Testing pairwise differences
In the previous tutorial, we computed marginal means at the 3 different Species levels from the iris dataset. However, one might also want to statistically test the differences between each levels, which can be achieved through contrast analysis. Although the procedure is much more powerful, its aim is analogous to the post hoc analysis (pretty much consisting of pairwise t-tests), which are heavily utilized in behavioral sciences as a way to follow up on hypotheses about global differences tested by ANOVAs with more specific hypotheses about pairwise differences.
Let’s carry out contrast analysis on the simple model from the previous tutorial:
library(ggplot2) library(modelbased) data(iris) model <- lm(Sepal.Width ~ Species, data = iris) means <- estimate_means(model, by = "Species") plot(means, point = list(width = 0.1)) + theme_minimal()
Contrast analysis can be achieved through the estimate_contrasts function:
estimate_contrasts(model, contrast = "Species")> Marginal Contrasts Analysis > > Level1 | Level2 | Difference | SE | 95% CI | t(147) | p > ------------------------------------------------------------------------------ > versicolor | setosa | -0.66 | 0.07 | [-0.79, -0.52] | -9.69 | < .001 > virginica | setosa | -0.45 | 0.07 | [-0.59, -0.32] | -6.68 | < .001 > virginica | versicolor | 0.20 | 0.07 | [ 0.07, 0.34] | 3.00 | 0.003 > > Variable predicted: Sepal.Width > Predictors contrasted: Species > p-values are uncorrected.We can conclude that all pairwise differences are statistically significant.
Complex model
Again, as contrast analysis is based on marginal means, it can be applied to more complex models:
model <- lm(Sepal.Width ~ Species * Petal.Width, data = iris) contrasts <- estimate_contrasts(model, contrast = "Species") contrasts> Marginal Contrasts Analysis > > Level1 | Level2 | Difference | SE | 95% CI | t(144) | p > ------------------------------------------------------------------------------ > versicolor | setosa | -1.59 | 0.39 | [-2.37, -0.81] | -4.04 | < .001 > virginica | setosa | -1.77 | 0.41 | [-2.59, -0.96] | -4.29 | < .001 > virginica | versicolor | -0.18 | 0.15 | [-0.47, 0.10] | -1.27 | 0.205 > > Variable predicted: Sepal.Width > Predictors contrasted: Species > Predictors averaged: Petal.Width (1.2) > p-values are uncorrected.For instance, if we add Petal.Width in the model, we can see that the difference between versicolor and virginica becomes not significant (and even changes sign).
Note that we can plot simple contrast analysis through lighthouse plots:
plot(contrasts, estimate_means(model, by = "Species")) + theme_minimal()
These represent the estimated means and their CI range (in black), while the grey areas show the CI range of the difference (as compared to the point estimate). One easy way to interpret lighthouse plots is that if the whole beam goes up or down (i.e., the upper limit and the lower limit are of the same direction), the difference is likely significant.
Changes in difference
Interestingly, we can also see how these differences are modulated by another continuous variable. Based on the model above (including the interaction with Petal.Width), we will compute the contrasts at 100 equally-spaced points of Petal.Width, that we will then visualise.
contrasts <- estimate_contrasts( model, contrast = "Species", by = "Petal.Width", length = 100, # we use a emmeans here because marginaleffects doesn't # generate more than 25 rows for pairwise comparisons backend = "emmeans" ) # Create a variable with the two levels concatenated contrasts$Contrast <- paste(contrasts$Level1, "-", contrasts$Level2) # Visualise the changes in the differences ggplot(contrasts, aes(x = Petal.Width, y = Difference)) + geom_ribbon(aes(fill = Contrast, ymin = CI_low, ymax = CI_high), alpha = 0.2) + geom_line(aes(colour = Contrast), linewidth = 1) + geom_hline(yintercept = 0, linetype = "dashed") + theme_minimal() + ylab("Difference")
As we can see, the difference between versicolor and virginica increases as Petal.Width increases.