TTests and Analysis of Variance

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T-Tests and Analysis of Variance

• Denisa Olteanu
• July 28th, 2009
• LISA Short-Course

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One Sample T-Test
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One Sample T-Test

• Used to test whether the population mean is
  different from a specified value.
• Example Is the mean amount of soda in a 20 oz.
  bottle different from 20 oz?

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Step 1 Formulate the Hypotheses

• The population mean is not equal to a specified
  value.
• H0 µ µ0
• Ha µ ? µ0
• The population mean is greater than a specified
  value.
• H0 µ µ0
• Ha µ gt µ0
• The population mean is less than a specified
  value.
• H0 µ µ0
• Ha µ lt µ0

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Step 2 Check the 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 the
  appropriate alternative hypothesis.
• Step 5 Write a conclusion.

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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 Hypotheses
• H0 µ 3.5 cm
• Ha µ ? 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 not
  significantly different from 3.5 cm.

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Two Sample T-Test
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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 Formulate the Hypotheses

• The population means of the two groups are not
  equal.
• H0 µ1 µ2
• Ha µ1 ? µ2
• The population mean of group 1 is greater than
  the population mean of group 2.
• H0 µ1 µ2
• Ha µ1 gt µ2
• The population mean of group 1 is less than the
  population mean of group 2.
• H0 µ1 µ2
• Ha µ1 lt µ2

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Step 2 Check the 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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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 µsetosa µversicolor
• Ha µsetosa ? µversicolor

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JMP

• Steps 2-4
• JMP Demonstration
• Analyze ? Fit Y By X
• Y, Response Sepal Width
• X, Factor Species

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JMP Output

• Step 5 Conclusion There is strong evidence
  (p-value lt 0.0001) that the mean sepal widths for
  the two varieties are different.

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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 Formulate the Hypotheses

• The population mean difference is not equal to
  zero.
• H0 µdifference 0
• Ha µdifference ? 0
• The population mean difference is greater than
  zero.
• H0 µdifference 0
• Ha µdifference gt 0
• The population mean difference is less than a
  zero.
• H0 µdifference 0
• Ha µdifference lt 0

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Step 2 Check the assumptions

• 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

• Step 3 Calculate the test Statistic

• Where d bar is the mean of the differences and sd
  is the standard deviation of the differences.
• Step 4 Calculate the p-value.
• Step 5 Write a conclusion.

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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 µAfter - Before 0
• Ha µ After - Before gt 0

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Paired T-Test Example

• Steps 2-4
• JMP Analysis
• Create a new column of After Before
• Analyze ? Distribution
• Y, Columns After Before
• Test Mean
• Specify Hypothesized Mean 0

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JMP Output
Step 5 Conclusion There is no evidence that the
fitness program increases flexibility.
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One-Way Analysis of 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 ANOVA F 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 Model
In other words, for each group the observed value
is the group mean plus some random variation.
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One-Way ANOVA Hypothesis

• Step 1 We test whether there is a difference in
  the means.

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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
A B C
A B
C Medical Treatment
Compare the variation within the samples to the
variation between the samples.
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ANOVA Test Statistic
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

• 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

• Mean square error, MSE, measures the variability
  within the groups.
• SSE stands for sums of squares error.

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Steps 4-5

• Step 4 Calculate the p-value.
• Step 5 Write a conclusion.

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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 µDrug A µDrug B µDrug C
• Ha The µi 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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Example 1 JMP Output and Conclusion

• Step 5 Conclusion There is strong evidence that
  the drugs are not all the same.

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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

for all pairs of factor levels. Tukeys HSD
controls the overall type I error.
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JMP Output

• The JMP output shows that drugs A and C are
  significantly different.

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Analysis of 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

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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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SAS and 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
  

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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.