** **

** **

**Binomial Distribution Sample
Confidence Intervals Estimation **

**5. Odds Ratio**

** **

Sorana BOLBOACĂ, Andrei ACHIMAŞ CADARIU

*“Iuliu
Haţieganu” **University** of **Medicine** and
Pharmacy, **Cluj-Napoca**, **Romania*

**Abstract**

Evaluation of the strength of association between predisposing or causal
factors and disease can be express as odds ratio in case-control studies. In
order to interpret correctly a point estimation of odds ratio we need to look
also to its confidence intervals quality. The aim of this paper is to introduce
three new methods of computing the confidence intervals, *R2AC, R2Binomial, *and
*R2BinomialC*, and compare the performances with the asymptotic method called
*R2Wald*.

In order to assess the methods a PHP program was develop. First, the
upper and lower confidence boundaries for all implemented methods were computes
and graphically represented. Second, the experimental errors, standard
deviations of the experimental errors and deviation relative to the imposed
significance level α = 5% were assessed. Estimating the experimental
errors and standard deviations at central point for given sample sizes was the
third criterion. The *R2Wald* and *R2AC* methods were assessed using
random binomial variables (*X, Y*) and sample sizes (*m, n*) from 4
to 1000.

The methods based on the original method *Binomial* adjusted for
odds ratio (*R2Binomial, R2BinomialC* functions) obtain systematically the
lowest deviation of the experimental errors percent relative to the expected
error percent and the *R2AC* method, the closest average of the
experimental errors percent to the expected error percent.

** **

**Keywords**

Confidence intervals estimation; Odds ratio; Case-control studies; Assessment of risk factors

** **

** **

**Introduction**

In medical studies if we look at the association between predisposing or causal factors and disease, the evaluation of the strength of the association can be express as the odds ratio in case-control studies or as the relative risk in cohort studies. In order to define the odds ratio, first we define the odds: "The concept of odds is similar to that of probability. The odds of having a disease are equal to the probability of contacting the disease divided by the probability of not contacting the disease". "The odds ratio compares the odds that exposed and nonexposed individual will have the disease, given by the relation: odds that exposed individual will have disease/odds that nonexposed individual will have the disease" [1]. Confidence intervals accompanying of odds ratio allowed a correct interpretations of the point estimation.

The Woolf procedure [2] describe the
steps necessary to obtain the confidence intervals for the odds ratio, methods
which was used in our experiment under the *R2Wald* name.

In the literature there we can find a series of article about the confidence
intervals for odds ratio but all the methods were based on the Woolf procedure
[3],
[4].
AGRESTI Allan proposed a correction of the well-known confidence intervals
asymptotic formula (Woolf method, named here *R2Wald*) [5]
replacing the z_{1-α/2} with 2.

The aim of this paper is to introduce three new methods of computing the confidence
intervals, represented by *R2AC, R2Binomial *and *R2BinomialC*, and comparing
theirs, performances with the asymptotic method name *R2Wald*.

**Materials and Methods**

In medical studies of evaluation of a risk factors, when most of the time
we have dichotomous variables the results can be organize in a 2 by 2
contingency table where four groups of cases can be define. First group is
represent by the patients with diseases which are expose to the investigate risk
factor (real positive cases), noted usually with *a.* Second group is
represent by patients which do not present the disease but are expose to the
investigate risk factor (false positive cases) usually noted with *b*.

Third group, the patients which present the disease but are not expose to
the risk factor (false negative cases), usually noted with *c* and the
fourth group, the patient which do not present the disease and not expose to
the risk factor (true negative cases) usually noted with *d*.

Using the definition of the odds ratio and the next substitution on square
two contingency table: a = *X*, b = *Y*, c = *m-X*, d = *n-Y*
(*X *and *Y* are independent binomial distribution variables of sizes
*m* and *n*) the odds ratio become:

_{} (1)

Thus, we can say that from mathematical point of view the odds ratio
parameter is of *X(n-Y)/Y/(m-X)* function type, function named *ci4*
in our program [6].

Based on the classical definition of confidence intervals [7] and on our
experiences in confidence intervals estimation, we defined four functions called
*R2Wald*, *R2AC*, *R2Binomial*, and *R2BinomailC*:

_{} (2)

_{} (3)

_{} (4)

_{} (5)

where R2 function is computes by the formula (see also ref. [6]):

_{} (6)

The *R2Wald *is the standard method used in medical studies, known
as Woolf procedure. *R2AC* and *R2Binomial* methods are new
implemented methods. The *R2AC* method was obtained by introducing an
adjustment parameter represented by the _{}parameter for the binomial variables (*X,
Y*) and _{} for
sample sizes (*m, n*), adjustment introduce by the Agresti and Coull to
the one-dimensional confidence intervals expression (CI = CI(X,n)) [8].
The *R2Binomial* method is base on the original method, *Binomial*,
method used for the first time in confidence intervals estimation of a
proportion [8].

The above-described functions were implements into a PHP program. The PHP source codes for the functions are:

*function
R2Wald($X,$m,$Y,$n,$z,$a){*

* if($m-$X) $t1=1/($m-$X); else
return array( 0 , (float)"INF" );*

* if($Y) $t4=1/$Y; else return
array( 0 , (float)"INF" );*

* if((!$X)&&(!($n-$Y)))
return array( 0 , pow($a/2,1/$m) * pow($a/2,1/$n) );*

* if($X) $t2=1/$X; return
array( 0 , pow($a/2,1/$m)*$Y ); *

* if($n-$Y) $t3=1/($n-$Y); else
return array( 0 , pow($a/2,1/$n)*($m-$X) ); *

* $t5=pow($t1+$t2+$t3+$t4,0.5)*$z;*

* return array(
$X*($n-$Y)*exp(-$t5)/$Y/($m-$X) , $X*($n-$Y)*exp($t5)/$Y/($m-$X));}*

*function
R2AC($X,$m,$Y,$n,$z,$a){*

* if($m==$X) return array( 0 ,
(float)"INF" ); if($Y==0) return array( 0 , (float)"INF" );
*

* $ci = R2Wald($X+pow($z,2)/4/sqrt(2),$m+pow($z,2)/2/sqrt(2),
*

*
$Y+pow($z,2)/4/sqrt(2),$n+pow($z,2)/2/sqrt(2),$z,$a);*

* return array( $ci[0] , $ci[1]
);}*

*function
R2Binomial($X,$m,$Y,$n,$z,$a){*

*$ciX =
Rap("Binomial",$X,$m,$z,sqrt($a/2));*

* $cinY = Rap("Binomial",$n-$Y,$n,$z,sqrt($a/2));*

* return array (
$ciX[0]*$cinY[0] , $ciX[1]*$cinY[1] );}*

*function
R2BinomialC($X,$m,$Y,$n,$z,$a){*

* $ciX =
Rap("Binomial",$X,$m,$z,(1/(1-0.125/log($m)))*sqrt($a/2));*

* $cinY =
Rap("Binomial",$n-$Y,$n,$z,(1/(1-0.125/log($n)))*sqrt($a/2));*

* return array
( $ciX[0]*$cinY[0] , $ciX[1]*$cinY[1] );}*

In order to obtain a 100·(1-α) = 95% confidence intervals (is most
frequently used confidence intervals) the experiments were runs at a
significance level α = 5% (noted with *a* in our program).
Corresponding to choused significance level was used its normal distribution
percentile *z _{1-α/2}* = 1.96 (noted with

*define("z",1.96);
define("a",0.05);*

The performance of each method for different sample sizes (*m, n*)
and different values of binomial variables (*X, Y*) was compares using a
set of criterions.

First were computed and graphical represented the lower and upper
confidence intervals limits for equal sample sizes (*m = n = 50*) using
the *R2BinomialC *method:

*$c_i=array("R2BinomialC");*

*define("N_min",50);
define("N_max",51); est_ci2_er(z,a,$c_i,
"ci4","ci");*

Second, were analyzed the experimental errors and standard deviations of
the experimental errors using the binomial distribution hypothesis as quantitative
and qualitative criterion of the assessment and equal sample sizes (*m = n =
5, 10, 20, and 40*):

*$c_i=array("R2Wald","R2AC","R2Binomial","R2BinomialC");*

·
For *m = n = *5:

*define("N_min",5);
define("N_max",6);
est_ci2_er(z,a,$c_i,"ci4","er");*

·
For *m = n = *10 was modified:

*define("N_min",10);
define("N_max",11); *

·
For *m = n = *20 was modified:

*define("N_min",20);
define("N_max",21); *

·
For *m = n = * 40 was
modified:

*define("N_min",40);
define("N_max",41);*

The standard deviation of the experimental error (*StdDev*) was
computed using the next formula:

_{} (7)

where *StdDev(X)*
is standard deviation, *X _{i}* is the experimental errors for a
given

If we have a sample of *n* elements with a known (or expected) mean
(equal with 100α), the deviation around α = 5% (imposed significance
level) is giving by:

_{} (8)

Third, the assessment of the confidence intervals methods was carried on
with a particular situation, represented by estimation of the experimental
errors at central point (*X = Y*) and equals sample sizes *m = n = 4, 6,
8..200 * (*m = n = *even numbers):

*$c_i=array("R2Wald","R2AC","R2Binomial","R2BinomialC");*

*define("N_min",
2); define("N_max",205); est_C2(z,a,$c_i,"ci4");*

Fourth, the average of the experimental errors, standard deviation and
deviation relative to the imposed significance level (α = 5%) were compute
for sample sizes which vary from 4 to 14 (*m=4..14, n = 4..14*):

*$c_i=array("R2Wald","R2AC","R2Binomial","R2BinomialC");*

*define("N_min",
4); define("N_max",15); est_C2(z,a,$c_i,"ci4", "mv");*

The last part of the experiment consisted on assessing the performance of
methods in 100 random numbers for binomial variables *X, Y* (4 ≤ *X*
< *n, *4 ≤ *Y* < *m*) and random sample sizes *m,
n* ( 4 ≤ *m, n* ≤ 1000):

*$c_i=array("R2Wald","R2AC");*

*define("N_min",
4); define("N_max",1000);
est_ci2_er(z,a,$c_i,"ci4","ra");*

The *R2Binomial* and *R2BinomialC* methods were not included in
this part of the experiment because we did not have the resources needed to
perform the experiment for the methods which used the hypothesis of binomial distribution.

** **

**Results**

The confidence boundaries for odds ratio were computes, the results imported in SlideWrite Plus program (figure 1), and Microsoft Excel (figure 2) where the graphical representations were create.

The Slide representations (figure 1) were created using a 3D-Mesh graph
type with 80% perspective, 60° tilt angle,
and 75° rotation angle. On X-axis were
represented the values of variable *X*, on the Y-axis the values *Y* variable
and on the Z-axis were represented the odds ratio, lower or upper confidence
intervals limits. There were represented with red color the experimental values
from 0 to 2, with green the values from 2 to 4, with blue the values from 4 to 6,
with cyan the values from 6 to 8, and with magenta the values from 8 to 10.

** **

*Figure2. The
representation of the odds ratio values and its confidence limits obtained with
R2BinomialC method at 0 < X, Y < m = n = 50*

The lower and upper confidence limits in logarithmical scale with *R2BinomialC*
method at equal sample sizes (*m = n = 50*) were graphical represented in
figure 2. On horizontal axis were represented the *m = n* values
(logarithmical scale), depending on *X, Y* and on the vertical axis the
values of the confidence intervals limits (logarithmical scale).

*Figure 2. The
upper and lower confidence limits (logarithmical scale) for odds ratio *

*at 0 < X, Y
< m = n = 50*

The contour plots of percentages of the experimental errors are in figure
3-6. On X-axis were represented the values of *X* binomial variable, on
Y-axis the values of *Y *binomial variable, and on Z-axis the values of
the percentage of experimental errors for each specified method. The graphical
representations were created using a 3D-Mesh graph type with 80% perspective,
60° tilt angle and 75° rotation angle. On the plots were
represented the percentages of the experimental errors with red color (0-2%),
green (2-4%), blue (4-6%), cyan (6-8%), and magenta (8-10%).

The graphical representations of the percentages of the experimental
errors using specified method for *m = n = 5* were presented in figure 3;
for *m = n = 10* in figure 4; for *m = n = 20* in figure 5; and for *m
= n = 40* in figure 6.

*Figure 3. The OR
experimental errors with R2Wald, and R2AC at 0<X,Y<m=n=5*

*Figure 3. The OR experimental
errors with R2Binomial, and R2BinomialC at 0<X,Y<m=n=5*

*Figure 4. The OR experimental
errors with R2Wald, R2AC R2Binomial, and R2BinomialC at 0 < X, Y < m = n
= 10*

*Figure 5. The OR experimental
errors with R2Wald, R2AC, R2Binomial, and R2BinomialC *

*at 0 < X, Y
< m = n = 20*

*Figure 6. The OR
experimental errors with R2Wald and R2AC at 0 < X, Y < m = n = 40*

*Figure 6. The OR experimental
errors, R2Binomial and R2BinomialC at 0<X,Y<m=n=40*

The averages (*MErr*) and standard deviations (*StdDev*) of the
experimental errors for specified equal (*m = n*) samples sizes were
presented in table 1.

n |
R2Wald |
R2AC |
R2Binomial |
R2BinomialC |

5 |
5.0 (6.1) |
1.7 (1.5) |
2.1 (1.2) |
2.3 (1.4) |

10 |
2.9 (2.0) |
3.1 (1.6) |
2.6 (0.8) |
2.9 (1.0) |

20 |
3.3 (1.2) |
3.9 (1.4) |
3.5 (1.0) |
3.9 (1.1) |

40 |
3.9 (1.0) |
4.5 (1.2) |
4.3 (0.8) |
4.5 (0.6) |

*Table 1. The MErr
and StdDev (parentheses) for odds ratio at m = n = 5, 10, 20, and 40*

The confidence intervals for central point (*X = Y*) were calculated
and the graphical representations was creates (figure 7). In the graphical
representation, on horizontal axis were represented the values of samples sizes
(*m = n = 4,6..200* (even numbers)) depending on *X = Y* values; on
the vertical axis were represented the percentage of the experimental errors.

*Figure 7. The
percentages of the experimental errors for odds ratio at central point X = Y*

*and at m= n =
4,6,..200*

*Figure 7. The percentages
of the experimental errors for odds ratio at central point X = Y*

*and at m= n = 4,6,..200*

The average of the percentages of experimental errors (*MErr*) and
the standard deviations (*StdDev*) of them for central point estimation (*X
= Y*) are in table 2.

Method |
R2Wald |
R2AC |
R2Binomial |
R2BinomialC |

MErr |
4.91 |
4.60 |
4.69 |
4.89 |

StdDev |
0.76 |
0.81 |
0.62 |
0.60 |

*Table 2. The MErr
and StdDev for OR at central point (X=Y) and m = n = 4,6..200*

The surface plots of dependences of averages of the experimental errors (left side) and of the deviations relative to the imposed significance level α = 5% (right side) for sample sizes varying in the range 4..14 were graphically represented in figure 8.

The dependency surface plots were created with 80% perspective, 40° tilt angle and 45° rotation angle (for experimental errors average) and with 15° rotation angle (for standard deviations). For the graphical representation of averages of the experimental errors (left side graphics), with red color were represented the experimental values from 0 to 2, with green the values from 2 to 4, with blue the values from 4 to 6, with cyan the values from 6 to 8, and with magenta the values from 8 to 10.

In the graphical representation of dependency of the deviations relative
to the significance level (*α* = 5%) (right side), with red color
were represented the experimental values from 2 to 2.5, with green color the
values from 2.5 to 3, with blue the values from 3 to 3.5, with cyan the values
from 3.5 to 4, with magenta the values from 4 to 4.5, and with yellow the
values from 4.5 to 5.

*Figure 8.
Dependences of the averages of experimental errors and of deviations relative
to imposed significance level for OR with R2BinomialC at m, n = 4..14*

The averages of the means of experimental errors (*MMErr*) and of
the deviations relative to the imposed significant level α = 5% (*MDev5*)
for sample sizes which vary in 4..14 domain are in table 3.

Method |
R2Wald |
R2AC |
R2Binomial |
R2BinomialC |

MMErr |
3.33 |
2.64 |
2.57 |
2.95 |

MDev5 |
3.90 |
2.92 |
2.70 |
2.41 |

*Table 3. The
averages and deviations relative to α = 5% of experimental errors for OR when
sample size m, n vary in 4..14 domain*

Using the results obtained from the 100 random binomial variable (*X*,
*Y; 0 < X, Y < m, n*) and samples size (*n, m*) from 4 to 1000
domain (*4 ≤ m, n ≤ 1000*), a set of calculations as are
described in paper [6] are done and presented in tables 4-7 and represented in figure 9.

In the figure 9 were represented with black dots the frequencies of the
experimental error for each specified method; with green line the best errors
interpolation curve with a Gauss curve (*dIG(er)*). The Gauss curves of
the average and standard deviation of the experimental errors (*dMV(er)*)
was represents with red line. The Gauss curve of the experimental errors
deviations relative to the significance level (*d5V(er)*) was represented
with blue squares. The Gauss curve of the standard binomial distribution from
the average of the errors equal with 100·α (*pN(er,10)*) was
represented with black line.

*Figure 9. The
pN(er, 10), d5V(er), dMV(er), dIG(er) and the frequencies of the experimental
errors for each specified method and random X, m, Y, n (0 < X, Y < m, n;
4 ≤ m, n ≤ 1000)*

Table 4 contain the average of the deviation of the experimental errors
relative to significance level α = 5% (*Dev5*), the absolute
differences of the average of experimental errors relative to the imposed
significance level (*|5-M|*), and standard deviations (*StdDev*).

No |
Method |
Dev5 |
Method |
|5-M| |
Method |
StdDev |

1 |
R2AC |
0.44 |
R2AC |
0.13 |
R2AC |
0.42 |

2 |
R2Wald |
0.56 |
R2Wald |
0.29 |
R2Wald |
0.47 |

*Table
4. Methods ordered by performance according to Dev5, |5-M| and StdDev
criterions*

Table 5 contains the absolute differences of the averages that result
from Gaussian interpolation curve to the imposed significance level (*|5-MInt|*),
the deviations that result from Gaussian interpolation curve (*DevInt*),
the correlation coefficient of interpolation (*r2Int*) and the Fisher
point estimator (*FInt*).

No |
Method |
|5-MInt| |
Method |
DevInt |
Method |
r2Int |
FInt |

1 |
R2AC |
0.00 |
R2AC |
0.42 |
R2Wald |
0.76 |
60 |

2 |
R2Wald |
0.02 |
R2Wald |
0.43 |
R2AC |
0.77 |
62 |

*Table 5. The
methods ordered by |5-MInt|, DevInt, r2Int and FInt criterions*

The superposition of the standard binomial distribution curve and
interpolation curve (*pNIG*), the superposition of standard binomial
distribution curve and the experimental error distribution curve (*pNMV*),
and the superposition of standard binomial distribution curve and the error
distribution curve around significance level (α = 5%) (*pN5V*) were
presented in table 6.

No |
Method |
pNIG |
Method |
pNMV |
Method |
pN5V |

1 |
R2AC |
0.44 |
R2AC |
0.44 |
R2AC |
0.45 |

2 |
R2Wald |
0.44 |
R2Wald |
0.47 |
R2Wald |
0.54 |

*Table 6. Methods
ordered by the pNIG, pNMV, and pN5V criterions*

In table 7 were presented the percentages of superposition of the
interpolation Gauss curve and of the Gauss curve of error around experimental
mean (*pIGMV*), between the interpolation Gauss curve and the Gauss curve
of error around imposed mean (α = 5%) (*pIG5V*), and between the
Gauss curve experimental error around experimental mean and the error Gauss
curve around imposed mean α = 5% (*pMV5V*).

No |
Method |
pIGMV |
Method |
pIG5V |
Method |
pMV5V |

1 |
R2AC |
0.87 |
R2AC |
0.98 |
R2AC |
0.88 |

2 |
R2Wald |
0.76 |
R2Wald |
0.87 |
R2Wald |
0.77 |

*Table 7. The
confidence intervals ordered by the pIGMV, pIG5V, and pMV5V criterions*

** Discussions**

** **

For the equal
sample sizes, if we look after a method which to obtained an average of the
experimental error closest to the imposed significance level (*α* =
5%), the *R2Wald* method can be chouse if *m *=* n *=* 5*. On
the other hand, the *R2Wald* method obtained the greatest standard
deviation compared with the *R2AC*, *R2Binomial*, and *R2BinomialC*
methods. The best standard deviation for *m *=* n *=* 5* was
obtains by the *R2Binomial*. For the *R2AC*, *R2Binomial* and *R2BinomialC*
methods the averages of the experimental errors increase with sample sizes but never
exceed the significance level (*α* = 5%). The best performance in confidence
intervals estimation for *m = n* = 40, as well as for *m = n* = 20, is
obtain by the *R2BinomialC* and the *R2AC *methods.

The best average of the experimental errors for central point (*X=Y*)
was obtained by the asymptotic method (*R2Wald*) and the best performance
at the central point was obtained with a binomial method (*R2BinomialC, *closely*
*followed by the* R2Binomial* method).

If we looked at the special experiment when sample sizes vary in 4..14 domain,
for all methods, the averages of experimental errors were less than the
expected value (*α* = 5%); the *R2Wald* method was the one which
obtained the greatest average of the errors (3.33%). The lowest standard
deviation was obtains by the *R2Binomial* method.

Looking at the results from random binomial variables (*X, Y*) and random
samples (*m, n*) we can remarked that the *R2AC* method obtains the
closest experimental error average to the significance levels (table 4), the
lowest experimental standard deviation and the closest deviation of the
experimental errors to the significance level (*α* = 5%).

The *R2AC* method obtains the closest interpolation average to the significance
level (table 5), the lowest interpolation deviation, and the best correlation
between theoretical curve and experimental data.

The *R2Wald *method (as well as the *R2AC* method) obtains the maximum
superposition between the curve of interpolation and the curve of standard
binomial distribution, the maximum superposition between the curve of standard
binomial distribution and the curve of experimental errors. Again the *R2Wald*
method obtained the maximum superposition between the curve of standard
binomial distribution and the curve of error distribution around the significance
level (*α* = 5%).

The maximum superposition between the Gauss curve of interpolation and
the Gauss curve of errors around experimental mean obtains by the *R2AC *method.
The *R2AC* method obtained again the maximum superposition between the Gauss
curve of interpolation and the Gauss curve of errors around significance level
(*α* = 5%), and the maximum superposition between the Gauss curve of experimental
errors and the Gauss curve of errors around imposed mean (*α* = 5%).

** **

** **

**Acknowledgements**

The first author is thankful for useful suggestions and all software implementation to Ph. D. Sci., M. Sc. Eng. Lorentz JÄNTSCHI from Technical University of Cluj-Napoca.

**References **

** **

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