Comparing more than two means Analysis of Variance

1
Comparing more than two means(Analysis of
Variance)

• Chapter 22

2
Example

• Using data from an alumni survey, the starting
  salaries (in thousands of dollars) and the area
  of specialization of recent graduates of
  Tulipville University are studied.
• Areas of specialty include computer science,
  business, liberal arts, and education.
• Does it appear that starting salary varies
  according to area of specialty?

3
Data
4
Side by Side Box Plots

5
Sample Size

• There are between three to seven data points in
  each box plot.
• Suppose instead there are 100 data points in each
  box plot. In this case, would the observed
  variation be more or less significant?

6
Comparing variability.

• The box plots on right are box plots of our data.
  The box plots on left are box plots for another
  university.
• Assuming the sample sizes are the same, in which
  graph are differences between the mean more
  significant?

7
Comparing means.

• To determine if differences in means are
  significant, one must
• Consider what the sample sizes are.
• Consider what the variation is between data in
  the same sample (underlying variation) and
  compare this to the variation between sample
  means.
• Analysis of variance (ANOVA) is a procedure which
  does this. The main idea is to compare
  underlying variation to variation between sample
  means. Furthermore, sample size is taken into
  account.

8
Hypotheses

• Analysis of variance is an overall test. It
  tests the hypothesis
• Null Hypothesis.
• Alternate Hypothesis.
• If we reject the null, we will do follow up
  analysis.

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Notation

• N
• I
• Sample means.
• For each group.
• Overall
• Sample standard deviations.
• For each group.

10
Measuring variation.

• MSE measures the variation between data that come
  form the same sample.
• MSG measures the variation between sample means.

11
The test statistic

• The two types of variation are compared by taking
  their ratio.
• What is F statistic for our example?
• Sketch of F distribution. The larger F is, the
  more significant the variation between sample
  means is. (P-value is area to the right.)

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Degrees of freedom

• The f statistic has two degrees of freedom.
• Numerator degrees of freedom are I-1.
• Denominator degrees of freedom are N-I.
• What are the degrees of freedom in our example?

13
P-value / Conclusion.

• Use table D to estimate P-value.
• What is conclusion?

14
Minitab output
Analysis of Variance Source DF SS
MS F P Factor 3 508.99
169.66 17.37 0.000 Error 16 156.32
9.77 Total 19 665.31
Individual 95 CIs For Mean
Based on Pooled
StDev Level N Mean StDev
--------------------------------- Computer
6 33.633 3.583
(---------) Business 7 34.586 2.827
(--------) Liberal
3 23.567 3.550 (--------------) Early
Ch 4 23.625 2.514 (-----------)

--------------------------------- Pooled
StDev 3.126 20.0 25.0
30.0 35.0
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Follow up analysis.

• Using the original box plots and/or the
  confidence intervals in the Minitab output, what
  do the variations from the null hypothesis appear
  to be?

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Assumptions for ANOVA

• SRS are taken from each population. These SRS
  are independent (like in two sample t).
• Each population has a normal distribution. The
  test is robust against this assumption since what
  really matters is that the sampling distribution
  of the means be normal. If there are no outliers
  and the distributions are roughly symmetric you
  can safely use ANOVA with sample sizes as small
  as 4 or 5. 
• All of the populations have the same standard
  deviation s, whose value is unknown. Check
  reasonableness by showing the largest sample
  standard deviation is no more than two times the
  smallest sample standard deviation.

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Summary

• ANOVA is an overall test used to test the null
  that all the populations means are equal against
  the alternative that not all of the means are
  equal.
• Idea is to compare variation bewteen means (MSG)
  to variation of data within the same sample
  (MSE). The test statistic is the ratio fF
  MSG/MSE. (You will not need to compute by hand,
  but should be able to interpret the various
  pieces of Minitab output.)
• The larger the F test statistic, the smaller the
  P-value. If the P-value is small and the null
  hypothesis is rejected, follow up analysis should
  be done.
• One should always check the assumptions of a
  statistical test.