1. Independent variables correlate with each other
- a possibility for mediated correlation,
direct and mediated effect;
- shoe size and food consumption – we know;
- body size and stomach volume – we do not know;
- two continuous variables;
- two categorical – unbalanced ANOVA:
black
white
males
20 observ.
2 obs.
females
3 obs.
24 obs.

2. Statistics cannot answer the question which variable has a
direct effect and which has an indirect (mediated) effect,
and to which extent but we can divide the variance into components.
1) the part which can certainly be ascribed to one variable;
2) the part which can certainly be ascribed to the other variable;
3) the part which we do not know how to divide;
the last part is the larger the stronger is the correlation.

3. Height of the plants as dependent on
1) temperature;
2) humidity
Let’s asume that temperature and humidity correlate with each other,
... we are on the southern margin of a desert:
temperature
humidity

4. temperature
plant height
plant height
humidity
Studying one by one there is no doubt that plant height
depends on both humidity and temperature,
but does humidity have such an effect which cannot be
explained by its correlation with temperatuure? (is direct, not
mediated)?
but does temperatuure have such an effect?
.... we include them both to an ANOVA model as
independent variables, but there are several ways... dividing SS

5. Type I analysis or type I sums of squares
all the grey area is assigned to this variable which appears
first in the model, estimates the maximal effect;
conservative with respect to the second variable – estimates what is “certainly its own effect“, estimates the minimal effect.
Humidity as the first variable:
DF Type I SS F P
humidity
temperature
.... we cannot claim that temperature has a direct effect.

6. Temperature as the first variable:
Source DF Type I SS F p
temperature
humidity
We cannot claim a direct (not mediated by temperature) effect of
humidity.
And now type III – conservative with respect to both!
DF Type III SS F P
temperatuur
niiskus

7. In Type I, the order is important, in type III is not!
The danger with type III – joint explanatory power may remain
undetected!
Do it for yourself in diferent ways,
present type III if it reflects the reality!
Otherwise you must explain.
If the variables are not correlated (ANOVA is balanced) then there
is no difference!
and also for a one-way analysis.
Type II and IV also exist.
Avoid when possible, but always you cannot!

8. Covariates in an experiment-
direct and indirect effect
Effect of crowding on moth fecundity,
via body weight, or is there something else?
Taking weight as a covariate!
Including the covariate changes the interpretation!
manipulation:
rearing in groups
pupal
weight
fecundity of
the moths

9. Multiple regression
height = 0,597*temp + 0,089*light + 0,196*humidity
Non-linear regression
y = sin(ax + c(log(x))bx
- which function to choose?
- know from theory?
- what can we conclude?
- properties of the function, not supported by the data.
- OK to describe.
Usually enough to study if it deviates from linearity,
- including the squared term: positive or negative;
- fitting a parabola.

11. Type II regression:
when we want to conclude something from the value of the slope;
type I regression for predicting, not to estimate the „real“ relationship,
to evaluate the relationship;
- different equations for different directions of the prediction;
Equation does not depend on switching the axes!
Geometric mean regression
- geometric mean of slopes both ways;
- does not matter if only the existence of a relationship is of interest!

12. Variations of ANOVA:
- hierarchical (nested);
- random factors;
- repeated measures;

13. Nested (hierarchical) ANOVA
the effect of a factor is „allowed“ to differ at different levels of another
factor; one factor is nested within another:
in brackets B(A):
Does tail length depend on sex?
An usual ANOVA
DF Type III SS F P
sex
species
but now sex nested within species:  
sex
sex(species)
classes in schools: class(school);
.... subpopulations, experimental design, covariates.

14. ANOVA with random factors
a random factor is such an independent variable, the levels of which
can be seen as a sample from a large population of levels.
- brood
For fixed factors, all levels are represended in our sample:
- treatment;
- sex.
We can also say that, for random factors, the error variance is
at two levels:
- the variation of individual observations around brood means;
- the variation of brood means around the grand mean.

18. We study the dependence of the size of oak leaf on temperatuure.
Which are random factors:
- tree individual;
- branch in a tree;
- season (summer etc.);
- grove (forest fragment);
- habitat (forest/ open land);
- year.
Can depend on question asked:
do we study the difference between these populations or do we
want to generalize the results to all populations (of Estonia?)
When is random, can be generalized!

19. Fixed effects ANOVA, random effects ANOVA, mixed ANOVA.
no difference for one-way ANOVA.
A two-way ANOVA (one factor mixed, one random),
difference is modest when there are no interactions;
with interactions the difference can be large!
- 6 broods, 2:4: how in the whole population? Do not know.
the difference is mostly in the effect of the fixed factor!

20. brood fixed: manipulation brood effect Type 3 Tests of Fixed Effects
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
trea
brood <.0001
trea*brood <.0001

21. brood random: manipulation brood effect Type 3 Tests Num Den
Type 3 Tests
  Num Den
Effect DF DF F Value Pr > F
trea
brood
trea*brood <.0001

22. brood fixed: manipulation brood effect Type 3 Tests of Fixed Effects
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
trea
brood <.0001
trea*brood

23. brood random: manipulation brood effect Type 3 Tests Num Den
Type 3 Tests
Num Den
Effect DF DF F Value Pr > F
trea
brood
trea*brood

24. not always there is a right and a wrong way to analyze;
should be a number of levels;
- should group the observations, not the observation itself!

25. weight
population

26. Repeated measures ANOVA
- one individual (or something else) has been measured
several times;
Should not be treated in an usual way :
- overestimate the number of df – pseudoreplications!;
- do not take the individuality into account.
REPEATED measurements ANOVA will help!
1. dependent samples t-test;
2. dependence of the weight of lice on bear fur thickness;
3. birds are fed with different food, parasitism index are counted,
all birds measured four times, time*trea interaction is of interest.