- Email: [email protected]
Ontario Bulk Milk Somatic Cell Count Reduction Program. 2. Dynamics of Bulk Milk Somatic Cell Counts
Ontario Bulk Milk Somatic Cell Count Reduction Program. 2. Dynamics of Bulk Milk Somatic Cell Counts YNTE H. SCHUKKEN,i K. E. LESUE,i A. J. WEERSINK,2 and S. W. MARTINi University of Guelph Guelph, ON, Canada N1G 2W1 ABSTRACT
INTRODUCTION
The objective of this study was to study the dynamics of bulk milk see in order to assist the individual farmer and the dairy industry to lower farm and population Sec. Data on milk quality, milk components, and kilograms of milk produced were collected monthly from January 1985 through September 1991 on approximately 9500 farms in the province of Ontario, eanada. A log-normal distribution was fitted to the see data. The log-normal distribution fitted the data reasonably well. This distribution was used to define performance goals for specified regulatory limits. Dynamics of farms were studied using a modified Markov model. Farms with bulk milk see from 300 x 103 to 599 x 103 were mostly responsible for a decrease, and farms with bulk milk see from 0 to 299 x 103 were mostly responsible for an increase in mean see. The see contribution, a parameter based on the number· of cells that are produced on a farm, was calculated for each farm. Most farms with high see did not have high see contributions. Farms with high see contributions typically had bulk milk see from 500 x 103 to 750 x 103. In order to keep population see low, an incentive should be offered to farms with low see. (Key words: somatic cell counts, milk quality)
The dairy producer is asked to produce a product that is free of inflammatory and inhibitory substances. The dairy industry and government agencies recognize these consumer demands and are making the regulatory limits of milk quality measurements more stringent. In recent years, regulatory limits for bulk milk see (BMSeC) have been imposed on the dairy industry worldwide. In a companion paper (8), we established that such a limit had an important impact on the mean BMSee in Ontario. An understanding of the dynamics of BMSee is important to guide the industry toward better milk quality. The BMSee are expected to follow a lognormal distribution (7, 9), which implies relatively large variability, especially with higher mean BMSee. When the inherent variability is understood, specific recommendations can be made to dairy producers to adhere to the regulations. The BMSee (or any other milk quality index) is not a random measurement; farms exhibit behavior in certain recognizable patterns (6, 7). Studying these patterns of BMSee may also aid in fine-tuning penalty programs or programs to assist producers to meet the regulatory limits. When the farms that are at risk for an increase in BMSee can be identified, specific efforts can be directed to these farms (3). The goal of this study was to investigate the dynamics of BMsee in order to assist the individual farmer and the dairy industry to lower farm and population BMSee.
Abbreviation key: BMSCC =bulk milk see, FP = freezing point, PLC = plate loop count.
MATERIAL AND METHODS
Data Received April 6. 1992. Accepted July la, 1992. IDepartment of Population Medicine. 2Department of Agricultural Economics. 1992 J Dairy Sci 75:3359-3366
The data are described in a companion paper (8). Briefly, data on milk production, BMSee, and farm characteristics were col3359
3360
SCHUKKEN ET AL.
lected from approximately 9500 Ontario dairy farms. These data were obtained from the Central Milk Testing Laboratory and the Ontario Milk Marketing Board. For each farm, monthly data were from January 1985 through September 1991 and included monthly kilograms of milk sold; monthly milk components measurements, fat, protein, and lactose percentages; and milk quality measurements, such as SCC, plate loop count (PLC), freezing point (FP), and inhibitor presence. Additional data included the location of the herd, milking system, the manufacturer of the milking equipment, the main breed of cows, and an estimate of the number of cows. Farms were classified according to their BMSCC (x 1
The mean and the variance of the natural logarithm of BMSCC were estimated from the data. These parameters were used to simulate a true log-normal distribution using a random number generator. The parameters (mean and variance) of the log-normal distribution are calculated as mean
= exp(~+·5a2)
where 1.1. and (J are the estimated mean and variance from In(BMSCC). The goodness of fit of the observed versus the generated distribution was evaluated using a chi-square statistic. The relation between the mean and the variance was used to define the performance goal for a farm. The performance goal was chosen such that a farmer had a defined small probability of crossing a penalty limit. The mean BMSCC performance was then chosen such that mean BMSCC + 2.65 x SD(BMSCC) < penality limit. Journal of Dairy Science Vol. 75, No. 12, 1992
Because the standard deviation is a function of the mean, this inequality can be solved for various penalty limits (2.65 corresponds to a 1% chance of crossing the penalty limit). BMSCC Class Transitions
The dynamics in the monthly BMSCC of a farm were modeled with a state transition model, in special cases also called a Markov chain model (1). A state transition model has two components: states and transitions. The transition represents a process that moves between a number of states, SI to Si, in such a way that, if the current state is Sj, then there is a certain probability that the next state is Sk (j 1.. i, k 1 .. i). The probability of transition from Sj to Sk is represented by Pjk' The transition probabilities together form a transition matrix; the rows represent the state j, and the columns present the state k. 1be summation of all probabilities in a row must equal 1, and each individual probability ranges from 0 to 1. The present state transition model used the seven BMSCC classes as just defined. In all months, the probability is calculated that a farm in one of these classes in month i moves to another class in month (year) i + 1 (i = January 1985. September 1991). With these transition matrices, the movement of farms between SCC levels was studied (6). The transition matrices were studied in three types of situations: periods when the average BMSCC increased, decreased, and remained equal.
=
=
Contribution of a Farm to the Provincial SCC
In order to evaluate the contribution of an individual farm to the overall number of somatic cells in the milk supply, a parameter called SCC contribution was calculated for each farm. This parameter utilizes the number of cells produced by a farm over an acceptable limit of 250,000 cells/mi. This contribution parameter is influenced by the actual monthly mean BMSCC and the amount of milk shipped in that month and is calculated as SCC contribution 12
~ (BMSCCj - 250)
=
x productioni
j-t
mean annual production
where
=
month
indicator,
1,2,..., 12;
3361
DYNAMICS OF BULK MILK SOMATIC CELL COUNTS
BMseCj = monthly mean BMSee; productioni = monthly kilograms of milk shipped; and mean annual production = mean annual milk shipment in kilograms of all fanns in Ontario. A fann with an annual production at exactly the Ontario mean and a BMSee of 251,000 has an sec contribution of 1. The performance of BMSee as a diagnostic tool for sec contribution is evaluated using standard test evaluation procedures, such as sensitivity, specificity, and a receiver-operator characteristic curve (4). RESULTS Mean-Variance Relationship of BMSCC
The annual mean BMSee of Ontario fanns in 1990 is plotted in Figure 1. A log-normal
distribution was fitted to the data. The estimated mean and variance of the log-normal distribution were 340 and 32,053, respectively, which fitted the data reasonably well (chisquare = 39.8, 31 df, P > .10). Because the BMSee apparently follows a log-normal distribution, the variability of the sec measurement increases with increasing mean sec (Figure 2). In the Ontario BMSee data, the relationship between the mean and the standard deviation followed a straight line: SD = 5.61 + .36 x mean. The observed data points were exactly on the estimated regression line. The coefficient of variation in the Ontario data showed a slightly decreasing pattern. The regression equation was the following: coefficient of variation = .43 - .00014 x mean + .00000oo62 x mean2 . When the defined relation between the mean and standard deviation
1 400
14
1 200
12
"-:' 1 000
10
o
s:::::
-U)
E
I-
CO LL.
...m Q)
8
:::s
600
6
.J:
400
4
200
2
800
C'" t/) I
Bulk milk
U
see
Figure 1. Observed bulk milk SCC distribution (0) and fitted distribution (*) based on a log-normal density function with mean of 340 and variance of 32,053. The goodness of fil chi-square values are presented in the bars. Journal of Dairy Science Vol. 75, No. 12, 1992
3362 500
SCHUKKEN ET AL. I~'
,.. ,
------,.5
!
400~~.
§!
~
~ 300
rJ
----)-----~_=
--
c,_
,.4
~;;
3
~ '0
I :
~2ool I 100/ ~
. /
~
/~
,/
;.2 ,
i ~
1.1 8
,.~o
ot/ 50
1.
and 2 contributed to this increase, and a slight increase occurred in class 3. Classes 2 and 1 contributed 50 and 27%, respectively, of the increase in mean BMSCC. Generally, classes 3 and 4 (300 x loJ to 599 x loJ) were mostly responsible for the decrease in mean BMSCC. Classes 1 and 2 (0 to 299 x 1()3) were responsible for increases in mean BMSCC. Fanns in the highest BMSCC classes always contributed to a decreasing mean BMSCC.
150 250 350 450 550 650 750 850 Annual mean
sec
Figure 2. The relation between the mean bulk milk SCC and two measures of variation, coefficient of variation (0), and standard deviation (0).
is used, advisable perfonnance goals (Figure 3) can be defined for specified regulatory limits. For example, when the regulatory limit is 500,000 cells/ml and a producer wants all BMSCC measurements to stay under this limit, the perfonnance goal should be set at approximately 250,000 cells/ml. Dynamics of
sec
(C.... Tranaitlons)
Typical transition matrices for periods in which BMSCC increased, decreased, and remained unchanged for 2 subsequent mo are shown in Table 1. In Table 2, the mean performance of fanns per SCC class is shown during the three 2-mo periods. which are representative of other periods that show increased or decreased BMSCC. The fITSt period shows 2 mo in which the mean BMSCC was similar. Fanns in class 1 «149 x loJ SCC) in mo 1 increased an average of 61 x loJ cells/ml during mo 2. Fanns in class 7 ~ x loJ SCC) in mo 1 decreased an average of 181 x loJ cells/mI. The second period shows 2 mo in which mean BMSCC decreased from 325 x loJ to 284 x 1oJ. As shown in Table 1, the classes contributing to this decrease were classes 3, 4, 5, 6, and 7, whereas classes 1 and 2 actually showed slightly increased mean BMSCC. Classes 3 and 4 contributed the most (each 28%) to the decrease in mean BMSCC. The third period included 2 mo in which mean BMSCC increased from 345 x loJ to 429 x loJ; classes 1 Journal of Dairy Science Vol. 75, No. 12, 1992
Contribution of a Farm to the Provincial SCC
The relationship between mean annual BMSCC and the SCC contribution is shown in Figure 4; most high BMSCC fanns did not have high SCC contributions. This difference can be explained by the lower annual shipment of the fanns with high BMSCC. Fanns with mean BMSCC ~750 x I ()3 had a mean shipment of 148,000 kg of milk in 1990; the mean shipment of all fanns in 1990 was 256,496 kg of milk. The farms with a high SCC contribution (>500 x loJ) usually had a mean annual BMSCC between 500 x loJ and 750 x loJ. The number of fanns with high SCC contributions that would be detected at various cutoff limits for mean BMSCC is shown in Table 3. At low BMSCC, a large number of falsepositive fanns are detected; at high BMSCC,
600
Ii 500 -
o D
..
.
iE
I
'-"--T'" ,i
g 400 '-:- - - f -
..
~ 3001r---t-r-ff~'F:lY~" Q.
'
'i 2001-;-'r-"-:=-=-...,----;----.-.-.. - - _. , - .! >
I
,
--,-, -
~100~ 01
-'-~.
-_.. _-,
! .
~
_L -'-
300 350 400 450 500 550 600 650 700 750 800 850 Regulatory limit for Bulk milk
see
Figure 3. Advisable performance goal with variable regulatory limits. 1be perfonnance goals are shown for the case in which penalties are based on t or the mean of n consecutive samples: n = I (+), 2 (0), 3 (0), and 10 (a).
3363
DYNAMICS OF BULK Mll..K SOMATIC CELL COUNTS
only very few true-positive farms are detected. In all instances, mean BMSee is a poor selection criterion for identifying problem farms, given that the goal of this farm selection process is to influence the population sec level. In Figure 5, the performance of BMSee as a diagnostic tool for sec contribution is evaluated using a receiver-operator characteristic curve (5). Several cutoff values for BMSCC are used to detect farms with a contribution ~500. This graph indicates that use of a very high BMSeC cutoff point resulted in very low sensitivity (Le., very few herds with high sec contribution were detected). However, with a low BMSee cutoff point, the specificity was low. The surface of the graph on the left-hand side of the receiver-operator characteristic curve represents the amount of diagnostic error. Based on this error rate, BMSeC was not an efficient tool to detect farms with high sec contributions.
DISCUSSION Mean-Variance Relationship
Both sec and BMSeC seem to follow a log-normal distribution (7, 9) (Figure 1). The goodness of fit of such a distribution is of academic value and also has important applications. This finding verified that the log transformation of BMSCC is necessary to use linear regression models based on the assumption of a normal distribution (2, 9). As shown in Figure 2, mean and variance were strongly related (in our data, the simple correlation was .76); this relationship can be modeled using the moments of the log-normal distribution. The observed relationship between mean and variance was then utilized to define performance goals for dairy farms, which is especially useful when regulatory limits are in place. Given the large inherent variability. when the regulatory limit for BMSeC is defined at
TABLE I. State transition matrices for bulk milk SCC during an increase, a decrease, or a stable l mean SCC between 2 subsequent mo. To class
From class
2
3
52.6 2 70.43 15.84
41.7 26.7 57.7
4.4 2.3 23.3
10.5 24.7 4.5
60.1 59.0 32.7
3
1.9 7.0 4.6
4
4
5
7
6
.1
.5 2.7
.1 .1 .4
.1 .I .I
0
23.1 13.3 50.3
4.8 2.4 10.0
1.3 .6 1.9
.1 .I .4
0
29.8 44.3 9.8
44.3 34.3 24.1
17.8 10.5 40.9
4.4 2.8 24.4
1.3 .6 13.0
.6 .5 5.2
.4 2.7 .1
11.5 21.5 4.7
31.7 41.2 21.2
32.5 23.0 36.4
15.6 8.4 25.1
5.7 2.0 9.4
2.6 1.2 3.0
5
.5 .8 .4
3.7 12.8 2.9
20.5 24.0 8.9
30.6 35.6 26.0
24.1 14.5 30.8
15.6 7.0 18.9
5.0 5.3 12.3
6
0 1.4 0
3.3 5.5 1.1
14.5 16.5 7.7
19.2 31.7 11.4
26.0 22.5 28.2
19.0 8.7 24.2
18.1 13.8 27.5
7
.5
.8 5.0 .6
7.2 13.8 2.6
9.7 20.9 7.9
19.6 18.4 11.3
15.3 14.6 21.2
47.0 27.2 56.4
2
0 0
1.1
.1 .1 .2
.Stable perfonnance was in August 10 September 1990, decrease was in November 10 December 1990 (41 x 103), and increase was in November to December 1985 (84 x 103). 2Transition probability August to September 1990; SCC is stable. 3Transition probability November to December 1990; SCC decreased. 4Transition probability November to December 1985; SCC increased. Journal of Dairy Science Vol. 75, No. 12, 1992
3364
SCHUKKEN ET AL.
TABLE 2. Dynamics of SCC classes during an increase, a decrease, or a stableI mean SCC between 2 subsequent mo. SCC Performance from month i to month i + 1 Stable
Decreases
~Mean2
SCC Class
~ean
(x 1()3)
o to 150 300 450 600 750
to to to to to
Increases Contr3
~ean
149 299 449 599 749 899
~900
+61,000 +45,000 +1000 -28,000 -83,000 -118,000 -181,000
+36,000 +5400 -54,000 -108,800 -155,000 -217,000 -287,000
Contr (%)
(%)
+140,000 +113,000 +78,700 +25,600 +12,500 -14,800 -122,000
28 28 17 11 16
27 50 19 3 1
IStable performance was in August to September 1990, decrease was in November to December 1990 (41 x 103), and increase was in November to December 1985 (84 x 1()3). 2~Mean
= The change in mean SCC between month i and month i + 1 of farms that were in month i in this SCC
class. 3Cootr = Contribution to the change in mean
sec.
600 X 103, a perfonnance goal of 599 x 103 cells/ml is not advisable because a farm with an average performance of 599 x 103 can be expected to receive several penalties. Figure 3 shows that a mean of approximately 300 x 1
Class Transitions
The results of this transition analysis indicate that the main effort in decreasing the population BMSee should focus on the farms with medium high sec (300 x 1
TABLE 3. Evaluation of the diagnostic value of mean bulk milk SCC (BMSCC) when the goal is to detect problem, farms, defined as having an SCC contribution >500.
sec BMSCC Cutoff point 400 450 500 550 600 650 700 750
Farms above cutoff 2537 1875 1392 1005 716 502 359 274
Contribution of farms <500 SCC (n = 8657)
>500 SCC (0 = 193) n 190 182 166 142 119 90 72 55
Correct I
n
Correct
98 94 86 74 62 47 37 28
2347 1693 1226 863 597 412 287 219
73 81 86 90 93 95 97 98
IThe percentage of farms that are correctly classified as with or without SCC problems. Journal of Dairy Science Vol. 75, No. 12, 1992
3365
DYNAMICS OF BULK Mll..K SOMATIC CELL COUNTS 1-22
n
= 138
n
= 55
Q
8
.8 _
18
..
------/0 ~/'-400
/
::
0::
:8
12
'E
8
Z
8
6
)' 550
'>
..,
E .6
1
~500
/
16
(600
0::
III
4>500
;J
I,
.4
(.) (.)
ta::
C/)
- 2 -
n
= 219
In(BMSCC)
Figure 4. The relation between the natural logarithm of the mean bulk milk SCC [In(BMSCC)] and SCC contribution in 1990. Reference lines are located at SCC contribution = 500, and In(BMSCC) = 6.6 (750 x 103 cellslml).
This approach is quite different from current practice, in which the farms with the highest BMsee receive most of the encouragement to perform better. The data from the class transitions lead to the importance of encouraging farms with low BMSee (classes I and 2) to maintain low BMSee. It may be advisable to offer an incentive to such farms. If successful, an incentive program would prevent the increases in mean provincial sec, as we observed in our data.
650
g
.2
J 900 ~ 1000
o
-;}------,----,------,----,--------,----,-------,------1
o
.05
.1
.15
.2 .25 .3 1 - Specificity
.35
.4
.45
.5
Figure 5. The relation between sensitivity and I - specificity when the bulk milk SCC is used as a diagnostic tool for SCC contribution ~500 (receiver-operator characteristic curve). Numbers in graph are SCC cutoff levels. Sensitivity is the probability that a problem farm is detected, I - specificity is the probability that a farm without a problem is incorrectly flagged as a problem farm.
BMsce on farms in classes 3 and 4 and to maintain low SCC on farms in classes I and 2. Attention should focus on farms with a high SCC contribution, which are generally not the farms with the highest BMSCC. ACKNOWLEDGMENTS
SCC Contribution
Consumers use dairy products made from all of the milk that farmers produce; thus, intake of milk components depends on the mean milk quality of a farm multiplied by its production. The relatively small impact of the farms with very high BMSee became apparent when the sec contribution of each farm to the total SCC production was calculated. The farms with the highest SCC contribution were typically not the farms with high BMSeC, but, rather, were the farms in sec classes 3, 4, and 5. Higher returns are expected when resources are utilized to influence management on farms with high SCC contributions instead of farms with high BMSCC. CONCLUSIONS
In order to affect the population sec, efforts should be directed to decrease the
This study was financially supported by the Bureau of Veterinary Drugs, the Ontario Ministry of Agriculture and Food, and the Ontario Milk Marketing Board. Great appreciation is extended to the staff of the Central Milk Testing Laboratory for support. Computing tasks were performed by Peter Sijed at the Ontario Ministry of Agriculture and Food, Computing Branch, Richard Canton at the Ontario Milk Marketing Board, and Steve Howie at the University of Guelph Computing Services. REFERENCES
I Carpenter, T. E. 1988. Microcomputer programs for Markov and modified Markov chain models. Prevo Vet. Med. 5:169. 2 Emanuelson, D., and H. Funke. 1991. Effect of milk yield on the relationship between bulk milk somatic cell count and prevalence of mastitis. 1. Dairy Sci. 74: 2479. Journal of Dairy Science Vol. 75, No. 12, 1992
3366
SCHUKKEN ET AL.
3 Leslie. K. E., I. Dohoo. and A. H. Meek. 1983. Somatic cell counts in bovine milk. Compned. Contino Educ. Pract. Vet. 5:S601. 4 Martin. S. W.. A. H. Meek. and P. Willeberg. 1987. Veterinary Epidemiology, Principles and Methods. Iowa State Univ. Press, Ames. 5 SASI& User's Guide: Statistics, Version 6 Edition. 1990. SAS Inst., Inc.• Cary, NC. 6 Scholl, D., and M. Den Ouden. 1991. Dynamics of farm bulk-tank: milk protein concentration. Page 637 in Proc. 6th Int. Soc. Vet. Epid. Econ. Symp., Ottawa, ON, Can. Univ. Guelph. Guelph. ON, Can.
Journal of Dairy Science Vol. 75, No. 12, 1992
7 Schukken, Y. H.• J. Buurman, A. Brand, D. van de Geer, and F. J. Gronuners. 1990. Population dynamics of bulk milk somatic cell counts. 1. Dairy Sci. 73: 1350. 8 SChukken, Y. H., K. E. Leslie, A. 1. Weersink, and S. W. Martin. 1992. Ontario bulk milk somatic cell count reduction program. I. Impact on somatic cell counts and milk quality. J. Dairy Sci. 75:3352. 9 Shook, G. E. 1982. Approaches to summarizing somatic cell count which improve interpretability. Page 150 in Proc. 21st Annu. Mtg. Nat!. Mastitis Counc.• Nat!. Mastitis Counc., Arlington, VA.
INTRODUCTION
The objective of this study was to study the dynamics of bulk milk see in order to assist the individual farmer and the dairy industry to lower farm and population Sec. Data on milk quality, milk components, and kilograms of milk produced were collected monthly from January 1985 through September 1991 on approximately 9500 farms in the province of Ontario, eanada. A log-normal distribution was fitted to the see data. The log-normal distribution fitted the data reasonably well. This distribution was used to define performance goals for specified regulatory limits. Dynamics of farms were studied using a modified Markov model. Farms with bulk milk see from 300 x 103 to 599 x 103 were mostly responsible for a decrease, and farms with bulk milk see from 0 to 299 x 103 were mostly responsible for an increase in mean see. The see contribution, a parameter based on the number· of cells that are produced on a farm, was calculated for each farm. Most farms with high see did not have high see contributions. Farms with high see contributions typically had bulk milk see from 500 x 103 to 750 x 103. In order to keep population see low, an incentive should be offered to farms with low see. (Key words: somatic cell counts, milk quality)
The dairy producer is asked to produce a product that is free of inflammatory and inhibitory substances. The dairy industry and government agencies recognize these consumer demands and are making the regulatory limits of milk quality measurements more stringent. In recent years, regulatory limits for bulk milk see (BMSeC) have been imposed on the dairy industry worldwide. In a companion paper (8), we established that such a limit had an important impact on the mean BMSee in Ontario. An understanding of the dynamics of BMSee is important to guide the industry toward better milk quality. The BMSee are expected to follow a lognormal distribution (7, 9), which implies relatively large variability, especially with higher mean BMSee. When the inherent variability is understood, specific recommendations can be made to dairy producers to adhere to the regulations. The BMSee (or any other milk quality index) is not a random measurement; farms exhibit behavior in certain recognizable patterns (6, 7). Studying these patterns of BMSee may also aid in fine-tuning penalty programs or programs to assist producers to meet the regulatory limits. When the farms that are at risk for an increase in BMSee can be identified, specific efforts can be directed to these farms (3). The goal of this study was to investigate the dynamics of BMsee in order to assist the individual farmer and the dairy industry to lower farm and population BMSee.
Abbreviation key: BMSCC =bulk milk see, FP = freezing point, PLC = plate loop count.
MATERIAL AND METHODS
Data Received April 6. 1992. Accepted July la, 1992. IDepartment of Population Medicine. 2Department of Agricultural Economics. 1992 J Dairy Sci 75:3359-3366
The data are described in a companion paper (8). Briefly, data on milk production, BMSee, and farm characteristics were col3359
3360
SCHUKKEN ET AL.
lected from approximately 9500 Ontario dairy farms. These data were obtained from the Central Milk Testing Laboratory and the Ontario Milk Marketing Board. For each farm, monthly data were from January 1985 through September 1991 and included monthly kilograms of milk sold; monthly milk components measurements, fat, protein, and lactose percentages; and milk quality measurements, such as SCC, plate loop count (PLC), freezing point (FP), and inhibitor presence. Additional data included the location of the herd, milking system, the manufacturer of the milking equipment, the main breed of cows, and an estimate of the number of cows. Farms were classified according to their BMSCC (x 1
The mean and the variance of the natural logarithm of BMSCC were estimated from the data. These parameters were used to simulate a true log-normal distribution using a random number generator. The parameters (mean and variance) of the log-normal distribution are calculated as mean
= exp(~+·5a2)
where 1.1. and (J are the estimated mean and variance from In(BMSCC). The goodness of fit of the observed versus the generated distribution was evaluated using a chi-square statistic. The relation between the mean and the variance was used to define the performance goal for a farm. The performance goal was chosen such that a farmer had a defined small probability of crossing a penalty limit. The mean BMSCC performance was then chosen such that mean BMSCC + 2.65 x SD(BMSCC) < penality limit. Journal of Dairy Science Vol. 75, No. 12, 1992
Because the standard deviation is a function of the mean, this inequality can be solved for various penalty limits (2.65 corresponds to a 1% chance of crossing the penalty limit). BMSCC Class Transitions
The dynamics in the monthly BMSCC of a farm were modeled with a state transition model, in special cases also called a Markov chain model (1). A state transition model has two components: states and transitions. The transition represents a process that moves between a number of states, SI to Si, in such a way that, if the current state is Sj, then there is a certain probability that the next state is Sk (j 1.. i, k 1 .. i). The probability of transition from Sj to Sk is represented by Pjk' The transition probabilities together form a transition matrix; the rows represent the state j, and the columns present the state k. 1be summation of all probabilities in a row must equal 1, and each individual probability ranges from 0 to 1. The present state transition model used the seven BMSCC classes as just defined. In all months, the probability is calculated that a farm in one of these classes in month i moves to another class in month (year) i + 1 (i = January 1985. September 1991). With these transition matrices, the movement of farms between SCC levels was studied (6). The transition matrices were studied in three types of situations: periods when the average BMSCC increased, decreased, and remained equal.
=
=
Contribution of a Farm to the Provincial SCC
In order to evaluate the contribution of an individual farm to the overall number of somatic cells in the milk supply, a parameter called SCC contribution was calculated for each farm. This parameter utilizes the number of cells produced by a farm over an acceptable limit of 250,000 cells/mi. This contribution parameter is influenced by the actual monthly mean BMSCC and the amount of milk shipped in that month and is calculated as SCC contribution 12
~ (BMSCCj - 250)
=
x productioni
j-t
mean annual production
where
=
month
indicator,
1,2,..., 12;
3361
DYNAMICS OF BULK MILK SOMATIC CELL COUNTS
BMseCj = monthly mean BMSee; productioni = monthly kilograms of milk shipped; and mean annual production = mean annual milk shipment in kilograms of all fanns in Ontario. A fann with an annual production at exactly the Ontario mean and a BMSee of 251,000 has an sec contribution of 1. The performance of BMSee as a diagnostic tool for sec contribution is evaluated using standard test evaluation procedures, such as sensitivity, specificity, and a receiver-operator characteristic curve (4). RESULTS Mean-Variance Relationship of BMSCC
The annual mean BMSee of Ontario fanns in 1990 is plotted in Figure 1. A log-normal
distribution was fitted to the data. The estimated mean and variance of the log-normal distribution were 340 and 32,053, respectively, which fitted the data reasonably well (chisquare = 39.8, 31 df, P > .10). Because the BMSee apparently follows a log-normal distribution, the variability of the sec measurement increases with increasing mean sec (Figure 2). In the Ontario BMSee data, the relationship between the mean and the standard deviation followed a straight line: SD = 5.61 + .36 x mean. The observed data points were exactly on the estimated regression line. The coefficient of variation in the Ontario data showed a slightly decreasing pattern. The regression equation was the following: coefficient of variation = .43 - .00014 x mean + .00000oo62 x mean2 . When the defined relation between the mean and standard deviation
1 400
14
1 200
12
"-:' 1 000
10
o
s:::::
-U)
E
I-
CO LL.
...m Q)
8
:::s
600
6
.J:
400
4
200
2
800
C'" t/) I
Bulk milk
U
see
Figure 1. Observed bulk milk SCC distribution (0) and fitted distribution (*) based on a log-normal density function with mean of 340 and variance of 32,053. The goodness of fil chi-square values are presented in the bars. Journal of Dairy Science Vol. 75, No. 12, 1992
3362 500
SCHUKKEN ET AL. I~'
,.. ,
------,.5
!
400~~.
§!
~
~ 300
rJ
----)-----~_=
--
c,_
,.4
~;;
3
~ '0
I :
~2ool I 100/ ~
. /
~
/~
,/
;.2 ,
i ~
1.1 8
,.~o
ot/ 50
1.
and 2 contributed to this increase, and a slight increase occurred in class 3. Classes 2 and 1 contributed 50 and 27%, respectively, of the increase in mean BMSCC. Generally, classes 3 and 4 (300 x loJ to 599 x loJ) were mostly responsible for the decrease in mean BMSCC. Classes 1 and 2 (0 to 299 x 1()3) were responsible for increases in mean BMSCC. Fanns in the highest BMSCC classes always contributed to a decreasing mean BMSCC.
150 250 350 450 550 650 750 850 Annual mean
sec
Figure 2. The relation between the mean bulk milk SCC and two measures of variation, coefficient of variation (0), and standard deviation (0).
is used, advisable perfonnance goals (Figure 3) can be defined for specified regulatory limits. For example, when the regulatory limit is 500,000 cells/ml and a producer wants all BMSCC measurements to stay under this limit, the perfonnance goal should be set at approximately 250,000 cells/ml. Dynamics of
sec
(C.... Tranaitlons)
Typical transition matrices for periods in which BMSCC increased, decreased, and remained unchanged for 2 subsequent mo are shown in Table 1. In Table 2, the mean performance of fanns per SCC class is shown during the three 2-mo periods. which are representative of other periods that show increased or decreased BMSCC. The fITSt period shows 2 mo in which the mean BMSCC was similar. Fanns in class 1 «149 x loJ SCC) in mo 1 increased an average of 61 x loJ cells/ml during mo 2. Fanns in class 7 ~ x loJ SCC) in mo 1 decreased an average of 181 x loJ cells/mI. The second period shows 2 mo in which mean BMSCC decreased from 325 x loJ to 284 x 1oJ. As shown in Table 1, the classes contributing to this decrease were classes 3, 4, 5, 6, and 7, whereas classes 1 and 2 actually showed slightly increased mean BMSCC. Classes 3 and 4 contributed the most (each 28%) to the decrease in mean BMSCC. The third period included 2 mo in which mean BMSCC increased from 345 x loJ to 429 x loJ; classes 1 Journal of Dairy Science Vol. 75, No. 12, 1992
Contribution of a Farm to the Provincial SCC
The relationship between mean annual BMSCC and the SCC contribution is shown in Figure 4; most high BMSCC fanns did not have high SCC contributions. This difference can be explained by the lower annual shipment of the fanns with high BMSCC. Fanns with mean BMSCC ~750 x I ()3 had a mean shipment of 148,000 kg of milk in 1990; the mean shipment of all fanns in 1990 was 256,496 kg of milk. The farms with a high SCC contribution (>500 x loJ) usually had a mean annual BMSCC between 500 x loJ and 750 x loJ. The number of fanns with high SCC contributions that would be detected at various cutoff limits for mean BMSCC is shown in Table 3. At low BMSCC, a large number of falsepositive fanns are detected; at high BMSCC,
600
Ii 500 -
o D
..
.
iE
I
'-"--T'" ,i
g 400 '-:- - - f -
..
~ 3001r---t-r-ff~'F:lY~" Q.
'
'i 2001-;-'r-"-:=-=-...,----;----.-.-.. - - _. , - .! >
I
,
--,-, -
~100~ 01
-'-~.
-_.. _-,
! .
~
_L -'-
300 350 400 450 500 550 600 650 700 750 800 850 Regulatory limit for Bulk milk
see
Figure 3. Advisable performance goal with variable regulatory limits. 1be perfonnance goals are shown for the case in which penalties are based on t or the mean of n consecutive samples: n = I (+), 2 (0), 3 (0), and 10 (a).
3363
DYNAMICS OF BULK Mll..K SOMATIC CELL COUNTS
only very few true-positive farms are detected. In all instances, mean BMSee is a poor selection criterion for identifying problem farms, given that the goal of this farm selection process is to influence the population sec level. In Figure 5, the performance of BMSee as a diagnostic tool for sec contribution is evaluated using a receiver-operator characteristic curve (5). Several cutoff values for BMSCC are used to detect farms with a contribution ~500. This graph indicates that use of a very high BMSeC cutoff point resulted in very low sensitivity (Le., very few herds with high sec contribution were detected). However, with a low BMSee cutoff point, the specificity was low. The surface of the graph on the left-hand side of the receiver-operator characteristic curve represents the amount of diagnostic error. Based on this error rate, BMSeC was not an efficient tool to detect farms with high sec contributions.
DISCUSSION Mean-Variance Relationship
Both sec and BMSeC seem to follow a log-normal distribution (7, 9) (Figure 1). The goodness of fit of such a distribution is of academic value and also has important applications. This finding verified that the log transformation of BMSCC is necessary to use linear regression models based on the assumption of a normal distribution (2, 9). As shown in Figure 2, mean and variance were strongly related (in our data, the simple correlation was .76); this relationship can be modeled using the moments of the log-normal distribution. The observed relationship between mean and variance was then utilized to define performance goals for dairy farms, which is especially useful when regulatory limits are in place. Given the large inherent variability. when the regulatory limit for BMSeC is defined at
TABLE I. State transition matrices for bulk milk SCC during an increase, a decrease, or a stable l mean SCC between 2 subsequent mo. To class
From class
2
3
52.6 2 70.43 15.84
41.7 26.7 57.7
4.4 2.3 23.3
10.5 24.7 4.5
60.1 59.0 32.7
3
1.9 7.0 4.6
4
4
5
7
6
.1
.5 2.7
.1 .1 .4
.1 .I .I
0
23.1 13.3 50.3
4.8 2.4 10.0
1.3 .6 1.9
.1 .I .4
0
29.8 44.3 9.8
44.3 34.3 24.1
17.8 10.5 40.9
4.4 2.8 24.4
1.3 .6 13.0
.6 .5 5.2
.4 2.7 .1
11.5 21.5 4.7
31.7 41.2 21.2
32.5 23.0 36.4
15.6 8.4 25.1
5.7 2.0 9.4
2.6 1.2 3.0
5
.5 .8 .4
3.7 12.8 2.9
20.5 24.0 8.9
30.6 35.6 26.0
24.1 14.5 30.8
15.6 7.0 18.9
5.0 5.3 12.3
6
0 1.4 0
3.3 5.5 1.1
14.5 16.5 7.7
19.2 31.7 11.4
26.0 22.5 28.2
19.0 8.7 24.2
18.1 13.8 27.5
7
.5
.8 5.0 .6
7.2 13.8 2.6
9.7 20.9 7.9
19.6 18.4 11.3
15.3 14.6 21.2
47.0 27.2 56.4
2
0 0
1.1
.1 .1 .2
.Stable perfonnance was in August 10 September 1990, decrease was in November 10 December 1990 (41 x 103), and increase was in November to December 1985 (84 x 103). 2Transition probability August to September 1990; SCC is stable. 3Transition probability November to December 1990; SCC decreased. 4Transition probability November to December 1985; SCC increased. Journal of Dairy Science Vol. 75, No. 12, 1992
3364
SCHUKKEN ET AL.
TABLE 2. Dynamics of SCC classes during an increase, a decrease, or a stableI mean SCC between 2 subsequent mo. SCC Performance from month i to month i + 1 Stable
Decreases
~Mean2
SCC Class
~ean
(x 1()3)
o to 150 300 450 600 750
to to to to to
Increases Contr3
~ean
149 299 449 599 749 899
~900
+61,000 +45,000 +1000 -28,000 -83,000 -118,000 -181,000
+36,000 +5400 -54,000 -108,800 -155,000 -217,000 -287,000
Contr (%)
(%)
+140,000 +113,000 +78,700 +25,600 +12,500 -14,800 -122,000
28 28 17 11 16
27 50 19 3 1
IStable performance was in August to September 1990, decrease was in November to December 1990 (41 x 103), and increase was in November to December 1985 (84 x 1()3). 2~Mean
= The change in mean SCC between month i and month i + 1 of farms that were in month i in this SCC
class. 3Cootr = Contribution to the change in mean
sec.
600 X 103, a perfonnance goal of 599 x 103 cells/ml is not advisable because a farm with an average performance of 599 x 103 can be expected to receive several penalties. Figure 3 shows that a mean of approximately 300 x 1
Class Transitions
The results of this transition analysis indicate that the main effort in decreasing the population BMSee should focus on the farms with medium high sec (300 x 1
TABLE 3. Evaluation of the diagnostic value of mean bulk milk SCC (BMSCC) when the goal is to detect problem, farms, defined as having an SCC contribution >500.
sec BMSCC Cutoff point 400 450 500 550 600 650 700 750
Farms above cutoff 2537 1875 1392 1005 716 502 359 274
Contribution of farms <500 SCC (n = 8657)
>500 SCC (0 = 193) n 190 182 166 142 119 90 72 55
Correct I
n
Correct
98 94 86 74 62 47 37 28
2347 1693 1226 863 597 412 287 219
73 81 86 90 93 95 97 98
IThe percentage of farms that are correctly classified as with or without SCC problems. Journal of Dairy Science Vol. 75, No. 12, 1992
3365
DYNAMICS OF BULK Mll..K SOMATIC CELL COUNTS 1-22
n
= 138
n
= 55
Q
8
.8 _
18
..
------/0 ~/'-400
/
::
0::
:8
12
'E
8
Z
8
6
)' 550
'>
..,
E .6
1
~500
/
16
(600
0::
III
4>500
;J
I,
.4
(.) (.)
ta::
C/)
- 2 -
n
= 219
In(BMSCC)
Figure 4. The relation between the natural logarithm of the mean bulk milk SCC [In(BMSCC)] and SCC contribution in 1990. Reference lines are located at SCC contribution = 500, and In(BMSCC) = 6.6 (750 x 103 cellslml).
This approach is quite different from current practice, in which the farms with the highest BMsee receive most of the encouragement to perform better. The data from the class transitions lead to the importance of encouraging farms with low BMSee (classes I and 2) to maintain low BMSee. It may be advisable to offer an incentive to such farms. If successful, an incentive program would prevent the increases in mean provincial sec, as we observed in our data.
650
g
.2
J 900 ~ 1000
o
-;}------,----,------,----,--------,----,-------,------1
o
.05
.1
.15
.2 .25 .3 1 - Specificity
.35
.4
.45
.5
Figure 5. The relation between sensitivity and I - specificity when the bulk milk SCC is used as a diagnostic tool for SCC contribution ~500 (receiver-operator characteristic curve). Numbers in graph are SCC cutoff levels. Sensitivity is the probability that a problem farm is detected, I - specificity is the probability that a farm without a problem is incorrectly flagged as a problem farm.
BMsce on farms in classes 3 and 4 and to maintain low SCC on farms in classes I and 2. Attention should focus on farms with a high SCC contribution, which are generally not the farms with the highest BMSCC. ACKNOWLEDGMENTS
SCC Contribution
Consumers use dairy products made from all of the milk that farmers produce; thus, intake of milk components depends on the mean milk quality of a farm multiplied by its production. The relatively small impact of the farms with very high BMSee became apparent when the sec contribution of each farm to the total SCC production was calculated. The farms with the highest SCC contribution were typically not the farms with high BMSeC, but, rather, were the farms in sec classes 3, 4, and 5. Higher returns are expected when resources are utilized to influence management on farms with high SCC contributions instead of farms with high BMSCC. CONCLUSIONS
In order to affect the population sec, efforts should be directed to decrease the
This study was financially supported by the Bureau of Veterinary Drugs, the Ontario Ministry of Agriculture and Food, and the Ontario Milk Marketing Board. Great appreciation is extended to the staff of the Central Milk Testing Laboratory for support. Computing tasks were performed by Peter Sijed at the Ontario Ministry of Agriculture and Food, Computing Branch, Richard Canton at the Ontario Milk Marketing Board, and Steve Howie at the University of Guelph Computing Services. REFERENCES
I Carpenter, T. E. 1988. Microcomputer programs for Markov and modified Markov chain models. Prevo Vet. Med. 5:169. 2 Emanuelson, D., and H. Funke. 1991. Effect of milk yield on the relationship between bulk milk somatic cell count and prevalence of mastitis. 1. Dairy Sci. 74: 2479. Journal of Dairy Science Vol. 75, No. 12, 1992
3366
SCHUKKEN ET AL.
3 Leslie. K. E., I. Dohoo. and A. H. Meek. 1983. Somatic cell counts in bovine milk. Compned. Contino Educ. Pract. Vet. 5:S601. 4 Martin. S. W.. A. H. Meek. and P. Willeberg. 1987. Veterinary Epidemiology, Principles and Methods. Iowa State Univ. Press, Ames. 5 SASI& User's Guide: Statistics, Version 6 Edition. 1990. SAS Inst., Inc.• Cary, NC. 6 Scholl, D., and M. Den Ouden. 1991. Dynamics of farm bulk-tank: milk protein concentration. Page 637 in Proc. 6th Int. Soc. Vet. Epid. Econ. Symp., Ottawa, ON, Can. Univ. Guelph. Guelph. ON, Can.
Journal of Dairy Science Vol. 75, No. 12, 1992
7 Schukken, Y. H.• J. Buurman, A. Brand, D. van de Geer, and F. J. Gronuners. 1990. Population dynamics of bulk milk somatic cell counts. 1. Dairy Sci. 73: 1350. 8 SChukken, Y. H., K. E. Leslie, A. 1. Weersink, and S. W. Martin. 1992. Ontario bulk milk somatic cell count reduction program. I. Impact on somatic cell counts and milk quality. J. Dairy Sci. 75:3352. 9 Shook, G. E. 1982. Approaches to summarizing somatic cell count which improve interpretability. Page 150 in Proc. 21st Annu. Mtg. Nat!. Mastitis Counc.• Nat!. Mastitis Counc., Arlington, VA.