Download Random effects - Lorenzo Marini

Introduction to Biostatistical Analysis Using R
Statistics course for PhD students in Veterinary Sciences
Session 3
Lecture: Analysis of Variance (ANOVA)
Practical: ANOVA
Lecturer: Lorenzo Marini, PhD
Department of Environmental Agronomy and Crop Production,
University of Padova, Viale dell’Università 16, 35020 Legnaro, Padova.
E-mail: [email protected]
Tel.: +39 0498272807
http://www.biodiversity-lorenzomarini.eu/
Statistical modelling: more than one parameter
Nature of the response variable
NORMAL
(continuous)
POISSON, BINOMIAL …
Generalized
Linear Models
GLM
General Linear Models
Nature of the explanatory variables
Categorical
Continuous
Categorical + continuous
ANOVA
Regression
ANCOVA
Session 3
Session 4
ANOVA: aov()
ANOVA tests mean differences between groups defined by
categorical variables
One-way ANOVA
ONE factor with 2 or
more levels
Multi-way ANOVA
2 or more factors, each
with 2 or more levels
Diet: 4 levels
drug: 4 doses
Sex
♀
♂
ASSUMPTIONS
Independence of cases - this is a requirement of the design.
Normality - the distributions in each cells are normal [hist(),
qq.plot(), shapiro.test()]
Homogeneity of variances - the variance of data in groups should be
the same (variance homogeneity with fligner.test()).
One-way ANOVA step by step
1. Test normality (also after model fitting)
2. Test homogeneity of variance
3. Run the ANOVA and fit the maximal model
4. Reject/accept the H0 that all the means are equal
2 approaches
5A. Multiple comparison to test
differences between the
level of factors
5B. Model simplification
working with contrasts
(merge similar factor levels)
6. Minimum Adequate Model
MAM
One-way ANOVA
Body weight: 4 diets (k)
One-way ANOVA is
used to test for
differences among two
or more independent
groups
y: body weight (NORMAL CONTINUOS)
x: 4 diets (CATEGORICAL: four levels: x1, x2, x3, x4)
Ho: µ1= µ2= µ3= µ4
H1: All the means are not equal
ANOVA model
yi = a + bx2 + cx3 + dx4
a=µ1
b=µ1-µ2
c=µ1-µ3
d=µ1-µ4
c
y
b
a
d
One-way ANOVA
ni
µi
Diet 1
6.08
5.7
6.5
5.86
6.17
5
6.06
Diet 2
6.87
6.77
7.4
6.63
6.98
5
6.93
Diet 3
10.26
10.21
10.02
9.65
Diet 4
8.79
8.42
8.31
8.57
X
9.03
4
5
10.03 8.62
Number of groups: k = 4
Number of observations N = 19
Grand mean = 7.80
Sum of squares (SS): deviance
Degree of freedom (df)
SS Total = Σ(yi – grand mean)2
SS Factor = Σ ni(group meani – grand mean)2
SS Error (within group) = Σ(yi – group meani)2
Total: N – 1
Group: k – 1
Error: N – k
One-way ANOVA: SS explanation
SS Total
SS Error
Grand mean
SS Factor
Variability within groups
mean3
mean4
SS Total = SS Factor + SS Error
mean2
mean1
Variability among groups
One-way ANOVA
SSTotal=SSFactor
SSTotal=SSFactor + SSError
µ3
µ4
µ2
Grand mean
µ1
SS can be divided by the respective df to get a variance
MS = SS /df Mean squared deviation
MSFactor
MSError
The pseudo-replication would work here!!!
One-way ANOVA: F test (variance)
F = Factor MS
Error MS
If the k means are not equal, then the
Factor MS in the population will be
greater than the population’s Error MS
How to define the correct F test can be a difficult task with
complex design (BE EXTREMELY CAREFUL!!!!)
If F calculated > F critic, then we can reject Ho
All we conclude is that
all the k population means are not equal.
I.e. at least one mean differs!!!
A POSTERIORI MULTIPLE
COMPARISONS
MODEL SIMPLIFICATION
WORKING WITH CONTRASTS
One-way ANOVA: contrasts
Contrasts are the essence of hypothesis testing and model
simplification in analysis of variance and analysis of covariance.
They are used to compare means or groups of means with other
means or groups of means
We used contrasts to carry out t test AFTER having found out a
significant effect with the F test
- We can use contrasts in model simplification (merge similar factor
levels)
- Often we can avoid post-hoc multiple comparisons
- We need to specify contrasts a priori
One-way ANOVA: multiple comparisons
If F calculated > F critic, then we can reject Ho
All we conclude is that all the k population means are not equal.
At least one mean differs!!!
A POSTERIORI MULTIPLE COMPARISONS
(lots of methods!!!)
Multiple comparison procedures are then used to determine
which means are different from which.
Comparing K means involves K(K − 1)/2 pairwise comparisons.
E.g. Tukey-Cramer, Duncan, Scheffè, LSD
One-way ANOVA: multiple comparisons
Ordinarily, there is a problem with conducting three comparisons
in a row. The problem is that with each additional test, it becomes
more likely that you will obtain one statistically significant result
just by chance. Think of it as a slot machine. If you pull the slot
machine arm 4 times, you are 4 times as likely to hit the jackpot
given a completely random process. If you do 4 tests of statistical
significance, you are 4 times as likely to obtain one p<0.05 result
when there is no real difference between your means.
What every method does is to make an adjustment to the
obtained significance level (p-value) to make it harder for you to
obtain a p<.05. This is like pulling the slot machine handle 4 times
and having the slot machine say "I know you just tried 4 times, so
I'm making the odds of winning harder."
One-way ANOVA: nonparametric
If the assumptions are seriously violated, then one can opt for a
nonparametric ANOVA
However
One-way ANOVA is quite strength even in condition of non-normality
and non-homogeneity of variance
Kruskal-Wallis test kruskal.test()
(if k = 2, then it corresponds to the Mann-Whitney test)
ANOVA by ranks
If there are tied ranks a correction term must be applied
Multi-way ANOVA
Factorial ANOVA is used when the experimenter wants to
study the effects of two or more treatment variables.
ASSUMPTIONS
Independence of cases - this is a requirement of the design
Normality - the distributions in each of the groups are normal
Homogeneity of variances - the variance of data in groups should
be the same
+ Equal replication (BALANCED AND ORTHOGONAL DESIGN)
Dose 1
Dose 2
Dose 3
X
10 obs
10 obs
10 obs
8 obs
Low temp
-
High temp
10 obs
X
If you use traditional general linear models just one
missing data can affect strongly the results
Fixed vs. random factors
If we consider more than one factor we have to distinguish two kinds
of effects:
Fixed effects: factors are specifically chosen and under control, they
are informative
(E.g. sex, treatments, wet vs, dry, doses, sprayed or not sprayed)
Random effects: factors are chosen randomly within a large
population, they are normally not informative
(E.g. fields within a site, block within a field, split-plot within a plot, family,
parent, brood, individuals within repeated measures)
Random effects occur in two contrasting kinds of circumstances
1. Observational studies with hierarchical structure
2. Designed experiments with different spatial or temporal scales
(dependence)
Fixed vs. random factors
Why is it so important to identify fixed vs. random effects?
They affect the way to construct the F-test in a multifactorial
ANOVA. Their false identification leads to wrong conclusions
You can find how to construct your F-test with different combinations
of random and fixed effects and with different hierarchical structures
(choose a well-known sampling design!!!)
In the one-way ANOVA only the way to formulate our hypothesis
changes but not the test
If we have both fixed and random effects, then we are working on
MIXED MODELS
yi = µ + αi (fixed) + ri (random) + ε
Factorial ANOVA: two or more factors
Factorial design: two or more factors are crossed. Each
combination of the factors are equally replicated and each factor
occurs in combination with every level of the other factors
3 diets
4 supplements
10
10
10
10
10
10
10
10
10
10
10
10
Orthogonal sampling
Factorial ANOVA: Why?
Why use a factorial ANOVA? Why not just use
multiple one-way ANOVA’s?
With n factors, you’d need to run n one-way ANOVA’s, which
would inflate your α-level
– However, this could be corrected with a Bonferroni
Correction
The best reason is that a factorial ANOVA can detect
interactions, something that multiple one-way ANOVA’s
cannot do
Factorial ANOVA: Interactions
E.g. We are testing two factors, Gender (male
and female) and Age (young, medium,
and old) and their effect on performance
If males performance differed as a function of
age, i.e. males performed better or
worse with age, but females
performance was the same across
ages, we would say that Age and
Gender interact, or that we have an Age
x Gender interaction
Performance
Interaction: When the effects of one independent variable
differ according to levels of another independent variable
Female
Male
Young
Old
Age
It is necessary that the slopes differ from one another
Factorial ANOVA: Main effects
Main effects: the effect of a factor is independent from any other
factors
This is what we were looking at with one-way ANOVA’s – if we have
a significant main effect of our factor, then we can say that the
mean of at least one of the groups/levels of that factor is
different than at least one of the other groups/levels
Medium
Old
Male
Performance
Performance
Young
Sex
It is necessary that the intercepts differ
Female
Age
Factorial ANOVA: Two-crossed factor design
Two crossed fixed effects: every level of each factor occurs in combination
with every level of the other factors
Model 1: two fixed effects
Model 2: two random effects (uncommon situation)
Model 3: one random and one fixed effect
We can test main effects and interaction:
1. The main effect of each factor is the effect of each factor
independent of (pooling over) the other factors
2. The interaction between factors is a measure of how the
effects of one factor depends on the levels of one or more
additional factors (synergic and antagonist effect of the factors)
Factor 1 x Factor 2
We can only measure interaction effects in factorial (crossed) designs
Factorial ANOVA: Two-crossed fixed factor design
Two crossed fixed effects:
Response variable: weight gain in six weeks
Factor A: DIET (3 levels: barley, oats, wheat)
Factor B: SUPPLEMENT (4 levels: S1, S2, S3, S4)
DIET* SUPPLEMENT= 3 x 4 = 12 combinations
We have 48 horses to test our two factors: 4 replicates
barley+S1
barley+S2
barley+S3
barley+S4
oats+S1
oats+S3
oats+S2
oats+S4
wheat+S1
wheat+S3
wheat+S2
wheat+S4
Factorial ANOVA: Two-crossed fixed factor design
The 48 horses must be independent units to be replicates
Barley
Oats
wheat
DIET
SUPPLEMENT
DIET*SUPPLEMENT
26.34
23.29
19.63
23.29
20.49
17.40
22.46
19.66
17.01
Barley
Oats
Wheat
F test for main effects and interaction
DIET
SUPPLEMENT
MS D
FD 
MS error
FS 
DIET*SUPPLEMENT
MS S
MS error
FDxS
S1
MS DxS

MS error
S2
S3
S4
25.57
21.86
19.66
Factorial ANOVA: Two-crossed fixed factor design
Examples of ‘good’ ANOVA results
2.0
3.0
3.0
Treatment
25
35
45
< 0.05
n.s.
n.s.
15
1.0
2.0
3.0
Treatment
1.0
2.0
Treatment
10
15
Two-crossed factor design
CloneA
CloneB
5
Mean y
n.s.
n.s
n.s
2.0
20
Treatment
Effect
Clone
Treat
CxT
Mean y
10
1.0
Worst case
20
Mean y
15
25
Mean y
25
15
1.0
n.s.
< 0.05
n.s.
Effect
Clone
Treat
CxT
30
< 0.05
< 0.05
n.s.
Effect
Clone
Treat
CxT
35
< 0.05
< 0.05
< 0.05
Effect
Clone
Treat
CxT
5
Mean y
Effect
Clone
Treat
CxT
1.0
2.0
Treatment
3.0
Treatment: 3 levels
3.0
Mixed models
Mixed Models
Mixes What?
– Fixed Effects
– Random Effects
– E.g. maybe the residuals are not independent
– Common if repeated measurements of same individual and
hierarchical sampling
Mixed models
1) Hierarchical structure
- Example: if collecting data from different medical
centers, center might be thought of as random.
- Example: if surveying animals, they can be clustered
into cohorts, cohort is random
2) Longitudinal studies
- Example: Repeated measurements are taken over time
for each subject. Subject is random.
In all these cases, it is not generally reasonable to assume that
observations within the same group are independent.
Mixed models: split-plot
We can consider random factors to account for the variability
related to the environment in which we carry out the experiment
Mixed models can deal with spatial or temporal dependence
E.g. SPLIT-PLOT design is one of the most useful design
The different treatment are applied to plot with different
size organized in a hierarchical structure
Mixed model: split-plot
Treatment (control, dose1, dose2)
8 cages
Organ (liver, heart or kidney)
Response variable:
Metabolite concentration
Fixed effects:
Treatment (control, dose1, dose2)
Organ (liver, heart or kidney)
Random effects:
Rats within cage
Organs within rat
Mixed model: aov()
Model formulation and simplification
Y~ fixed effects + error terms
y ~ a*b*c + Error(a/b/c)
Growth
Here you can
specify your
sampling hierarchy
Cages
Uninformative
Treatment
Organ
Informative!!!
Growth ~ treatment*organ+
Error(cage/treatment/organ))
Mixed models: tradition vs. REML
Mixed models using traditional ANOVA requires
perfect orthogonal and balanced design
(THEY WORK WELL WITH THE PROPER SAMPLING)
avoid to work with multi-way ANOVA in
non-orthogonal sampling designs
If something has gone wrong with the sampling
In R you can run Mixed models with missing data and
unbalanced design (non orthogonal design) using the
REML estimation lme4()
Mixed models: tradition vs. REML
Mixed model: REML
• REML: Residual
Maximum
Likelihood
– vs. Maximum
Likelihood
– Unbalance, nonorthogonal, multiple
sources of error
• Packages
– NLME (quite old)
• New Alternative
– lme4
Mixed model: REML
When should I use REML?
For balanced data, both ANOVA and REML will give the same answer.
However, ANOVA’s algorithm is much more efficient and so should be
used whenever possible.
Are all factor
Can you identify blocks
combinations
(or a hierarchy of blocks)
present and sample sizes of similar experimental
equal?
units?


Most efficient analysis


Fixed effects ANOVA
(or regression)
Mixed effects ANOVA


Regression


REML
Generalized Linear Models (GLM)
We can use GLMs when the variance is not constant, and/or when the
errors are not normally distributed.
A GLM has three important properties
1. The error structure
2. The linear predictor
3. The link function
Count data
Proportion data
2
20
40
Variance
3
2
1
Variance
6
4
Variance
3.0
0
2
4
6
Mean
8
10
0
0
0
2.0
Variance
60
80
4
Gamma
8
4.0
10
Normal
0
2
4
6
Mean
8
10
0
2
4
6
Mean
8
10
0
2
4
6
Mean
8
10
Generalized Linear Models (GLM)
Error structure
In GLM we can specify error structure different from the normal:
- Normal (gaussian)
- Poisson errors (poisson)
- Binomial errors (binomial)
- Gamma errors (Gamma)
glm(formula, family = …, link=…, data, …)
Generalized Linear Models (GLM)
Linear predictor (η)= predicted value output from a GLM
The model relates each observed y to a predicted value
The predicted value is obtained by a TRANSFORMATION of
the value emerging from the linear predictor
   xi j  j
p
β are the parameters estimated for
the p explanatory variables
x are the values measured for the p
explanatory variables
Fit of the model = comparison between the linear predictor and
the TRANSFORMED y measured
The transformation is specified in the LINK FUNCTION
Generalized Linear Models (GLM)
Link functions (g)
The link function related the mean value of y to its linear
predictor
  g ( )
The value of η is obtained by
transforming the value of y by the link
function
The predicted value of y is obtained by applying the inverse
link function to the η value of y by the link function
Typically the output of a GLM is η
Need to transform η to get the
predicted values
Known the distribution start with the canonical link function
Generalized Linear Models (GLM) glm()
Model fit
Deviance is the measure of goodness-of-fit of GLM
Deviance=-2(log-likelihoodcurrent model -log-likelihood saturated model)
We aim at reducing the residual deviance
Error
Deviance
normal
 ( yi  mean)
Variance
2
1
poisson
2 /  y ln( y / mean)  ( y  mean)
mean
binomial
2 /  y ln( y / mean)  (n  y) ln( n  y) /( n  mean)
mean(n  mean)
n
Gamma
2 /  ( y  mean) / y  ln( y / mean)
mean2
n is the size of the sample
Proportion data and binomial error
Proportion data and binomial error
1. The data are strictly bounded (0-1)
2. The variance is non-constant (∩-shaped relation with the mean)
3. Errors are non-normal
 p
ln    a  bx
q
Link function: Logit
Proportion
0.8
e abx
p
1  e abx
0.4
p
2.0
1.0
0.0
0.0
Variance
Proportion data
0.0
0.2
0.4
0.6
p
0.8
1.0
-100
-50
0
x
50
100
Proportion data and binomial error
After fitting a binomial-logit GLM must be:
Residual df ≈ residual deviance
NO
YES
The model is adequate
Fit a quasibinomial
Check again
YES
The model is adequate
NO
Change distribution
2 Examples
1. The first example concerns sex ratios in insects (the proportion of all
individuals that are males). In the species in question, it has been observed
that the sex ratio is highly variable, and an experiment was set up to see
whether population density was involved in determining the fraction of males.
2. The data consist of numbers dead and initial batch size for five doses of
pesticide application, and we wish to know what dose kills 50% of the
individuals (or 90% or 95%, as required).
Count data and Poisson error
Count data and Poisson error
1. The data are non-negative integers
2. The variance is non-constant (variance = mean)
3. Errors are non-normal
The model is fitted with a log link (to ensure that the fitted values are bounded
below) and Poisson errors (to account for the non-normality).
Count data
3
1
2
log(x)
6
4
2
0
0
Variance
8
4
10
Count
0
2
4
6
Mean
8
10
0
20
40
60
x
80
100
Count data and Poisson error
After fitting a Poisson-log GLM must be:
Residual df ≈ residual deviance
NO
YES
The model is adequate
Fit a quasipoisson
Check again
YES
The model is adequate
NO
Change distribution
Example
1. In this example the response is a count of the number of plant species
on plots that have different biomass (continuous variable) and different
soil pH (high, mid and low).