1. ANOVA notes NR 245 Austin Troy
Based primarily on material accessed from Garson, G. David Univariate GLM, ANOVA, and ANCOVA. Statnotes: Topics in Multivariate Analysis.

2. Central tendency refresher
Mean
Median
Variance
Standard Deviation
Variance for population
For sample

3. ANOVA
“main effect” vs interactive effects of categorical independent variables (factors) on a continuous/interval dependent
Tests for overall differences in means, not variances
Pairwise comparisons

4. Whether difference exists depends on:
Size of differences between group means.
Sample sizes in each group.
Variances of dependent variable by group

5. Fixed vs. random effects ANOVA
Fixed: data are collected on all categories of independent variables.
Factors with all category values included are "fixed factors."
Random effect ("Model II"), data collected only for a partial sample of categories.
One-way ANOVA: computation of F is the same for fixed and random effects
Mixed effects models with both types

6. Research design: between groups
Dependent variable is measured for independent groups of sample members, where groups represent different condition, or categories.
Experimental mode: conditions assigned randomly to subjects, or subjects assigned randomly to conditions
Non-experimental mode, conditions are measures of the independent variable for each group.

7. Full factorial ANOVA More than one factor: two-way or higher
In this approach, each cell becomes a “group”
Source:

8. ANOVA assumptions Interval data. Homogeneity of variances.
nonparametric Kruskal-Wallace
Homogeneity of variances.
Box plots
Multivariate normality.

9. ANOVA assumption Adequate sample size. Random sampling
Equal or similar sample sizes.

10. Interpretting 2+way ANOVA
Profile plots
Color= 2nd factor (e.g. gender)
Parallel lines= lack of interaction effects
Separate lines = different means based on gender
Triangle or X= different groups means based on region
Bottom row: both effects plus interaction
Source:

11. F test for comparing group means
For most designs, F is between-groups mean square variance divided by within-groups MSV
If F >1, then there is more variation between groups than within groups, = the grouping variable does make a difference.
Significance of F stat: using df=k-1 (between group; df for numerator) and df=N-k-1 (within group; df for denominator)
Larger the ratio of between-groups variance (signal) to within-groups variance (noise), the less likely that the null hypothesis is true=more variation between groups than within groups

12. Pairwise comparisons Assess group differences
The possible number of comparisons is k(k-1)/2.
For two comparisons use standard t test
For more comparisons:
Bonferroni test: like t-test but adjust the significance level by multiplying by the number of comparisons being made.
Tukey’s HSD test: like a t-test, but corrects for experiment-wise error rate; gives conservative results when group sizes are unequal; good with large number of categories.

13. ANOVA outputs Example: pain level by analgesic drug group
Rsquare is model of goodness of fit and is often referred to as “partial eta squared” for ANOVA= ratio of the between-groups sum of squares to the total sum of squares
Adjusted R-square adjusts the Rsquare value to penalize for more parameters by using the degrees of freedom in its computation R-adj= 1 - SSE(n-1)/SST(v)
Root MSE gives the standard deviation of the random error
Mean of response gives sample mean

14. ANOVA outputs This section gives F test results DF= degrees of freedom
Sum of Squares C. Total= sum of squared distances of each response from the sample mean; Error is the residual or unexplained SS after fitting the model
Mean square is a sum of squares divided by its associated df
F score is ratio of the Model mean square to the error mean square
Prob>F is p value; observed significance probability of obtaining a greater F-value by chance alone or less considered evidence of a regression effect.
Also get Mean, Std Error (in this case is the root mean square error divided by square root of the number of values used to compute the group mean. and confidence intervals for each group

15. ANOVA outputs-comparisons
Top: LSD Threshold matrix: absolute difference in the means minus the LSD, which is the minimum difference that would be significant. Pairs with a positive value are significantly different. The q statistic is a scaling variable
Next table: significant differences based on the letters that apply to them (A, B)
Final table gives all pairwise comparisons. Gives differences, confidence intervals and p value. Those with significant differences are starred
Also gives diamond and circle plots. If two circles overlap significantly, those groups are not different

16. Box plot
Outliers are either 3×inter-quartile range (the width of the box in the box-and-whisker plot) or more above the third quartile or 3×IQR or more below the first quartile