ANOVA3/19/12

• Mini Review of simulation versus formulas and theoretical distributions

• Analysis of Variance (ANOVA) to compare means:testing for a difference in means between multiple groups

Section 8.1 Professor Kari Lock MorganDuke University

• Anonymous Midterm Evaluation (due TODAY, 5pm)

• Project 1 (due Thursday, 3/22, 5pm)

• Homework 7 (due Monday, 3/26)NO LATE HOMEWORK ACCEPTED!Turn in by Friday, 3/23, 5pm to get it graded

before Exam 2.

To Do

Two Options for p-valuesWe have learned two ways of calculating p-values:

The only difference is how to create a distribution of the statistic, assuming the null is true:

1)Simulation (Randomization Test): • Directly simulate what would happen, just by

random chance, if the null were true

2)Formulas and Theoretical Distributions: • Use a formula to create a test statistic for which

we know the theoretical distribution when the null is true, if sample sizes are large enough

Two Options for IntervalsWe have learned two ways of calculating intervals:

1)Simulation (Bootstrap): • Assess the variability in the statistic by

creating many bootstrap statistics

2)Formulas and Theoretical Distributions: • Use a formula to calculate the standard error

of the statistic, and use the normal or t-distribution to find z* or t*, if sample sizes are large enough

Pros and Cons1)Simulation Methods

PROS:• Methods tied directly to concepts, emphasizing conceptual

understanding• Same procedure for every statistic• No formulas or theoretical distributions to learn and

distinguish between• Works for any sample size• Minimal math needed

CONS:• Need entire dataset (if quantitative variables)• Need a computer• Newer approach, so different from the way most people do

statistics

Pros and Cons2)Formulas and Theoretical Distributions

PROS:• Only need summary statistics• Only need a calculator• The approach most people take

CONS:• Plugging numbers into formulas does little for conceptual

understanding• Many different formulas and distributions to learn and

distinguish between• Harder to see the big picture when the details are different for

each statistic• Doesn’t work for small sample sizes• Requires more math and background knowledge

Two Options

• If the sample size is small, you have to use simulation methods

• If the sample size is large, you can use whichever method you prefer

• It is redundant to use both methods, unless you want to check your answers

Accuracy• The accuracy of simulation methods depends on the number of simulations (more simulations = more accurate)

• The accuracy of formulas and theoretical distributions depends on the sample size (larger sample size = more accurate)

• If the sample size is large and you have generated many simulations, the two methods should give essentially the same answer

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

Hypotheses

To test for a difference in means across k groups:

0 1 2: ...: At least one

k

a i j

HH

Test Statistic

Why can’t use the familiar formula

to get the test statistic?

•More than one sample statistic•More than one null value

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

XX

s

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?

Notation•k = number of groups

•nj = number of units in group j

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

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

VariabilityBetween Groups

VariabilityWithin Groups

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

VariabilityBetween Groups

VariabilityWithin Groups

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)

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

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?

F-Statistic

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

•The F-statistic is the test statistic for testing for a difference in means across more than 2 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

Jumping and Bone DensityDoes jumping improve bone density?30 rats were randomized to three treatment groups:

No jumping (10 rats - group 1) 30 cm jump (10 rats - group 2)60 cm jump (10 rats - group 3)

Rats performed 10 jumps per day, 5 days per week. Bone density was measured after 8 weeks.

Jumping and Bone Density

Control Highjump Lowjump

560

580

600

620

640

660

mean sd nControl 601.1 27.36360 10Lowjump 612.5 19.32902 10Highjump 638.7 16.59351 10

0 :: At least one

NJ LJ HJ

a i j

HH

Jumping and Bone Density

2

1

20013.4n

ii

SST X X

2

1

7433.9k

j jj

SSG n X X

2

,11

12579.5jnk

i j jij

X XSSE

Source

Groups

Error

Total

df

k-1 = 2

n-k =27

n-1=29

Sum ofSquares7433.9

12579.5

20013.4

MeanSquare

7433.9/2 =3716.9

12579.5/27 = 465.9

FStatistic

3716.9465.9

=7.98

ANOVA Table

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?

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

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

If the null hypothesis is true, what kind of F-statistics would we observe just by random chance?

Simulation•Rerandomize (reallocate) the rats to treatment groups, keeping response values fixed

•Calculate the F-statistic

•Repeat this simulation many times to form a randomization distribution

•Calculate the p-value as the proportion as extreme or more extreme than the observed F-statistic

F-distributionRandomization Distribution

F-statistic

Frequency

0 2 4 6 8 10

0100

200

300

400

500

600

p-value = 0.002

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

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)http://www.capdm.com/demos/software/html/capdm/qm/fdist/usage.html

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 jumping and bone density F-statistic?

a) Yesb) Noc) I need more information

mean sd nControl 601.1 27.36360 10Lowjump 612.5 19.32902 10Highjump 638.7 16.59351 10

Jumping and Bone Density

Control Highjump Lowjump

560

580

600

620

640

660

F-distribution p-values1)Online applet: http://www.danielsoper.com/statcalc3/calc.aspx?id=7

2)RStudio:

> pf(7.98,2,27,lower.tail=FALSE) [1] 0.001892532

3) TI-83:2nd DISTR 9:Fcdf( 7.98, 9999, 2, 27

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

Source

Groups

ErrorTotal

df

2

27

29

Sum ofSquares7433.9

12579.5

20013.4

MeanSquare3716.9

465.9

F-Statistic

7.98

ANOVA Table

p-value

0.0019

We have strong evidence that jumping does increase bone density, at least in rats.

Study Hours by Class YearIs 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

Source

Groups

ErrorTotal

df

2

195

197

Sum ofSquares

318

24984

20013.4

MeanSquare

159

128.1

F-Statistic

1.24

ANOVA Table

p-value

0.29

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