STAT 512 Homework Assignment 4: Due in class, Friday September 25, 2015
Verma et al. (1999) carried out an experimental study to compare the effects of different hydration treatments on a number of physiological variables in human beings subjected to a standard
(common) level of physical activity. (See their paper, attached.)
1. Does it appear that the investigators used an appropriate randomization procedure in this
study? Explain your answer.
2. Table 3 of the paper displays an ANOVA table for one of the responses, pulse rate. In each
of the following questions, assume that the sums-of-squares reported in this table ... but
not necessarily the other components ... resulted from data collected under the indicated
alternative designs. For each of the alternative designs described, compute the F -statistic
appropriate for testing equality of treatments, along with the associated degrees of freedom
for that test.
(a) Suppose that the study was conducted using 5 treatments and 5 subjects as reported,
but had been carried out in 5 days, rather than 10, i.e. that the design had been an
unreplicated Latin Square.
(b) Suppose that the study was conducted using 5 treatments and 10 days as reported, but
that 10 subjects, rather than 5, were included. Five of the subjects participated in Days
1-5, and the other participated in Days 6-10. Here, interpret the sum of squares for
subjects as being subjects-within-weeks, i.e. assume it does not reflect any difference
between the 5 subjects who participated in the first week and the 5 who participated in
the second.
3. For each of the alternative experiments described below, use R or another computer package
that supports numerical linear algebra, to determine whether this design is “Condition E”
equivalent to a completely randomized design with the same number of treatments and units
assigned to each treatment. Compute the design information matrix, X02|1 X2|1 , for each design.
(a) Suppose that it was discovered, after the fact, that subject 1 had actually carried additional water to drink during the hike, and that the amount of water he drank on each day
could not be determined. As a result, it was decided that data taken from this individual
should be excluded from the analysis.
(b) Suppose that the experiment was planned as reported, but at the last minute treatment
A (the one that excludes all water intake) was determined to be too stressful, so it was
eliminated. No other changes were made to the design; 5 subjects were used and the
experiment took place over 10 days, but only 4 treatments were used each day. Individual
subjects got a “day off” on days they had been assigned treatment A.
(c) Suppose again that the experiment was planned as reported, but that a major storm
occurred on day 10, cancelling the activities for that day, so only data from days 1 - 9
were available for analysis.
Reference: Verma, S.S., Y.K. Sharma, G. Pichan (1999). “An Application of Replicated Latin
Square Design in Physiological Research,” E. Schweizerbart’sche Verlagsbuchhandlung 82, 241-247.
Z. Morph. Anthrop.
82
2/3
of Biostatistics
Department
Defence
Institute of Physiology
and Allied
241-247
Sciences,
An application of replicated
in physiological research
S.S. Verma,
Y. K. Sharma
Stuttgart, Mai
Delhi,
1999
India
latin square design
and G. Pichan, Delhi
With 5 tables inthe text
of complex
in physiological
statistical designs
research are quite
Summary:
Applications
difficult due to unavoidable
faced by physiologists.
are,
practical
Physiologists
problems
use of simple stastistical
to make
in most of the physio
therefore, accustomed
designs
is to illustrate an application
of
aim of the present investigation
logical studies. The main
latin square design
for obtaining
the
highly informative results for evaluating
replicated
on physiological
It has
effect of dehydration
in hot environment.
responses
during work
on physiological
been concluded
that there was a significant effect of dehydration
respon
ses of the subjects during work in hot environment.
trifft in der phy
Die Anwendung
statistischer Verfahren
komplexer
auf Schwierigkeiten,
da hier unvermeidbare
Probleme
praktische
Forschung
siologischen
Studien relativ einfache
statisti
auftreten. Aus diesem Grunde werden bei physiologischen
Zusammenfassung:
Studie
ist es zu zeigen, wie beson
bevorzugt. Das Ziel der vorliegenden
mit
Hilfe
des lateinischen
Informationen
der
Quadrats
aussagekraftige
Anwendung
der
des Einflusses
werden
konnen. Als Beispiel
dient hierbei die Ermittlung
gewonnnen
sche Verfahren
ders
der Arbeit bei hohen Tempera
Reaktionsmuster
wahrend
auf physiologische
Dehydration
turen. Die Untersuchung
auf die phy
zeigt, daB es einen signifikanten Dehydrationseffekt
hohen
wahrend
der
Arbeit
bei
der
Reaktionen
Temperatu
Versuchspersonen
siologischen
ren gibt.
Introduction
The theory of design of experiments has played a major role in the field of agri
cultural research for better statistical interpretationof results (Das & Giri 1979).
Yates (1936) has given a good account of work on the application of incomplete
latin square design in agricultural research. Emmens (1960) emphasized on the
possibility of obtaining highly informative results by using the factorial design in
animal experiments. Bliss (1967) showed the applications of various statistical
designs to solve practical problems in biomedical sciences. The applications of
latin square design have also been made in the ergonomic evaluation of various
types of digging tools (Sengupta et al. 1974, 1975). When small latin squares
are used, it is frequently desirable to replicate them to increase the error degrees
of freedom.Montgomery
(1991) illustrated several ways to analyze a replicated
latin square design. Attempts have been made to apply replicated latin square
?
1999 E. Schweizerbart'sche
0044-314X/99/0082-0241
Verlagsbuchhandlung,
D-70176
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$1.75
Stuttgart
242
S. S. Verma, Y. K. Sharma and G. Pichan
design in various branches of biomedical sciences (Harari & Hosey 1979, Kaj
din 1980, 1984, Johnson & Remillard
1988, Cro
1983, Clark & Petersen
signani et al. 1991, Tranquilli
et al. 1992, Grant &
et al. 1991, Colombo
Weidner
1992, Sahlu & Fernandez
1992) to achieve more reliable findings.
Recently Abolin et al. (1993) used replicated latin square design in biomedical
sciences to compare the bioavailability of four temazepam 7.5 mg capsules with
thatof a single temazepam 30 mg capsule. Perhaps, the application of this design
has not been made in physiological research. Thus, themain aim of the present
investigation is to illustrate an application of replicated latin square design in the
field of heat physiology to examine the effect of differentdegree of dehydration
on certain physiological responses of the subjects working under heat stress and
to evaluate the effect of dehydration on certain physiological responses during
work in hot environment for obtaining highly informative results.
Material
and methods
The study was conducted with five human male subjects 25-33 years of age.
These subjects have been staying inDelhi for the last two years and were accli
matised to heat. The subjects were made to march nine miles daily between
1000-1300 hours 1ST carrying a weight of 12 kg. After every fiftyminutes of
march, rest pauses of tenminutes were given enroute, and this yielded two rest
pauses during the march. During the remaining part of the day, subjects were
engaged in normal active duties which were of sedentary nature. The study was
conducted in two series of trials of five days' duration in the summer.
The subjects were given five treatmentsA, B, C, D and E. One subject was
being put on one treatment each day. The latin square design used in this study
was necessary as the environmental temperatures on different days of the trial
were expected to vary. By this design, each subject had to undergo all the treat
ments on differentdays of the trial.
The five treatmentsA, B, C, D and E were defined as
A = No water was allowed to be taken by the subjects during the period of
stress.
B = The subjects were allowed to take 360 ml of water during the stress period.
This quantity was given in two equal amounts during the rest pauses.
C = 720 ml of water was given during the stress in two equal amounts during
the stress
pauses.
D = Prehydration 1. 540 ml of water was given before the commencement of
the stress and another 720 ml of water during the stress as in treatmentC.
E = Prehydration 2. The subjects were made to drink 900 ml of water before
the startof the stress and were given 720 ml of water during the heat stress
as in treatmentC.
These treatmentswere assigned to produce differentdegrees of dehydration as
well as to assess the effect of drinking water before and during the stress. The
four physiological variables measured using the standard physiological tech
niques for each subject were: sweat loss, rise in oral temperature, rise in pulse
rate and dehydration as percent of body weight. In order to increase the efficien
cy of the latin square design, the trialwas replicated for another five days with
the same subjects having the same treatments.Thus a replicated latin square de
sign was formed to obtain highly informative results (see below for pattern).
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An application of replicated
latin square design
243
Days
Subject
10
B
D
D
D
D
D
C
D
B
B
D
D
B
D
The statisticalmodel for this type of replicated latin square design is given by
=
+
+
+ eijkl
+
Yyki P< oil Pj 7k
=
n and
where, i=l,
p; j l,
np; k=l,
p; 1=1,
Yijkl is the effect of
kth treatment in the ithrow, jth column and 1threplication, |x is general mean, cq
is effect of ithrow,
(3j is effect of jth column, 7k is effect of kth treatment,e^i is
random errorwhich is distributed normally with zero mean and variance a2.
In the present investigation, n = 2, p = 5. Various comparisons between treat
ments have been made by using Newman Keuls Multiple Range Test (Woolf
1968).
Results
1, 2, 3 and 4 show the analysis of variance tables of replicated latin
square design for four physiological responses (viz. sweat loss, rise in oral tem
perature, rise in pulse rate and dehydration as percent of body weight).
Tables
Table
1. Analysis
of variance
table of replicated
latin square
design
for sweat
loss during
exercise.
Source of variation
Between
F P SS
MS
4
1.0505
0.2626
9
5.7857
0.6429
14.92
< 0.001
treatments
4
2.2699
0.5675
13.17
< 0.001
Error
32
Total
49
1.3795
10.4856
of variance
table of replicated
2. Analysis
rature at the end of route march.
Table
Source of variation
Between subjects
Between days
Between treatments
Total
< 0.01
subjects
Between days
Between
df
df
SS
4
9
4
32
Error
49
0.2912
2.0912
1.2592
2.2856
5.9272
6.09
0.0431
latin square
for rise in oral
design
MS
0.0728
0.2324
0.3148
0.0714
P
tempe
F
1.02
NS
3.25
4.41
< 0.01
< 0.01
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S. S. Verma, Y. K. Sharma and G. Pichan
244
of variance
3. Analysis
at the end of route march.
Table
Source of variation
Between subjects
Between days
9
Between treatments
Table
table of replicated
design
for rise
in pulse
rate
FPMS
df
SS
4
5261.28
1528.08
1928.48
3348.64
12126.48
4
Error
32
Total
49
latin square
4. Analysis
of variance
table of replicated
at the end of route march.
12.56
NS
1.62
4.75
1315.32
169.78
497.12
104.64
latin square
design
< 0.001
< 0.01
for percentage
body
dehydration
FP
MS
Source of variation
df
SS
Between subjects
Between days
Between treatments
Error
Total
4
9
4
32
49
2.9961
15.1354
26.7417
4.1496
49.0228
Table
5. Mean
physiological
Physiological
responses
responses
of replicated
0.7490
1.6817
6.6854
0.1297
latin square
5.77
12.96
51.54
< 0.01
< 0.001
< 0.001
design.
Treatments
A
B
C
D
Sweat loss
duringexercise (kg)
2.89
2.83
2.81
3.06
Rise in oral temperature
0.46
0.32
0.14
0.00
50
44
4246
3.7
3.2
3.2
3.38
-0.04
at the end of
route march
(?C)
Rise inpulse rate 60
at the end of route
march
(Beats/min)
Percentage body dehydration 5.1
4.4
at the end of route march
Evidently the changes in the sweet loss during exercise are highly significant
(P< 0.001, Table 1) between treatments.Table 2 shows the significant (P<0.01)
rise in oral temperature at the end of the routemarch due to treatments.The rise
in pulse rate at the end of route march between treatments is significant at 1%
level of significance (P<0.01, Table 3). Changes in the percentage of body dehy
dration at the end of the route march between treatments are highly significant
(P< 0.001, Table 4). Table 5 shows the mean physiological responses of the
replicated latin square design at different treatments.
During the first replication, the thermal stress was very severe during the
hours of stress, and themaximum temperature of the day varied from 42.3 to
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An application of replicated
latin square design
245
44.4 ?C. The condition remained hot throughout the five days of the trial and
therewas no remarkable day to day variation in the environmental temperature.
During the second replication, the environmental stress was comparatively less
severe during the hours of work, and themaximum temperature of the day varied
between 35.0 to 38.0?C. The maximum temperature of 44.4?C encountered du
ring these trials represents the highest temperature normally encountered during
summer in this part of the country.
Tables 1, 2, 3 and 4 show the effect of five treatments on four physiological
responses (viz. sweat loss, rise in oral temperature, rise in pulse rate and change
in percentage body dehydration). Evidently the treatments are having a signifi
cant effect on these four physiological responses. The level of significance varies
from 0.01 to 0.001. The mean physiological responses consisting of sweat loss,
dehydration as percent of body weight, rise in pulse rate and rise in oral temper
ature are presented inTable 5 by pooling the values of two replications.
Discussion
Itwas observed (Table 5) that the sweat loss for treatmentA, B and C was found
to be the same with themean value of about 2.84 kg in three hours, indicating
thatwater intake up to 720 ml during stressmade no significant difference on
sweat loss, compared to that of no intake of water. However, with treatmentD
and E, where respectively 540 ml and 900 ml of water were given before the
stress and 720 ml during the stress, sweat loss was higher than thatwith the
other three treatments, the mean value being 3.06 and 3.38 kg, respectively.
Sweat loss with treatmentE which had a higher intake of water initially was
significantly (P<0.01) higher than that of treatmentD. The difference in sweat
loss between treatmentD and E was almost equal to the difference in theirwater
intake.
Table 5 shows that the level of body dehydration varied in each treatmentas it
was influenced by the water intake during as well as immediately before the
stress.With treatmentsD and E, where 540 ml and 900 ml of water were taken
before
the stress,
the mean
dehydration
was
3.2
percent.
The
higher
initial
water
intakewith treatmentE resulted in an amount of extra sweat loss almost equal to
it.Dehydration with treatmentA, B and C which had almost the same amount of
sweat loss was highest with treatmentA which had no water intake and there
after it decreased with treatmentB and C, equal to the difference inwater intake.
The mean highest dehydration was 5.1 percent of body weight. The difference in
dehydration is statistically significant (P< 0.001) between various treatments ex
cept between D and E.
From Table 5, it is clear that the rise in pulse rate at the end of stress was
highest with treatmentA and averaged about 60 beats per minute. With treat
ment B, themean rise was 50 beats per minute, the difference between the two
being significant (P<0.05). The rise in pulse ratewith treatmentC, D and E was
almost the same with an average value of 44, 46 and 42 beats per minute.
Table 5 shows that the rise in oral temperaturewas also highest with treatment
A where themean rise was found to be 0.46?C. With treatmentB, C, D and E,
this rise was 0.32, 0.14, 0.00 and -0.04 ?C, respectively. The rise with treatment
A and B was higher than thatwith treatmentC, D and E. In both the prehydra
tion groups D and E, therewas no rise in the oral temperature during the three
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S. S. Verma, Y. K. Sharma and G. Pichan
246
hours of the trial.Thus highly informative results were obtained by using a repli
cated latin square design in physiological studies on human subjects.
Acknowledgements
authors are grateful to Dr. W. Selvamurthy,
Institute of Physio
Director, Defence
are
for his keen interest and continuous
Thanks
Sciences,
encouragement.
logy and Allied
due toMiss
Shikha
Arora
for computer processing
of the data.
The
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