Psychology 407 Assignment #6

1) To assess the effects of anxiety on test performance, three sets of instructions to examinees were used: (1) instructions designed to reduce anxiety,

(2) neutral (standard) directions, or (3) directions designed to produce anxiety. Students were classified by sex, then randomly assigned to one of three test conditions: A_{1} (anxiety-reducing), A_{2} (neutral), A_{3} (anxiety-producing). Within each level of factor A, 10 boys and 10 girls took a standardized verbal ability test (B_{1}) and 10 boys and 10 girls took a standardized math ability test (B_{2}) previously calibrated to be of the same difficulty level as the verbal test. Assume factor C is gender (C_{1} and C_{2} are boys and girls in some order).

The resulting means based on the 10 observations per cell are given below along with the various sums of squares:

\begin{matrix}A_{1}B_{1}C_{1} & A_{1}B_{1}C_{2} & A_{1}B_{2}C_{1} & A_{1}B_{2}C_{2} & A_{2}B_{1}C_{1} & A_{2}B_{1}C_{2} & A_{2}B_{2}C_{1} & A_{2}B_{2}C_{2} & A_{3}B_{1}C_{1} & A_{3}B_{1}C_{2} & A_{3}B_{2}C_{1} & A_{3}B_{2}C_{2} \\ 12.7 & 14.0 & 16.0 & 10.5 & 13.0 & 13.2 & 12.6 & 10.2 & 12.1 & 16.3 & 12.2 & 13.2 \end{matrix}

Sum of Squares

A  34.20
B  36.30
C  1.20
AB  15.20
AC  122.60
BC  132.30
ABC  25.80
Error  2947.50

i) Complete the appropriate ANOVA table and carry out the relevant significance tests.

ii) Interpret the results of the 3-factor interaction test in (i) (whether significant or not) by graphing sets of specific 2-factor interactions. In how many ways (i.e., sets of specific 2-factor interactions) could you do this?

iii) Given the results of (i), explain what is occurring substantively by graphing appropriate sets of means (e.g., think specific effects and interactions).

iv) Indicate schematically how this three-way ANOVA could be rephrased as a regression model and indicate explicitly what the design matrix would look like. (You only need to provide representative rows in the design matrix that would be repeated for all of the 10 observations in each cell).

2) An experiment is performed in the twenty elementary schools of a large school district. Ten of the schools are randomly designated to be the sites for adoption of an innovative science curriculum, “Science: A Process Approach (SAPA)”. The SAPA material were bought and placed in the ten schools; teachers were trained to use them. The other ten elementary schools continue to use the district’s traditional textbook-based science curriculum. After two years of study in the respective programs, sixth-grade pupils in all twenty schools are given the Science test (a 45-item measure of scientific methods, reasoning, and knowledge) of the Sequential Tests of Educational Progress (STEP). Each student’s score is expressed as a percentage. There are 50 to 120 sixth-grade pupils in each school; but since the school itself (along with its teachers, administrators, surrounding neighborhoods, and the like) was randomly designated as either SAPA or Traditional (the two experimental conditions, E versus C), the school is the experimental unit. The twenty schools’ means of sixth-grade pupils STEP-Science scores will be used as the observational unit in the statistical analysis. The data collected in the experiment are reproduced below (denoted as Y) along with average Scholastic Aptitude Scores (X) obtained prior to the treatment.

E Schools  C Schools
X Y X Y
105.7  77.63% 101.2 64.10%
100.3  74.13  97.6  43.67
94.3  67.20  96.4  50.40
108.7  78.23  109.6  84.33
93.1  57.93  94.0  44.93
96.7  57.65  105.4  71.43
106.9  83.30  102.4  71.10
100.3  73.90  100.6  44.57
86.5  45.90  104.2  68.23
96.1  64.83  112.6  68.47
Means: 98.86 68.07 102.40 61.12

In terms of Kutner et al.’s notation, you may find the following quantities helpful.

SSTO_{Y} = 3266.24 SSTR_{Y} = 241.30 SS_{Y} = 3024.94 SSTO_{X} = 796.48 SSTR_{X} = 62.66 SSE_{X} = 733.82 SPTO = 1145.97 SPTR = −122.91 SPE = 1268.88

i) Carry out two separate one-way ANOVAs on X and Y. Interpret.

ii) Carry out an analysis-of-covariance on Y treating X as the covariate. Interpret any differences from what you found in (i). Graph the data (X by Y) and impose the within-group regression lines of Y on X.

iii) What are the “adjusted” means; interpret these in terms of the plot in (ii).

iv) Schematically indicate how the regression approach could be used to carry out the analysis-of-covariance. Relate the various adjusted mean squares to what you would obtain in comparing certain Full and Reduced models.

v) Discuss how you would evaluate the assumption of equal within-group slopes using the regression.

vi) If one were interested in comparing adjusted means post-hoc (even though there are only two here, and, thus, it is redundant to do so), what standard error of the difference between the two means would be used? Compute it.