The descriptive statistics can be used to determine the inferential statistics.
> Results <- aov(Outcome ~ Factor)
> model.tables(Results, "means")
Tables of means
Grand mean
5
Factor
Factor
1 2 3
2 6 7
> tapply(Outcome, Factor, function(x) c(length(x), mean(x), sd(x)))
$`1`
[1] 4.00000 2.00000 2.44949
$`2`
[1] 4.00000 6.00000 2.44949
$`3`
[1] 4.00000 7.00000 2.44949
The values of the group statistics are calculated separately for each group or condition. They are not identical to the values obtained from analyzing the variable as a whole.
The inferential statistics show the key elements to be calculated.
> TukeyHSD(Results)
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = Outcome ~ Factor)
$Factor
diff lwr upr p adj
2-1 4 -0.8358956 8.835896 0.1055254
3-1 5 0.1641044 9.835896 0.0431300
3-2 1 -3.8358956 5.835896 0.8352889
For the 1 vs. 2 comparison, HSD = 2.309 and p = .106. This would not be considered a statistically significant diffeence.
For the 1 vs. 3 comparison, HSD = 2.887 and p = .043. This would be considered a statistically significant difference.
For the 2 vs. 3 comparison, HSD = 0.577 and p = .835. This would not be considered a statistically significant difference.
For each comparison, the researcher estimates the true population mean difference adjusted for multiple comparisons (knowing that the estimates could be incorrect).
For the 1 vs. 2 comparison, the 95% CI is [-0.836, 8.836].
For the 1 vs. 3 comparison, the 95% CI is [0.164, 9.836].
For the 2 vs. 3 comparison, the 95% CI is [-3.836, 5.836].
Cohen’s d for each comparison can be calculated but is not shown here.
Post hoc tests build on the ANOVA results and provide a more focused comparison among the groups and usually follows a presentation of the ANOVA (which already includes the descriptive information).
A series of Tukey’s HSD comparisons revealed that the first group (n = 3, M = 2.00, SD = 2.45) scored substantially and significantly lower Outcome scores than the third group (n = 3, M = 7.00, SD = 2.45), 95% CI [0.16, 9.84], t(9) = 2.89, p = 043. However, the other comparisons revealed effectively little and not significant differences between the other groups (ps > .05).
Alternatively, the means, standard deviations, and confidence intervals could be presented in a table or figure associated with this text.