ANOVA.short .Course

7/27/2019 ANOVA.short .Course

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T-TESTSAND

ANALYSISOF VARIANCEJennifer Kensler


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ONE SAMPLE T-TEST


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STEP 2: CHECKTHE ASSUMPTIONS

The sample is random.

The population from which the sample is drawn is

either normal or the sample size is large.


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STEPS 3-5

Step 3: Calculate the test statistic:

Where

Step 4: Calculate the p-value based on theappropriate alternative hypothesis.

Step 5: Write a conclusion.

ns

yt

/

0

1

1

2

n

yy

s

n

i

i


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IRIS EXAMPLE

A researcher would like to know whether the mean

sepal width of a variety of irises is different from 3.5

cm.

The researcher randomly measures the sepal width

of 50 irises.

Step 1: HypothesesH0: = 3.5 cm

Ha: 3.5 cm


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JMP

Steps 2-4:

JMP Demonstration

Analyze Distribution

Y, Columns: Sepal Width

Test Mean

Specify Hypothesized Mean: 3.5


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JMP OUTPUT

Step 5 Conclusion: The sepal width is notsignificantly different from 3.5 cm.


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TWO SAMPLE T-TEST

Two sample t-tests are used to determine whether

the mean of one group is equal to, larger than or

smaller than the mean of another group.

Example: Is the mean cholesterol of people taking

drug A lower than the mean cholesterol of people

taking drug B?


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STEP 1: FORMULATETHE HYPOTHESES

The population means of the two groups are not

equal.

H0: 1 = 2

Ha

: 1

2

The population mean of group 1 is greater than the

population mean of group 2.

H0: 1 = 2

Ha:

1>

2 The population mean of group 1 is less than the

population mean of group 2.

H0: 1 = 2

Ha: 1 < 2


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STEP 2: CHECKTHE ASSUMPTIONS

The two samples are random and independent.

The populations from which the samples are drawn

are either normal or the sample sizes are large.

The populations have the same standard deviation.


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STEPS 3-5

Step 3: Calculate the test statistic

where

Step 4: Calculate the appropriate p-value.

Step 5: Write a Conclusion.

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21

11

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s

yyt

p

2

)1()1(

21

2

22

2

11

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snsnsp


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TWO SAMPLE EXAMPLE

A researcher would like to know whether the mean

sepal width of a setosa irises is different from the

mean sepal width of versicolor irises.

Step 1 Hypotheses:

H0: setosa = versicolor

Ha: setosa versicolor


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JMP

Steps 2-4:

JMP Demonstration:

Analyze Fit Y By X

Y, Response: Sepal WidthX, Factor: Species


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PAIRED T-TEST


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PAIRED T-TEST

The paired t-test is used to compare the means of

two dependent samples.

Example:A researcher would like to determine if background

noise causes people to take longer to complete

math problems. The researcher gives 20 subjects

two math tests one with complete silence and one

with background noise and records the time each

subject takes to complete each test.


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STEP 1: FORMULATETHE HYPOTHESES

The population mean difference is not equal to zero.

H0: difference = 0

Ha: difference 0

The population mean difference is greater than zero.H0: difference = 0

Ha: difference > 0

The population mean difference is less than a zero.

H0: difference = 0

Ha: difference < 0


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STEP 2: CHECKTHEASSUMPTIONS

The sample is random.

The data is matched pairs.

The differences have a normal distribution or the

sample size is large.


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STEPS 3-5

ns

dt

d /

0

Where d bar is the mean of the differences and sd is

the standard deviations of the differences.

Step 4: Calculate the p-value.


Step 3: Calculate the test Statistic:


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PAIRED T-TEST EXAMPLE

A researcher would like to determine whether a

fitness program increases flexibility. The researcher

measures the flexibility (in inches) of 12 randomly

selected participants before and after the fitness

program.

Step 1: Formulate a Hypothesis

H0: After-Before = 0

Ha: After-Before > 0


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PAIRED T-TEST EXAMPLE

Steps 2-4:

JMP Analysis:

Create a new column of After Before

Analyze DistributionY, Columns: After Before

Test Mean

Specify Hypothesized Mean: 0


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JMP OUTPUT

Step 5 Conclusion: There is not evidence that thefitness program increases flexibility.


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ONE-WAY ANALYSISOF VARIANCE


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ONE-WAY ANOVA

ANOVA is used to determine whether three or more

populations have different distributions.

A B C

Medical Treatment


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ANOVA STRATEGY

The first step is to use the ANOVAF test to

determine if there are any significant differences

among means.

If the ANOVA F test shows that the means are not

all the same, then follow up tests can be performed to

see which pairs of means differ.


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ONE-WAY ANOVA HYPOTHESIS

Step 1: We test whether there is a difference in the

means.

equal.allnotareThe:

: 210

ia

r

H

H


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STEP 2: CHECK ANOVA ASSUMPTIONS

The samples are random and independent of each

other.

The populations are normally distributed.

The populations all have the same variance.

The ANOVA F test is robust to the assumptions of

normality and equal variances.


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STEP 3: ANOVA F TEST

Compare the variation within the samples to the

variation between the samples.

A B C A B C

Medical Treatment


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ANOVA TEST STATISTIC

MSE

MSG

GroupswithinVariation

GroupsbetweenVariationF

Variation within groups small

compared with variation

between groups

Large F

Variation within groups large

compared with variation

between groups Small F


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MSG

1-r

)(n)(n)(n

1-r

SSGMSG

2

1r

2

22

2

11

yyyyyy

The mean square for groups, MSG, measures the

variability of the sample averages.

SSG stands for sums of squares groups.


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MSE

1

)(

s

Where

r-n

1)s-(n1)s-(n1)s-(n

r-n

SSEMSE

1

i

2

rr

2

22

2

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i

n

j

iij

n

yyi

Mean square error, MSE, measures the variabilitywithin the groups.

SSE stands for sums of squares error.


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STEPS 4-5

Step 4: Calculate the p-value.



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ANOVA EXAMPLE

A researcher would like to determine if three drugs

provide the same relief from pain.

60 patients are randomly assigned to a treatment

(20 people in each treatment).

Step 1: Formulate the Hypotheses

H0: DrugA = Drug B = Drug C

Ha : The i are not all equal.


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STEPS 2-4

JMP demonstration

Analyze Fit Y By X

Y, Response: Pain

X, Factor: Drug


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FOLLOW-UP TEST

The p-value of the overall F test indicates that level

of pain is not the same for patients taking drugs A,

B and C.

We would like to know which pairs of treatments

are different.

One method is to use Tukeys HSD (honestly

significant differences).


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TUKEY TESTS

Tukeys test simultaneously tests

JMP demonstration

Oneway Analysis of Pain By Drug

Compare Means All Pairs, Tukey HSD

'a

'0

:H

:H

ii

ii

for all pairs of factor levels. Tukeys HSD controls

the overall type I error.


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ANALYSISOF COVARIANCE


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ANALYSIS OF COVARIANCE (ANCOVA)

Covariates are variables that may affect the

response but cannot be controlled.

Covariates are not of primary interest to the

researcher.

We will look at an example with two covariates, the

model is

ijiijy covariates


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ANCOVA EXAMPLE

Consider the previous example where we tested

whether the patients receiving different drugs

reported different levels of pain. Perhaps age and

gender may influence the efficacy of the drug. We

can use age and gender as covariates.

JMP demonstration

Analyze Fit Model

Y: Pain

Add: Drug

Age

Gender


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JMP OUTPUT


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CONCLUSION

The one sample t-test allows us to test whether the

mean of a group is equal to a specified value.

The two sample t-test and paired t-test allows us to

determine if the means of two groups are different.

ANOVA and ANCOVA methods allow us to

determine whether the means of several groups are

statistically different.


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SASAND SPSS

For information about using SAS and SPSS to do

ANOVA:

http://www.ats.ucla.edu/stat/sas/topics/anova.htm

http://www.ats.ucla.edu/stat/spss/topics/anova.htm
http://www.ats.ucla.edu/stat/sas/topics/anova.htmhttp://www.ats.ucla.edu/stat/spss/topics/anova.htmhttp://www.ats.ucla.edu/stat/spss/topics/anova.htmhttp://www.ats.ucla.edu/stat/sas/topics/anova.htm


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REFERENCES

Fishers Irises Data (used in one sample and two

sample t-test examples).

Flexibility data (paired t-test example):

Michael Sullivan III. Statistics Informed Decisions

Using Data. Upper Saddle River, New Jersey:

Pearson Education, 2004: 602.

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ANOVA.short .Course