Results

ANOVA

First let’s check that the predictor variable (drink) and the covariate (drunk) are independent. To do this we can run a one-way ANOVA. The output shows that the main effect of drink is not significant, F(2, 12) = 1.36, p = 0.295, which shows that the average level of drunkenness the night before was roughly the same in the three drink groups. This result is good news for using this model to adjust for the variable drunk.

ANOVA - drunk
Cases Sum of Squares df Mean Square F p
drink 8.400 2 4.200 1.355 0.295
Residuals 37.200 12 3.100  
Note.  Type III Sum of Squares

ANCOVA

The output below shows that the covariate significantly predicts the outcome variable, so the drunkenness of the person influenced how well they felt the next day F(1,11) = 27.89, p < .001. What’s more interesting is that after adjusting for the effect of drunkenness, the effect of drink is significant F(2, 11) = 4.318, p = 0.041.

ANCOVA - well
95% CI for ω²ₚ
Cases Sum of Squares df Mean Square F p ω²ₚ Lower Upper
drink 3.464 2 1.732 4.318 0.041 0.307 0.000 0.613
drunk 11.187 1 11.187 27.886 < .001 0.642 0.209 0.820
Residuals 4.413 11 0.401  
Note.  Type III Sum of Squares

Descriptives

The group means should be used for interpretation. The means below show that the significant difference between the water and the Lucozade groups reflects people feeling better in the Lucozade group (than the water group).

Descriptives - well
drink N Mean SD SE Coefficient of variation
Water 5 5.000 1.225 0.548 0.245
Lucozade 5 5.800 1.483 0.663 0.256
Cola 5 5.800 0.447 0.200 0.077

Descriptives plots

To interpret the covariate we can create a Descriptives plot of the outcome (well, y-axis) against the covariate ( drunk, x-axis). The resulting plot below shows that there is a negative relationship between the two variables: that is, high scores on one variable correspond to low scores on the other, whereas low scores on one variable correspond to high scores on the other. The more drunk you got, the less well you felt the following day.

drunk - well

Post Hoc Tests

In order to investigate the pairwise group differences while taking into account the covariate, we can conduct a post hoc analysis (since that's based on the marginal means). This analysis makes pairwise comparisons between each adjusted group mean while correcting the p-values for multiple comparisons. Here, we can see that the only significant difference is between Lucozade and Water. The comparison is Water (1st column) vs. Lucozade (2nd column) and is negative, which indicates that Lucozade had the higher group mean. In human words, that means that people felt better in the Lucozade group than the Water group, after accounting for their drunkenness the night before.


The low sample size is a tad worrisome here though, and is reflected in the very wide confidence intervals we obtain - these data are probably not informative enough to confidently conclude one drink is better than the other in curing hangovers.

Standard (HSD)

Post Hoc Comparisons - drink
95% CI for Mean Difference 95% CI for Cohen's d
Mean Difference Lower Upper SE df t Cohen's d Lower Upper ptukey
Water Lucozade -1.129 -2.224 -0.034 0.405 11 -2.785 -1.783 -3.882 0.317 0.043
  Cola -0.142 -1.275 0.991 0.420 11 -0.338 -0.224 -2.097 1.649 0.939
Lucozade Cola 0.987 -0.207 2.181 0.442 11 2.233 1.558 -0.621 3.738 0.109
Note.  P-value and confidence intervals adjusted for comparing a family of 3 estimates (ci for mean difference corrected using the tukey method; ci for effect size corrected using the bonferroni method).