**HW#1 **

1. (8 points) Prove that the sum of the interaction affects in a between-subjects two-way ANOVA is equal to zero. In other words, prove that

2. (8 points) A student in one of my previous classes stated: “A treatment is a treatment, whether the study involves a single factor or multiple factors. The number of factors has little effect on the interpretation of the results.” Discuss.

3. Write up a statistics problem that describes a single-factor design, preferably involving unicorns, glitter, zombies, aliens, or some combination of these things. Design your scenario so that your theoretically-driven hypotheses take the form of three or more orthogonal single-df planned comparisons. You get 4 points for writing a scenario that meets these criteria. Please be sure to explain the motivation for your hypotheses.

a. (4 points) Demonstrate that the contrasts are orthogonal.

b. (8 points) Generate your data for this problem in SAS for full credit, or using Excel for a 4-point penalty. Generate your data in a manner that is consistent with the assumptions and linear model associated with the one-way ANOVA. If you used SAS to generate your data, turn in all SAS files that are necessary to generate the data (your program, any datafiles, etc.). If you use Excel then turn in the spreadsheet.

c. (4 points) Run your planned comparisons using SAS. Turn in the script and results file as your answer to this part.

4. (8 points) In a study of intentions to get flu-vaccine shots in an area threatened by an epidemic, 90 people were classified into three groups of 30 according to the degree of risk of getting the flu. The experimenter brought each group one at a time into a room and verbally asked each member of the group about their likelihood of getting a flu shot, on a probability scale ranging from 0 to 1. Unavoidably, most participants heard the responses of nearby participants. An analyst wishes to test whether the mean intent scores are the same for the three risk groups. Consider each assumption for the ANOVA procedure and explain whether this assumption is likely to hold in the present situation. For any assumption that is unlikely to hold, suggest a remedy if one exists.

5. (12 points) Consider a 1-way between-subjects ANOVA with 6 treatment levels and 4 subjects per treatment. You have k pairwise comparisons to make amongst the treatment means. You have two choices to accomplish this:

**Choice A**: You can treat them as planned comparisons. You would conduct k two-tailed t-tests using the standard t-test formula (meaning that the pooled variance is in the denominator of the formula), and use the Bonferroni correction to control the familywise error rate.

**Choice B:** You can treat the comparisons as post hoc and use Tukey’s HSD procedure. In this case you would use

## Sheet1

25 35 3 5

7

32 8 9

4 6 8 10

Female Cues | Male Cues | Originals | ||||||||||||||

Female Subjects | 2 | 9 | 3 | 6 | 1 | 4 | 5 | 22 | 25 | |||||||

35 | 33 | 8 | 7 | 20 | 30 | |||||||||||

28 | 38 | 10 | 16 | 23 | 32 | |||||||||||

Male Subjects | 18 | |||||||||||||||

31 | 15 | 11 | ||||||||||||||

26 | 34 |

**HW#1 **

1. (8 points) Prove that the sum of the interaction affects in a between-subjects two-way ANOVA is equal to zero. In other words, prove that

2. (8 points) A student in one of my previous classes stated: “A treatment is a treatment, whether the study involves a single factor or multiple factors. The number of factors has little effect on the interpretation of the results.” Discuss.

3. Write up a statistics problem that describes a single-factor design, preferably involving unicorns, glitter, zombies, aliens, or some combination of these things. Design your scenario so that your theoretically-driven hypotheses take the form of three or more orthogonal single-df planned comparisons. You get 4 points for writing a scenario that meets these criteria. Please be sure to explain the motivation for your hypotheses.

a. (4 points) Demonstrate that the contrasts are orthogonal.

b. (8 points) Generate your data for this problem in SAS for full credit, or using Excel for a 4-point penalty. Generate your data in a manner that is consistent with the assumptions and linear model associated with the one-way ANOVA. If you used SAS to generate your data, turn in all SAS files that are necessary to generate the data (your program, any datafiles, etc.). If you use Excel then turn in the spreadsheet.

c. (4 points) Run your planned comparisons using SAS. Turn in the script and results file as your answer to this part.

4. (8 points) In a study of intentions to get flu-vaccine shots in an area threatened by an epidemic, 90 people were classified into three groups of 30 according to the degree of risk of getting the flu. The experimenter brought each group one at a time into a room and verbally asked each member of the group about their likelihood of getting a flu shot, on a probability scale ranging from 0 to 1. Unavoidably, most participants heard the responses of nearby participants. An analyst wishes to test whether the mean intent scores are the same for the three risk groups. Consider each assumption for the ANOVA procedure and explain whether this assumption is likely to hold in the present situation. For any assumption that is unlikely to hold, suggest a remedy if one exists.

5. (12 points) Consider a 1-way between-subjects ANOVA with 6 treatment levels and 4 subjects per treatment. You have k pairwise comparisons to make amongst the treatment means. You have two choices to accomplish this:

Choice A: You can treat them as planned comparisons. You would conduct k two-tailed t-tests using the standard t-test formula (meaning that the pooled variance is in the denominator of the formula), and use the Bonferroni correction to control the familywise error rate.

Choice B: You can treat the comparisons as post hoc and use Tukey’s HSD procedure. In this case you would use 𝑑𝑓𝑆|𝐴 to determine critical values.

Find the maximum value of k for which the planned comparison approach is more powerful than the post hoc approach. Use a familywise alpha of 0.05.

6. (8 points) You design an experiment with 2 IVs and 1 DV and are deciding between 2-way between-subjects ANOVA and multiple regression for the eventual data analysis. Discuss the situations in which one analysis method is more appropriate than the other. You only need to consider the variants of the two approaches that were discussed in 507/508 (that is, standard multiple regression and the 2-way between-subjects ANOVA with both main effects and the interaction in the model).

7. In the construction of a projective test, 40 pictures of two or more human figures were used. In each picture, the sex of at least one of the figures was only vaguely suggested. In a study of the influence of the introduction of extra cues into the pictures, the original images were photoshopped (altered) so that the vague figure looked slightly more female; in another set, each original was retouched to make the figure look slightly more male. A third set was made up of the original pictures as a control. The images were administered to a group of 18 male college students and an independent group of 18 female college students. Six members of each group saw the pictures with the female cues, six the pictures with the male cues, and six the original pictures. Each participant was scored according to the number of pictures in which the indistinct figure was interpreted as female. The data are included in the Excel file accompanying the HW (Problem8.xslx).

a. (8 points) Conduct the appropriate ANOVA using SAS. For your answer to this part please turn in your SAS code, the results file, and a written interpretation of the main effects and interaction. For the interpretation part of your answer, write as if you are presenting your statistics in the Results section of a journal article.

b. (6 points) Estimate the treatment effects and interaction effects from the data. There should be a treatment effect estimates for Factor A, b estimates for Factor B, and ab estimates of the interaction effects.

c. (8 points) Imagine that in this experiment everything had turned out as shown, except that the distinction between male and female participants was not made. Instead, imagine that the picture manipulation was the only factor, and a one-way analysis had been carried out. What would happen to the within-groups sum of squares in such an analysis, relative to the

within-groups sum of squares in part a? What would this new within-groups sum of squares actually include? Explain.