Psychology 406 Assignment #13

Suppose that an experimenter is interested in “level of aspiration” as the dependent variable in an experiment. An experimental task has been developed consisting of a difficult game apparently involving motor skill, yielding a numerical score that can be attached to a person’s performance. But this appearance is deceptive: unknown to the subject, the game is actually under the control of the experimenter, so that each subject is made to obtain exactly the same score. After a fixed number of trials, during which the subject unknowingly receives the preassigned score, the individual is asked to predict what the score will be on the next group of trials. However, before this prediction, the subject is given “information” about how the score compares with some fictitious norm group. In one experimental condition, the subject is told that the first performance is above average for the norm group; in the second that it is average; and in the third that it is below average. There are thus three possible experimental “standings” that might be given to any subject. (Of course, after the experiment, each subject is full informed of this little ruse by the experimenter.)

The dependent score value Y is based on the report the subject makes about anticipated performance in the next group of trials. Because each subject has obtained the same score, this anticipated score on the next set of trials is treated as equivalent to a level of aspiration that the subject has set. Each subject is tested privately, and no communication is allowed between subjects until the entire experiment is completed. Each of the three groups contains 20 randomly assigned subjects.

In addition to the dependent measure, Y, prior to the experiment each subject had been tested on a game very similar to that used in the experiment proper, and a “skill score”, X_{1}, obtained for each. The data that resulted from this experiment can be represented in the following form:

above average average below average
Y  X_{1}  Y X_{1} Y  X_{1}
52  44  28 38 15  23
48  47  35 26 14  17
43  30  34 36 23  31
50  38  32 30 21  25
43  40  34 36 14  27
44  45  27 23 20  35
46  36  31 45 21  25
46  41  27 28 16  28
43  40  29 34 20  30
49  43  25 37 14  37
38  48  43 40 23  32
42  24  34 36 25  32
42  39  33 41 18  34
35  36  42 29 26  48
33  46  41 39 18  39
38  33  37 37 26  38
39  38  37 47 20  30
34  26  40 34 19  24
33  41  36 47 22  31
34  36  35 31 17  19

In addition to Y and X_{1}, define three “dummy” variables, X_{2}, X_{3}, and X_{4}: X_{j} = 1, if the subject belongs to group j − 1; = 0, otherwise. The SYSTAT output that is attached gives the basic statistics in addition to information on fitting a variety of models. (In giving the basic statistics, Groups 1, 2, and 3 are above average, average, and below average, respectively.) The models that are fitted are given as (a) through (f) below:

Model (a):

Y = β_{0} + β_{1}(X_{1}X_{2}) + β_{2}(X_{1}X_{3}) + β_{3}(X_{1}X_{4}) + ϵ

Model (b):

Y = β_{0} + β_{1}X_{1} + ϵ

Model (c):

Y = β_{0} + β_{1}X_{2} + β_{2}X_{3} + ϵ

Model (d):

Y = β_{0} + β_{1}(X_{1}X_{2}) + β_{2}(X_{1}X_{3}) + β_{3}(X_{1}X_{4}) + β_{4}X_{2} + β_{5}X_{3} + ϵ

Model (e):

Y = β_{0} + β_{1}X_{1} + β_{2}X_{2} + β_{3}X_{3} + ϵ

Model (f):

X_{1} = β_{0} + β_{1}X_{2} + β_{2}X_{3} + ϵ

Questions:

i) Model (c) performs a “one-way analysis of variance” on the dependent measure Y in relation to the 3 groups. Show explicitly the relation between the means on Y within each of the three groups (given in the basic statistics) and the estimated means on Y for all the various combinations of values that X_{2} and X_{3} can take on. What does the analysis-of-variance table say about the “effectiveness” of the 3 treatments? Why isn’t an X_{4} term included in model (c)?

ii) Looking at model (f), carry out a similar interpretation as in (c). Are the results surprising? Why?

iii) Plot the regression lines of Y on X_{1} implied by model (d) for the three separate groups. Superimpose on this plot the regressions of Y on X_{1} implied by model (e) for the three separate groups. Carry out a test of model (d) versus model (e) and interpret.

iv) Plot the regression lines of Y on X_{1} implied by model (e) and model (b). Carry out a test of model (e) versus model (b) and interpret. This is called “analysis-of-covariance”, and supposedly is a way of assessing the effectiveness of the three treatments. How is it different than what was done in (i)?

(It may help to interpret what was done in (i) as a comparison of model (c) against a restricted model, Y = β_{0} + ϵ.) Analysis of covariance is based on an assumption that model (e) is the “Full Model”. How does this relate to what was done in (iii)?

v) Suppose I have some given value on X_{1}, say P. Using model (e), what are the expected values on Y for the three separate groups. Suppose I have a second given value on X_{1}, say Q. What are the expected values on Y for the three separate groups, again using model (e), and what are the relationships between the two sets of expected values. Now, do the same for model (d) and comment on the differences from using model (e). Using these interpretations, why is it argued that one cannot compare the effectiveness of treatments merely by comparing model (d) and (a) (when model (e) cannot be assumed correct)? Also, why is it argued that we can actually “control” for the effect of X_{1} in assessing treatment effectiveness when model (e) is “true” but not if model (d) is “true”?

vi) Carry out a test of model (d) versus (b). What is this a test of any how does it differ from a comparison of model (d) versus model (a) and of model (e) versus model (b)?

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