Deck 1: Multiple Comparison Procedures

Question 1

A one-factor fixed-effects ANOVA is performed on data for 5 groups of unequal sizes, and H₀ is rejected at the .05 level of significance. Using the Scheffe' procedure, test the contrast that
at the .05 level of significance given the following information:

Answer 1

Because |t| = 2 < 3.28, we fail to reject the null hypothesis and conclude that the contrast is not significant at the .05 level of significance.

Question 2

A one-factor fixed-effects ANOVA is performed on data from three groups of equal size (n = 15) and equal variances, and H₀ is rejected at the .05 level. The following values were computed: MS_with = 60 and the sample means are.
$\bar{Y}_{.1}$ = 25, $\bar{Y}_{.2}$ = 20, and $\bar{Y}_{.3}$ = 12. Use the Tukey HSD method to test all possible pairwise contrasts ( $\alpha$ = .05)

Answer 2

$\begin{array}{l}J=3, n_{1}=n_{2}=n_{3}=15 ; d f_{\text {with }}=n_{1}+n_{2}+n_{3}-J=42, M S_{\text {with }}=60 ; \alpha=.05 \\\bar{Y}_{.1}=25, \bar{Y}_{.2}=20, \bar{Y}_{.3}=12\end{array}$
Because $J=3$ , there are three possible pairwise contrasts: 1 vs. 2,1 vs. 3,2 vs. 3 .
For each contrast, $q=\frac{\bar{P}_{. j}-P_{i j}}{s_{\psi}}$ , where $\bar{Y}_{i j}$ and $\bar{Y}_{i j}$ are two group means to be compared.
Because standard error $s \Psi=\sqrt{\frac{M s_{\text {with }}}{n}}=\sqrt{\frac{60}{15}}=2$
$q_{1}=\left(\frac{P_{-1}-P_{t 2}}{s_{\psi}}\right)=(25-20) / 2=2.5$
- $q_{2}=\left(\frac{P_{s 1}-P_{s s}}{s_{\psi}}\right)=(25-12) / 2=6.5$
- $q_{1}=\left(\frac{P_{2}-P_{s}}{s_{\psi}}\right)=(20-12) / 2=4$ Using Tukey HSD, the critical values are $\pm$ _$\alpha$ q_{df
(with), J} = _.05q_{60, 3} = $\pm$ 3.40.
q₁ < critical q, so $\mu$ ₁ - $\mu$ ₂ is not statistically significant at $\alpha$ = .05.
q₂ > critical q, q₃ > critical q, so $\mu$ ₁ - $\mu$ ₃, $\mu$ ₂ - $\mu$ ₃ are statistically significant at $\alpha$ = .05.

Question 3

Which of the following linear combinations of population means is a legitimate contrast?

A) $\mu$ ₁- $\mu$ ₂/2+ $\mu$ ₃/2
B) ( $\mu$ ₁+ $\mu$ ₂/2+ $\mu$ ₃
C) ( $\mu$ ₁+ $\mu$ ₂)+ $\mu$ ₃
D) $\mu$ ₁- $\mu$ ₂- $\mu$ ₃

Answer 3

( $\mu$ ₁+ $\mu$ ₂)+ $\mu$ ₃

Question 4

If J = 3, which of the following sets of contrasts is orthogonal?

A) $\mu$ ₁- $\mu$ ₂, $\mu$ ₁- $\mu$ ₃
B) $\mu$ ₂- $\mu$ ₃, $\mu$ ₂+ $\mu$ ₃
C) $\mu$ ₁- $\mu$ ₃, $\mu$ ₁-( $\mu$ ₂+ $\mu$ ₃)/2
D) $\mu$ ₁- $\mu$ ₃, $\mu$ ₂-( $\mu$ ₂+ $\mu$ ₃)/2

Question 5

A researcher used Fisher's LSD to test three contrasts: $\mu$ ₁- $\mu$ ₂, $\mu$ ₁- $\mu$ ₃, $\mu$ ₁-( $\mu$ ₂+ $\mu$ ₃)/2. Evaluate this practice.

A) This practice is problematic because Fisher's LSD is always too liberal.
B) This practice is problematic because Fisher's LSD can only be used to test simple
Contrasts.
C) This practice is problematic because the contrasts are not orthogonal.
D) I do not see any problem with this practice. It is great!

Question 6

In an experiment, $\bar{Y}_{1}$ =10, $\bar{Y}_{2}$ =20, and $\bar{Y}_{3}$ =40. Tukey HSD shows that $\mu$ ₂- $\mu$ ₁ is a significant contrast. What will you find out about $\mu$ ₃- $\mu$ ₁ and $\mu$ ₃- $\mu$ ₂ if the same procedure and same $\alpha$ level are used?

A) Only $\mu$ ₃- $\mu$ ₁ is significant.
B) Only $\mu$ ₃- $\mu$ ₂ is significant.
C) Both contrasts are significant.
D) None of the contrasts are necessarily significant.

Question 7

In an experiment,? $\bar{Y}_{1}$ =10, $\bar{Y}_{2}$ =200, and $\bar{Y}_{3}$ =4000. Which pairwise contrast will necessarily be significant

A) $\mu$ ₂- $\mu$ ₁
B) $\mu$ ₃- $\mu$ ₁
C) $\mu$ ₃- $\mu$ ₂
D) None of the contrasts will necessarily be significant.