1. 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

2. 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

3. 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)

4. SAS Box Plots Look For Caution Outliers Skewness Common Variance
Mean
Median
Outlier (>1.5*IQR)
Quartile (25%)
Min
IQR
Look For
Outliers
Skewness
Common Variance
IQR = interquartile range (25% - 75%)
Caution
Not many observations per group

5. Additivity Yij =  + i + ij CRD Yij =  + i + j + ij RBD
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

6. When the assumption would not be correct...
Two nitrogen treatments applied to 3 blocks
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

7. SAS interaction plot

8. 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 ( ), block means and block effects ( )
Not responsible for the mechanics of Tukey’s test or Bartlett’s test – just for illustration. Know when to use them.
Compute SS for nonadditivity =
(Q2*N)/(SST*SSB) with 1 df
The error term is partitioned into nonadditivity and residual and can be tested with F
N = t*r
Test can also be done with SAS

9. Residuals Yij =  + i + ij CRD Yij =  + i + ij CRD
Residuals are the error terms – what is left over after accounting for all of the effects in the model
Yij =  + i + ij CRD
Yij =  + i + ij CRD
Yij =  + i + j + ij RBD

10. 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)

11. Normality Look at stem leaf plots, boxplots of residuals
Normal probability plots
Minor deviations from normality are not generally a problem for the ANOVA

12. Normality
Data should fall on a straight line. Quantile = proportion of data below that point.

13. Normality
Moderate departures from normality are not serious. F tests and t tests are robust in this regard.

14. Homogeneity of Variances
Replicates
Treatment Total Mean s2 A B C D
Logic would tell us that differences required for significance would be greater for the two highly variable treatments

15. If we analyzed together:
Source df SS MS F
Treatments **
Error
LSD=4.74
Analysis for A and B
Source df SS MS F
Treatments *
Error
Conclusions would be different if we analyzed the two groups separately:
Source df SS MS F
Treatments
Error
Analysis for C and D

16. Relationships of Means and Variances
Most common cause of heterogeneity of variance
What the ANOVA assumes
Test the effect of a new vitamin on the weights of animals.
What you see

17. 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 =
Block effect =
Treatment effect =
so ...
then ...
Finally ...

18. Looking at the error components
Trt. I II III IV Mean A B C D
Mean
There may be differences among the means for each treatment, or among the means for each block, but the residuals after removing all of the other effects in the model are of a similar magnitude for all groups.
Trt. I II III IV Mean A B C D
Mean
e11 = 47 – 52 – = 0

19. Looking at the error components
Trt. I II III IV Mean A B C D E F
Mean
In this case some treatment groups, and some block groups, have a larger range of residuals.
Trt. I II III IV A B C D E F
e11 = 0.18 – = 8.88

20. Predicted values

21. 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

22. Residual Plots – Outlier Detection
Recheck data input and correct obvious errors
If an outlier is suspected, could look at studentized residuals (ij)
Residuals (eij)
For a CRD
Treat as a missing plot if too extreme (e.g. ij > 3 or 4)
Outliers
Predicted Values

23. Residual Plot of Stand Ratings
Visual Scores
Values are discrete
Do not follow a normal distribution
Range of possible values is limited
Alternatives?
Residual Plot of Stand Ratings
Could get several people to make a rating and take the average for each plot.
Find a more quantitative and objective measure of stand establishment.
Use a nonparametric test.
Predicted Values

24. 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
Errors are not randomly distributed above and below zero. This situation could result if you tried to fit a straight line regression when the response was really curvilinear. The line would overpredict Y at small and large values of X, and underpredict the response at X values in the middle of the range.

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
shows relationship between means and variances

26. 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.
Called F max test in Kuehl – Table VIII
Can also be done with an F test, but use /2, and df associated with each mean square
Where t = number of independent variances (mean squares) that you are comparing
v = degrees of freedom associated with each mean square

27. An Example
An RBD experiment with four blocks to determine the effect of salinity on the application of N and P on sorghum
/9.03 =
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)

28. 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.

29. 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

30. 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

31. 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

32. Comparing Ratios - Which Transformation?
Trt Mean Var SDev Var/M SDev/M
M-C
M-V
C-C
C-V
S-C
S-V
SDev roughly proportional to the means

33. 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

34. 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)

35. 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

36. 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

37. Arcsin or Angular Transformation
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
Yij should be expressed as a decimal fraction – gives results in radians

38. Arcsin or Angular Transformation
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 %, 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 = degrees
Yij should be expressed as a decimal fraction – gives results in radians

39. Summary of Transformations

40. 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”