM + /EMMEANS syntax to perform such analyses. Tutorials (in Chinese) for the SPSS syntax can be found in: Tutorial #1, Tutorial #2, and Tutorial #3.

Here, the R function EMMEANS can do the same thing as in SPSS and can do much better and easier (just see the section "Examples").

To note, simple effects per se do NOT need any form of p-value adjustment, because what we test in simple-effect analyses are still "omnibus F-tests".

2. Post-Hoc Test

The term "post-hoc" means that the tests are performed after ANOVA. Given this, some may (wrongly) regard simple-effect analyses also as a kind of post-hoc tests. However, these two terms should be distinguished. In many situations and softwares, "post-hoc tests" only refer to "post-hoc comparisons" using t-tests and some p-value adjustment techniques. We need post-hoc comparisons only when there are factors with 3 or more levels. For example, we can perform the post-hoc comparisons of mean values (1) across multiple levels of one factor in a pairwise way or (2) particularly between the two conditions "A1B1" and "A2B2".

Post-hoc tests are totally independent of whether there is a significant interaction effect. It only deals with factors with multiple levels. In most cases, we use pairwise comparisons to do post-hoc tests. See the next part for details.

3. Multiple Comparison

As mentioned above, multiple comparisons are post-hoc tests by its nature but do NOT have any relationship with simple-effect analyses. In other words, "(post-hoc) multiple comparisons" are independent of "interaction effects" and "simple effects". What's more, when the simple main effect is of a factor with 3 or more levels, we also need to do multiple comparisons (e.g., pairwise comparisons) within the simple-effect analysis. In this situation (i.e., >= 3 levels), we need p-value adjustment methods such as Bonferroni, Tukey's HSD (honest significant difference), FDR (false discovery rate), and so forth.

There are many ways to do multiple comparisons. All these methods are included in the current EMMEANS function. If you are familiar with SPSS syntax, you may feel that the current R functions MANOVA and EMMEANS are a nice combination of the SPSS syntax MANOVA and GLM + /EMMEANS. Yes, they are. More importantly, they outperform the SPSS syntax, either for its higher convenience or for its more fruitful results.

"pairwise" - Pairwise comparisons (default is "higher level - lower level")
"seq" or "consec" - Consecutive (sequential) comparisons
"poly" - Polynomial contrasts (linear, quadratic, cubic, quartic, ...)
"eff" - Effect contrasts (vs. the grand mean)

Examples

Run this code

# NOT RUN {
#### Between-Subjects Design ####

between.1
MANOVA(data=between.1, dv="SCORE", between="A") %>%
  EMMEANS("A")
MANOVA(data=between.1, dv="SCORE", between="A") %>%
  EMMEANS("A", p.adjust="tukey")
MANOVA(data=between.1, dv="SCORE", between="A") %>%
  EMMEANS("A", contrast="seq")
MANOVA(data=between.1, dv="SCORE", between="A") %>%
  EMMEANS("A", contrast="poly")

between.2
MANOVA(data=between.2, dv="SCORE", between=c("A", "B")) %>%
  EMMEANS("A") %>%
  EMMEANS("A", by="B") %>%
  EMMEANS("B") %>%
  EMMEANS("B", by="A")

between.3
MANOVA(data=between.3, dv="SCORE", between=c("A", "B", "C")) %>%
  EMMEANS("A", by="B") %>%
  EMMEANS(c("A", "B"), by="C") %>%
  EMMEANS("A", by=c("B", "C"))
## just to name a few
## you can test many other combinations of effects


#### Within-Subjects Design ####

within.1
MANOVA(data=within.1, dvs="A1:A4", dvs.pattern="A(.)",
       within="A") %>%
  EMMEANS("A")

within.2
MANOVA(data=within.2, dvs="A1B1:A2B3", dvs.pattern="A(.)B(.)",
       within=c("A", "B")) %>%
  EMMEANS("A", by="B") %>%
  EMMEANS("B", by="A")  # singular error matrix

within.3
MANOVA(data=within.3, dvs="A1B1C1:A2B2C2", dvs.pattern="A(.)B(.)C(.)",
       within=c("A", "B", "C")) %>%
  EMMEANS("A", by="B") %>%
  EMMEANS(c("A", "B"), by="C") %>%
  EMMEANS("A", by=c("B", "C"))


#### Mixed Design ####

mixed.2_1b1w
MANOVA(data=mixed.2_1b1w, dvs="B1:B3", dvs.pattern="B(.)",
       between="A", within="B", sph.correction="GG") %>%
  EMMEANS("A", by="B") %>%
  EMMEANS("B", by="A")

mixed.3_1b2w
MANOVA(data=mixed.3_1b2w, dvs="B1C1:B2C2", dvs.pattern="B(.)C(.)",
       between="A", within=c("B", "C")) %>%
  EMMEANS("A", by="B") %>%
  EMMEANS(c("A", "B"), by="C") %>%
  EMMEANS("A", by=c("B", "C"))

mixed.3_2b1w
MANOVA(data=mixed.3_2b1w, dvs="B1:B2", dvs.pattern="B(.)",
       between=c("A", "C"), within="B") %>%
  EMMEANS("A", by="B") %>%
  EMMEANS("A", by="C") %>%
  EMMEANS(c("A", "B"), by="C") %>%
  EMMEANS("B", by=c("A", "C"))


#### Other Examples ####
air=airquality
air$Day.1or2=ifelse(air$Day %% 2 == 1, 1, 2) %>%
  factor(levels=1:2, labels=c("odd", "even"))
MANOVA(data=air, dv="Temp", between=c("Month", "Day.1or2"),
       covariate=c("Solar.R", "Wind")) %>%
  EMMEANS("Month", contrast="seq") %>%
  EMMEANS("Month", by="Day.1or2", contrast="poly")
# }