Education 257 HW1 Jan 13 2005 ("Due" Jan 22 2005)
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Preliminaries:
Due date indicates approximate date for posting solutions
Data sets: unless otherwise indicated (or part of web examples)
data reside in the class HW directory
URL is http://www-stat.stanford.edu/~rag/ed257/hw/[file]
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1.
From Neter Wasserman Kutner
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.)
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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.
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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2.
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.
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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3.
Prior to conducting a clinical trial that involves a subjective evaluation
of a patient's progress, the participating physicians are asked to agree
on certain criteria for reaching an evaluation.
To examine the consistency in their evaluations before the initiation of a
particular clinical trial, a pilot study was conducted on four patients who
had been treated with a drug that was to be included in the trial.
Each of the five physicians who were to participate in the study was
asked to evaluate (on a 0-to-l0-point scale) the degree of cure
after a two-week treatment period. Since the clinical evaluations of a
patient's cure were to be based on the results of a bacterial culture
analysis, each physician analyzed two cultures from each patient. This
feature was unknown to the physicians, who were merely told they would be
analyzing eight separate bacterial cultures. The evaluations based
on these cultures are recorded here.
a. Treating physicians as fixed and patients as random, write an appropriate
model. Identify all terms in the model.
b. Construct the AOV table. Show the expected mean squares.
Test hypotheses for main effects and interactions.
c. For the fixed factor carry out pairwise comparisons using Tukey method.
Patient
physician 1 2 3 4
1 7.2 4.2 9.5 5.4
9.6 3.5 9.3 3.9
2 8.5 2.9 8.8 6.3
9.6 3.3 9.2 6.0
3 9.1 1.8 7.6 6.1
8.6 2.4 7.1 5.6
4 8.2 3.6 7.3 5.0
9.0 4.4 7.0 5.4
5 7.8 3.7 9.2 6.5
8.0 3.9 8.3 6.9
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4. Unbalanced 2-way designs: Why pay 'em anything?
A sociologist selected a random sample of 45
adjunct professors who teach in the evening division of a large
metropolitan university for a study of special problems associated
with teaching in the evening division. The data collected include the
amount of payment received by the faculty member for teaching a course
during the past semester. The sociologist classified the faculty
members by subject matter of course (C2 factor A; i = 1,...4
{Humanities, Social Sciences, Engineering, Management}) and highest
degree earned (C3 factor B; j = 1,2,3 {Bachelor, Master, Ph.D.}). The
earnings per course (in thousand dollars) are given in C1 in these
data. output given below.
The full data are given in NWK.
The data are also in file adjprof.dat
a. Construct a profile (cell mean) plot for the cell means for this 4 x 3
data structure.
Comment on the appearance of main effects and interactions.
b. From the GLM output below (some entries obscured by &&&&) carry out
a test of Ho: alpha(i) = 0 vs. Ha: not all alpha(i) = 0
using Type I error rate .01.
c. These data reside in file adjprof.dat in the class directory.
Replicate the full GLM analysis which is given in abbreviated form below.
Compare the GLM results--e.g. the anova Table and test statistics-- with
the approximate solution (unweighted means) described in class
(adapted from Miller).
MTB > table c2 c3;
SUBC> mean c1; SUBC> count.
ROWS: C2 COLUMNS: C3
1 2 3 ALL
1 1.8000 1.9500 2.7000 2.4250
2 2 8 12
2 2.4500 2.5200 3.4500 2.7846
4 5 4 13
3 2.7500 2.8500 3.7400 3.2364
2 4 5 11
4 2.5500 2.5500 3.4200 3.0333
2 2 5 9
ALL 2.4000 2.5385 3.2364 2.8489
10 13 22 45
CELL CONTENTS --C1:MEAN
COUNT
MTB > glm c1 = c2|c3
Factor Levels Values
C2 4 1 2 3 4
C3 3 1 2 3
Analysis of Variance for C1
Source DF Seq SS Adj SS Adj MS F P
C2 && 4.1676 4.2326 &&&&&& &&&&& &&&&&
C3 && 8.3825 8.2287 &&&&&& &&&&&& &&&&&
C2*C3 && 0.0444 0.0444 &&&&&& &&&& &&&&&
Error && 0.7180 0.7180 &&&&&&
Total 44 13.3124
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5. In the rehabililitation center example in problem #1
Obtain the power of the test in 1(c) if the population
group mean are 37, 35, 28 and the within-cell variance is 4.5
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Randomized Blocks Designs
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6.
We revisit the in class example of
the experiment on treating depression by the Imipramine,
an anti-depressant drug.
The example is taken from the text "The Design and Analysis
of Clinical Experiments" by J L Fleiss. A total of 60 patients
were paired on age sex time of entry in study to form 30 matched
pairs or blocks. One member of each pair was randomly asssigned to
receive Imipramine and the other to receive a placebo. The outcome
measure was the score on the Hamilton rating scale for depression
(higher score = more severe depression) after 5 weeks of treatment.
The file depress.dat in the class directory contains the outcome
scores for each of the 30 pairs, c1 Imipramine, c2 Placebo.
a. For this randomized block design, carry out an anova to test the
equality of the treatment outcomes. State your conclusion.
b. Give a measure of the efficiency of the randomized block design
relative to the completely randomzed design. What is the importance
of this efficiency measure?
c. Analyze these same data using the paired t-test. Compare your results.
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7. Randomized Block Designs
Statistical training, Tax audits, scary stuff?
These data come from the "Auditor Training" example, NWK
An accounting firm, prior to introducing in the firm widespread
training in statistical sampling for auditing, tested three training
mathods:
(1) study at home with programmed training materials
(2) training sessions conducted at local offices conducted by
local staff
(3) training sessions in Chicago conducted by national staff
Thirty auditors were grouped into 10 blocks of 3, according to time
elapsed since college graduation, and the three auditors in each block
were randomly assigned to the three training methods. At the end of
the training, each auditor was asked to analyze a complex case involving
statistical applications; a proficiency measure based on this analysis
was obtained for each auditor.
In file audit.dat the columns are
proficiency measure; block; training method
a. Write an appropriate linear statistical model for this randomized
blocks design. List the assumptions and identify terms.
b. Construct a profile plot by plotting proficiency by blocks.
What does this plot suggest
about the appropriateness of the no interaction assumption here?
c. Obtain an anova table for this design; carry out a test of the
null hypothesis that mean proficiency is the same for all 3
training methods. Use Type I error rate .01 and state your
conclusion.
d. Follow-up the testing of the null hypothesis of no difference
between training methods by forming interval estimates for all the
pairwise comparisons. Use the Tukey method with family-wise
confidence coefficient .99.
e. Obtain an indication of how useful blocking on 'time since college'
was in increasing the precision of this study compared to a simple
completely randomized (one-way) design. For example, this study uses
a total of 30 subjects; how many subjects would be neccessary to
achieve the same precision if a completely randomized (i.e. no
blocking) design were employed?