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Experimental Design One-way ANOVA
One-way ANOVA
● Method to compare more than two samples simultaneously without inflating Type I Error rate (α)
● Simplicity
● Few assumptions
● Adequate for highly complex hypothesis testing
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Experimental Design One-way ANOVA
Outline of this class●Data organization and layout
●Repartitioning of variance
●Definition of a linear model
●Combine the linear model with the repatitioning of variances
●Definition of a statistic (F-test)
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Experimental Design One-way ANOVA
Data organization
Suppose that we want to investigate the average length of a fish species in three different lakes because we suspect that there might be some form of local adaptation
We sample 5 fish (replicates) at each lake
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Experimental Design One-way ANOVA
Data organization
First we establish how to measure “length”
Lenght
This is an important part of experimental design!
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Experimental Design One-way ANOVA
Data organization
Then we collect the data
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Experimental Design One-way ANOVA
Data organization
Factor “Lake” has three levels: 1, 2 and 3
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Experimental Design One-way ANOVA
Data organization
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Experimental Design One-way ANOVA
Data organization
We may represent it as
Note that Lake is a classification criteria, that is, we can classify each fish according to the lake where it belongs
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Experimental Design One-way ANOVA
Total variation =
Ronald Aylmer Fisher (1890-1962)
Repartitioning the variance
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Experimental Design One-way ANOVA
Total variation =
Sum of all the squared differences between each individual value and the grand mean (overall mean)
But why squaring the differences?
Why this formula?
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Experimental Design One-way ANOVA
Total variation =
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Experimental Design One-way ANOVA
= 0
Total variationWithin treatments variation
Among (between) treatments variation
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Experimental Design One-way ANOVA
Repartitioning the variance
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Experimental Design One-way ANOVA
What do these quantities measure?
Repartitioning the variance
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Experimental Design One-way ANOVA
Why use analysis of variance to test hypothesis about the means?
Repartitioning the variance
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Experimental Design One-way ANOVA
Defining a linear model
Any single measurement can be predicted if we know the mean (μ) of the treatment or sample where it belongs (i) and the error (e) associated with that particular replicate (j) in the sample i
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Experimental Design One-way ANOVA
An interesting propertyTake sample 1 (Lake 1)
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Experimental Design One-way ANOVA
An interesting property
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Experimental Design One-way ANOVA
An interesting property
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Experimental Design One-way ANOVA
An interesting propertyWe can represent any sample in terms of its errors
We will make use of this property later on...
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Experimental Design One-way ANOVA
Back to the linear model
H0: μ
1 = μ
2 = μ
3 = ... = μ
i ... = μ
a = μ
If the null hypothesis is true, all samples (treatments or levels) came from the same population
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Experimental Design One-way ANOVA
Defining the linear model
If the null hypothesis is false, some samples will deviate from the grand mean by an amount called A
H0: A
1 = A
2 = A
3 = ... = A
i ... = A
a = 0
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Experimental Design One-way ANOVA
Defining the linear model
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Experimental Design One-way ANOVA
Defining the linear model
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Experimental Design One-way ANOVA
Defining the linear model
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Experimental Design One-way ANOVA
Joining the linear model andthe repartitioning of variances
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
Where do we know this from?
We know that a sample can also be represented by the deviations of each replicate to the sample mean (errors)
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
COVARIANCE
1st Assumption: individual observations are independent from each other (that is, no particular observation influences any other observation in the same or other sample)
INDEPENDENCE OF OBSERVATIONS
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Experimental Design One-way ANOVA
COVARIANCE
If observations are independent, covariance is null (zero)
INDEPENDENCE OF OBSERVATIONS
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Experimental Design One-way ANOVA
If observations are independent, covariance is null (zero)
INDEPENDENCE OF OBSERVATIONS
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Experimental Design One-way ANOVA
Let’s focus on this term...
This is the deviation of sample means from the grand mean (Remember the Central Limit Theorem?)
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Experimental Design One-way ANOVA
The central limit theorem says
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Experimental Design One-way ANOVA
2nd Assumption: sample variances are equal (homogeneous or homoscedastic)
HOMOGENEITY OF VARIANCES
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Experimental Design One-way ANOVA
2nd Assumption: sample variances are equal (homogeneous or homoscedastic)
HOMOGENEITY OF VARIANCES
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Experimental Design One-way ANOVA
2nd Assumption: sample variances are equal (homogeneous or homoscedastic)
HOMOGENEITY OF VARIANCES
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Experimental Design One-way ANOVA
2nd Assumption: sample variances are equal (homogeneous or homoscedastic)
HOMOGENEITY OF VARIANCES
Using the same argument
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Experimental Design One-way ANOVA
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Experimental Design One-way ANOVA
Change the order of “Between” and “Within” samples since this is the most common layout for an ANOVA
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Experimental Design One-way ANOVA
Introducing degrees of freedom
● For a factor with a levels: a-1
● For the within samples variation: a(n-1)
● For the Total variation: an-1
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Experimental Design One-way ANOVA
Introducing degrees of freedom
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Experimental Design One-way ANOVA
Introducing degrees of freedom and Mean Squares
Mean Square (MS) = Sum of Squares / Degrees of Freedom (SS/DF)
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Experimental Design One-way ANOVA
Introducing degrees of freedom and Mean Squares
Mean Square (MS) = Sum of Squares / Degrees of Freedom (SS/DF)
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Experimental Design One-way ANOVA
Revisiting the null hypothesisIf the null hypothesis is true, sample means will be the same as the grand mean and deviations from the latter (A
i) will be zero
H0: A
1 = A
2 = A
3 = ... = A
i ... = A
a = 0
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Experimental Design One-way ANOVA
If the null hypothesis is true
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Experimental Design One-way ANOVA
If the null hypothesis is true
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Experimental Design One-way ANOVA
Choosing a statistical test
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Experimental Design One-way ANOVA
The adequate statistical test
3th Assumption: the variable being sampled follows a normal distribution (often stated as: the population being sampled follows a normal distribution)
NORMALITY OF SAMPLED POPULATION
If this is true, the ratio between two variances follows a F-distribution
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Experimental Design One-way ANOVA
The F distribution
F ≈ 1: H0 true
F > 1: H0 false
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Experimental Design One-way ANOVA
ANOVA in action
Source of variation
SS DF MS F P
Lakes 48.933
Error 50.000
Total 98.933
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Experimental Design One-way ANOVA
ANOVA in action
Source of variation
SS DF MS F P
Lakes 48.933 2
Error 50.000 12
Total 98.933 14
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Experimental Design One-way ANOVA
ANOVA in action
Source of variation
SS DF MS F P
Lakes 48.933 2 24.467
Error 50.000 12 4.167
Total 98.933 14
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Experimental Design One-way ANOVA
ANOVA in action
Source of variation
SS DF MS F P
Lakes 48.933 2 24.467 5.872
Error 50.000 12 4.167
Total 98.933 14
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Experimental Design One-way ANOVA
ANOVA in actionSource of variation
SS DF MS F P
Lakes 48.933 2 24.467 5.872
Error 50.000 12 4.167
Total 98.933 14
F > Fcrit
H0 rejected
HA accepted
Average length of fish species differs among lakes
0.017
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