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ANOVA notes
NR 245
Austin Troy
Based primarily on material accessed from Garson, G. David
2010. Univariate GLM, ANOVA, and ANCOVA. Statnotes:
Topics in Multivariate Analysis.
http://faculty.chass.ncsu.edu/garson/PA765/statnote.htm
Central tendency refresher
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Mean
Median
Variance
Standard Deviation
Variance for
population
For sample
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
Whether difference exists depends
on:
• Size of differences between group
means.
• Sample sizes in each group.
• Variances of dependent variable by
group
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
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.
Full factorial ANOVA
• More than one factor: two-way or higher
• In this approach, each cell becomes a “group”
Source: http://faculty.chass.ncsu.edu/garson/PA765/anova.htm
ANOVA assumptions
• Interval data.
– nonparametric Kruskal-Wallace
• Homogeneity of variances.
– Box plots
• Multivariate normality.
ANOVA assumption
• Adequate sample size.
• Random sampling
• Equal or similar sample sizes.
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: http://faculty.chass.ncsu.edu/garson/PA765/anova.htm
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
Pairwise comparisons
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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.
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
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. 0.05 or less
considered evidence of a
•Also get Mean, Std Error (in this case is the root
regression effect.
mean square error divided by square root of the
number of values used to compute the group
mean. and confidence intervals for each group
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
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