Statistics: Unlocking the Power of Data Lock 5 STAT 101 Dr. Kari Lock Morgan ANOVA SECTION 8.1 Testing for a difference in means across multiple categories

Statistics: Unlocking the Power of Data Lock5

STAT 101Dr. Kari Lock Morgan

ANOVA

SECTION 8.1• Testing for a difference in means across

multiple categories


Review: Chi-Square TestsThe χ2 goodness-of-fit tests if one categorical

variable differs from a null distribution

The χ2 test for association tests for an association between two categorical variables

For both, you compute the expected counts in each cell (assuming H0) and the χ2 statistic:

Find the proportion above the χ2 statistic in a randomization or χ2-distribution (if all expected counts > 5)

2

2 observed - expected

expected


Multiple Categories

•So far, we’ve learned how to do inference for a difference in means IF the categorical variable has only two categories

•Today, we’ll learn how to do hypothesis tests for a difference in means across multiple categories


Hypothesis Testing

1.State Hypotheses

2.Calculate a statistic, based on your sample data

3.Create a distribution of this statistic, as it would be observed if the null hypothesis were true

4.Measure how extreme your test statistic from (2) is, as compared to the distribution generated in (3)

test statistic


Cuckoo Birds•Cuckoo birds lay their eggs in the nests of other birds

•When the cuckoo baby hatches, it kicks out all the original eggs/babies

•If the cuckoo is lucky, the mother will raise the cuckoo as if it were her own

http://opinionator.blogs.nytimes.com/2010/06/01/cuckoo-cuckoo/

•Do cuckoo birds found in nests of different species differ in size?





Length of Cuckoo Eggs


Notation

•k = number of groups

•nj = number of units in group j

•n = overall number of units = n1 + n2 + … + nk


Cuckoo Eggs

k = 5n1 = 15, n2 = 60, n3 = 16, n4 = 14, n5 = 15n = 120

Bird Sample Mean

Sample SD

SampleSize

Pied Wagtail 22.90 1.07 15

Pipit 22.50 0.97 60

Robin 22.58 0.68 16

Sparrow 23.12 1.07 14

Wren 21.13 0.74 15

Overall 22.46 1.07 120


Hypotheses

To test for a difference in means across k groups:


Test Statistic

Why can’t use the familiar formula

to get the test statistic?

We need something a bit more complicated…

sample statistic null value

SE


Difference in Means

Whether or not two means are significantly different depends on

• How far apart the means are

• How much variability there is within each group


Difference in Means

group1 group2

02

46

810

group1 group2

02

46

810

1214

group1 group2

4.5

5.0

5.5

6.0

6.5

group1 group2

02

46

810

group1 group2

02

46

810

1214

group1 group2

4.5

5.0

5.5

6.0

6.5

group1 group2

02

46

810

group1 group2

02

46

810

1214

group1 group2

4.5

5.0

5.5

6.0

6.5

1

2

1 2

65

2

X

ssX

1

2

1 2

95

2

X

ssX

1

2

1 2

5

0.6

2s

XXs


Analysis of Variance

•Analysis of Variance (ANOVA) compares the variability between groups to the variability within groups

Total Variability

VariabilityBetween Groups

VariabilityWithin Groups


Analysis of Variance

If the groups are actually different, then

a) the variability between groups should be higher than the variability within groups

b) the variability within groups should be higher than the variability between groups


Discoveries for Today

•How to measure variability between groups?

•How to measure variability within groups?

•How to compare the two measures?

•How to determine significance?








Sums of Squares

•We will measure variability as sums of squared deviations (aka sums of squares)

•familiar?


Sums of Squares

2

1

n

ii

X X

Total Variability



2

1

k

j jj

n X X

2

,11

jnk

i j jij

X X

overall mean

data value i

overall mean

mean in group j mean in

group j

ith data value in group j

Sum over all data values Sum over all groups Sum over all data values


Deviations

Group 1

Group 2

X

Total iX X

Overall Mean

1X

Group 1 Mean

,

Within i j jX X

1

BetweenX X


Sums of Squares

2

1

n

ii

X X

Total Variability



2

1

k

j jj

n X X

2

,11

jnk

i j jij

X X

SST (Total sum of squares)

SSG(sum of squares due to groups)

SSE(“Error” sum of squares)


Cuckoo Birds

2

1

137.19n

ii

SST X X

2

1

35.90k

j jj

SSG n X X

2

,11

101.29jnk

i j jij

X XSSE


Source

Groups

Error

Total

df

k-1

n-k

n-1

Sum ofSquares

SSG

SSE

SST

MeanSquareMSG =

SSG/(k-1)MSE =

SSE/(n-k)

ANOVA TableThe “mean square” is the

sum of squares divided by the degrees of freedom

variability

average variability


ANOVA Table•Fill in the beginnings of the ANOVA table based on the Cuckoo birds data.

Source

Groups

Error

Total

df

k-1

n-k

n-1

Sum ofSquares

SSG

SSE

SST

MeanSquare

MSG = SSG/(k-1)

MSE = SSE/(n-k)

Bird Sample Mean

Sample SD

SampleSize


Pipit 22.50 0.97 60

Robin 22.58 0.68 16

Sparrow 23.12 1.07 14

Wren 21.13 0.74 15

Overall 22.46 1.07 120

SSG = 35.9SSE = 101.20


Source

Groups

Error

Total

df Sum ofSquares

MeanSquare

ANOVA Table•Fill in the beginnings of the ANOVA table based on the Cuckoo birds data.








F-Statistic

•The F-statistic is a ratio of the average variability between groups to the average variability within groups

average between group variability

average within group variability

MSGF

MSE


Source

Groups

Error

Total

df

k-1

n-k

n-1

Sum ofSquares

SSG

SSE

SST

MeanSquareMSG =

SSG/(k-1)MSE =

SSE/(n-k)

FStatistic

MSGMSE

ANOVA Table


Cuckoo Eggs

Source

Groups

Error

Total

df

4

115

119

Sum ofSquares35.90

101.29

137.19

MeanSquare

35.9/4 = 8.97

101.29/115 = 0.88

FStatistic

8.97/0.88= 10.19


F-statisticIf there really is a difference between the groups, we would expect the F-statistic to be

a) Higher than we would observe by random chance

b) Lower than we would observe by random chance








How to determine significance?

We have a test statistic. What else do we need to perform the hypothesis test?

A distribution of the test statistic assuming H0 is true

How do we get this? Two options:1) Simulation2) Distributional Theory


www.lock5stat.com/statkey Simulation

Because a difference would make the F-statistic higher, calculate proportion in the upper tail

An F-statistic this large would be very unlikely to happen just by random chance if the means were all equal, so we have strong evidence that the mean lengths of cuckoo birds in nests of different species are not all equal.

http://www.lock5stat.com/statkey


F-distributionRandomization Distribution

F-statistic

Frequency

0 2 4 6 8 10

0100

200

300

400

500

600

Randomization Distribution

F-statistic

Frequency

0 2 4 6 8 10

0100

200

300

400

500

600

F-distribution


F-DistributionIf the following conditions hold,

1.Sample sizes in each group are large (each nj ≥ 30) OR the data are relatively normally distributed

2.Variability is similar in all groups

3.The null hypothesis is true

then the F-statistic follows an F-distribution

•The F-distribution has two degrees of freedom, one for the numerator of the ratio (k – 1) and one for the denominator (n – k)


Equal Variance•The F-distribution assumes equal within group variability for each group

•As a rough rule of thumb, this assumption is violated if the standard deviation of one group is more than double the standard deviation of another group


F-distributionCan we use the F-distribution to calculate the p-value for the Cuckoo bird eggs?

a) Yesb) Noc) Need more information

Bird Sample Mean

Sample SD

SampleSize


Pipit 22.50 0.97 60

Robin 22.58 0.68 16

Sparrow 23.12 1.07 14

Wren 21.13 0.74 15

Overall 22.46 1.07 120


Length of Cuckoo Eggs


Source

Groups

Error

Total

df

k-1

n-k

n-1

Sum ofSquares

SSG

SSE

SST

MeanSquareMSG =

SSG/(k-1)MSE =

SSE/(n-k)

FStatistic

MSGMSE

p-value

Use Fk-1,n-k

ANOVA Table


Cuckoo Eggs


Source

Groups

Error

Total

df

4

115

119

Sum ofSquares35.90

101.29

137.19

MeanSquare

8.97

0.88

FStatistic10.19

p-value

4.3 × 10-7

ANOVA Table

We have very strong evidence that average length of cuckoo eggs differs for nests of different species

Equal variability Normal(ish) data


Can we use the F-distribution to calculate the p-value for whether there is a difference in average hours spent studying per week by class year at Duke?

a) Yesb) Noc) Need more information

Study Hours by Class Year

Year Sample Mean

Sample SD

SampleSize

First Year 16.06 10.33 72

Sophomore 17.51 9.29 74

Upperclass 19.31 14.74 52


Study Hours by Class Year

Is there a difference in the average hours spent studying per week by class year at Duke?

(a)Yes(b)No(c)Cannot tell from this data(d)I didn’t finish

318

24984

SSG

SSE

Year Sample Mean

Sample SD

SampleSize

First Year 16.06 10.33 72

Sophomore 17.51 9.29 74

Upperclass 19.31 14.74 52


Source

Groups

ErrorTotal

df Sum ofSquares

MeanSquare

F-Statistic

ANOVA Table

p-value


Summary• Analysis of variance is used to test for a difference in means between groups by comparing the variability between groups to the variability within groups

• Sums of squares are used to measure variability

• The F-statistic is the ratio of average variability between groups to average variability within groups

• The F-statistic follows an F-distribution, if sample sizes are large (or data is normal), variability is equal across groups, and the null hypothesis is true


To DoRead Section 8.1 (we are skipping 8.2)

Do Homework 6 (due Monday, 3/24)

Documents

Statistics: Unlocking the Power of Data Lock 5 STAT 101 Dr. Kari Lock Morgan ANOVA SECTION 8.1 Testing for a difference in means across multiple categories