Copyright (c) Bani K. Mallick1 STAT 651 Lecture # 12.

Copyright (c) Bani K. Mallick 1

STAT 651

Lecture # 12


Topics in Lecture #12 The ANOVA F-test, and the basics of

the F-table


Book Sections Covered in Lecture #12

Chapter 8.1-8.2


Relevant SPSS Tutorials ANOVA-GLM

Post hoc tests


ANalysis Of VAriance

We now turn to making inferences when there are 3 or more populations

This is classically called ANOVA

It is somewhat more formula dense than what we have been used to.

Tests for normality are also somewhat more complex


ANOVA

Suppose we form three populations on the basis of body mass index (BMI):

BMI < 22, 22 <= BMI < 28, BMI > 28

This forms 3 populations

We want to know whether the three populations have the same mean caloric intake, or if their food composition differs.


ANOVA

If you do lots of 95% confidence intervals, you’d expect by chance that about 5% will be wrong

Thus, if you do 20 confidence intervals, you expect 1 = 20 x 5% will not include the true population parameter

This is a fact of life


ANOVA

One procedure that is often followed is to do a preliminary test to see whether there are any differences among the populations

Then, once you conclude that some differences exist, you allow somewhat more informality in deciding where those differences manifest themselves

The first step is the ANOVA F-test


ANOVA

Consider the ACS data, with 3 BMI groups and measuring the % calories from fat (first FFQ)

What is your preliminary conclusion about differences in means/medians?

About differences in variability?

About massive outliers?

388462N =

ACS Data: % Calories from Fat

BMI Group

High BMIMedium BMILow BMI

Ba

selin

e F

FQ

70

60

50

40

30

20

10

0

8

148


ANOVA

Consider the ACS data, with 3 BMI groups and measuring the % calories from fat (first FFQ)

Between-Subjects Factors

Low BMI 62

MediumBMI

84

High BMI 38

1.00

2.00

3.00

BMIGroup

Value Label N


ANOVA

The F-test is easy to compute, and provided in all statistical packages

The populations are 1, 2, … t

The sample sizes are

The population means are

The hypothesis to test is

1 tn , ,n

1 tμ ,...,μ

0 1 2 tH :μ =μ =μ


ANOVA

The data from population i are

The sample mean from population i is

The sample mean of all the data is

The total sample size is the total number of observations, called

ii1 i2 inY ,Y , ,YiY

Y

Tn


ANOVA

The ANOVA Table (demo in SPSS in class of ACS data) “Analyze” “Compare Means” “1-way ANOVA”: I’ll now show you what each item is

ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.


ANOVA

The sample mean from population i is

The sample mean of all the data is

The idea of the F-test is based on distances

The distance of the data to the overall mean is

TSS = Total Sum of Squares

iY

Y

2ij

ij

TSS = (Y Y )


ANOVA

The distance of the data to the overall mean is

TSS = Total Sum of Squares

This has degrees of freedom

2ij

ij

TSS = (Y Y )

Tn 1


ANOVA

Next comes the “Between Groups” row

ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total



ANOVA

The sum of squares between groups is

It has t-1 degrees of freedom, so the number of populations is the degrees of freedom between groups + 1.

2ii

i

n (Y Y )


ANOVA

Next comes the “Within Groups” row

ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total



ANOVA

The distance of the observations to their sample means is

This is the Sum of Squares Within

It has degrees of freedom

2iij

ij

SSW = (Y Y )

Tn t


ANOVA

Next comes the “Mean Squares”

These are the different sums of squares divided by their degrees of freedom

ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total



ANOVA

Next comes the F-statistic

It is the ratio of the mean square between groups to the mean square within groups

ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total



ANOVA

The F-statistic is

T

SSB/(t-1) F =

SSW/(n -t)2

iii

2iij T

ij

n (Y Y ) /(t 1) =

(Y Y ) /(n -t)


ANOVA

The F-statistic is

What values (large or small) indicate differences?

Clearly large, since if the population means are equal, the sample means will be close, and the top will be near 0

2ii

i2

i Tijij

n (Y Y ) /(t 1) =

(Y Y ) /(n -t)


Why do they call it ANOVA?

ANOVA = ANalysis Of VAriance

I want to show you the concept in graphs, because these become important in STAT 652

I will illustrate the idea with samples from two populations

The first population will be in red

The second in green

When pooled I will use blue



The data from the first population

The total distance of the observations to their sample mean is

Y Y Y Y Y Y

1Y

1

21j

j

(Y Y )



I will use this funny symbol to denote total distance

The total distance of the observations to their sample mean is

Y Y Y Y Y Y

1

21j

j

(Y Y )



Consider two similar populations

Summing the two symbols is the SSW = Sum of Squared distances Within the two samples

Y Y Y Y Y Y

Y Y Y Y Y Y



Now pool the two similar populations

Y Y Y Y Y Y

Y Y Y Y Y Y

YYYY YY YY Y Y Y Y

The Blue symbol represents the sum of squared distances of the total sample to the total sample mean

The is the Total Sum of Squares, or TSS



Now pool the two similar populations

Y Y Y Y Y Y

Y Y Y Y Y Y

YYYY YY YY Y Y Y Y

The Blue symbol represents the sum of squared distances of the total sample to the total sample mean

The is the Total Sum of Squares, or TSS

Note how the SSW and the TSS are about the same: when this happens, it indicates equal means for the populations



Now note what happens if the population means are different

Y Y Y Y Y Y

Y Y Y Y Y Y

Y Y Y Y YYYY Y Y Y Y Note how the TSS has greatly increased

Note how the SSW are the same as before (remember, we add the squared distances separately for the two populations

It is this phenomenon that the F-test measures


ANOVA

The F-statistic is compared to the F-distribution with t-1 and degrees of freedom.

See Table 8 ,which lists the cutoff points in terms of . If the F-statistic exceeds the cutoff, you reject the hypothesis of equality of all the means.

SPSS gives you the p-value (significance level) for this test

Tn t


ANOVA I have used the language of the book up

to this point, which coincides with the SPSS 1-way ANOVA output.

For our purposes, the general linear model is more useful.

It allows us to compare all the populations simultaneously

It also allows us to check assumptions about normality

Unfortunately, it uses slightly different terminology


ANOVA


ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total


Tests of Between-Subjects Effects

Dependent Variable: Baseline FFQ

960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183

SourceCorrected Model

Intercept

BMIGROUP

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .059 (Adjusted R Squared = .049)a.

ANOVA Procedure

General Linear ModelProcedure: note how F-statistics and p-valuesare identical


ANOVA


ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between GroupsWithin Groups

Total




960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183

Source

Corrected ModelIntercept

BMIGROUPError

Total

Corrected Total



Between Groups

Corrected model or variable name (BMIGROUP)


ANOVA


ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within GroupsTotal




960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

ErrorTotal

Corrected Total



Within Groups

Error


ANOVA


ANOVA

Baseline FFQ

960.287 2 480.143 5.689 .004

15275.639 181 84.396

16235.925 183

Between Groups

Within Groups

Total




960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total



Total

Corrected Total


ANOVA

If the populations have a common variance 2, the Mean squared error estimates it

For 2-populations, the mean squared error equals the square of the pooled sd:

Hence, estimated common s.d.

2Ps



960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total



84.396 9.19


ANOVA

There appears to be some difference

95% CI for difference in population means between high and low BMI groups is 2.26 to 10.20

95% CI for difference in population means between medium and low BMI groups is from -1.59 to 4.31

High & Medium: 1.08 to 8.65

Conclusions?


ANOVA

The ANOVA Table (demo in SPSS in class of ACS data)



960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total




ANOVA

The ANOVA Table

Number of populations is t=3, so degrees of freedom for the model (BMIGROUP) is

t-1 = 2 Tests of Between-Subjects Effects


960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total




ANOVA

The ANOVA Table

Total sample size is 184, so the degrees of freedom for the corrected total is = 183

Tn 1Tests of Between-Subjects Effects


960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total




ANOVA

The mean square between groups is 480.143

The mean square within groups is 84.396

The F-statistic is the ratio: 5.689 Tests of Between-Subjects Effects


960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total




ANOVA

The p-value is 0.004: what does this mean?

What was the null hypothesis?



960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total




ANOVA

The p-value is 0.004: what does this mean?

What was the null hypothesis? That the populations means are all equal



960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total




ANOVA

The p-value is 0.004: what does this mean? We have more than 99% confidence that the null hypothesis is false

What was the null hypothesis? That the populations means are all equalTests of Between-Subjects Effects


960.287a 2 480.143 5.689 .004

196009.919 1 196009.919 2322.508 .000

960.287 2 480.143 5.689 .004

15275.639 181 84.396

226223.216 184

16235.925 183


Intercept

BMIGROUP

Error

Total

Corrected Total



Copyright (c) Bani K. Mallick1 STAT 651 Lecture # 12.

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Transcript of Copyright (c) Bani K. Mallick1 STAT 651 Lecture # 12.