Education 257 HW2 Feb 2 2005 ("Due" Feb. 8 2005)
A. Nested Designs
1.
Group Decision Making Example (see course ex "cross-nested")
NWK v4 Section 28.9; NWK v5 sec 26.9
For this design with crossed and nested factors use the data in
course example (NWK Table 28.12 26.12)
a. obtain cell means for the Nationality X Team Size 2x2 factorial
design
b. recreate the (Minitab) anova table shown in NWK
Carry out tests for the Nationality and Team Size
Main effects and the interaction between these two factors.
How would these tests change if observer was treated as a
fixed rather than random factor?
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2.
Health awareness data,
NWK ver4 p.1157 problem 28.9 onward, ver 5 prob 26.9;
Three states (factor A) participated in a health awareness study.
Each state independently devised a health awareness program.
Three cities (factor B) within each state were selected for
participation and 5 households within each city were randomly
selected to evaluate the effectiveness of the program. All
members of the selected households were interviewed before and
after participation in the program and a composite index was
formed for each household measuring the impact of the health
awareness program (larger the index, greater the awareness)
Data p.1157 (p.1121) exist in file haware.dat
Do problems:
28.9 c (26.9 c)
28.10 parts a b c d (26.10 abcd)
28.11 c (26.11c)
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3.
Revisit Training Data from NWK Ch 28 (ch26) & Course Example
a. For this data analysis, recreate the anova table for
this nested design using only basic analysis of variance
methods (i.e. not the nested capability of the anova
command) by obtaining SS etc for the nested factor
separately at each level of School.
b. Recreate the construction of the Tukey pairwise
comparisons on p.1135-6 (p1101) of NWK.
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B. Repeated Measures Designs
4. Hearing Tests Story
Hearing aids must be fit individually. A common way to test
whether a particular hearing aid is right for a patient is to
play a tape on which 25 words are pronounced clearly but at low
volume, and ask the patient to repeat the words as heard.
Different lists are available that are supposed to be of equal
difficulty to understand correctly. However, a major problem for
those wearing hearing aids is that the aids amplify background
noise as well as the desired sounds. The question here is,
Are the test lists still equally difficult to understand in the
presence of background noise?
In this experiment, 24 subjects with normal hearing listened to
standard audiology tapes of English words at low volume, with a
noisy background. They repeated the words and were scored correct
or incorrect in their perception of the words. The order of list
presentation was randomized. "Hearing" is the dependent variable
and "List" and "Subject" as factors
An objective of the data collection is to assess whether the
different lists are equally difficult to understand .
This design results in only one observation per cell since there
are 96 observations, 24 subjects and 4 lists (24x4 = 96).
In file hearing.dat, c1 is the subject indicator (1,...24), c2
is the list indicator (1,...4), and c3 is the hearing score.
For these data
a. carry out the repeated measures anova and test for
list main effect
b. do Tukey pairwise comparisons among the 4 lists (as was done
in the wine-judging example NWK Section 29.2 (27.2)
c. obtain an estimate of the variance component for subjects.
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5. REPEATED MEASURES "WALK-THROUGH"
a. shoes from NWK , the example used in Sec 29.4 (sec27.4) of NWK;
the series of class example files is shoes.*
(i) Use these data to replicate the information in
NWK Fig 29.8 (27.6)
(ii) replicate the Minitab anova output in shoes.lis also shown
in NWK Fig 29.9 (27.7). Note that there are two parts to this
anova model: a crossing of the fixed effects ad-type and time
and a nesting of the random factor sites (serving as subjects)
within ad-type.
(iii) The between-subjects portion of the analysis.
verify that the test statistic for the between subjects
factor ad-type usues as its error term sites nested within ad-
type. Refer to NWK Tables 29.8 and 29.9 (27.5,27.6) to justify
this error term. What are the relevant degrees of freedom for
this test statistic?
(iv) The within-subjects portion of the analysis.
verify that test statistics for the repeated measures factor time
and its interaction with ad-type use MSerror in the denominator
of their test statistics and justify with Tables 29.8 and
29.9. What are the degrees of freedom for the test statistic
for time if the compound symmetry assumptions of this mixed
model are assumed to hold perfectly? In class, we discussed a
conservative correction first suggested by Box to multiply degrees
of freedom in tests of within-subjects factors by "epsilon", where
the lower bound for epsilon is 1/(r - 1) where r is the number of
levels of the repeated measures factor (here the number of time
points). What would epsilon be and what would the degrees of freedom
associated with the test statistics for time if the Box
correction were used? (A less conservative epsilon value is
given by the Greenhouse-Geisser factor )
b. Winer dial example. Now that we have revisited the shoes example,
consider a famous (though not scintillating) example from Winer's
traditional text (Chap 7 esp p.525). The data description
(p.525) "Consider a factorial experiment in which the levels of
factor A [between-subjects factor] are two methods for
calibrating dials and the levels of factor B are four shapes for the
dials. Outcome measure are accuracy scores on a series of trials on
each of the dials." For each calibration method, three randomly
chosen subjects complete the trial for each of the four dial shapes.
To summarize: dial is a 2 (calibration method) x 4 (shape of dial)
design with repeated measues on the second factor. Outcome variable
is accuracy in callibrating dials veiwed as a function of calibration
method (between subjects) and dial shape (within subjects).
The data are presented in the following form in Winer's text
dial.data:
subj|dial 1 dial 2 dial 3 dial 4
1 | 0 0 5 3
method 1 2 | 3 1 5 4
3 | 4 3 6 2
----|--------------------------
4 | 4 2 7 8
method 2 5 | 5 4 6 6
6 | 7 5 8 9
(i) create a data file dial.dat that mimics shoes.dat in structure
(ii) use minitab to run the repeated measures anova to mimic
the analysis in shoes.lis
(iii) carry out tests for the between subjects and within-subjects
factors. State the values of the test statistics and their associated
degrees of freedom. For test statistics for within-subjects factors
use the Box correction to the degrees of freedom.
c. Implementation in SAS. the following set of commands
will produce the repeated measures analysis for the shoes
data
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data shoes;
input ad m1 m2 m3;
datalines;
1 958.00 1047.00 933.00
1 1005.00 1122.00 986.00
1 351.00 436.00 339.00
1 549.00 632.00 512.00
1 730.00 784.00 707.00
2 780.00 897.00 718.00
2 229.00 275.00 202.00
2 883.00 964.00 817.00
2 624.00 695.00 599.00
2 375.00 436.00 351.00
;
proc glm data=shoes;
class ad;
model m1--m3 = ad /nouni;
repeated Time 3 (1 2 3) /summary printe;
run;
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you can execute one of two ways.
with SAS on a PC just paste these commands into the editor
window and run
on leland system make these commands into a shoes.sas text file
then simply and the command line:
sas shoes.sas
will create a shoes.lst file containg the output
From SAS output use the value for the Huynh-Feldt Epsilon
to adjust the critical values for the within 'subjects' tests.
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6. Polynomial trends (picking up from class 1/31)
Consider the bock text reading data from class handout
The vocabulary learning data bock*.* comes from Bock, MSMBR, p.454.
"Data are drawn from test results on file in the Records Office of the
Laboratory School of the University of Chicago. They consist of
scores, obtained from a cohort of pupils at the eigth through
eleventh gade level on alternative forms of the vocabulary section of
the Cooperative Reading Tests." There are 64 students in all, 36
male, 28 female (ordered). In bock.dat the first column is ID, the next
four the vocab scores and the last gender (Male = 1).
A repeated measures anova (as in the handout) will give an
occasions effect with 3 degress of freedom. As discussed in
class the shape of the overall growth curve is often examined
via orthogonal polynomials (here 1df for each of linear, quadratic,
cubic trends). This is a common example of the use of planned
orthogonal contrasts.
The SS for each component is given by
n*(SUM[Y-bar(j)*c(j)])^2/SUM[c(j)^2]
where c(j) are the components of the contrast and
Y-bar(j) components of the mean vector.
Reproduce the 1df decomposition of the occasions effect for these
data (ref handout).
Alternatively have SAS or other program compute the decomposition
(eg /polynomial)
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7. Student supplied repeated measures experimental data
(discussed in class)
Description
The Expectation Study
This experiment was designed to study the effects of expectation of item
format on reading comprehension performance. Students were randomly
placed into treatment groups. They were either told to expect
multiple-choice questions after reading a passage or to expect
constructed-response questions. After reading the passage, both groups
answered the same 12 questions. Questions were classified by type
(factual or relational) and format (multiple-choice or constructed
response).
questions.dat data file
c1 number of questions correct (0-3)
c2 treatment group : question expectation
(1 for multiple-choice, 2 for constructed-response)
c3 student number (nested within treatment)
c4 question type (1 for factual, 2 for relational)
c5 question format (1 for multiple-choice, 2 for constructed-response)
Carry out a repeated measures anova analysis for these data. You
can either view the within-subjects 2x2 structure as a 1x4 to
match prior examples, or much better, use the within subjects
factorial structure.
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C Nonparametric Alternatives
NWK ver 5 sec 18.7, p.900, 1138-9
Nonparametrics
A good review of basic nonparametric procedures is in
Minitab student handbook/primer Chaps 12 or 13 (depending on
version)
8. 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.
END