,
•
e
STATISTICAL METHODS FOR EVALUATION OF MASTITIS TREATMENT DATA
by
Gary G. Koch, James E. Grizzle, Kofi Semenya, and P.K. Sen
Department of Biostatistics
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 1156
February 1978
~
~
Running Head:
(Ed
Symposium:
Multivariate and Categorical Data Analysis
n 319)
Statistical Methods for Evaluation
of Mastitis Treatment Data
Gary G. Koch, James E. Grizzle, Kofi Semenya, and P. K. Sen
Department of Biostatistics
University of North Carolina
Chapel Hill, N.C. 27514
Abstract
Categorical data on the response to treatment for mastitis are analyzed
by several methods.
The most important aspect of these analyses is the em-
phasis that 'the cow rather than a quarter of the udder is the basic experimental unit.
Thus, the untenable assumption of independence among the qU3r-
ters of an udder is not required.
Given this framework, it is shown by example
that many questions of interest can be investigated by weighted regression
analysis.
In particular, methodological strategies are outlined for
1.
assessing the extent of interaction among experimental
factors and its implications to model fitting for describing relationships,
2.
accounting for pre-treatment scores as a covariable,
3.
undertaking multivariate analyses with respect to the
four quarters simultaneously.
The conclusion from these analyses is that the various active treatments under study are homogeneous and are superior to no treatment.
Cows
on any of the active treatments completed the trial with approximately one
less quarter being infected than when the study began.
In particular,. the
decline in the average number of infected quarters was from 1.8 to 0.8.
Key Words:
Categorical Data, Weighted Analysis of Variance, Discrete Data,
Statistics, Mastitis
2
Introduction
A study of the treatment of mastitis in dairy cows presents many of
the same problems encountered in clinical trials of therapies that use human
subjects.
In this regard, the design and analysis issues that should re-
ceive attention can be summarized as follows:
1.
Statement of the problem and study objectives.
2.
Specification of what class of animals are eligible
for the study. It is important to be specific about
this because it defines the population to which the
findings of the study apply.
3.
Description of the therapies under study and indications for their modification and for when other supportive measures are permitted.
4.
Specification of criteria for evaluation of the outcome of treatment. This is sometimes called the end
point in clinical trials.
5.
Specification of sample size and the extent to which
it is large enough to detect clinically important
differences.
6.
Description of the randomization scheme and/or other
technical aspects of the research design. This should
indicate whether blocking is to be used to control for
heterogeneity and whether there are important covariables
to be included in the analysis.
7.
Description of the methods to be used for data analysis.
The questions on this list require knowledge of physiology, the natural
history of the disease, cli.nical practice and statistics for their resolution.
This paper deals mostly with statistical methods for data analysis specifically
with respect to the results of a clinical trial for the treatment of mastitis.
The Problem
Heald et ale (2) described a clinical trial to
i.
optimize the dose of penicillin for the treatment of
Streptoc.occus agalactiae at drying-off when the drug
e·
3
was used in combination with novobiocin for the treatment of Staphylococcus aureus, and
ii.
determine if a combination of penicillin and novobiocin
was justified for treatment of both infections at drying
off.
Cows from sixteen Southwestern Virginia Holstein dairy farms were used
in the study, including both milking parlor and stanchion barn herds.
These
herds also reflected a range of subjective management scores of poor, fair, good,
and excellent, as judged by the technicians who visited the farms on a weekly
basis.
Cows were assigned sequentially as they were identified, rather than
randomly, to one of the following nine treatments:
1.
.
2.
e
3.
4.
5.
6.
7.
8..
9.
No treatment (Control)
1x10 5 I. u. Penicillin
2x10 5
5
4xlO
2
4x10
5
1x10
5
2x10
5
4xlO
2
6x10
I. U. Penicillin
I. U. Penicillin
Novobiocin
I.
u.
Penicillin + 4x10
I. U. Penicillin + 4x10
I. U. Penicillin
2
2
Novobiocin
Novobiocin
2
+ 4x10 Novobiocin
Novobiocin
In addition, dairymen sometimes requested that certain cows be treated.
These cows were given one of the eight active (non-control) treatments.
Other specific details concerning the research design for this investigation
are given in (2).
The infections present at the beginning and at the end of treatment
were identified and constituted the basic data for analysis.
Although many
different organisms were potentially involved, they have been pooled here in
order to simplify the discussion of alternative statistical methods.
Thus,
the presence or absence of any infection at the beginning and end of the
4
study for each of the four quarters of the cow's udder are the variables to
be used for treatment evaluation.
The Method
As indicated by Miller (5), the infection state variables for the
four quarters of the cows udder are generally not independent.
Thus, the
usual types of contingency table analyses cannot be validly applied to the
raw data for the respective quarters.
However, certain related methods which
are both more powerful and more comprehensive can be adapted to this situation
without great changes in concept because they provide a framework which permits the quarter data to be appropriately linked to the corresponding cows.
Consider each quarter of a cow's udder as being infected or not, before
and after treatment.
This combination of conditions can be represented in a
2x2 table.
post-treatment state
.
Pretreatment
state
where I denotes infected and
I
I
I
TIll
TI
-I
TI
TI
2l
I, not infected; n . .
Ji J2
12
22
denotes the probability of
the (jl' j2)-th state combination for a given quarter and experimental condition; and the {TI
j
j } sum to 1.
1 2
Tables can be constructed for each quarter.
Similarly, the marginal
table obtained by summing over the quarters can also be used as a basis for
inferences about experimental conditions.
This does not, however, imply that
the quarters can be regarded as independent sampling units.
basic sampling unit.
The cow is the
5
To analyze the marginal table for sums over quarters, it is necessary
to construct a covariance matrix which does not assume independence among the
quarters.
For this purpose, the pattern of response for a given cow is
represented as shown below:
Pattern of Response
Quarter
Left front
Left rear
Right rear
Right front
II
II
II
II
(1)
Total
(2)
(3)
1
1
1
(4)
1
Each row of the table has 1 as its entry under II, II, II, or II.
total for each row is 1 and the total for the table is 4.
Thus, the
The total for each
column is a random variable which can take on values 0, 1, 2, 3, 4.
distribution of the entries in the table is multinomial with 44
e
= 256
The joint
para-
meters, but many of them may be equal if the quarters are exchangeable (a1though exchangeability is not a necessary assumption for analysis).
For
example,
1
Total
1
1
1
1
4
II
II
II
Total
1
0
0
0
0
2
1
1
1
0
0
0
1
0
0
0
0
0
1
1
1
4
II
1
2
1
0
0
3
4
Total
II II
0
0
0
0
0
Quarter
0
1
0
0
0
0
1
0
1
II
0
1
1
2
II
is equivalent to
Quarter
1
2
3
4
Total
1
Thus, in such cases, the different interior patterns with identical totals
do not need to be distinguished because they can be viewed as providing the
same information.
The relevance of this consideration to alternative methods
6
of constructing valid estimates for covariance matrices which do not require
exchangeability is described for various types of analyses in subsequent sections
of this paper.
Other methods which are potentially more efficient when certain
exchangeability assumptions are applicable are still under investigation.
Once these column totals and their corresponding estimated covariance
matrix are computed, their analysis can be undertaken by the weighted least
squares (WLS) methods discussed in Grizzle et al. (1) via the computer program
GENCAT discussed in Landis et a1. (4).
can be fit to a vector of functions
In particular, models of the form
~(!)
of the multinomial cell probabilities!
where X is a design matrix of known coefficients and § is a vector of parameters
to be estimated.
The usual rank constraints for linear models are assumed to
hold in this situation.
This model can be fitted and S estimated by WLS where
the weight matrix is the asymptotic covariance matrix of Fep), where F(p)
--
--
is the same as F(TI), except the observed probabilities p are substituted
for TI.
Any linear hypothesis, H :
O
on C, can be tested by
where b is the WLS estimate of S and
~
§ = 0, with suitable rank constraints
~b
is the estimated covariance matrix
of~. This quadratic form has, asymptotically, a central X2 distribution
wit~
degrees of freedom equal to the rank of
f
if H is true.
O
Prelimina9' Analyses
For descriptive purposes, average probabilities of each pattern of
infection response within each treatment group are shown in Table 1.
The
average probabilities in the table are the totals over all quarters for cows
in all herds, standardized to add to 1 by division by 4 times the corresponding
~
e
Table 1-
e
Probability of eachpsttern of response.•.
Novobiocin
Units of PeniCillin 'x '105·.
' 1
2
0
(Mg)
II
0
.27
II
II
(26)
.15 .18
II
.39
II.
.14
600
II
.11
(42)
.32 .05
(37)
400
e
II
.48
. II
.• 10
II
.34
.05
.47
.05
(20)
.31 .11
.53
.18
.32
The sample size in each cell is shown in (
II
II
.11
II
II
II
(35)
.33 .06
.49
(27)
.08
)
II
(34)
.33 .13, .43
(27)
.14
4
.42
.18
.31
.05
(26)
.46
.06
.40
.08
.45
.
"
8
sample size.
Thus, it can be seen that the average probabilities for II and
II dominate the distributions for the eight active treatments and thereby suggest that the extent of infection is less post-treatment than pre-treatment
for these groups.
This pattern is
ment (control) group.
not
apparent for
the no
treat-
Further discussion of the statistical significance of
these results is given later.
Average probabilities for each herd are shown in Table 2.
These quan-
tities are more difficult to interpret because of their variability and the
small sample sizes.
However, to get an idea of whether there are herd effects,
the herd management scores are considered.
Table 2.
These data are also shown in
From this point of view, it can be seen that both the average proba-
bility of being infected at the beginning of the study period and at
depend somewhat on the quality of management.
th~
end
In this regard, there seems to
be a trend toward smaller probabilities of infection as the herd management
score varies from poor to excellent, although there is substantial overlap
among them; e.g., poor and fair management scores have similar state distributions, as do good and excellent.
Before proceeding to more detailed analyses, a check of the comparability of the cows assigned to different treatments is worth making, particularly since they were not randomly assigned.
One way of examining this issue
~
is to combine the II and II classes within treatments and then to test for
homogeneity across treatments, i.e., to determine whether the average probability of being infected at the beginning of the study is the same for each
treatment.
These quantities are shown in Table 3 and can be seen to be
reasonably homogeneous.
pretation is given later.
A formal test statistic which supports this interAnother check of interest is to examine the dif-
ference between the average probabilities of being infected before treatment
~
9
Table 2.
Herd
number
e
11
12
4
6
9
10
16
13
14
15
25
1
2
3
7
8
Probability of each pattern of response tabulated by herds.
N
II
II
II
II
21
8
48
21
14
22
7
14
10
10
25
27
10
13
17
7
.19
.22
.13
.09
.46
.14
.18
.05
.10
.10
.12
.20
.00
.04
.03
.07
.42
.44
.31
.37
.36
.32
.18
.41
.40
.22
.43
.26
.30
.27
.12
.14
.05
.04
.09
.06
.00
.09
.14
.04
.05
.07
.10
.19
.05
.10
.12
.04
.33
.31
.47
.48
.18
.45
.50
.50
.45
.60
.35
.34
.65
.60
.74
.75
Milking Management
system
score
P
P
P
P
S
P
P
P
P
S
S
S
S
S
P
S
Hl1king system: S = Stanchion, P = Parlor
Management score: P = Poor, F = Fair, G = Good, E.-= Excellent
--
P
P
F
F
F
F
F
G
G
G
G
G
G
G
E
E
10
Table 3.
Probability of being infected at the beginning of the study.
Units of Penicillin x 105
Mg. Novobiocin
0
400
600
0
1
2
.42
.48
.36
.46
.43
.49
.50
4
.44
.46
e-
11
~
and after treatment in the control group.
This difference which corresponds
to (II - II) should be approximately zero because no effective treatment was
applied to these cows.
Since the actual difference was .03 with a standard
error of .07, this check also appears to be satisfied.
Thus, it is considered
reasonable to assume for purposes of analysis that the sequential treatment
assignment scheme produced comparable groups of cows.
Detailed Analyses
Analvsis of post-treatment number of infected quarters with adjustment for
herd management score
Since herd management score was thought to have an effect on an
animal's infection status, it should be taken into account in the analysis.
Let F
~
iki
be the total number of post-treatment infected quarters for the i-th
animal in the group from the sub-population corresponding to the i-th treatment and k-th management score.
Thus, this application involves a single
response variable with r = 5 levels (i.e., 0, 1, 2, 3, 4 infected quarters)
and two factors, treatment and herd management score, for which the joint
classification defines s
= (9x2) = 18
sub-populations.
score is collapsed into two levels, poor or fair versus
Herd management
good
or
excellent, in order to maintain sample sizes large enough for asymptotic
distributions to hold (see Koch et al. (3».
The corresponding contingency
table is displayed in Table 4.a.
The remainder of the analysis in this section will be directed at the
.~
12
Analysis of post-treatment number of-infected quarters with respect to treatment and herd management
Table 4.
ecore.
e.
~
Observed contingency table, mean scores, and standard errors.
Sub-populations
Herd
management
Observed
frequencies for the response
Treatment
variable post-treatment number
of infected quarters
Sample
Penicillin
Novobiocin
2
4
0
1
3
Size
(IU)
(mg)
Poor
or
Fair
Good
or
Excellent
b.
None
100,000
200,000
400,000
None
100,000
200,000
400,000
None
1
10
5
11
12
8
5
9
5
2
9
6
2
5
2
3
4
3
3
2
4
2
1
1
2
1
2
7
1
1
1
1
2
3
1
0
0
0
0
1
1
1
1
0
0
13
22
16
17
20
14
14
15
10
1.06
.76
.70
1.00
1.43
.60
.70
None
None
None
None
400
400
400
400
600
None
100,000
200,000
400,000
None
100,000
200,000
400,000
None
3
10
8
10
11
6
11
5
5
7
5
3
2
1
2
2
3
4
2
2
0
3
0
2
0
1
0
0
1
2
2
1
0
0
1
1
13
20
18
18
17
13
13
11
10
1.38
.85
.83
.67
.76
1.08
.31
.55
.60
7
7
1
0
1
0
0
0
1
Main Effect Model
Xb
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
~
.27
.17
.22
.29
.25
.36
.36
.23
.25
.73
.32
.24
.24
.21
.31
.:)2
.23
.24
.38
Final Model X =
=~
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 1
1.96 +
- .20
-1.12 :+:
- .91
-1.13 +
-1.16 :;:- .80:+:
-1.19 :+:
-1.29
-1. 23
+
+
+
+
.21
.12
.25
.26
.27
.28
.32
.28
.26
.29
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
X b
Estimate + s.e.
1
1
1
1
1
1
1
The parameters for this model correspond respectively
to the following: predicted value for no treatment
for cows with poor or fair management, (negative)
increment effect for good or excellent management,
eight separate (negative) increments for the successive active treatments as arranged in table 4.3.
c.
Estimated
s.e.
2.23
None
None
None
None
400
400
400
400
600
Fitted models, estimated parameters, and standard errors.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Mean
Score
1-1.88 :t-1.13 +
-
-
~
.21-'
.22_,
Estimate + s.e.
1
1
1
1
1
1
The parameters for this model correspond
respectively to the following: predicted
value for no treatment without regard to
management, and common (negative) increment
for any of the eight active treatments.
Chi-square tests of significance for weighted analysis of variance (WANOVA)
Source of Variation
Management
Treatments
Active Treatments
Residual Goodness of Fit
D.F.
1
8
7
8
Q
2.50
31.81
4.91
9.40
Source of Variation
Active Treatments
D.F.
Q
1
27.51
Residual Goodness of Fit
16
16.B?
Total
17
44.33
~.
13
e
mean scores computed by
. ·1
Fik ~ --n ik
n
1
4
= --j n
n ik j=O
ijk
I
where n
ik
ik
I
~=l
Fiko
JV
'
4
=
I
j=O
j PiJ'k
denotes the number of animals in the group corresponding to the
i-th treatment and the k-th management score, n, 'k denotes the number of such
~J
animals with j infected quarters at post-treatment evaluation, and p
denotes
-ik
the (5x1) vector of response proportions Pijk
In matrix terms, the
{F ik }
can
be
= (nijk/nik) for
j
= 0, 1, 2, 3, 4.
calculated by applying the linear
transformation operation
~
= [0,
1, 2, 3, 4] ® !18
to the overall (90x1) compound vector _p of the sub-population vectors p
-ik
for i = 1, 2, ••• , 9; k = 1, 2. Here ® denotes Kronecker matrix products,
which represent a compact notation for describing
~
as an (18 x90) block
diagonal matrix with the 18 diagonal blocks being sub-matrices of the
type (0, 1, 2, 3, 4).
by division
by
4)
right of Table 4.a.
The resulting mean scores (which have not been standardized
and their estimated standard errors are shown at the
In this case, the standard errors are the square roots
of the diagonal elements of the estimated covariance
matrix
YF
for f which is
obtained via the matrix operation
--
A V A'"
"'" -e
-v
where V is the estimated covariance matrix for the vector of estimated pro-
-£
portions
£.
For more complete computational details, see (1), (3), and (4).
14
The effects of treatment and management on the post-treatment means
will be examined by weighted least squares regression.
X
-~
A main effect model
containing treatment and management effects is used to test for treat-
ment differences after adjustment for the effects of herd management score.
The specific structure of this model together with estimated parameters is
shown in Table 4.b.
Since its goodness of fit test statistic in Table 4.c
is non-significant (a=.25), further consideration of its implications is
warranted.
The other test statistics in the weighted analysis of variance (WANOVA)
Table 4.c
show that differences among treatments are significant (a=.05) ,
and the main effect due to herd management score is non-significant (a=.25).
Also, the differences among the active (non-control) treatments are clearly
non-significant (a=.25).
final model
~FM
Thus, the main effect model X
-~
is reduced to the
which has only two parameters; namely, a baseline overall
estimated mean of (1.88 + .21) infected quarters for the control treatment
and a common estimated effect of (-1.13 + .22) infected quarters for the
active treatments.
For the sake of completeness, it is of interest to consider the treatment main effect model
~~
shown in Table 5 as an alternative to
~FM.
How-
ever, the goodness of fit test statistic for this model is significant (a=.05)
and thus implies that its use is not appropriate.
between the goodness of fit test statistic for
In addition, the difference
~TME
and the goodness of
fit test statistic for the preliminary main effect model
~.
~ME
corresponds to
the test statistic for the (Penicillin x Novobiocin) interaction in the
model.
~
_ME
e-
Since the corresponding test statistic
Q = 23.97 - 9.40
=
14.57
with D.F. = 3 is significant (a=.OS), there is definitely interaction between
15
Table 5.
Alternative fitted model (~ = ~TME) for post-treatment number of
infected quarters, estimated parameters, standard errors and chi-square tests
of significance.
Xb=
100 0 0 0 0
1010000
100 1 0 0 0
100 0 1 0 0
100 0 0 1 0
1 0 1 0 0 1 0
1 0 0 1 0 1 0
100 0 110
100 000 1
1100000
III 0 0 0 0
1 1 0 1 0 0 0
110 010 0
110 0 0 1 0
1 1 1 0 010
1101010
1 1 0 0 1 1 0
110 000 1
1.49 + .16
Source of Variation
-.17 + .12
Management
1
Q
1.95
-.51 + .19
Penicillin
3
11. 76
-.45 + .19
Novobiocin
2
11.27
-.60 + .18
Residual Goodness
of Fit
11
23.97
-.30 + .13
D.F.
-.77 + .26
Estimate
± s .e.
The parameters for this model correspond respectively
to the following: predicted value for no treatment for
cows with poor or fair management, (negative) increment
for good or excellent management, (negative) increments
for 100,000 units, 200,000 units, and 400,000 units of
Penicillin, and (negative) increments for 400 and 600
mg of Novobiocin.
16
the effects of the two drugs.
for
However, in view of the other results given
and subsequently eFM' its interpretation is straightforward.
~ME
The
lowest dosage under study for either drug considered separately gives the
maximal reduction in the number of infected quarters, and higher dosages or
drug combinations provide no further reduction.
Bivariate analysis of pre-treatment and post-treatment total numbers of
infected quarters
Let
~ii
denote the total number of infected quarters at the h-th
time for the i-th cow in the i-th treatment group where h = 1, 2, for (pre,
post); i
= 1, 2,
... , 9;
and
.t = 1,
2,
... , n i·
There are d
= 2 response
variables, each of which has 5 outcome levels (Le. , 0, 1, 2, 3, 4 infected
quarters), so that there are 52
= 25 possible multivariate response profiles.
There are s = 9 sub-populations corresponding to the respective treatments. Thus,
the bivariate analysis of pre-treatment and post-treatment total numbers of
infected quarters can be formulated as matrix operations on the underlying
(9x25) contingency table displayed in Table 6 for the cross-classification
of treatment with the bivariate pre-treatment versus post-treatment response
profiles.
Estimated pre-treatment and post-treatment mean numbers of infected
=9
quarters for the s
treatment groups can be computed by applying the
linear transformation matrix
A=
~
00000 11111 22222 33333
4444~rC\
'6'1
[ 01234 01234 01234 01234 01234
-9
to the overall (225 Xl) compound vector 12 of cell proportions.
The resulting
17
e
Table 6. Contingency table for cross-classification of treatment with the bivariate pre-treatment vs.
post-treatment joint distribution for numbers of infected quarters.
Novobiocin
(mg)
Penicillin
(IU)
Pre-treatment
number of
infected
quarters
0
1
None
None
2
3
4
None
100.000
0
1
2
3
4
None
e
None
200.000
400,000
400
400
-
600
200.000
400.000
None
1
0
5
5
4
2
6
1
3
8
6
3
4
4
7
6
8
1
5
0
1
0
0
0
5
0
0
0
1
0
0
0
0
0
1
9
7
14
4
8
20
14
6
0
3
0
0
3
2
1
4
3
1
3
2
1
0
2
2
1
0
0
0
0
1
1
0
0
0
0
11
7
3
6
7
13
13
6
1
5
1
1
5
4
7
2
3
4
2
0
0
1
0
1
1
1
1
0
0
0
1
1
0
0
0
0
1
9
7
8
4
7
21
1
7
4
5
4
3
0
4
1
2
1
0
1
0
0
0
0
0
0
0
3
1
0
0
0
1
8
9
6
6
8
23
8
1
3
2
6
2
1
1
4
0
2
0
1
1
0
1
2
1
0
1
0
1
0
2
0
0
0
0
1
7
5
14
4
4
4
1
4
6
3
3
1
3
1
2
0
0
1
0
0
0
1
1
0
0
2
0
2
0
0
0
0
1
0
5
3
1
3
4
1
1
1
0
3
2
1
0
0
0
0
0
1
0
-0
0
0
0
0
0
8
1
0
4
0
0
1
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
6
7
1
4
2
2
3
4
2
3
4
1
100.000
2
0
1
1
0
1
0
0
400
1
3
0
2
5
2
2
Post-treatment
marginal
total
4
0
1
3
1
None
2
2
0
0
0
Pre-treatment
marginal
total
0
1
2
3
4
0
400
Post-treatment number of
Infected Quarters
0
1
2
3
4
2
3
4
0
1
2
3
2
3
4
0
1
2
3
4
5
2
1
4
0
4
3
8
7
5
5
2
8
5
3
3
7
1
7
4
2
1
16
4
2
4
1
16
6
3
1
0
12
5
2
0
1
18
mean score functions
in Table 7. a.
{~i}
and their estimated standard errors
are given
The standard errors are the square roots of the diagonal ele-
ments of
v = A""
-m
-
V
A
-£-
Otherwise, it should be recognized that although the
formation of
~
matrix linear trans-
£ is a straightforward way to define the functions
{~i}' they
are most readily computed as raw data mean scores as described in (3) and (4).
The variation among the estimated pre-treatment and post-treatment
mean numbers of
squares.
infect~d
quarters will be investigated by weighted least
Initially, the preliminary cell mean identity model
~M =
!18 '
is used to test hypotheses about the treatment differences for the pre and
post mean scores.
These tests are given in Table 7.b.
These results show that
the differences among the pre-treatment mean scores are clearly non-significant (a=.2S); and that the differences among the post-treatment mean scores
are significant (a=.OS).
Differences among the post-treatment mean
scores
for the active (non-control) treatments are non-significant (a=.2S).
Equality
of pre-treatment mean scores should be expected from the nature of the experimental design.
~CM
Thus, this constraint is reflected in the reduced model
in a spirit analogous to covariance analysis; i.e.,
~CM
permits post-
treatment comparisons to be investigated in a framework where the respective pre-treatment scores have been standardized to a common baseline.
structure of X
-CM
The
,corresponding estimated parameters and estimated standard
errors, and tests of significance are given in Table
7.c~
Since the residual goodness of fit test statistic for this model is
clearly non-significant (a=.2S), further consideration of its implications
is warranted.
As with
~M
' these results indicate that differences among
19
Table 7.
a.
Bivariate analysis of pre-treatment and post-treatment numbers of infected quarters.
Observed mean scores and standard errors.
None
100,000
200,000
400,000
None
100,000
200,000
400,000
None
None
None
None
None
400
400
400
400
600
b.
Pre-treatment
number
infected quarters
Mean
s.e.
Total
sample
size
Treatment
Novobiocin Penicillin
(IU)
(mg)
1.69
1.88
1.74
1.80
1.92
2.00
1.96
1.85
1.45
26
42
34
35
37
27
27
26
20
e
1.81
.79
.94
.30
.21
.27
.25
.24
.31
.30
.32
.30
.71
.73
1.04
.89
.58
.65
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
Pre-treatment means
2.60
Post-treatment me"ans
8
23.81
Post-treatment means
for active treatment
7
4.31
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
I
0
0
±
±
- .90 ±
-1.13 ±
-1.02 ±
-1. 27 ±
- .85 ±
1. 82
.09
1.85
.20
'1
1
.26
.26
.29
.25
P.!-
.31
-1.14 :!; .2~
-1.21 ± .30
-1.08
±
Estimate
1
1
1
1
1
1
1
1
1
1
1
1
.24
1
1
1
1
± s.e.
= -18
I
•
0
0
0
1
0
1
0
1
0
1
0
1.
0
1
0
1
0
1
~ 1.81 ± .o~
-1.05
±
.09
Estimate + s.c.
The parameters for this model correspond
to the following: predicted value for
common pre-treatment status of all cows
and common (negative) increment for the
post-treatment status of cows on any of
the eight active treatments.
The parameters for this model correspond respectively
tQ the following: predicted value for common pretreatment status of all cows, predicted value for
post-treatment status of cows with no treatment (control) , eight separate (negative) increments for the
successive active treatments as arranged in Table 7.a.
Chi-square tests of significance for weighted analysis of variance (WANOVA) for covariance models.
Source of Variation
D.F.
Q
Source of Variation
D.F.
Treatments (post)
-
.28
.21
.30
.25
.26
.33
.29
.38
.38
Q
8
Fitted covariance models, estimated parameters, and standard errors.
Covariance Model ~ = ~Q1
!eO'
d.
-.12
1.10
.79
1.09
1.19
.96
1.07
1.27
.80
"-I'M
D.F.
Pre-treatment VB.
pos t-trea tmcnt
difference score
Mean
s.e.
.22
.14
.17
.18
.19
.24
.24
.16
.23
Chi-square tests of significance for preliminary cell mean identity model X_
Source of Variation
c.
Post-treatment
number
infected quarters
Mean
s.e.
L
8
30.63
Active treatments (pos t)
1
121. 38
Active treatments (post)
7
4.08
Residual goodness of fit
16
6.72
Control pre vs. post
1
.02
Residual goodness of fit
Total
8
2.60
17
128.10
20
the post-treatment mean scores are significant (a=.OS) and that differences
among the post-treatment mean scores for the active (non-control) treatments are non-significant (a=.2S).
Also, the pre-treatment versus post-
treatment difference for the no treatment (control) group is non-significant
(a-.25).
Thus, the model
~CM
can be reduced to the final model
~FM
whose
parameters can be interpreted as a common overall pre-treatment mean for
all treatments (as well as the post-treatment mean for control) and a
common post-treatment decrement in number of infected quarters for the
active (non-control) treatments.
tistics are given in Table 7.c.
This model, the estimates and test staAs
expected from the steps in its formula-
tion, the goodness of fit test statistic for this model is clearly nonsignificant (a=.2S), and so the final model predicted values are of particular interest.
These quantities indicate that the estimated average number
of infected quarters for each of the treatment groups prior to treatment
is (1.81 ± .08); and that the estimated average number of infected quarters
for each of the active treatment groups after treatment is (.76 + .06),
which corresponds to a common net reduction of (1.05 + .09) infected quarters
for the cows receiving any of the active treatments.
Analysis of pre-treatment versus post-treatment differences for number
of infected quarters
In the previous section, a bivariate analysis of pre-treatment and
post-treatment numbers of infected quarters was performed.
Tais analysis
permitted the simultaneous examination of the pre-treatment and post-treatment mean numbers of infected quarters and, hence, their corresponding differences.
21
An alternative approach for evaluating treatment effectiveness is to
analyze the estimated mean pre-treatment versus post-treatment difference
scores.
The difference
reflects the extent of improvement for the t-th cow in the i-th treatment
group in that it is the reduction in the number of infected quarters between
the pre-treatment and post-treatment evaluations.
Its mean is calculated via
Since the range of variation for both m and m
are 0, 1, 2, 3, 4, the
2it
l1i
range of variation for the differences {fit} is -4, -3, -2, -1, 0, 1, 2, 3, 4.
Thus, this application involves one response variable with 9 outcome levels
and 9 sub-populations corresponding to the respective treatments.
The re-
suIting 9x9 contingency table is displayed in Table 8.
The estimated mean pre-treatment versus post-treatment difference
scores for the 9 treatment groups can be computed from the contingency table
in Table 8 by applying the linear transformation matrix
A = [-4, -3, -2, -1, 0, 1, 2, 3, 4] ® !9
to the overall (8lxl) compound vector p of cell proportions,
The estimated
mean difference scores, and their estimated standard errors are given in
Table 8.
These same quantities could also have been computed from the con-
tingency table in Table 6 by applying the transformation
A=
[0,-1,-2,-3,-4,1,0,-1,-2,-3,2,1,0,~1,-2,3,2,1,0,-1,4,3,2,1,0]
® !9
to the (225xl) compound vector of proportions considered there, or by
computing them directly from the raw data corresponding to the {fit} as described in (3) and (4).
e
22
Table 8.
••
Analysis of pre-treatment vs. post-treatment difference scores.
Observed contingency table, difference scores, and standard errors.
Pre-treatment vs. post-treatment difference
for number of infected quarters
1
4
-4
-3
-1
0
2
3
-2
Treatment
Novobiocin Penicillin
(IU)
(mg)
None
100,000
200,000
400,000
None
100,000
200,000
400,000
None
None
None
None
None
400
400
400
400
600
b.
2
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
1
4
0
1
0
0
0
0
2
0
2
4
4
5
1
2
3
3
1
7
12
10
10
12
11
9
6
9
9
12
8
6
8
5
6
4
2
2
7
4
8
7
2
4
1
2
Fitted models, estimated parameters. and standard errors.
Preliminary Model X c ~M
~c
1
1
1
1
1
1
1
1
1
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
-.12.±. .28
0
0
0
0
0
0
0
1
1. 21 .±. .35
.91 .±. .41
1.20 .±. .37
1. 30 .±. .38
1.08.±. .43
0
4
3
3
5
2
2
6
5
0
3
3
3
3
4
3
4
0
Mean
score
Estimated
s.e.
-.12
1.10
.79
1.09
1.19
.96
1.07
1.27
.80
.28
Final Model X
o
1
1
1
1
1
1
1
1
.21
.30
.25
.26
.33
.29
.38
.38
c
~FM
[1.05 + .10]
-
1.19.±. .40
1.38 .±. .47
.92.±. .48
Estimate.±. s.e.
The parameters for this model correspond respectively
to the following: predicted value for difference
score status of cows ~ith no treatment (control).
eight separate increments of improvement for th.l
successive active treatments as arranged in Tabte 8.a.
e.
The parameter for this model corresponds
to the common value for the extent of improvement between pre-treatlllcnt and posttreatment for cows on any of the eight
active treatments.
Chi-square tests of significance for weighted analysis of variance (WANOVA)
Source of variation
D.F.
Treatments
8
Q
17.41
Active treatments
1
Q
110.16
Active treatments
7
1.93
Residual goodness of fit
8
2.11
Unadjusted total
9
112.27
Control treatment mean
1
0.17
Unadjusted total
9
112.27
Source of variation
D.F.
e-
.(
23
The variation among the estimated mean difference scores {fi} is investigated by weighted least squares.
!PM =
where 0
"'8
For this purpose, the incremental model
~
1
O~
~8
:s ... 8
is an (8 X1) vector of O's, 1
...8
is an (8 X1) vector of l's and I
... 8
is
an (8 x 8) identity matrix is used as a preliminary model for evaluating the extent
to which the active treatments have incremental effects over the control
treatment.
This hypothesis can be tested by comparing the corresponding
estimated treatment effects (b , b , ••• , b ) to their estimated standard
9
3
2
errors. All are significant (a=.05), with (Novobiocin=600 and Penici11in=
None) being the only borderline exception.
The test statistic for comparing
the mean difference scores for the treatments is significant (a=.05), a1though the comparison of active treatments is non-significant (a=.25) as
indicated in the analyses in previous sections.
In addition, the control
treatment estimated mean difference score is not significantly (a=.25) different from zero.
Thus, for summary purposes; the preliminary model X
is
;"'l'M
reduced to the final model
whose parameter can be interpreted, as the common estimated pre-treatment
versus post-treatment mean difference, which is equivalent to the mean improvement for the active (non-control) treatments.
Thus, the conclusion
which emerges from the analysis performed in this section is that there is
no reduction in the number of infected quarters between the pre-treatment
and post-treatment evaluation for the control treatment and a common net re-
24
duction of (1.05
±
.10) for the active treatments, which is essentially the
same as that obtained previously.
Mul tivariate analysis of quarter difference scores
Let f
git
denote the pre-treatment vs post-treatment infected quarter
difference score for the g-th quarter of the t-.th cow in the i-th treatment
group where
g = 1, 2, 3, 4;
i=l, 2, ... ,9; and t=l, 2, ... , n ..
1
More
specifically,
1 if g-th quarter is infected before treatment and not
infected after
f
git
=
0 i f g-th quarter has same infection status after treatment
as before
-1 i f g-th quarter is not infected before treatment but infected
after
As a result, there are
4 response variables, each of which has 3 outcome
levels, so that there are 34 = 81 possible multivariate response profiles.
There are 9 sub-populations corresponding to the respective treatments.
Thus, the multivariate analysis of quarter difference scores conceptually
involves the consideration of the 9x8l contingency table for the crossclassification of treatment vs the multivariate quarter difference profile.
If the questions of substantive interest primarily pertain to the extent
of differences among treatments for the four quarters both separately and
simultaneously, then attention can be restricted to the four first-order
quarter difference mean scores
f gi •
where g
= 1,
2, 3, 4 and i
= 1,
2,
=-
1
ni
ni
R.=l
... ,
L
f
gH
9, via arguments given in (3).
Although these estimates are most easily computed as mean scores from the
25
~
raw data matrix, they are identical to analogous linear functions of the cell
proportions in the underlying 9x8l contingency table.
Moreover, as indicated
in (3) and (4), the corresponding contingency table based estimated covariance matrix for the {f
gi
} can be computed in terms of sums of squares and
cross-products of deviations matrices with elements
s gg ~ , i
for the separate i
= I,
2, ••• , 9 treatment groups.
Given these considera-
tions, the mean scores {f . } and their estimated standard errors {Is gg, i}
g~.
are given in Table 9.
They were obtained via the computer program MISCAT,
which is an extension of GENCAT with better raw data capabilities (see
Stanish et al. (7».
The variation among the estimated pre-treatment vs post-treatment
infected quarter difference scores {f i } is investigated by weighted
g •
least squares asymptotic regression.
mean identity model X
-PM
=I
-36
In this regard, the preliminary cell
is used as a framework for testing hypotheses
pertaining to quarter differences within treatments and treatment differences within quarters.
Corresponding test statistics are given in Table 9.
These results indicate that significant (a=.05) treatment differences are
most apparent for quarter 2 and that significant (a=.05)
quarter differences
are most apparent for treatment 6 and treatment 7 (or more generally Novobiocin
= 400
mg).
Otherwise, the quarter differences for the control treat-
ment are clearly non-significant.
In view of the results in the previous sec-
tion for the overall difference score (summed over quarters), test statistics
are also provided for the comparison of the active treatments within each
quarter.
Although these are individually all non-significant (a=.25), the
26
Table 9.
a.
Multivariate analysis of quarter difference scores.
Observed mean scores and standard errors.
Treatment
Novobiocin
Penicillin
(IU)
(mg)
Total
-.08
(.11)
.21
(.09)
.29
(.11)
-.12
(.28)
.21
(.07)
.15
(.11)
-.08
(.11)
.31
(.10)
.03
(.11)
.20
(.10)
.34
(.10)
.23
(.08)
.31
(.11)
1.09
(.25)
37
.46
(.10)
.24
(.08)
.16
(.09)
.32
(.09)
1.19
(.26)
100,000
27
.41
(-.11)
.26
(.11)
.00
(.12)
.30
(.10)
.96
(.33)
400
200,000
27
.37
(.09)
.19
(.12)
.11
(.08)
.41
(.11)
1.07
(.29)
400
400,000
26
.27
.46
(.10)
.19
(.12)
.35
(.13)
(.13)
1.27
(.38)
.35
(.13)
.05
(.15)
.25
(.12)
.15
(.15)
.80
(.38)
None
None
26
.12
(.15)
-.08
(.08)
None
100,000
42
None
200,000
34
.36
(.08)
.32
(.11)
None
400,000
35
400
None
400
None
600
b.
Pre-treatment vs post-treatment
difference score for infected quarters
Q2
Q3
Q4
Ql
Total
sample
size
20
1.10
(.21)
.79
(.30)
Chi-square tests of significance for prelirrJ.nary cell mean iden tHy model.
Source of variation
Quert~rs_(Q)
in
Q in
Q in
Q in
Q in
Q in
Q in
Q in
Q in
Q in
treatment (T)
Tl
T2
T3
T4
T5
T6
T7
T8
T9
Quarters (Q) in active
treatments (T2-T9)
D.F.
Q
27
3
3
3
3
3
3
3
3
3
53.03
1.75
3.16
4.93
1.76
7.43
14.02
10.46
5.39
4.13
24
51.28
Source of variation
.
. Treatm~nts (T) jn quarterE (Q)
T in left front (Ql)
(Q2)
T in left rear
T in right rear (Q3)
T in right front (Q4)
Active treatments (AT) in
quarters (Q)
AT in left front
AT in left rear
AT in right rear
AT in right front
(Ql)
(Q2)
(Q3)
(Q4)
D.F.
Q
32
41.05
8
8
6.71
8
8
24.58
11.80
13.96
28
7
7
7
7
39.29
4.40
8.78
7.19
3.24
e
27
interpretation of their simultaneous test statistic is not so straightforward
since its significance is borderline (.OS < a < .10).
Thus, it becomes of
interest to investigate the types of models which can be fitted to the quarter
difference scores for both the case where this result is regarded as nonsignificant as well as the case where it is regarded as significant.
In this
way, the nature of any conflict between them can be used as a basis for
judging which approach is most sensible in terms of the conclusions which
it implies.
If the simultaneous test statistic for within quarter equality of
active (non-control) treatments is interpreted to be non-significant, then
the same final model which was found to be appropriate for the overall difference scores can be jointly applied to the separate quarters.
...
multivariate model has the form:
JM
•
!4 ® [:8]'
This joint
where® denotes Kronecker
product matrix notation; i.e., the first four rows of
~
are (l X4) vectors
of D's, and the remaining 32 rows are 8 successive repetitions of the rows
of the 4x4 identity matrix !4.
factory fit
This model is regarded as providing a satis-
since the difference between its goodness of fit statistic and
the simultaneous test statistic for within quarter equality of active treatments (which here is being assumed true) is non-significant (a=.2S); i.e.,
Q = 41.80 - 39.29 = 2.51 with D.F.=4 is non-significant.
The parameters of
this model can be interpreted as a measure of the common extent of improvement for the active (non-control) treatments for each of the respective
quarters.
The corresponding estimates and their standard errors are shown
in Table 10.a.
On examining these quantities more closely, it can be seen
that the estimated improvement is greater for the two quarters on the left
28
Table 10.
a.
Multivariate models for quarter difference scores
Fitted multivariate models, estimated parameters, and standard errors
Joint Multivariate Model
e=
Final Model
~JM
.25 + .03
Xc:
...
~ =
1
.17 + .03
1
1
.27 + .03
The parameters for this model correspond to the common value for the
extent of improvement between pretreatment and post treatment for the
respective four quarters for cows on
any of the active (non-control) treatments.
1
_~
= X
-FM
[26
0
-4,2
.33 + .03
...X
+
.o~
.04 + .01
x
18
The parameters for this model correspond to the common value for the
active (non-control) treatments for
the improvement of Q2 and Q4 and a
combined side and front-rear increment
(decrement) for Q1(Q3).
--------------e
b.
Chi-square tests of significance for weighted analysis of variance (WANOVA)
Source of variation
D.F.
Q
Left vs right
Front vs rear
1
1
8.23
8.81
Side vs front-rear
1
.04
Quarters
3
16.41
32
41.80
Residual goodness
of fit
.
Source of variation
Quarters
Residual goodness
of fit
D. F.
Q
1
16.25
34
41. 96
29
hand side than their two counterparts on the right and is greater for the
front two quarters than their two counterparts at the rear.
Moreover, as
indicated by the test statistics shown in Table 10.b, these quarter location
differences are both significant (a=.OI); and the interaction between
them is non-significant (a=.25).
On the basis of these conclusions and the
fact that side differences and front-rear differences are approximately the
same
,
the model X
~M
is reduced to the final model X
-m whose parameters cor-
respond to a common value for the active (non-control) treatments for the equal
improvement of the left rear and right front quarters and a combined side
and front-rear parameter that represents an increase in improvement
for the left front quarter and a decrease in improvement for the right rear
quarter.
4It
The specific structure of this model, estimated parameters and
standard errors,
and test statistics are given in Tables 10.a and 10.b.
Thus, it can be seen that for each of the active (non-control) treatments, the
most improvement occurred for the left front quarter and the least for the
right rear. Finally, an overall estimate of improvement can be obtained from
this analysis by adding the separate estimates for the four quarters together.
For the model X
-m
,this approach leads to an overall estimate of (1.02 + .09)
-
for the common net reduction of infected quarters for the cows which received
any of the active treatments.
In this
sense, it essentially yields the same
conclusion concerning treatment differences as the simpler analyses given in
previous sections for the total numbers of infected quarters at pre-treatment
and post-treatment, but at the same time provides additional interesting: information about the differences in improvement for the four separate quarters.
Alternatively, if the simultaneous test statistic for within quarter
equality of active treatments is interpreted as significant, a somewhat dif-
30
ferent analytical strategy is needed.
More specifically, it becomes necessary
to identify which of the four quarters is responsible for the significance
of this test statistic.
In this regard, quarter 2 (left rear) appears to be
a strong possibility because it has a somewhat wider range of difference scores
for the active treatments than the other three quarters.
9 (Novobiocin=600
m~seems
Moreover, treatment
to be somewhat less effective for this quarter
than the other active treatments which again seem to be relatively similar
with respect to their respective estimates for extent of improvement.
Putting
all of these considerations together yields the model
x-JM =
~4 ® [t]
lggo~
~
o0
o
1 0
0 0 1
which is exactly the same as ~JM except that an estimated value of 0 improvement is implied by it for treatment 9 at quarter 2.
Since its goodness of
fit statistic Q = 33.21 with D.F.=32 is non-significant (a=.25), this model
provides a satisfactory framework for characterizing the variation among the
quarter difference scores.
In addition, it also would seem to be a better
model than X ,for which the corresponding goodness of fit test statistic
-JM
is larger, and less
non-significant (0=.10).
However, X
-JM
is more difficult
to interpret than X because there does not seem to be any sensible reason
-JM
why treatment 9 would have no effect on quarter 2 and an effect equal to the
other active treatments on the other three quarters and why it would have
less effect on quarter 2 than treatment 5 which is its smaller dosage counterpart.
In other words, the relatively small estimated value for improvement
31
of quarter 2 for cows given treatment 9 would seem to be either a consequence
of data collection or transcription errors or simply a relatively unexpected
random event.
Since the latter interpretation is considered more reasonable,
the analysis based on X and X is judged to be more preferable than that
-JM
- FM
based on ~JM.
Other types of analyses
As stated at the beginning of this paper, the analyses which have been
undertaken here are intended only to be illustrative of the corresponding
methodology.
Other types of applications are definitely feasible and may
be of greater substantive interest.
In this regard, separate and/or joint
analyses can be undertaken for specific organisms like Streptococcus
~galactiae
and Staphylococcus aureus.
Also, pre-treatment scores and post-
treatment scores can be incorporated in multivariate analyses for the separate
quarters as well as for analyses of their overall total.
Other statistical
issues which pertain to this type of multivariate data, including strategies for
dealing with such problems as incomplete data, adjustments for blocking
variables like herd, and tests of significance for moderate as opposed to
large sample sizes are discussed in (6) and (7).
32
Acknowledgements
This research was, in part, supported by Burroughs Wellcome Company.
In addition, the authors would like to thank Dr. William Vinson for providing
the data used in the example; Ingrid Ann Amara, Jane Beth Markley, and
William Stanish, for computational assistance; and Jean McKinney and Jean
Harrison for typing the manuscript.
References
(1)
Grizzle, J.E., C.F. Starmer, and G.G. Koch. 1969.
data by linear models. Biometrics 25:489.
Analysis of categorical
(2)
Heald, C.W., G.M. Jones, S. Nickerson, and T.L. Bibb. 1977. Mastitis
control by penicillin and novobiocin at drying off. Canadian
Veterinary Journal 18:171.
(3)
Koch, G.G., J.R. Landis, J.L. Freeman, D.H. Freeman, and R.G. Lehnen.
1977. A general methodology for the analysis of experiments with
repeated measurement of categorical data. Biometrics 33:133.
(4)
Landis, J.R., W.M. Stanish, J.L. Freeman, and G.G. Koch. 1976. A
computer program for the generalized chi-square analysis of categorical data using weighted least squares (GENCAT). Computer
Programs in Biomedicine 6(4):196.
(5)
Miller, C.C., 1975. Considerations in the evaluation of a mastitis
treatment product. Unpublished manuscript presented at the Annual
Meeting of National Mastitis Council, Inc. February 10-12, 1975.
(6)
Stanish, W.M., D.B. Gillings, and G.G. Koch. 1978. An Application of
Multivariate Ratio Methods for the Analysis of a Longitudinal
Clinical Trial with Missing Data. To appear in Biometrics 34.
(7)
Stanish, W.M., G.G. Koch, and J.R. Landis. 1977. A computer program for
multivariate ratio analysis (MISCAT). Submitted to Computer
Programs in Biomedicine.
33
e
'"
APPENDIX
Data format specification
CID = Cow Identification Number.
HERDS = Each cow belonged to one of 16 herds.
TRT = Treatment:
1 = 200,000 IU penicillin
2 = 400,000 IU penicillin
3 = 200,000 IU penicillin + 400 mg. novobiocin
4 = 400,000 IU penicillin + 400 mg. novobiocin
5 = 100,000 IU penicillin + 400 mg. novobiocin
6 = 400 mg. novobiocin
7 = 600 mg. novobiocin
8 = 100,000 IU penicillin
9 = No treatment (control)
Qll = Pre-treatment infection state of the left front quarter of the cow's
udder.
Q21
= Pre-treatment infection state of the left rear quarter of the cow's
udder.
Q31 = Pre-treatment infection state of the right rear quarter of the cow's
udder.
Q41 = Pre-treatment infection state of the right front quarter of the cow's
udder.
e
Q12
Post-treatment infection state of the left front quarter of the cow's
udder.
Q22
= Post-treatment infection state of the left rear quarter of the cow's
udder.
Q32 = Post-treatment infection state of the right rear quarter of the cow's
udder.
Q42
= Post-treatment infection state of the right front quarter of the cow's
udder.
Organisms:
0 = None (infection free)
1 = Staphylococcus Aureus
2 = Staphylococcus Epidermis
3 = Streptococcus Aga1actiae
6 = Coliform
7 = Other
HMS
= Herd Management Score (1 = Poor, 2 = Fair, 3 = Good, 4 = Excellent).
HMT
= Herd
Milking Type (0
= Stanchion,
1
= Milking
parlor)
34
e
~
Data listing
TI\8LE 11
CIa
8
:s
HEROS
TRT
Qll
1
1
1
1
1
1
2
2
2
:5
3
0
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7
0
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0
122
1
1
1
1
1
1
'+
1
12'+
123
120
2
275
125
6
11'3
187
5
121
10
'3
11
1
1
1
1
1
1
1
'+
'+
5
5
5
6
6
6
7
7
7
118
117
1
188
1
1
1
1
1
1
126
1
e
7
1
1
D
9
182
1/jl
16
15
1
1
C)
2
2
111
13
2
1
2
J
18~
12
18'3
19
H
190
%
194
26
193
25
1'Jl
20
27
23
21
192
22
2/1
18 1•
2
2
2
2
2
a
8
')
'+
5
6
6
8
2
e
O?
CJ
.3
.3
.3
1
1
~
.3
.3
.3
.3
.5
.3
3
.3
130
3
'+
205
I'
213
212
31
1 112
133
'+0
'+
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lj
'+
'+
217
II
<>08
20£,'
II
'+
2
2
3
3
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5
6
6
7
8
CJ
1
1
1
1
1
1
2
2
2
2
2
MtI'.i"'IS
Q21
Q31
0
0
1
1
0
0
2
7
2
0
7
1
l)
0
IJ
0
0
0
2
1
0
0
7
7
2
0
0
0
7
7
0
0
0
0
0
7
0
0
2
0
0
0
1
0
7
7
0
0
0
3
1
2
0
7
0
0
0
0
0
7
7
2
2
0
0
0
0
1
0
1
0
1
0
0
2
7
0
1
0
7
7
0
0
7
0
0
0
0
7
0
0
0
0
0
1
7
0
7
0
0
0
0
7
0
0
2
0
0
0
0
2
0
0
u
7
2
7
0
0
1
0
0
0
0
·0
0
7
0
:5
1
0
0
7
0
2
0
0
0
0
0
0
0
0
0
7
0
7
0
0
0
6
1
0
1
lREl\l~ErJT
QLH
Q12
7
1
1
2
7
7
0
0
0
0
0
0
0
2
1
0
0
1
0
7
0
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0
7
0
2
.,
7
0
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0
7
0
0
0
0
0
0
0
2
0
0
1
7
0
7
0
0
0
0
0
3
0
7
DATA
Q22
:5
Q32
Q42
0
2
0
0
0
0
·1
0
0
0
0
0
0
2
2
2
2
0
2
0
2
7
0
0
2
7
0
2
0
2
0
0
1
0
2
0
2
0
0
0
2
0
2
0
2
0
1
0
0
0
0
0
0
0
0
1
2
2
2
0
2
7
2
0
0
0
0
0
0
0
0
0
0
7
2
0
0
0
0
6
2
7
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
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1
0
0
0
6
0
0
0
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