WP3 Food quality and safety
T1 On farm risk analysis
Dairy Farm Risk Analysis 2
Basic epidemiological measures used in summarising animals health
monitoring data
[E-J 1]
The disease incidence is defined as number of new cases of illness commencing, or of
persons falling ill, during a specified time period in a given population. In animal
husbandry, where the animals disease status is usually monitored during long time period
and one observation corresponds to the animal’s disease status in unit period, the disease
incidence can be defined also as the number of observations with disease.
For example the results about disease incidences in Estonian dairy cows are presented in
Table 1. Note, that here the total number of disease incidents does not mean the number of
different diseased cows, but this number shows the number of months where the cows had
a certain disease summed over whole time period and studied farms.
Table 1. Disease incidence
Disease or disease group
Disease incidence
Udder diseases
4708
Uterine infection
2389
Metabolic diseases
1190
Retained placenta
751
Foot diseases
349
Other injuries
207
Enteritis
164
Disorders of rumen or abomasum
140
Ovulatory dysfunction
137
Dystocia
129
Abortion
50
Prolapse of uterus
52
Skin diseases
29
Diseases of respiratory tract
17
TOTAL
10312
Based on disease incidences the proportions of certain diseases in whole number of disease
incidents can be found.
For example the panorama of multifactorial diseases in Estonian dairy cows is given in
Figure 1. The most common diseases of dairy cows in Estonia are udder diseases (45.7%),
uterine infection (23.2%), metabolic diseases (11.5%), retained placenta (7.3%) followed
by foot diseases (3.0%), other injuries (2.0%), enteritis (1.6%), disorders of rumen or
abomasum (1.4%), ovulatory dysfunction (1.3%) and dystocia (1.3%). Relatively fewer
cases were registered on abortion (0.5%), prolapse of the uterus (0.5%), skin diseases
(0.3%) and diseases of the respiratory tract (0.2%).
Figure 1. The panorama of multifactorial diseases in Estonian dairy cows
Uterine infection
23.2%
Metabolic diseases
11.5%
Retained placenta
7.3%
Foot diseases
3.4%
Other injuries
2.0%
Enteritis
1.6%
Disorders of rumen
or abomasum
1.4%
Ovulatory
dysfunction
1.3%
Abortion
0.5%
Udder diseases
45.7%
Diseases of
respiratory tract
0.2%
Dystocia
1.3%
Prolapse of uterus
0.5%
Skin diseases
0.3%
The prevalence rate (PR) is defined as the proportion of diseased animals in fixed time
moment or period to population size in that moment or period.
The incidence rate (IR) is defined as the proportion of new cases of illness commencing
during a specified time period in a given population to the population at risk (the total
number of observations).
The estimate of where the true value of a result lies is usually expressed in terms of a 95%
confidence interval (CI), or confidence limits. The calculation of confidence limits bases
on the (asymptotic) distribution of studied characteristic. In case of large sample, usually
the normal distribution is used to get approximate confidence intervals.
For incidence rate the asymptotic confidence limits can be found using the following
formula:
95% CIIR  IR 1.96 IR (number of observations) .
For example, the disease incidences, incidence rates in percents (multiplied with 100%)
and the confidence intervals of incidence rates in Estonian milking cow’s study are
presented in Table 2. The total number of observations was 87332.
Table 2. Disease incidences in Estonian dairy cows (n = 87332)
Disease or disease group
Disease incidence Incidence rate (%)
Udder diseases
4708
5.39
Uterine infection
2389
2.74
Metabolic diseases
1190
1.36
Retained placenta
751
0.86
Foot diseases
349
0.40
Other injuries
207
0.24
Enteritis
164
0.19
Disorders of rumen or abomasum
140
0.16
Ovulatory dysfunction
137
0.15
Dystocia
129
0.15
Abortion
50
0.06
Prolapse of uterus
52
0.06
Skin diseases
29
0.03
Diseases of respiratory tract
17
0.02
TOTAL
10312
11.81
95% CI
5.37–5.41
2.73–2.75
1.35–1.37
0.85–0.87
0.40–0.40
0.24–0.24
0.19–0.19
0.16–0.16
0.15–0.15
0.15–0.15
0.06–0.06
0.06–0.06
0.03–0.03
0.02–0.02
11.78–11.82
To compare the disease status in different groups the measure called as relative risk (RR)
is used. The relative risk is defined as the ratio of the probability of developing, in a
specified period of time, an outcome among those receiving the treatment of interest or
exposed to a risk factor, compared with the probability of developing the outcome if the
risk factor or intervention is not present. The RR can be calculated as the ratio of incidence
rates.
For example, comparing the keeping conditions in Estonian milking cow’s study, there
were in total 6600 observations with free stall keeping and 80732 observations with tie
keeping. There were registered 190 and 4518 udder diseases, respectively in case of free
stall keeping and tie keeping. The corresponding incidence rates are 190/ 6600  0.0288
and 4518/ 80732  0.0560 . The risk for cow to get an udder disease in case of tie keeping
compared to free stall keeping is estimated as 0.0560/ 0.0288  1.94 .
For relative risk the asymptotic confidence limits can be found using the following
formula:
95% CIRR  eln(RR)1.96se[ln(RR)] 
e
RR
1.96se[ln(RR)]

; RR  e1.96se[ln(RR)] ,
where ln is the natural logarithm, e is the known constant (e = 2.71828… );
se[ln(RR)]

1
1

number of cases in exposed animals number of cases in non-exposed animals
and se denotes the standard error.
For example, comparing the keeping conditions in Estonian milking cow’s udder diseases
study, the 95% confidence limits for RR are approximately calculated as
95% CI RR 
e
1.94
1.96 1 190 1 4518
;1.94  e1.96
1 190 1 4518
  1.68; 2.24 .
As this interval does not include 1, then there is less than a 1 in 20 chance that the reported
difference between keeping conditions is solely due to chance.
If the risk factor has more than two levels, then the relative risks can be calculated in
relation to the different levels, usually the level of risk factor with the lowest incidence rate
is used as the base.
For example, the udder diseases incidences, incidence rates, relative risks and the
confidence intervals of relative risks in case of different dung removal methods and in case
of different types of bedding in Estonian milking cow’s study are presented in Table 3.
Table 3. Udder diseases – incidence and relative risks by risk factors in Estonian
dairy cows
Risk factor
Dung removal
manual
scaper
tractor
Type of bedding
straw
peat
sawdust
Number of
observations
Number of cases Incidence rate
Relative risk
95% CI
848
29735
52041
12
1548
3148
0.014
0.050
0.057
1.00
3.54
4.07
2.01–6.25
2.31–7.17
14952
47879
19793
540
3156
1012
0.035
0.062
0.049
1.00
1.77
1.39
1.62–1.94
1.25–1.54
Other commonly used disease status measures in epidemiological studies are odds and
odds ratios. The odds of an event are calculated as the number of events divided by the
number of non-events.
For example, on average 2000 drones are born in every 60000 births in beehive during
year, so the odds of any randomly chosen bee being that of a drone is:
number of drones number of queens  2000 58000  0.034 .
Equivalently we could have calculated the same answer as the ratio of the bee being a
drone (0.033) and it not being a drone (0.967). If the odds of an event are greater than one
the event is more likely to happen than not; if the odds are less than one the chances are
that the event won't happen.
When events are rare, risks and odds are very similar. For example, in the bee’s sex
example 2000 of 60000 born bees were drones: a risk of 0.033 [2000/60000] or an odds of
0.034 [2000/(60000-2000)].
Odds ratio (OR; synonyms: cross-product ratio, relative odds) is the probability of the
event divided by the probability of the nonevent. It is a measure of the degree of
association – for example, the odds of exposure among the cases (receiving the treatment of
interest or exposed to a risk factor) compared with the odds of exposure among the controls
(the risk factor or intervention is not present).
For odds ratio the asymptotic confidence limits can be found using the following formula:
95% CIOR  eln(OR)1.96se[ln(OR)] 
e
OR
1.96se[ln(OR)]

;OR  e1.96se[ln(OR)] ,
where
se[ln(OR)] 
1
number of cases in
exposed animals

1
number of cases in
non-exposed animals

1
number of controls in
exposed animals

1
number of controls in
non-exposed animals
.
When (disease) events are rare (which is usual in veterinary medicine), the estimates of RR
are similar to those of OR.
For example, comparing the keeping conditions in Estonian milking cow’s udder diseases
study, the OR and its approximated 95% CI are calculated as
OR 
95% CIOR 
e
4518
190
 0.0593  2.00 ,
(80732  4518) (6600  190) 0.0296
2.00
1.96 1 190 1 45181 6410 1 76214
; 2.00  e1.96
1 190 1 45181 6410 1 76214
  1.73; 2.32 .
This suggests that those cows who are not kept in free stalls, are almost 2 times more likely
diseased in udder diseases than those who are living in farms with tie keeping.
Similarly to the relative risk, in case of risk factors with more than two levels, the odds
ratios can be calculated in relation to the different levels.
Odds ratios are the main parameters used in hypothesis testing and model building in
epidemiological studies.
Epidemiological studies generally try to identify factors that cause harm – those with odds
ratios greater than one (contrary to clinical trials, where typically is looked for treatments
which reduce event rates, and which have odds ratios of less than one). For example, the tie
keeping can cause more udder diseases compared with free stall keeping.
The “logit” model
The "logit" model is used instead of standard regression and analysis of variance
(ANOVA) models, if the dependent variable is binary and is measured on the 0/1-scale. For
example the disease status (healthy/diseased), pregnancy status, treatment effect (no/yes).
The logistic regression model has a form
logit( )  ln[ (1   )]     x   ,
or
 (1   )  e   x ,
where ln is the natural logarithm, e is the known constant (e = 2.71828… );  is the
probability that the studied event occurs, for example, the animal is diseased;  (1   ) is
the odds ratio and ln[ (1   )] is the log odds ratio, or “logit”; x is the independent
variable (argument); ,  and  are respectively the regression coefficients and random
error term, like in standard regression analyses.
The logistic regression model is simply a non-linear transformation of the linear regression.
The "logistic" distribution is an S-shaped distribution function, which constrains the
estimated probabilities to lie between 0 and 1. From the logistic regression model the
estimated probability is expressed as:
   x
  e    x .
1 e
Now it is evident, that if you let    x  0 , then   0.5 ; as    x gets really big, 
approaches 1; and as    x gets really small,  approaches 0.
For example, assuming that all observations in Estonian milking cow’s study are
independent, then the udder diseases incidence (  ) is predictable by stall length (SL) with
simple logistic regression model of the form
7.546 0.028*SL
  e 7.5460.028*SL .
1 e
The graphical representation of last model is visible on Figure 2.
Figure 2. The udder diseases incidence in Estonian dairy cows predicted by stall
length
Mastitis incidence
0,06
0,05
π
0,04
e 0,591 0,015*X
1  e  0,591 0,015*X
0,03
0,02
0,01
140
160
180
200
220
240
Stall lenght
The exponent of regression coefficient  , e  , is interpreted as the change in odds ratio
corresponding to the one unit change in independent variable. For example, if e   2 , then
a one unit change in independent variable would make the event twice as likely to occur.
Negative regression coefficients lead to odds ratios less than one: if e   1 , then a one unit
change in independent variable leads to the event being less likely to occur.
For example, in Estonian milking cow’s study the regression coefficient allowing to predict
the changes in udder disease incidences based on stall length, is 0.028. Thus the odds ratio
corresponding to the 1 cm change in stall length is equal to e0.028  1.03 ; the odds ratio
corresponding to the 10 cm change in stall length is equal to e0.028*10  1.32 .
Usually the logistic regression analysis is performed with help of statistical analysis
programs (SAS, R, SPSS, …) and the output contains additionally information about the
exact confidence limits of estimated parameters and p-values corresponding to the tests of
parameters statistical significance. If only confidence intervals to regression coefficients
are printed out, then the confidence intervals to odds ratios can be calculated by applying
the exponent function to coefficient  CI’s.
The part of standard output of SAS procedure LOGISTIC is presented on Figure 3.
Figure 3. The part of standard output of SAS procedure LOGISTIC
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept
SL
1
1
-7.5463
0.0279
0.2824
0.00167
713.9724
278.1635
<.0001
<.0001
Odds Ratio Estimates
Effect
SL
Point
Estimate
1.028
95% Wald
Confidence Limits
1.025
1.032
Since usually the investigated diseases are multifactorial, i.e. many different factors
participate in their aetiology, it is natural to analyse different risk factors in the context of a
single complex model, also taking into consideration possible confounding influences. For
this the generalized linear models with logistic link function can be used.
For example, in Estonian milking cow’s udder diseases study, the complex model of the
following form was used:
logit() =  + Ki + DRj + BTk + Fl + YMm + b1*SLijklmno + Ln + ijklmno,
where  is disease incidence,  is the intercept, Ki is the influence of housing type i, DRj is
the influence of manure removal j, BTk is the influence of type of bedding k, Fl is the
influence of the farm l, YMm is the influence of year-month combination m, SLijklmno is stall
length and b1 is the corresponding regression coefficient, Ln describes the effect of the
repeated measurement of the nth cow, and ijklmno designates the portion of the value of the
investigated attribute that failed to be described by the factors (random error).
From such models, where the possible confounding influences are taken into the
consideration, adjusted odds ratios (AOR, sometimes named also as adjusted relative
risks) can be estimated by applying the exponent function to the assessments of the
parameters of the model issued by the computer.
The part of standard output of SAS procedure GENMOD, used to fit the abovementioned
model with udder diseases data in Estonian milking cow’s study, is presented on Figure 4.
From parameter estimates table in Figure 4 the adjusted odds ratios and their confidence
intervals can be calculated and in last table the Wald Type 3 test results about factors
significance are presented.
Figure 4. The part of standard output of SAS procedure GENMOD
Analysis Of GEE Parameter Estimates
Empirical Standard Error Estimates
Parameter
Estimate
Standard
Error
Intercept
-6.6032
0.8732
95% Confidence
Limits
Z Pr > |Z|
-8.3145
-4.8918
-7.56
<.0001
0.3711
0.0000
-2.1325
-1.7744
0.0000
1.1731
1.8845
0.0000
0.0006
0.9843
0.0000
-0.8214
-0.5673
0.0000
3.4647
4.2375
0.0000
0.0181
4.33
.
-4.42
-3.80
.
3.97
5.10
.
2.09
<.0001
.
<.0001
0.0001
.
<.0001
<.0001
.
0.0366
……
KEEP
KEEP
BE_TYPE
BE_TYPE
BE_TYPE
DU_REM
DU_REM
DU_REM
ST_LENGTH
1
3
1
2
3
1
2
3
0.6777
0.0000
-1.4770
-1.1709
0.0000
2.3189
3.0610
0.0000
0.0094
0.1564
0.0000
0.3345
0.3079
0.0000
0.5846
0.6003
0.0000
0.0045
Wald Statistics For Type 3 GEE Analysis
Source
DF
ChiSquare
Pr > ChiSq
KEEP
BE_TYPE
DU_REM
FARM
YEAR_MON
ST_LENGTH
1
2
1
10
38
1
18.77
19.50
93.52
504.74
253.67
4.37
<.0001
<.0001
<.0001
<.0001
<.0001
0.0366