STA 6166 UNIT 3 Section 2 Answers
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Social and Education

1. Using the data and analysis from the previous exercise (Unit 3 Section 1), perform a multiple comparison analysis, with experimentwise Type I error rate of 0.05, using each of the following procedures:
1. Fisher's LSD
2. Student-Newman-Keuls
3. Tukey's W
4. Duncan's MCP
5. Waller-Duncan

If you transformed the data prior to the analysis of variance, the multiple comparison procedures should be performed on the transformed data as well.

2. Using these same data, test the contrast where t=1 for age group A, t=2 for age group B, t=3 for age group C.

Only the appropriate statistics, critical values and results are given for each procedure. This analysis was run in SAS. Other packages only provide limited multiple comparison procedures. You can compare these results to those you obtained by hand calculations.

```
OVERVIEW
Means with the same letter are not significantly different
Level of            ----------attention----------
group         N             Mean          Std Dev  LSD* TUKEY SNK DUNCAN WALLER*
A            10       22.9000000       13.1271559   A     A    A     A     A
B            10       19.6000000       10.8545945   A     A    A     A     A
C            10       21.9000000       13.3287492   A     A    A     A     A
* Technically, since the overall F-test was not significant, we would never run the
Fisher's LSD or the WALLER procedure.

t Tests (LSD)
Alpha                            0.05
Error Degrees of Freedom           27
Error Mean Square            155.9333
Critical Value of t           2.05183 <- From T table.
Least Significant Difference   11.458

Tukey's Studentized Range (HSD) Test
Alpha                                   0.05
Error Degrees of Freedom                  27
Error Mean Square                   155.9333
Critical Value of Studentized Range  3.50643  <- From Table 10 (q)
Minimum Significant Difference        13.846

Student-Newman-Keuls Test
Alpha                        0.05
Error Degrees of Freedom       27
Error Mean Square        155.9333
Number of Means              2              3
Critical Range       11.458488      13.846314  <- By equation using values
from Table 10  q=(2.9017, 3.506

Duncan's Multiple Range Test
Alpha                        0.05
Error Degrees of Freedom       27
Error Mean Square        155.9333
Number of Means          2          3
Critical Range       11.46      12.04  <- By equation using values from Duncan's table
q'=(2.90211,3.048997)

Waller-Duncan K-ratio t Test
Kratio                              100
Error Degrees of Freedom             27
Error Mean Square              155.9333
F Value                            0.18  <- From AOV
Critical Value of t             2.62873  <- From Waller Table
Minimum Significant Difference    14.68

No Transformation needed.

Contrast                  DF   Contrast SS   Mean Square  F Value  Pr > F
l1                         1    2.81666667    2.81666667     0.02  0.8941
```
• Let us take the output apart and see what we have.
• The contrast SS is the contrast sums of squares.
• This is the equation at the bottom of page 433 in the book.
• The mean square is the contrast SS divided by the degrees of freedom which is one, since there is only one equation in the contrast being tested.
• The F-value is the F-statistic as calculated from the equation on page 436, ie. the Contrast mean square divided by the Mean Square error from the AOV table.
• If you divide the Contast SS by the MSE you get the F-statistic. The pr > F value is the p-value associated with the observed F-value from an F table with ndf=1, ddf=error df .
• Hence everything you need to perform the requested test was here.