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AN alysis O f VA riance (ANOVA). Comparing > 2 means Frequently applied to experimental data Why not do multiple t-tests? If you want to test H 0 : m 1 = m 2 = m 3 Why not test: m 1 = m 2 m 1 = m 3 m 2 = m 3.

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slide1

ANalysis Of VAriance

(ANOVA)

Comparing > 2 means

Frequently applied to experimental data

Why not do multiple t-tests? If you want to test

H0: m1 = m2 = m3

Why not test:

m1 = m2

m1 = m3

m2 = m3

For each test 95% probability to correctly fail to reject (accept?) null, when null is really true

0.953 = probability of correctly failing to reject all 3 = 0.86

slide2

Probability of if incorrectly rejecting at least one of the (true) null hypotheses = 1 - 0.86 = 0.14

As you increase the number of means compared, the probability of incorrectly rejecting a true null (type I error) increases towards one

Side note: possible to correct (lower)  if you need to do multiple tests (Bonferroni correction)- unusual

slide3

ANOVA: calculate ratios of different portions of variance of total dataset to determine if group means differ significantly from each other

Calculate ‘F’ ratio, named after R.A. Fisher

1) Visualize data sets

2) Partition variance (SS & df)

3) Calculate F (tomorrow)

slide4

Pictures first

3 fertilizers applied to 10 plots each (N=30), yield measured

How much variability comes from fertilizers, how much from other factors?

8

7

6

5

Overall mean

Yield (tonnes)

4

3

2

Fert 1

Fert 3

Fert 2

1

0

0

10

20

30

Plot number

slide5

Thought Questions

What factors other than fertilizer (uncontrolled) may contribute to the variance in crop yield?

How do you minimize uncontrolled factors contribution to variance when designing an experiment or survey study?

If one wants to measure the effect of a factor in nature (most of ecology/geology), how can or should you minimize background variability between experimental units?

slide6

Fertilizer (in this case) is termed the independent or predictor variable or explanatory variable

Can have any number of levels, we have 3

Can have more than one independent variable. We have 1, one way ANOVA

Crop yield (in this case) is termed the dependent or response variable

Can have more than one response variable…. multivariate analysis (ex MANOVA). Class taught by J. Harrell

slide7

Pictures first

-calculate deviation of each point from mean

-some ‘+’ and some ‘-’

-sum to zero (remember definition of mean)

8

7

6

5

Overall mean

Yield (tonnes)

4

3

2

Fert 1

Fert 3

Fert 2

1

0

0

10

20

30

Plot number

slide8

Sum the squared values

Square all values

slide9

** why (n-1)?? Because… all deviations must sum to zero, therefore if you calculate n-1 deviations, you know what the final one must be. You do not actually have n independent pieces of information about the variance.

=

n-1

SS not useful for comparing between groups, it is always big when n is big. Using the mean SS (variance) allows you to compare among groups

calculate ~mean SS

a.k.a. variance

slide10

Partitioning Variability

Back to the question:

How much variability in crop yield comes from fertilizers (what you manipulated), how much from other factors (that you cannot control)?

Calculate mean for each group, ie plots with fert1, fert2, and fert3 (3 group means)

But first imagine a data set where…………

slide11

-Imagine case were the group (treatment) means differ a lot, with little variation within a group

-Group means explain most of the variability

Group means

8

7

6

5

Overall mean

Yield (tonnes)

4

3

2

Fert 1

Fert 3

Fert 2

1

0

0

10

20

30

Plot number

slide12

Now…. imagine case were the group (treatment) means are not distinct, with much variation within a group

-Group means explain little of the variability

-3 fertilizers did not affect yield differently

Group means

8

7

6

5

Overall mean

Yield (tonnes)

4

3

2

Fert 1

Fert 3

Fert 2

1

0

0

10

20

30

Plot number

slide13

H0: mean yield fert1= mean yield fert2 = mean yield fert3

Or

Fertilizer type has no effect on crop yield

-calculating 3 measures of variability,

start by partitioning SS

slide14

Sum of squares of deviations of data around the grand (overall) mean

(measure of total variability)

Total SS =

Within group SS =

(Error SS)

Sum of squares of deviations of data around the separate group means

(measure of variability among units given same treatment)

Unfortunate word usage

Sum of squares of deviations of group means around the grand mean

(measure of variability among units given different treatments)

Among groups SS =

slide15

k = number experimental groups

Xij = datum j in experimental group I

Xbari = mean of group I

Xbar = grand mean

Sum of squares of deviations of data around the grand (overall) mean

(measure of total variability)

Total SS =

2

ni

k

Total SS =

Xij - X

j=1

i=1

Sum of deviations of each datum from the grand mean, squared, summed across all k groups

Total SS =

slide16

k = number experimental groups

Xij = datum j in experimental group I

Xbari = mean of group I

Xbar = grand mean

Sum of squares of deviations of data around the separate group means

(measure of variability among units given same treatment)

Within group SS =

2

ni

k

Xij - Xi

Within group SS =

j=1

i=1

Sum of deviations of each datum from its group mean, squared, summed across all k groups

Within group SS =

slide17

k = number experimental groups

Xij = datum j in experimental group I

Xbari = mean of group I

Xbar = grand mean

Sum of squares of deviations of group means around the grand mean

(measure of variability among units given different treatments)

Among groups SS =

2

k

Among groups SS =

Xi - X

ni

i=1

Sum of deviations of each group mean from the grand mean, squared

Among groups SS =

slide18

partitioning DF

Total number experimental units -1

In fertilizer experiment, n-1= 29

Total df =

units in each group -1, summed for all groups

In fertilizer experiment, (10-1)*3; 9*3=27

Within group df =

(Error df)

Unfortunate word usage

Number group means -1

In fertilizer experiment, 3-1=2

Among groups df =

slide19

SS and df sum

Total SS = within groups SS + among groups SS

Total df = within groups df + among groups df

slide20

Mean squares

Combine information on SS and df

Total mean squares = total SS/ total df

total variance of data set

Within group mean squares = within SS/ within df

variance (per df) among units given same treatment

Error MS

Unfortunate word usage

Among groups mean squares = among SS / among df

variance (per df) among units given different treatments

slide21

Tomorrow:

the big ‘F’

example calculations

slide22

Mean squares

Combine information on SS and df

Total mean squares = total SS/ total df

total variance of data set

Within group mean squares = within SS/ within df

variance (per df) among units given same treatment

Error MS

Unfortunate word usage

Among groups mean squares = among SS / among df

variance (per df) among units given different treatments

slide23

 Back to the question: Does fitting the treatment mean explain a significant amount of variance?

In our example…. if fertilizer doesn’t influence yield, then variation between plots with the same fertilizer will be about the same as variation between plots given different fertilizers

Among groups mean squares

F =

Within group mean squares

Compare calculated F to critical value from table (B4)

slide24

If calculated F as big or bigger than critical value, then reject H0

But remember…….

H0: m1 = m2 = m3

Need separate test (multiple comparison test) to tell which means differ from which

slide25

Remember… Shape of t-distribution approaches normal curve as sample size gets very large

But….

F distribution is different…

always positive skew

shape differs with df

See handout

slide26

Two types of ANOVA: fixed and random effects models

Calculation of F as:

Among groups mean squares

F =

Within group mean squares

Assumes that the levels of the independent variable have been specifically chosen, as opposed to being randomly selected from a larger population of possible levels

slide27

Exs

Fixed: Test for differences in growth rates of three cultivars of roses. You want to decide which of the three to plant.

Random: Randomly select three cultivars of roses from a seed catalogue in order to test whether, in general, rose cultivars differ in growth rate

Fixed: Test for differences in numbers of fast food meals consumed each month by students at UT, BG, and Ohio State in order to determine which campus has healthier eating habits

Random: Randomly select 3 college campuses and test whether the number of fast food meals per month differs among college campuses in general

slide28

In random effects ANOVA the denominator is not the within groups mean squares

Proper denominator depends on nature of the question

***Be aware that default output from most stats packages (eg, Excel, SAS) is fixed effect model

slide29

Assumptions of ANOVA

Assumes that the variances of the k samples are similar (homogeneity of variance of homoscedastic)

robust to violations of this assumption, especially when all ni are equal

Assumes that the underlying populations are normally distributed

also robust to violations of this assumption

slide30

Model Formulae

Expression of the questions being asked

Does fertilizer affect yield?

(word equation)

yield = fertilizer

response var

explanatory var

Right side can get more complicated

slide31

General Linear Models

Linear models relating response and explanatory variables and encompassing ANOVA (& related tests) which have categorical explanatory variables and regression (& related tests) which have categorical explanatory variables

In SAS proc glm executes ANOVA, regression and other similar linear models

Other procedures can also be used, glm is most general

slide32

data start;

infile'C:\Documents and Settings\cmayer3\My Documents\teaching\Biostatistics\Lectures\ANOVA demo.csv'dlm=','DSD;

input plot fertilizer yield;

options ls=80;

procprint;

data one; set start;

procglm;

class fertilizer;

model yield=fertilizer;

run;

slide33

The SAS System 5

12:53 Thursday, September 22, 2005

The GLM Procedure

Class Level Information

Class Levels Values

fertilizer 3 1 2 3

Number of Observations Read 30

Number of Observations Used 30

The SAS System 6

12:53 Thursday, September 22, 2005

The GLM Procedure

Dependent Variable: yield

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 2 10.82274667 5.41137333 5.70 0.0086

Error 27 25.62215000 0.94896852

Corrected Total 29 36.44489667

R-Square Coeff Var Root MSE yeild Mean

0.296962 20.97804 0.974150 4.643667

Source DF Type I SS Mean Square F Value Pr > F

fertilizer 2 10.82274667 5.41137333 5.70 0.0086

Source DF Type III SS Mean Square F Value Pr > F

fertilizer 2 10.82274667 5.41137333 5.70 0.0086