Education 161 Winter 2000
Assignment 2 Due Feb 1, 2000
Note data files are available in one of two locations:
path: /usr/class/ed161/[data file]
or using web-services at URL
http://www.stanford.edu/class/ed161/hw/[data file]
1.
An experiment was conducted to examine the effects of different levels of
reinforcement and different levels of isolation on children's ability to
recall. A single analyst was to work with a random sample of 30 children
selected from a relatively homogeneous group of fourth-grade students.
Two levels of reinforcement (none and verbal) and three levels of
isolation (20, 40, and 60 minutes) were to be used. Students were randomly
assigned to the six treatment groups, with a total of six students being
assigned to each group.
Each student was to spend a 30-minute session with the analyst. During this
time the student was to memorize a specific passage, with reinforcement
provided as dictated by the group to which the student was assigned.
Following the 30-minute session, the student was isolated for the time
specified for his or her group and then tested for recall of the
memorized passage. These data appear in the accompanying table.
Time of Isolation (Minutes)
Level of
Reinforcement 20 40 60
26 19 30 36 6 10
None 23 18 25 28 11 14
28 25 27 24 17 19
15 16 24 26 31 38
Verbal 24 22 29 27 29 34
25 21 23 21 35 30
Clearly, both factors are fixed factors.
Create a file with the 36 times from the
table above (C1). Use Mintab SET command (or your editor) to add the
factorial design designations in C2 and C3.
a. Construct a profile plot and comment.
b. Write out the statistical model for this two-way classification
c. Carry out the series of hypothesis tests for the two-way anova.
Keep your overall error rate at or below .05 for the 3 tests.
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2.
NOTE: In HW1 #5
you were asked to do parts a-e of the problem below
with data in knee.dat.
For HW2 continue with these data and complete part(f)
A rehabililitation center researcher was interested in examining the
relationship between physical fitness prior to surgery of persons
undergoing corrective knee surgery and time required in physical therapy
until sucessful rehabilitation. 24 male subjects ranging in age from 18
to 30 years who had undergone similar corrective knee surgery during the
past year were selected for the study.
In the data file knee.dat
c1 contains the number of days required for sucessful completion of
physical therapy and c2 contains an indicator of prior physical fitness
status-- 1 = below average; 2 = average; 3 = above average.
(So this data set is of the form of a time-to-mastery study.)
a) obtain mean and variance of time to recovery for each group
b) present a graphical look at the scores for the three groups
by constucting aligned dotplots for the three groups
c) carry out an anova for this one-way classification and test the
omnibus null hypothesis of no differences between the group means
using Type I error rate .05.
d) display residuals from the fit of the anova model for each group.
e) carry out the post-hoc pairwise comparison procedure in order to
obtain interval estimates of each pairwise comparison using
experimentwise error rate .05.
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f) in planning a follow-up study which will have equal numbers of
subjects in each group, how many subjects should there be in each
group so that the interval estimate for these pairwise comparisons
will have width of 5 days (again using experimentwise error rate
.05)?
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3.
Nonparametrics
A good review of basic nonparametric procedures is in
Minitab student handbook/primer Chaps 12 or 13 (depending on
version)
The story below is kind of boring but you can think of the
data as the number of errors (e.g. reading or writing) made under
5 different conditions/protocols:
An experiment was conducted to compare the number of major defectives
observed along each of five production lines in which changes were
being instituted. Producton was monitored continuously during the
period of changes, and the number of major defectives was recorded
per day for each line. The data for the five lines are shown here
(you can cut-and-paste
Production line
1 2 3 4 5
34 54 75 44 8O
44 41 62 43 52
32 38 45 30 41
36 32 10 32 35
51 56 68 55 58
(i). Does the standard anova assumption of equal within-group variances
appear to hold here? Does it matter?
(ii). Conduct a standard one-way anova and test the omnibus null hypothesis
of equal group means. Type I error rate .05.
(iii). An alternative approach would be to turn nonparametric.
Try a Kruskal-Wallis procedure on these data.
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4.
"Contrast Statistics Class with Deprivation Torture"
The data in file
animal.dat represent outcomes in an animal learning
task. There are five animals in each of the five conditions.
Conditions 1 and 2 represent control conditions in which the animal
received "ad lib" food and water (data in C1) or food and water twice
a day (data in C2). In Condition 3 animals were food deprived (C3).
In Condition 4 animals were water deprived (in C4) and in Condition 5
animals were deprived of both food and water (in C5).
The planned comparisons (contrasts) proposed for use in the analysis
of the one-way classification are:
1. (1,2) vs (3,4,5)
2. (1) vs (2)
3. (3,4) vs (5)
4. (3) vs (4)
a. give a short interpretation for the question each contrast
is addressing
b. verify that these 4 contrasts are orthogonal or show that they
are not
c. construct point estimates of each contrast
d. construct a set of interval estimates for these 4 contrasts.
use overall error rate .10 and use alphatot.tab entry to
identify the error rate (confidence coefficient) for use
with each interval estimate. what do you conclude?
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5.
More on interactions for factorial designs.
A question that was raised and discussed at length in a previous incarnation
of this course centered on a question like, What more can be done
about describing (or drawing inferences) about the interaction terms
beyond just (rejecting or not) the omnibus null hypothesis of no
interaction? We had extensively discussed the importance of the
profile plot as the major descriptive technique for looking at
interactions. There is more that can be done inferentially in
addition.
In lecture Wednesday the college Mathematics learning example (a 2x3
design) brings up the question of estimating the row effects separately
for each level of the column factor.
One of the TA's at that time said that in psychological journals inferences
about the contrasts between methods at each level of math ability
would be carried out by separate two-sample t procedures
(which I very much hope have the total error rate divided up for the multiple
inferences). An alternative is constructing
either the Tukey multiple comparisons on all (axb) six cells to obtain the
inferences about the desired pairwise comparisons or use a Bonferroni multiple
comparison procedure which may offer some improvement over
Tukey since not all pairwise comps. are of interest.
To pursue this issue I constructed the following exercise.
I constructed data for a 2 x 3 design (two rows, 3 columns as in the
class discussion) with 5 replications per cell. Consider both factors
to be fixed (you could adapt the setting from the college mathematics
example). Carry out the standard analysis for the two-way design--
construct the profile plot from the cell means, obtain the anova
table and construct the test statistics for main effects and
interaction. Compare the method I suggested using a Tukey multiple
comparison procedure on the 1x6 array of cells to estimate the
contrasts of fiffences between the rows at each level of the column
with the "journal method" of separate two-sample t based inferences.
The data exist in file extraint.dat. In c1 are the outcome scores; in
c3 and c4 are the row and column indicators for the 2x3 array. In c2
are cell indicators (1,2,...6) useful in one-way adaptation for this
design.
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6.
Glass& Hopkins, Chap 18, page 527, problems 1 and 2 (parts b,c). You are
encouraged to do this on the computer.
note: for p.2 , the pairwise multiple comparisons can either be done
by hand, as in the in-class Castle bakery illustration, or for the
computationally adventurous, use the Minitab glm command in the same
way you would use anova. For inexplicable reasons the glm graphical
menu allows you to implement tukey in a two-way design, whereas
anova does not. This will be illustrated in solutions and in
section discussion.