STA 6166 UNIT 3 Section 2 Answers
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# Toxicology and Health

• 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.

• Using these same data, test the contrast where t=1 for group A, t=2 for group B, etc.

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            -------------fat-------------
treatment     N             Mean          Std Dev LSD  TUKEY  SNK  DUNCAN WALLER
A             5       2.62800000       0.41227418  B     A     A     B      B
B             5       3.19800000       0.30605555   A    A     A      A      A
C             5       3.10600000       0.38115614   A    A     A      A     BA
D             5       2.91600000       0.18942017  BA    A     A     BA     BA

t Tests (LSD)
Alpha                            0.05
Error Degrees of Freedom           16
Error Mean Square              0.1112  <- From AOV
Critical Value of t           2.11991  <- From t-table
Least Significant Difference   0.4471  <- From equation

Tukey's Studentized Range (HSD) Test
Alpha                                   0.05
Error Degrees of Freedom                  16
Error Mean Square                     0.1112
Critical Value of Studentized Range  4.04609  <- q From Table 10
Minimum Significant Difference        0.6034  <- From Equation

Student-Newman-Keuls Test
Alpha                        0.05
Error Degrees of Freedom       16
Error Mean Square          0.1112
Number of Means          2          3          4
Critical Range    0.447094  0.5441991  0.6033972 <- From Equation using q values
from Table 10 q=(3.00, 3.65, 4.05)

Duncan's Multiple Range Test
Alpha                        0.05
Error Degrees of Freedom       16
Error Mean Square          0.1112
Number of Means          2       3       4
Critical Range       .4471   .4688   .4824  <- From Equation using q' values from
Duncan's Table q'=(2.998, 3.1435, 3.2347)
Waller-Duncan K-ratio t Test
Kratio                              100
Error Degrees of Freedom             16
Error Mean Square                0.1112
F Value                            2.85  <- From AOV
Critical Value of t             2.36065  <- From Waller Table, k=100
Minimum Significant Difference   0.4979  <- From equation

Response was not transformed.

Contrast      DF    Contrast SS    Mean Square   F Value   Pr > F
l1             1     0.74370667     0.74370667      6.69   0.0199

```
• 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.