Assumptions of the ANOVA
 The error terms are randomly, independently, and
normally distributed, with a mean of zero and a common
variance.
– There should be no systematic patterns among the residuals
– The distribution of residuals should be symmetric (not skewed)
– there should be no relationship between the size of the error
variance and the mean for different treatments or blocks
– The error variances for different treatment levels or different
blocks should be homogeneous (of similar magnitude)
 The main effects are additive
– the magnitude of differences among treatments in one block
should be similar in all other blocks
– i.e., there is no interaction between treatments and blocks
If the ANOVA assumptions are violated:
 Affects sensitivity of the F test
 Significance level of mean comparisons may
be much different than they appear to be
 Can lead to invalid conclusions
Diagnostics
 Use descriptive statistics to test assumptions
before you analyze the data
– Means, medians and quartiles for each group
(histograms, box plots)
– Tests for normality, additivity
– Compare variances for each group
 Examine residuals after fitting the model in your
analysis
–
–
–
–
Descriptive statistics of residuals
Normal plot of residuals
Plots of residuals in order of observation
Relationship between residuals and predicted values
(fitted values)
SAS Box Plots
Look For
Mean
Outlier (>1.5*IQR)
 Outliers
 Skewness
 Common
Variance
Caution
Median
IQR
Quartile (25%)
Min
 Not many
observations
per group
IQR = interquartile range (25% - 75%)
Additivity
 Linear additive model for each experimental design

Yij =  + i + ij
CRD
Yij =  + i + j + ij
RBD
Implies that a treatment effect is the same for all
blocks and that the block effect is the same for all
treatments
When the assumption would not be correct...
Two nitrogen treatments applied to 3 blocks
1
2
3
Water Table
Differences between treatments might be greater in block 3
 When there is an interaction between blocks and
treatments - the model is no longer additive
– may be multiplicative; for example, when one treatment
always exceeds another by a certain percentage
SAS interaction plot
Testing Additivity --- Tukey’s test
 Test is applicable to any two-way classification such
as RBD classified by blocks and treatments
 Compute a table with raw data, treatment means,
treatment effects ( Y. j  Y.. ), block means and block
effects (Yi.  Y.. )
Q   Yij  Yi.  Y..   Y. j  Y.. 
 Compute SS for nonadditivity =
(Q2*N)/(SST*SSB)
N = t*r
with 1 df
 The error term is partitioned into nonadditivity and
residual and can be tested with F
Test can also be done with SAS
Residuals
 Residuals are the error terms – what is left over after
accounting for all of the effects in the model
Yij =  + i + ij
CRD
eij  Yij  Y..  Ti
Yij =  + i + j + ij
RBD
eij  Yij  Y..  Bi  Tj
Independence
 Independence implies that the error (residual) for
one observation is unrelated to the error for
another
– Adjacent plots are more similar than randomly
scattered plots
– So the best insurance is randomization
– In some cases it may be better to throw out a
randomization that could lead to biased estimates of
treatment effects
– Observations in a time series may be correlated (and
randomization may not be possible)
Normality
 Look at stem leaf plots, boxplots of residuals
 Normal probability plots
 Minor deviations from normality are not
generally a problem for the ANOVA
Normality
Normal Probability Plot from Original Data
350
300
250
Residuals
200
150
100
50
0
-3
-2
-1
0
1
-50
-100
-150
Quantiles of standard normal
2
3
Normality
Normal Probability Plot from the same Data
After Transformation
3
2
Residuals
1
0
-3
-2
-1
0
1
-1
-2
-3
-4
Quantiles of standard normal
2
3
Homogeneity of Variances
Replicates
Treatment
1
2
3
4
5 Total Mean
s2
A
3
1
5
4
2
15
3
2.5
B
6
8
7
4
5
30
6
2.5
C
12
6
9
3 15
45
9
22.5
D
20 14 11 17
70
14
22.5
8
 Logic would tell us that differences required for
significance would be greater for the two highly
variable treatments
If we analyzed together:
Source
df
SS
MS
F
Treatments 3
330
110
8.8**
Error
200
12.5
16
LSD=4.74
Analysis
for A and B
Conclusions would
be different if we
analyzed the two
groups separately:
Analysis
for C and D
Source
df
SS
MS
F
Treatments 1
22.5
22.5
9*
Error
8
20.0
2.5
Source
df
SS
MS
F
Treatments 1
62.5
62.5
2.78
Error
180
22.5
8
Relationships of Means and Variances
 Most common cause of heterogeneity of variance
Test the effect of a new
vitamin on the weights
of animals.
What you see
What the ANOVA assumes
Examining the error terms
 Take each observation and remove the general mean, the
treatment effects and the block effects; what is left will be
the error term for that observation
The model =
Yij  Y..  i   j  eij
Block effect =
i  Yi.  Y..
 j  Y.j  Y..
Treatment effect =

 

so ...
Yij  Y..  Yi.  Y..  Y. j  Y..  eij
then ...
Yij  Y i.  Y. j  Y..  eij
Finally ...
eij  Yij  Y i.  Y. j  Y..
Looking at the error components
Trt.
A
B
C
D
Mean
I
47
50
57
54
52
II
52
54
53
65
56
e11 = 47 – 52 – 53 + 58 = 0
III
62
67
69
74
68
IV
51
57
57
59
56
Trt.
I
A
0
B
-1
C
4
D
-3
Mean 0
Mean
53
57
59
63
58
II
1
-1
-4
4
0
III
-1
0
0
1
0
IV
0
2
0
-2
0
Mean
0
0
0
0
0
Looking at the error components
Trt.
A
B
C
D
E
F
Mean
I
.18
.32
2.0
2.5
108
127
40
II
.30
.4
3.0
3.3
140
153
50
III
.28
.42
1.8
2.5
135
148
48
e11 = 0.18 – 40 - 0.3 + 49 = 8.88
IV
Mean
.44
0.3
.46
0.4
2.8
2.4
3.3
2.9
165
137
176
151
58
49
Trt.
I
II
A
8.88 -1.00
B
8.92 -1.00
C
8.60 -0.40
D
8.60 -0.60
E -20.00
2.00
F -15.00
1.00
III
0.98
1.02
0.40
0.60
-1.00
2.00
IV
-8.86
-8.94
-8.60
-8.60
19.00
16.00
Predicted values
Remember
Yij  Y..  i   j  eij
 Y i.  Y. j  Y..  eij
Predicted value Ŷij  Y..  i   j
 Yi.  Y. j  Y..
Plots of
eij vs Ŷij should be random
Plots of
e ij vs Yij will be autocorrelated
Residual Plots
 A valuable tool for examining the validity of assumptions
for ANOVA – should see a random scattering of points on
the plot
 For simple
models, there
may be a limited
number of
groups on the
Predicted axis
 Look for random
dispersion of
residuals above
and below zero
Residual Plots – Outlier Detection
 Recheck data input and correct obvious errors
 If an outlier is suspected, could look at studentized
residuals (ij)
For a CRD
Residuals (eij)
 ri  1 
seij  MSE 

r
 i 
ij 
eij
Outliers
seij
 Treat as a missing
plot if too extreme
(e.g. ij > 3 or 4)
Predicted Values
Visual Scores
 Values are discrete
– Do not follow a normal distribution
– Range of possible values is limited
 Alternatives?
Residual Plot of Stand Ratings
Frequency
Plant Stand
Ratings
20
10
0
Predicted Values
3 4 5 6 7 8
Score
Are the errors randomly distributed?
 Residuals are not randomly distributed around zero
– they follow a pattern
 Model may not be adequate
– e.g., fitting a straight regression line when response is curvilinear
Model not adequate
1.5
Residuals
1
0.5
0
-0.5
0
5
10
15
-1
-1.5
Predicted values
20
25
Are variances homogeneous?
 In this example the variance of the errors increases with
the mean (note fan shape)
 Cannot assume a common variance for all treatments
160
Residuals
120
80
40
0
-40 0
20
40
60
80
100
-80
-120
-160
Predicted values
120
140
160
Homogeneity Quick Test (F Max Test)
 By examining the ratio of the largest variance to
the smallest and comparing with a probability
table of ratios, you can get a quick test.
 The null hypothesis is that variances are equal,
so if your computed ratio is greater than the
table value (Kuehl, Table VIII), you reject the
null hypothesis.
2
s (max)
2
s (min)
Where t = number of independent
variances (mean squares) that you are
comparing
v = degrees of freedom associated with
each mean square
An Example
Treatment
N
P
N+P
S
N+S
P+S
N+P+S
Variance
19.54
1492.27
98.21
5437.99
9.03
496.58
22.94
An RBD experiment with four blocks to
determine the effect of salinity on the
application of N and P on sorghum
5437.99/9.03 = 602.21
Table value (t=7, v=r-1=3) = 72.9
602.21>72.9
Reject null hypothesis and
conclude that variances are NOT
homogeneous (equal)
Other HOV tests are more sensitive
 If the quick test indicates that variances are not equal
(homogeneous), no need to test further
 But if quick test indicates that variances ARE
homogeneous, you may want to go further with a Levene
(Med) test or Bartlett’s test which are more sensitive.
 This is especially true for values of t and v that are
relatively small.
 F max, Levene (Med), and Bartlett’s tests can be adapted
to evaluate homogeneity of error variances from different
sites in multilocational trials.
Homogeneity of Variances - Tests
 Johnson (1981) compared 56 tests for
homogeneity of variance and found the Levene
(Med) test to be one of the best.
– Based on deviations of observations from the median
for each treatment group. Test statistic is compared to
a critical F,t-1,N-t value.
– This is now the default homogeneity of variance test in
SAS (HOVTEST).
 Bartlett’s test is also common
– Based on a chi-square test with t-1 df
– If calculated value is greater than tabular value, then
variances are heterogeneous
What to do if assumptions are violated?
 Divide your experiment into subsets of blocks or
treatments that meet the assumptions and conduct
separate analyses
 Transform the data and repeat the analysis
– residuals follow another distribution (e.g., binomial, Poisson)
– there is a specific relationship between means and variances
– residuals of transformed data must meet the ANOVA assumptions
 Use a nonparametric test
– no assumptions are made about the distribution of the residuals
– most are based on ranks – some information is lost
– generally less powerful than parametric tests
 Use a Generalized Linear Model (PROC GLIMMIX in SAS)
– make the model fit the data, rather than changing the data to fit the
model
Relationships between means and variances...
 Can usually tell just by looking. Do the variances
increase as the means increase?
 If so, construct a table of ratios of variance to
means and standard deviation to means
 Determine which is more nearly proportional - the
ratio that remains more constant will be the one
more nearly proportional
 This information is necessary to know which
transformation to use – the idea is to convert a
known probability distribution to a normal
distribution
Comparing Ratios - Which Transformation?
Trt
Mean
M-C
0.3
M-V
0.4
C-C
2.4
C-V
2.9
S-C 137.0
S-V 151.0
Var
SDev
0.01147
0.107
0.00347
0.059
0.3467
0.589
0.2133
0.462
546.0
23.367
425.3
20.624
Var/M SDev/M
0.04
0.01
0.14
0.07
3.98
2.82
0.36
0.15
0.24
0.16
0.17
0.14
SDev roughly
proportional to the
means
The Log Transformation
 When the standard deviations (not the variances) of
samples are roughly proportional to the means, the log
transformation is most effective
 Common for counts that vary across a wide range of
values
– numbers of insects
– number of diseased plants/plot
 Also applicable if there is evidence of multiplicative rather
than additive main effects
– e.g., an insecticide reduces numbers of insects by 50%
– e.g., early growth of seedlings may be proportional to current size
of plants
General remarks...
 Data with negative values cannot be
transformed with logs
 Zeros present a special problem
 If negative values or zeros are present, add 1 to
all data points before transforming
 You can multiply all data points by a constant
without violating any rules
 Do this if any of the data points are less than 1
(to avoid negative logs)
Recheck...
 After transformation, rerun the ANOVA on the
transformed data
 Recheck the transformed data against the
assumptions for the ANOVA
– Look at residual plots, normal plots
– Carry out Levene’s test or Bartlett’s for homogeneity of variance
– Apply Tukey’s test for additivity
 Beware that a transformation that corrects one
violation in assumptions may introduce another
Square Root Transformation
 One of a family of power transformations
 The variance tends to be proportional to the mean
– e.g., if leaf length is normally distributed, then leaf area may
require a square root transformation
 Use when you have counts of rare events in time or space
– number of insects caught in a trap
 May follow a Poisson distribution (for discrete variables)
 If there are counts under 10, it is best to use square root
of Y + 0.5
 Will be easier to declare significant differences in mean
separation
 When reporting, “detransform” the means – present
summary mean tables on original scale
Arcsin or Angular Transformation
arcsin Yij
 Counts expressed as percentages or proportions of
the total sample may require transformation
 Follow a binomial distribution - variances tend to be
small at both ends of the range of values ( close to
0 and 100%)
 Not all percentage data are binomial in nature
– e.g., grain protein is a continuous, quantitative variable
that would tend to follow a normal distribution
 If appropriate, it usually helps in mean separation
Arcsin or Angular Transformation
arcsin Yij
 Data should be transformed if the range of percentages is
greater than 40
 May not be necessary for percentages in the range of 30-
70%
 If percentages are in the range of 0-30% or 70-100%, a
square root transformation may be better
 Do not include treatments that are fixed at 0% or at 100%
 Percentages are converted to an angle expressed in
degrees or in radians
– express Yij as a decimal fraction – gives results in radians
– 1 radian = 57.296 degrees
Summary of Transformations
Type of data
Quantitative variable
where treatment
effects are
proportional
Positive integers that
cover a wide range of
values
Counts of rare events
Issue
Natural scale is not
normal (often rightskewed)
Lognormal distribution
Standard deviation
proportional to the mean
and/or nonadditivity
Variance = mean
Poisson distribution
Transformation
Log
Percentages, wide
range of values
including extremes
Variances are smaller
near zero and 100
Binomial distribution
ArcSin
Log or log(Y+1)
Square root or
sqrt(Y+0.5)
Reasons for Transformation
 We don’t use transformation just to give us
results more to our liking
 We transform data so that the analysis will be
valid and the conclusions correct
 Remember ....
– all tests of significance and mean separation should
be carried out on the transformed data
– calculate means of the transformed data before
“detransforming”