1. Basics of ANOVA Why ANOVA Assumptions used in ANOVA
Various forms of ANOVA
Simple ANOVA tables
Interpretation of values in the table
R commands for ANOVA
Exercises

2. Why ANOVA
If we have two samples then under mild conditions we can use t-test to test if difference between means is significant. When there are more than two sample then using t-test might become unreliable.
ANalysis Of VAriances – ANOVA is designed to test differences between means in many sample cases.
Examples of ANOVA: Suppose that we want to test effect of various exercises on weight loss. We want to test 5 different exercises. We recruit 20 men and assign for each exercises four of them. After few weeks we record weight loss. Let us denote i=1,2,3,4,5 as exercise number and j=1,2,3,4 person’s number. Then Yij is weight loss for jth person on the ith exercise programme. It is one-way balanced ANOVA. One way because we have only one category (exercise programme). Balanced because we have exactly same number of men on each exercise programme.
Another example: Now we want to subdivide each exercises into 4 subcategories. For each subcategory of the exercise we recruit four men. We measure weight loss after few weeks. i – exercise category
j – exercise subcategory
k – kth men.
Then Yijk is weight loss for kth men in the jth subcategory of ith category. Number of observations is 5x4x4 = 80. It is two-fold nested ANOVA.
We want to test: a) There is no significant differences between categories; b) there is no significant difference between different subcategories

3. Examples of ANOVA
One more example: We have 5 categories of exercises and 4 categories of diets. We hire for each exercise and category 4 persons. There will be 5x4x4=80 men. It is two way crossed ANOVA. Two-way because we have categorised men in two ways: exercises and diets. This model is also balanced: we have exactly same number of men for each exercise-diet.
i – exercise number
j – diet number
k – kth person
Yijk – kth person in the ith exercise and jth diet.
In this case we can have two different types of hypothesis testing. Assume that mean for each exercise-diet combination is ij. If we assume that model is additive, i.e. effects of exercise and diet add up then we have: ij = i+j. i is the effect of ith exercise and j is the effect of diet. Then we want to test following hypotheses: a) ij does not depend on exercise and b) ij does not depend on diet.
Sometimes we do not want to assume additivity. Then we want to test one more hypothesis: model is additive. If model is not additive then there might be some problems of interpretations with other hypotheses. In this case it might be useful to use transformation to make the model additive.
Models used for ANOVA can be made more and more complicated. We can design three, four ways crossed models or nested models. We can combine nested and crossed models together. Number of possible ANOVA models is very large.

4. Assumptions
ANOVA models are special cases of the linear models. We can write the model as:
Where Y is the observation vector,  -is vector of the means composed of the treatment means and  is the error vector. Basic assumptions in ANOVA models are:
Expected values of the errors are 0
Variance of all errors are equal to each other
Errors are independent
Errors are normally distributed
All ANOVA treatments are very sensitive to assumptions 1)-3). F-tests meant to be robust against the assumption 4). If assumptions 1)-3) are valid then 4) will always be valid at least asymptotically. I.e. for large number of the observations.

5. ANOVA tables Standard ANOVA tables look like
Where v1,,,vp are values we want to test if they are 0. df is degrees of freedom corresponding to this value. SSh is sum of the squares corresponding to this value (h denotes hypothesis). F is F-value used for F distribution. Its degrees of freedom is (di,de). Prob is corresponding probability. If probability is very low then we reject hypothesis that this value is 0. If the value for prob is small enough then we not reject null-hypothesis.
These values are calculated using likelihood ratio test. Let us say we want to test hypothesis:
H0: vi=0 vs H1:vi0
Then we maximise likelihood under null hypothesis find corresponding variance then we maximise the likelihood under alternative hypothesis and find corresponding variance. Then we calculate sum of the squares for null and alternative hypotheses and find F-statistics
effect df
SSh MS F
prob v1 d1
SS1
MS1=SS1/d1
MS1/MSe
pr1
... … vp dp
SSp
MSp=SSp/dp
MSp/MSe
prp
error de
SSe
MSe=SSe/de
total N
SSt

6. LR test for ANOVA Suppose variances are:
Then mean sum of the squares for the null and alternative hypotheses as:
Since first sum of the squares is 2 with degrees of freedom dfh and the second sum of squares is 2 with degrees of freedom dfe and they are independent then their ratio has F-distribution with degrees of freedom (dfh,dfe). Degrees of freedom of hypothesis is found using number of elements in the category-1 in the simplest case.
Using this type of ANOVA tables we can only tell if there is significant differences between means. It does not tell which one is significantly different.
This ratio has F distribution if null-hypothesis is true. Otherwise it has non-central F-distribution.
Degree of freedom of hypothesis is defined by number of constraints it implies. Degree of freedom of error is as usual number of observations minus number of parameters

7. Example: Two way ANOVA
Let us consider an example taken from Box, Hunter and Hunter. Experiment was done on animals. Survival times of the animals for various poisons and treatment was tested. Table is:
treatment
A B C D
poisons I II
III

8. ANOVA table ANOVA table produced by R:
Df Sum Sq Mean Sq F value Pr(>F)
pois e-07 ***
treat e-06 ***
pois:treat
Residuals
Most important values are F and Pr(>F).
In this table we have tests for pois. and treat. Moreover we have “interaction” between these two categories. Interaction means that it would be difficult to separate effects of these two categories. They should be considered simultaneously. Pr. for interaction is not very small and it is not large enough to discard interaction effects. In these situations transformation of the variables might help. Let us consider ANOVA table for the transformed observations. Let us use transformation 1/y. Now ANOVA table looks like:
Df Sum Sq Mean Sq F value Pr(>F)
pois e-13 ***
treat e-09 ***
pois:treat
Residuals

9. ANOVA table
According to this table Pr. corresponding to the interaction term is high. It means that interaction for the transformed variables is not significant. We could reject interaction terms. We can build the ANOVA table without the interactions. It will look like:
Df Sum Sq Mean Sq F value Pr(>F)
pois e-14 ***
treat e-10 ***
Residuals
Now we can say that there is significant differences between poisons as well as treatments.
Sometimes it is wise to use transformation to reduce effect of interactions. For this several different transformations (inverse, inverse square, log) could be used. For each of them ANOVA tables could be built. Then by inspection you can decide which transformation gives better results. Following argument could be used to justify transformation. If effects of two different categories is multiplicative then log of them will have additive effect. It is easier to interpret additive effects than others.

10. R commands for ANOVA
There are basically two type of commands in R. First is to fit general linear model and second is analyse results.
Command to fit linear model is lm and is used
lm(data~formula)
Formula defines design matrix. See help for formula. For example for PlantGrowth data (available in R) we can use
data(PlantGrowth) load data into R from standard package
lmPlant = lm(PlantGrowth$weight~PlantGrowth$group)
Then linear model will be fitted into data and result will be stored in lmPlant
Now we can analyse them
anova(lmPlant) will give ANOVA table.
If there are more than one factor (category) then for two-way crossed we can use
lm(data~f1*f2) - It will fit complete model with interactions
lm(data~f1+f2) - It will fit only additive model
lm(data~f1+f1:f2) - It will fit f1 and interaction between f1 and f2. It is used for nested models.
Other useful commands for linear model and analysis are
summary(lmPlant) – give summary after fitting
plot(lmPlant) plot several useful plots
Please let me know if any of the results is not clear then we can discuss and try sort out the problems.

11. R commands for ANOVA Another useful command for ANOVA is
confint(lmPlant)
This command gives confidence intervals for some of the coefficients and therefore differences between effects of different factors.
To find confidence intervals between any two given effects one can use bootstrap.

12. Bootstrap for ANOVA Algorithm for bootstrap:
Use lm to fit the model into the data
Resample residuals and add to the fitted values of observations
Use lm to fit model into the data
Save coefficients
Repeat steps 2-4 (for around times)
Build distributions for each coefficients and other statistics of interest
Then for confidence intervals one can use:
lm1 = sort(lmBoot$coefficients[2,])
l=length(lm1)
lmlow = lm1(round(lm1[0.025*l]))
lmhigh = lm1(round(lm1(0.975*l]))
It will give lower and upper limits of 95% confidence intervals
example of implementation is:

13. Exercise 2 Analyse these data using ANOVA
What do you think about the differences.
What do you think about differences?

14. References
Stuart, A., Ord, KJ, Arnold, S (1999) Kendall’s advanced theory of statistics, Volume 2A
Box, GEP, Hunter, WG, Hunter, JS (1978) Statistics for experimenters