Question

Conceptual Problems

1. In your own words describe how RM ANOVA can be used to test for differences between groups, differences between repeated observations, and differences in observations as a function of group membership.

2. Why do we say that in RM ANOVA time is treated as a categorical variable?

3. What is the difference between polynomial contrasts and comparisons of means between different observation periods?

Computer Problems

Researchers were interested in how student confidence fluctuates from the end of sophomore year through the end of senior year in college. Further, researchers were interested in whether students attended a private (i.e., Harvard, Stanford, and Yale) or public (i.e., UMass – Boston, San Francisco State University, and Southern Conneticut State University) university would influence end-of-year student confidence ratings, and possible trajectories.

Use the data set “confidence.csv” to answer the following questions. For your information, t1-t3 indicate time of observation 1-3; i.e., end of sophomore year, end of junior year, and end of senior year, respectively. The variable public indicates whether the student attended a public (public = 1) or private (public = 0) university.

1. Convert the wide format data set to a long format data set. Remember that you will need to create an ID variable, and an Observation/Time variable. Show syntax and the header and footer of the long format data set.

Syntax:

Data header and footer:

2. Test whether public and private unveristies differed in their confidence scores. Conduct any pairwise comparison necessary. Report your conclusions.

Syntax:

Results:

3. Test whether confidence differs over time. Do this by comparing all observations to each other. Also do this by testing the maximum allowable number of polynomial contrasts. Report your conclusions for both tests.

Syntax:

Results:

4. Test for a possible interaction between variables tested in question 2 (university type) and 3 (mean differences between observations and significant trends in confidence). Conduct any follow-up analyses necessary if there is a significant interactions. Report your conclusions for all tests.

Syntax:

Results:

Extra

1. (Continue with analysis from above.) Create a plot of the data that represents the trajectory for public and private students. Make sure the trajectories have some indication of variability around the average trajectory. Sufficiently label and describe your figure.

Syntax:

Figure and comments:

2. Report your conclusions in APA format as you would if you were writing them for a peer reviewed journal. You may use tables to help report your statistical findings if you think it is appropriate. Please make sure your conclusions are in the context of the scenario presented.

3. What are the assumptions of RM ANOVA? Describe, in your own words, what limitations these assumptions create for analyzing repeated measures data. (Hint: consider characteristics common to repeated measures data and why RM ANOVA might be ill-equipped to handle them)

Solution Preview

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#3. Test whether confidence differs over time. Do this by comparing all observations to each other. Also do this by testing the maximum allowable number of polynomial contrasts. Report your conclusions for both tests.
summary(aov(observation~time + Error(id), data = data_long))
#_Conclusion:_
#_From the above summary of ANOVA we can see that, there is significant difference between the means because the p-Value is significant for being less than 0.05. We will go for the contrasts and which contrasts differs


#_To perform Polynomial Contrasts let's set the stage_

contrasts(data_long$time) <- contr.poly(3)# Defining contrasts

aov.out = aov(observation ~ time + Error(id), data=data_long) # do the ANOVA

summary(aov.out, split=list(
time=list("t1 vs t2"=1,"t2 vs t3"=2)
)) # visualize the results

# Conclusion: From the ANOVA we clearly saw that there's a significant difference between grouping variable time, after doing contrasts we came to know that t1 and t2 significantly differs between the groups...

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