One-way ANOVA - FCUP · 09/30/12 2 Experimental Design One-way ANOVA Outline of this class Data...

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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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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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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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Data organization

First we establish how to measure “length”

Lenght

This is an important part of experimental design!

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Data organization

Then we collect the data

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Data organization

Factor “Lake” has three levels: 1, 2 and 3

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Data organization

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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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Total variation =

Ronald Aylmer Fisher (1890-1962)

Repartitioning the variance

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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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Total variation =

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= 0

Total variationWithin treatments variation

Among (between) treatments variation

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What do these quantities measure?


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Why use analysis of variance to test hypothesis about the means?


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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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An interesting propertyTake sample 1 (Lake 1)

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An interesting property

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An interesting property

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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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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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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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Joining the linear model andthe repartitioning of variances

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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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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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COVARIANCE

If observations are independent, covariance is null (zero)


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If observations are independent, covariance is null (zero)


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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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The central limit theorem says

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2nd Assumption: sample variances are equal (homogeneous or homoscedastic)

HOMOGENEITY OF VARIANCES

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Using the same argument

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Change the order of “Between” and “Within” samples since this is the most common layout for an ANOVA

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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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Introducing degrees of freedom

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Introducing degrees of freedom and Mean Squares

Mean Square (MS) = Sum of Squares / Degrees of Freedom (SS/DF)

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Introducing degrees of freedom and Mean Squares

Mean Square (MS) = Sum of Squares / Degrees of Freedom (SS/DF)

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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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If the null hypothesis is true

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If the null hypothesis is true

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Choosing a statistical test

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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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The F distribution

F ≈ 1: H0 true

F > 1: H0 false

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ANOVA in action

Source of variation

SS DF MS F P

Lakes 48.933

Error 50.000

Total 98.933

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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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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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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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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

One-way ANOVA - FCUP · 09/30/12 2 Experimental Design One-way ANOVA Outline of this class Data...

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