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 Scheffé procedure, test the contrast that $\bar{Y}_{.1}-\frac{\bar{Y}_{.2}+\bar{Y}_{.3}}{2}$ at the .05 level of significance given the following information: df_with = 30, = 5, n₁ = 6, = 7.5, n₂ = 6, = 8.5, n₃ = 6, and MS_with = 9.

Accepted Answer

For the contrast, c₁= 1, c₂= -1/2, c₃= -1/2; n₁= n₂ = n₃ = 6; df_with = 30; MS_with = 9; = 5, = 7.5, = 8.5; $\alpha$= .05; J = 5. Therefore, contrast $\Psi$= 5 -(7.5 + 8.5)/2 = -3; standard error of contrast: s_$\Psi$ = $\sqrt{M S_{\text {with }}\left(\frac{c_{1}^{2}}{n_{1}}+\frac{c_{2}^{2}}{n_{2}}+\frac{c_{3}^{2}}{n_{3}}\right)}=\sqrt{9\left(\frac{1}{6}+\frac{1}{24}+\frac{1}{24}\right)}=3 / 2$ t = $\Psi$_$\Psi$= -3/(3/2) = -2. critical values: 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

Dr. Guinea Pigg would like to test if the amount of caffeine intake would affect how well laboratory rats performed on a specific task. He randomly assigned 40 rats to five groups (n = 8). Each rat walked the same maze after drinking caffeinated water. The amount of caffeine the rats took and the number of mistakes they made in walking the maze were recorded in the following table.

$\begin{array}{crrrrr}
\text { Dosage } & 10 \mathrm{mg} & 20 \mathrm{mg} & 30 \mathrm{mg} & 40 \mathrm{mg} & 50 \mathrm{mg} \
\hline & 16 & 16 & 2 & 5 & 7 \
& 5 & 10 & 9 & 8 & 12 \
& 11 & 7 & 11 & 1 & 14 \
\text { Number of } & 23 & 12 & 13 & 5 & 16 \
\text { mistakes } & 18 & 7 & 10 & 8 & 11 \
& 12 & 4 & 13 & 11 & 9 \
& 12 & 23 & 9 & 9 & 19 \
& 19 & 13 & 9 & 9 & 24 \
\hline
\end{array}$
 Using the data, conduct a trend analysis to test if there is any linear, quadratic, or cubic trend ($\alpha$ = .05).

Accepted Answer

Procedure:
Create a data set with two variables, Dosage and Mistakes. The data set should have 40 cases, each representing one rat.

Go to Analyze $\rightarrow$General Linear Model $\rightarrow$ Univariate. 
Select Mistakes as the Dependent Variable and Dosage as the Fixed Factor. Go to Contrasts and select Polynomial in the pull-down menu under Change Contrasts. Click Change, then Continue. 
To plot the trend, go to Plots, select Dosage to Horizontal Axis. Click Add, then Continue.

Selected SPSS Output:

A trend analysis was conducted to determine if there was any significant linear, quadratic, or cubic trend in the number of mistakes made as the dosage of caffeine increases ($\alpha$ = .05). From the summary table, we see that the linear trend was not significant (p =.324), nor was the cubic trend significant (p =.131). However, the quadratic trend was statistically significant (p = .005). 
The profile plot suggested that the number of mistakes made decreased as the caffeine intake increased from 10 mg to 40 mg, yet increased drastically as the intake further increased to 50 mg.

Question 3

How many possible pairwise contrasts are there if J = 3?
A) 3
B) 4
C) 5
D) 6

Accepted Answer

The number of possible pairwise contrasts when J = 3 (where J represents the number of groups) is calculated using the formula J(J - 1) / 2. Substituting 3 for J gives 3(3 - 1) / 2 = 3.

Question 4

Which of the following linear combinations of population means is a legitimate contrast? A) $\mu$ ₁-$\mu$ S1U1B12S1U1B0/2+$\mu$ S1U1B13S1U1B0/2 B) ($\mu$ S1U1B11S1U1B0+$\mu$ S1U1B12S1U1B0)/2+$\mu$ S1U1B13S1U1B0 C) ($\mu$ S1U1B11S1U1B0+$\mu$ S1U1B12S1U1B0)/2-$\mu$ S1U1B13S1U1B0 D) $\mu$S1U1B1 1S1U1B0-$\mu$ S1U1B12S1U1B0-$\mu$ S1U1B13S1U1B0

Accepted Answer

For a legitimate contrast, the sum of the contrast coefficients must be equal to 0.)

Question 5

If J = 3, which of the following sets of contrasts is orthogonal?
A) $\mu$ S1U1B11S1U1B0-$\mu$ S1U1B12S1U1B0/2+$\mu$ S1U1B13S1U1B0/2
B) ($\mu$ 1+$\mu$ 2)/2+$\mu$ 3
C) ($\mu$ 1+$\mu$ 2)/2-$\mu$ 3
D) $\mu$ 1-$\mu$ 2-$\mu$ 3

Accepted Answer

For a, $\Sigma$c_jc_j'= 2. For b, $\mu$ ₂+$\mu$ S1U1B13S1U1B0is not a legitimate contrast. For c, $\Sigma$c_jc_j'= 3/2. For d, both contrasts are legitimate and $\Sigma$c_jc_j'= 0.)

Question 6

A one-factor fixed-effects ANOVA results in the following: F (1, 17) = .011, p ₌.918. Which post-hoc procedure should be used as a follow-up? A) Scheffé method B) Tukey HSD method C) Fisher LSD method D) None of the above

Accepted Answer

When the omnibus ANOVA F is not significant, there is no need to conduct post-hoc MCPs.)

Question 7

Applying the Dunn procedure, what is the per contrast alpha if the nominal alpha is 0.05 and there are three contrasts?
A) .02
B) .05
C) .10
D) .15

Accepted Answer

For the Dunn procedure, the per contrast alpha is calculated as the nominal alpha divided by the number of contrasts thus .05/3 = .0167 or approximately .02.)

Question 8

A researcher used Fisher's LSD to test three contrasts: $\mu$ ₁-$\mu$ ₂,$\mu$ S1U1B11S1U1B0-$\mu$ S1U1B13S1U1B0,$\mu$ S1U1B11S1U1B0-($\mu$ S1U1B12S1U1B0+$\mu$ S1U1B13S1U1B0)/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 be used only 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!

Accepted Answer

Fisher's LSD is designed to test pairwise contrasts. It has good control of Type I error when J = 3, but becomes too liberal when J > 3.)

Question 9

A researcher conducts a study of different instructional methods. The researcher wants to examine if student scores will increase as the instructional time increases. She randomly assigned 60 students to three classes. Each class spent 4 hours, 5 hours, and 6 hours per day, respectively, on instruction. Then she compared the average scores of the three classes. Which procedure is the most appropriate for this research question?
A) Dunnett method
B) Dunn (Bonferroni) method
C) Tukey HSD
D) Trend analysis

Accepted Answer

The groups represent different quantitative levels of a ratio-scaled variable, i.e., instructional time.)

Question 10

A researcher conducts a study of different instructional methods. The researcher finds that student scores increase significantly when instructional time increases, so she predicts that scores will be even higher if the instructional time is 12 hours per day. Evaluate the prediction.
A) The prediction is problematic because 4, 5, 6, and 12 are not equally spaced.
B) The prediction is problematic because the classes with longer instructional time may not have the same number of students.
C) The prediction is problematic because we have no way of knowing what the trend is outside of the range of 4 to 6 hours of instructional time.
D) The prediction is accurate because it is based on empirical evidence.

Accepted Answer

The trend may or may not be the same outside of the range of levels investigated.)

Question 11

A researcher finds a significant omnibus F test with J = 4. Which of the following statements is true? A) You should always do a priori comparisons before an ANOVA. B) You should always do post hoc tests after an ANOVA. C) You should always do post hoc tests to determine why H₀ was rejected. D) You should always do post hoc tests to determine why H₀ was not rejected.

Accepted Answer

A priori comparisons are usually conducted without regard to omnibus ANOVA. Post hoc tests are conducted when H₀ is rejected in the omnibus test and J > 2.)

Question 12

In an experiment, = 10, = 20, and = 40. Tukey HSD shows that $\mu$ S1U1B12S1U1B0-$\mu$ S1U1B11S1U1B0 is a significant contrast. What will you find out about $\mu$ S1U1B13S1U1B0-$\mu$ S1U1B11S1U1B0 and $\mu$ S1U1B13S1U1B0-$\mu$ S1U1B12S1U1B0 if the same procedure and same$\alpha$ level are used?
A) Only $\mu$S1U1B13S1U1B0-$\mu$S1U1B11S1U1B0 is significant.
B) Only $\mu$S1U1B13S1U1B0-$\mu$S1U1B11S1U1B0  is significant.
C) Both contrasts are significant.
D) None of the contrasts are necessarily significant.

Accepted Answer

Using the same procedure and $\alpha$level, if 11ef2a29_0441_03fe_add3_532f30d9c194_TBR1344_11= 10 is significant, any pairwise contrasts with larger group differences will be significant.)

Question 13

In an experiment, = 10, = 200, and = 4000. Which pairwise contrast will necessarily be significant?
A) $\mu$S1U1B12S1U1B0-$\mu$S1U1B11S1U1B0
B) $\mu$S1U1B13S1U1B0-$\mu$S1U1B11S1U1B0
C) $\mu$S1U1B13S1U1B0-$\mu$S1U1B12S1U1B0
D) None of the contrasts will necessarily be significant.

Accepted Answer

We do not know the values of the standard error, critical value, etc.)

Quiz 12: Multiple Comparison Procedures

How many possible pairwise contrasts are there if J = 3?

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

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

A one-factor fixed-effects ANOVA results in the following: F (1, 17) = .011, p ₌.918. Which post-hoc procedure should be used as a follow-up?

Applying the Dunn procedure, what is the per contrast alpha if the nominal alpha is 0.05 and there are three contrasts?

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 researcher conducts a study of different instructional methods. The researcher finds that student scores increase significantly when instructional time increases, so she predicts that scores will be even higher if the instructional time is 12 hours per day. Evaluate the prediction.

A researcher finds a significant omnibus F test with J = 4. Which of the following statements is true?

In an experiment, = 10, = 20, and = 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?