Download Lecture 8-2 - Notes - for Dr. Jason P. Turner

Analysis of Variance
(ANOVA) II
MARE 250
Dr. Jason Turner
Assumptions for One-Way ANOVA
Normality and Equal variance is more difficult to test
with multiple populations
Another way to assess:
Residual – the difference between the observation and
the mean of the sample containing it
IF Normality and equal variances assumptions are met
THEN normal probability plot should be roughly linear
THEN residuals plot should be centered and symmetric
about the x-axis
Assumptions for One-Way ANOVA
A. Residuals centered and symmetric about the x-axis
normally distributed, equal variances
B. Residuals curved
data not normal
C. Residuals cone shaped
variances not equal
Assumptions for One-Way ANOVA
Four-in-one Plot:
Probability plot, Residuals versus fitted
Histogram, Residuals versus order
Residual Plots for Otis
Normal Probability Plot of the Residuals
Residuals Versus the Fitted Values
99.9
30
Are residuals centered
and symmetric?
15
90
Residual
Percent
99
50
10
0
-15
1
0.1
-40
-20
0
Residual
20
-30
40
Histogram of the Residuals
80
90
100
Fitted Value
110
Residuals Versus the Order of the Data
30
Are residuals
distributed in a random
pattern?
15
15
Residual
Frequency
20
10
5
0
-22.5 -15.0
-7.5 0.0
7.5
Residual
15.0
120
22.5
0
-15
-30
1
20
40
60 80 100 120 140 160
Observation Order
Non-Parametric Version of ANOVA
Kruskal-Wallis
If samples are independent, similarly distributed data
Use nonparamentric test regardless of normality or
sample size
Is based upon median of ranks of the data – not the
mean or variance (Like Mann-Whitney)
If the variation in mean ranks is large – reject null
Uses p-value like ANOVA
Last Resort/Not Resort –low sample size, “bad” data
Non-Parametric Version of ANOVA
Kruskal-Wallis Test: _ Urchins versus Distance
Kruskal-Wallis Test on _ Urchins
Distance
Deep
Middle
Shallow
Overall
N
50
75
150
275
Median
Ave Rank
3.000000000 208.2
0.000000000 153.2
0.000000000 107.0
138.0
H = 64.49 DF = 2 P = 0.000
H = 103.96 DF = 2 P = 0.000 (adjusted for ties)
Z
6.90
1.94
-7.08
When Do I Do the What Now?
“Well, whenever I'm confused, I just check my underwear. It holds the answer
to all the important questions.” – Grandpa Simpson
If you are reasonably sure that the distributions
are normal –use ANOVA
Otherwise – use Kruskal-Wallis
1. Test all samples for
normality
Data Not
normal
Use Kruskal-Wallis test
Data
normal
2. Test samples for equal
variance (Bartlett’s test)
Variances
equal
Use single factor ANOVA
Variances
not equal
Use Kruskal-Wallis test