# Basic principles of probability theory

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```					                            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
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
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
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.
Assumptions
ANOVA models are special cases of the linear models. We can write the model as:
Y με

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:
1.   Expected values of the errors are 0
2.   Variance of all errors are equal to each other
3.   Errors are independent
4.   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
ANOVA tables
Standard ANOVA tables look like

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
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 we want to test. 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 high enough then we do 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
LR test for ANOVA
Suppose variances are:
ˆ
 2 for null hypothesis  2 for thealternative hypothesis
ˆ                      ˆ

Then mean sum of the squares for the null and alternative hypotheses as:
ˆ
 2  2
ˆ    ˆ
SSh               and for the alternative hypothesis
dfh
2
ˆ
SSe 
dfe

Since first sum of 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
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        0.31   0.82        0.43    0.45
0.45   1.10        0.45    0.71
0.46   0.88        0.63    0.66
0.43   0.72        0.76    0.62

II        0.36       0.92    0.44    0.56
0.29       0.61    0.35    1.02
0.40       0.49    0.31    0.71
0.23       1.24    0.40    0.38

III       0.22        0.30    0.23    0.30
0.21       0.37     0.25    0.36
0.18       0.38     0.24    0.31
0.23       0.29     0.22    0.33
ANOVA table
ANOVA table produced by R:
Df Sum Sq           Mean Sq      F value   Pr(>F)
pois        2 1.03828          0.51914      22.5135   4.551e-07 ***
treat      3  0.92569          0.30856      13.3814   5.057e-06 ***
pois:treat 6  0.25580          0.04263       1.8489    0.1170
Residuals 36 0.83013            0.02306

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        2    34.903     17.452        72.2347      2.501e-13 ***
treat      3     20.449      6.816        28.2131      1.457e-09 ***
pois:treat 6      1.579      0.263         1.0892       0.3874
Residuals 36      8.697      0.242
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      2 34.903        17.452           71.326       3.124e-14 ***
treat     3 20.449         6.816           27.858       4.456e-10 ***
Residuals 42 10.276       0.245

Now we can say that there is significant differences between poisons and 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.
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.
Exercise 3.
a)     Analyse these data using ANOVA
http://www.ysbl.york.ac.uk/~garib/mres_course/2004/exercise_3a.html
What do you think about the differences.

b) Analyse these data
http://www.ysbl.york.ac.uk/~garib/mres_course/2004/exercise_3b.html

What do you think about differences?
References

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

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