library(statforbiology)
library(emmeans)
library(multcomp)
library(tidyverse)
library(car)
dataset <- getAgroData("mixture") |>
  mutate(Treat = factor(Treat))
lm1 <- lm(Weight ~ Treat, data = dataset)
Anova(lm1)
Anova Table (Type II tests)

Response: Weight
           Sum Sq Df F value    Pr(>F)    
Treat     1089.53  3  23.663 2.509e-05 ***
Residuals  184.18 12                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
summary(lm1)

Call:
lm(formula = Weight ~ Treat, data = dataset)

Residuals:
   Min     1Q Median     3Q    Max 
-6.360 -2.469  0.380  2.567  6.025 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)            9.175      1.959   4.684 0.000529 ***
TreatMixture_378      -4.047      2.770  -1.461 0.169679    
TreatRimsulfuron_30    7.685      2.770   2.774 0.016832 *  
TreatUnweeded         17.598      2.770   6.352 3.65e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.918 on 12 degrees of freedom
Multiple R-squared:  0.8554,    Adjusted R-squared:  0.8193 
F-statistic: 23.66 on 3 and 12 DF,  p-value: 2.509e-05
groupMeans <- emmeans(lm1, ~ Treat)

These two are the same (adjust = “tukey” is the default) – Tukey’s HSD for balanced data and Tukey–Kramer test for unbalanced data:

contrast(groupMeans, method = "pairwise")
 contrast                         estimate   SE df t.ratio p.value
 Metribuzin__348 - Mixture_378        4.05 2.77 12   1.461  0.4885
 Metribuzin__348 - Rimsulfuron_30    -7.68 2.77 12  -2.774  0.0698
 Metribuzin__348 - Unweeded         -17.60 2.77 12  -6.352  0.0002
 Mixture_378 - Rimsulfuron_30       -11.73 2.77 12  -4.235  0.0055
 Mixture_378 - Unweeded             -21.64 2.77 12  -7.813 <0.0001
 Rimsulfuron_30 - Unweeded           -9.91 2.77 12  -3.578  0.0173

P value adjustment: tukey method for comparing a family of 4 estimates 
contrast(groupMeans, method = "pairwise", adjust = "tukey")
 contrast                         estimate   SE df t.ratio p.value
 Metribuzin__348 - Mixture_378        4.05 2.77 12   1.461  0.4885
 Metribuzin__348 - Rimsulfuron_30    -7.68 2.77 12  -2.774  0.0698
 Metribuzin__348 - Unweeded         -17.60 2.77 12  -6.352  0.0002
 Mixture_378 - Rimsulfuron_30       -11.73 2.77 12  -4.235  0.0055
 Mixture_378 - Unweeded             -21.64 2.77 12  -7.813 <0.0001
 Rimsulfuron_30 - Unweeded           -9.91 2.77 12  -3.578  0.0173

P value adjustment: tukey method for comparing a family of 4 estimates

And the same again with multivariate t-distribution

contrast(groupMeans, method = "pairwise", adjust = "mvt")
 contrast                         estimate   SE df t.ratio p.value
 Metribuzin__348 - Mixture_378        4.05 2.77 12   1.461  0.4886
 Metribuzin__348 - Rimsulfuron_30    -7.68 2.77 12  -2.774  0.0697
 Metribuzin__348 - Unweeded         -17.60 2.77 12  -6.352  0.0002
 Mixture_378 - Rimsulfuron_30       -11.73 2.77 12  -4.235  0.0056
 Mixture_378 - Unweeded             -21.64 2.77 12  -7.813 <0.0001
 Rimsulfuron_30 - Unweeded           -9.91 2.77 12  -3.578  0.0172

P value adjustment: mvt method for 6 tests

Very similar results. The mvt method is “not recommended for pairwise comparisons in balanced experiments, whereas it may prove useful in other situations, for example in the presence of strongly unbalanced data.”

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