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2-way ANOVA Example Problem:
Researchers have sought to examine the effects of various types of music on agitation levels in patients who are in the early and middle stages of Alzheimer’s disease. Patients were selected to participate in the study based on their stage of Alzheimer’s disease. Three forms of music were tested: Easy listening, Mozart, and piano interludes. While listening to music, agitation levels were recorded for the patients based on heart rate in beats per minute. Scores are recorded below.
Group Piano Interlude Mozart Easy Listening Early Stage Alzheimer’s
Mean: 103.
Mean: 77.
Mean: 104. Middle Stage Alzheimer’s
Mean: 106.
Mean: 79.
Mean: 107.
- Propose null hypotheses to test whether types of music or stage of Alzheimer’s have an effect on agitation level.
- Using the various equations for calculating a two-way ANOVA, explain how to calculate the values in each box of the following ANOVA table. Then, fill it out using the attached MATLAB script.
Source Sum of Sq. Degrees of Freedom
Mean of Squares
F-statistic P-value
Type of Music Degree of Alzheimer’s Alzheimer’s x Music type Within X X Total X X X
Sum of Squares :
A. Type of music: SSmusic=^2
3
1
nb ( y y ) i
i
- Where yi is the mean of a type of treatment, n represents the number of data values in a treatment, b represents the number of stages of Alzheimer’s, and y is the weighted average of all means.
B. Degree of Alzheimer’s: SSAlzheimers=
ra ( y (^) j j 1
2
^ y^ )
2
- Where yj is the mean of a stage of Alzheimer’s, n represents the number of data values in a stage, a represents the number of types of music, and y is the weighted average of all means.
C. Interaction: SSInteraction=
n i 1
3
( yij
j 1
2
^ yi ^ y^ j ^ y^ )^2
- Where yij is the mean of the data values of a type of music and a stage of Alzheimer’s, yi is the mean of the data values of that type of music yj is the mean of the data values of that stage of Alzheimer’s, and y is the weighted average of the all means.
D. Within: SSwithin=
i 1
3
k 1
4
( yijk
j 1
2
^ yij )^2
- Where yijk is a specific data value of a type of music and a stage of Alzheimer’s, k represents the index of a data value for a type of music and a stage of Alzheimer’s, and yij is the mean of the data values of a type of music and a stage of Alzheimer’s.
E. Total: SSTotal=
i 1
3
k 1
4
( yijk
j 1
2
^ y^ )
2
- Now that you have all the necessary components, using your F-statistic values that you calculated above, an appropriate α value, and the correct degrees of freedom, look up the p-values for each category.
- Interpret your F-statistic and P-values. Are there any significant main effects or an interaction effect?
- Now that you have all the information you need, come up with a conclusion statement and discuss the relationship between the factors.
Source: http://www2.webster.edu/~woolflm/8banswer.html
