Topic: Two-Way Anova
Data:  psych.sav
Example: Physical Therapy and Psychiatric Treatment

Goal: Determine if a difference in Physical Therapy Treatment affects time to recovery, determine if a difference in Psychiatric Therapy Treatment affects time to recovery, and determine if an interaction exists.

1. Background

Researchers at a trauma center wished to develop a program to help brain-damaged trauma victims regain an acceptable level of independence. An experiment involving 72 subjects with the same degree of brain damage was conducted. The objective was to compare different combinations of psychiatric treatment and physical therapy. Each subject was assigned to one of 24 different combinations of four types of psychiatric treatment and six physical therapy programs. There were three subjects in each combination. The response variable is the number of months elapsing between initiation of therapy and time at which the patient was able to function independently. The results were as follows:

	Psychiatric Treatment
Physical Therapy Program	1	2	3	4
I	11.0	9.4	12.5	13.2
	9.6	9.6	11.5	13.2
	10.8	9.6	10.5	13.5
II	10.5	10.8	10.5	15.0
	11.5	10.5	11.8	14.6
	12.0	10.5	11.5	14.0
III	12.0	11.5	11.8	12.8
	11.5	11.5	11.8	13.7
	11.8	12.3	12.3	13.1
IV	11.5	9.4	13.7	14.0
	11.8	9.1	13.5	15.0
	10.5	10.8	12.5	14.0
V	11.0	11.2	14.4	13.0
	11.2	11.8	14.2	14.2
	10.0	10.2	13.5	13.7
VI	11.2	10.8	11.5	11.8
	10.8	11.5	10.2	12.8
	11.8	10.2	11.5	12.0

Can one conclude on the basis of these data that the different psychiatric treatment programs have different effects? Can one conclude that the physical therapy programs differ in effectiveness? Can one conclude that there is interaction between psychiatric treatment programs and physical therapy programs?

2. Enter Data

Take care to enter data carefully. Make sure the dependent variable, the variable of interest, is in one column. Then make a column for each factor of interest. These columns will have numbers indicating what category for each factor. In the above example, the dependent variable is time (in months). Then factor one is physical therapy program. So that column (physical) will have numbers 1-6 in it. The third column is for psychiatric treatment (psych). It will have 1-4 in it.

3. View Data

To view line graph, click on Graphs\Line\Multiple\Define. Click time over to the circle labeled Other summary function. Click the first treatment variable (in this case physical) over to the Category Axis: box and then click the second treatment variable (in this case psych) over to the Define Lines: by box. Then hit OK.

Conclusion: It appears a difference is going on.

4. State Hypotheses

a. Physical Therapy Treatments: H₀: m ₁=m ₂=m ₃  v.s.  H_a: At least two of the means differ.

If rejected, level of physical therapy was affecting time to recovery.

b. Psychiatric Therapy Treatments: H₀: m ₁=m ₂=m ₃=m ₄ v.s. H_a: At least two of the means differ

If rejected, level of psychiatric therapy was affecting time to recovery.

c. Interaction: H₀: Interaction does not exist v.s. H_a: Interaction does exist.

5. Analysis

Click on Analyze\General Linear Model\Univariate. Place the variable of interest (time) in the Dependent Variable box. Put the factors of interest (physical, psych) in the Fixed Factors box. If one hits OK now, SPSS will test with interaction. That is okay. If the interaction is significant then you are done. If it is not, the model must be refit without the interaction term. To do this click the Model button. Place a check in the Custom circle. Click the factor variables (physical, psych) over to the Model box. In the Build terms box, click on Main effects.

Without intercept:

With intercept:

Conclusions: We reject the Null Hypothesis for all three since the P-values for PHYSICAL, PSYCH, and PHYSICAL*PSYCH (interaction) are below 0.05.

6. Multiple Comparison:   Do as above, but click the Post Hoc button. If interested in identifying when means differ for both treatments, then click both over to the box labeled Post Hoc Tests for:

PSYCH

PHYSICAL

Homogeneous Subsets

Conclusion: For Psych, reject the null for all 6 pairs using SNK, Tukey HSD, and Duncan. For Physical, there are 15 pairs. Fail to reject for pairs 1-6, 2-3, 2-4, 2-5, 2-6, 3-4, 3-5, 4-5. There are three subgroups.

Exercise

Background: Suppose the USGA tests four different brands (A,B,C,D) of golf balls and two different clubs (driver, five-iron) in a completely randomized design. Each of the eight Brand-Club combinations is randomly and independently assigned to four experimental units, each experimental unit consisting of a specific position in the sequence of hits by Iron Byron. The distance response is recorded for each of the 32 hits. The objective of this research to see if there is any difference in using balls from different brands.

	BRAND
Club	A	B	C	D
Driver	226.4	238.3	240.5	219.8
	232.6	231.7	246.9	228.7
	234.0	227.7	240.3	232.9
	220.7	237.2	244.7	237.6
Five-iron	163.8	184.4	179.0	157.8
	179.4	180.6	168.0	161.8
	168.6	179.5	165.2	162.1
	173.4	186.2	156.5	160.3

1. Create a data file.
2. State the hypothesis using the objective of this research.
3. View profile lines and make a conclusion.
4. Perform a two-way ANOVA and test the hypothesis and draw a conclusion.
5. Perform multiple comparison and draw a conclusion.