1. Introduction

In many applications, units can be classified as defective or non defective. Pool testing involves pooling such units into groups or pools, testing the groups, and classifying each group as defective or non-defective. A group is defined as non-defective if non of the unit in the group contains the characteristic of interest otherwise a group is said to be defective. A group testing design has been shown to be a compelling alternative to one-at-a-time testing in many areas where rare traits are of interest. Research has shown that pool studies can be used in plant pathology, genetics and reduction of cost in early stages of drug discovery (Hammick and Gastwirth, 1994; Swallow, 1985; Xie et al., 2001). Pool testing has also been applied in screening the population for the presence of HIV antibody (Kline et al., 1989 and Manzon et al., 1992). Computational testing that focuses on classifying subjects has been developed Maheswaran et al. (2008).

Recently more research work are focused on estimating the rate of trait. Thomson (1962) considered estimation problem using pool testing which was later considered by Brookmayer (1999) by introducing errors. Sufficiently accurate estimate of the prevalence can be obtained from testing pooled samples as demonstrated by Hammick and Gastwirth (1994) and their procedure provides greater protection of respondent’s identity which can be useful in improving the response rate. On the same year, Gastwirth and Johnson (1994) used pool testing to estimate HIV prevalence cost-effectively. Hardwick et al., (1998) considered sequentially deciding between two experiments for estimating a common success prevalence rate where he considered the individual Bernoulli trials or the product of k individual independent Bernoulli trials. Nyongesa (2011) used moment method to estimate the prevalence and he observed that his proposed testing procedure reduced misclassification, particularly the false positives. Computational statistics has been used in pool testing to compute the statistical measures when perfect and imperfect tests are used (Syaywa and Nyongesa, 2010; Tamba et al., 2012).

Benefits from group testing depend on size of the pools. Swallow (1985) showed that large group sizes can lead to estimators with enormous bias. In addition, there are biological issues to be considered. For example in HIV testing, enzyme-linked immunosorbent assay tests (ELISA) are commonly used in screening experiments to detect the presence of the virus. However, sensitivity levels for such tests are known to be poor when many blood samples are pooled together. With many of the standard enzyme-linked immunosorbent assay tests (ELISA), group sizes of up to 15 are typically used without experiment dilution effects (Behets et al., 1990; Cahoon-Young et al., 1989; Kline et al., 1989; Tu al., 1995). Recent studies have provided algorithm for the computation of pool sizes (Ding and Xiong, 2015).

This study has focused on estimation of proportions and in particular a better estimation of the prevalence rate. The essence of the study, is to device a method of selecting between three experiments namely:

i) individual testing of items of a population with a view to estimating prevalence rate p, with misclassification, this experiment will be denoted by ¹E,

ii) pool testing experiment as proposed by Dorfman (1943) but with errors in inspection, this experiment will be denoted by ²E and

iii) estimating the prevalence rate of the characteristic of interest by retesting the pools declared positive in the first pool test and this experiment will be denoted by ³E.

The rest of the paper is arranged as follows: in Section 2 we shall develop the models and formula for calculating Fisher information, in Section 3 we shall plot the graphs Fisher information against the value of p. In Section 4 we shall compute the cut off values. In Section 5 we shall develop the joint model and in section 6 we shall compute the MLE of p of the joint model. In section 7 we shall compare the variances of the models by plotting their graphs. In Section 8 we shall compute the ARE values and in section 9 we shall have conclusion of the study.

2. The Model

The model have been split into three, that is ¹E-, ²E-, and ³E-experiments. r, s and t have been assumed to be the total number of observations from ¹E-, ²E-, and ³E-experiments respectively. In typical sequential allocation problems, different experiments give information about different parameters. However, in this study, the three experiments i.e ¹E-, ²E-, and ³E-experiments give information about the same parameter (p), although one experiment have given more information than the other two experiments under consideration depending on the actual value of the parameter, pool size, sensitivity and specificity of the tests. For simplicity, it has been assumed that individual units being pooled are independent and identically distributed Bernoulli random variables.

2.1. The ¹E-experiment

If ¹E-experiment is to be used to estimate the prevalence rate , and if for is a sequence of identically independent distributed random variable, then where is the probability of declaring an individual as positive defined by the relation, given and are sensitivity and specificity of the tests respectively.

For a single experiment, the probability density function is

(1)

The Fisher information denoted by , on the prevalence rate contained in a single observation of the ¹E-experiment is

(2)

easily obtained by MLE method from (1). If observations from only the ¹E-experiment are used to estimate , then the maximum likelihood estimator of p, denoted by is

(3)

The asymptotic variance of is obtained from (2) which yields

(4)

2.2. The ²E-experiment

The ²E-experiment involves putting together items to form a pool and testing the pool rather than testing each individual for the evidence of a characteristic of interest. A negative reading indicates that the pool contains no defective item and a positive reading indicates at least one defective item in the pool. Pooling procedures have proved to reduce the cost of testing when the prevalence rate is low. In this experiment, the probability of declaring a pool of size positive is denoted by . If denote a sequence of identically independent distributed random variables for then

. For a single experiment equivalently the probability density function is

(5)

and the Fisher information denoted by contained in a single observation of the ²E-experiment is

(6)

Suppose there are pools for the ²E-experiment each of size k, available for estimating and suppose pool test positive on the test, then the maximum likelihood estimator of p, denoted by is

(7)

Noted is Thompson (1962) maximum likelihood estimator (MLE) of p i.e

is a special case of (7).

The asymptotic variance of is

(8)

2.3. The ³E-experiment

The ³E-experiment involves retesting of the pools declared positive in the first pool test inorder to approximate the prevalence rate. Retesting of already tested pools reduce misclassification (Nyongesa, 2011). In this experiment, the probability of declaring a pool of size positive is denoted by where . If denote a sequence of identically independent distributed random variables for , then . For a single experiment the probability density function is

(9)

Similarly the Fisher information contained in a single observation of the ³E-experiment is

(10)

If t observations from the ³E -experiment are used to estimate and pool tests positive, then the maximum likelihood estimator of p, denoted by is

(11)

Equivalently the asymptotic variance of obtained from (10) is

(12)

3. Comparison of of ¹E-, ²E- and ³E-experiments

This study compares the Fisher information of ¹E-, ²E- and ³E-experiments in this section by plotting the graphs of of ¹E-, ²E- and ³E-experiments for values of versus

As seen from Figures 1 to 6, the Fisher information for the ¹E -experiment is independent of the pool size hence it is not affected by change of the value of For the ²E - and ³E-experiments, the Fisher information is very high for small values of and it approaches zero as increases. As sensitivity and specificity of the test kits increases, the gap between the Fisher information of the ²E - and ³E -experiments shrinks. Holding constant, increasing sensitivity and specificity of the test kits, the region at which the Fisher information of the ¹E- and ³E-experiments is better shrinks while for ²E-experiment increases. Similarly as increases the region in which the Fisher information of ¹E - and ³E-experiments is better decreases while that of ²E-experiment increases. From Figures 1 to 6 it can be concluded that the ³E-experiment is better than ¹E- and ²E-experiments for values of relatively small, for values of relatively large, the ¹E-experiment is better and the ²E-experiment is better for some values of between 0 and 1. It is also noted that the region in which one experiment is better than the other experiments depends on sensitivity, specificity and the pool size.

Figure 1. A plot of Fisher information against the value of with and

Figure 2. A plot of Fisher Information against the value of with and

Figure 3. A plot of Fisher information against the value of with and

Figure 4. A plot of Fisher Information against the value of with and

Figure 5. A plot of Fisher Information against the value of with and

Figure 6. A plot of Fisher Information against the value of with and

4. Computation of Cut-Point Values

The cut-point value is defined as the value of p at the point where the Fisher information of one of the experiment surpasses the Fisher information of the other experiment while comparing any two of the ¹E-, ²E- and ³E-experiments. If is the cut-point value, then is a unique root in [0, 1] of the equation for and

4.1. Computation of Cut-Point Values of and

In this section the cut-point values of ¹E- and ²E-experiments are computed by equating the Fisher information of the two experiments. Therefore equating (2) and (6) and simplifying yields

(13)

(13) has no solution in closed form therefore the equation is solved iteratively using an R code that we developed which is presented in Appendix A.

4.2. Computation of Cut-Point Values of and

The cut-point values of ¹E- and ³E-experiments are computed in this section by equating (2) and (10) which yields

(14)

after simplifying. Similarly (14) is solved iteratively using an R code that we developed presented in Appendix B.

4.3. Computation of Cut-Point Values of and

Similarly the cut-point values of ²E- and ³E-experiments are computed by equating (6) and (10) which yields

(15)

after simplifying. Equivalently (15) is solved iteratively using an R code developed which is presented in Appendix C.

For various values of and the values the roots of (13), (14) and (15) are given in Table 1.

From Table 1 it can be noted that the cut-point values are sensitive to k (pool size). As specificity and sensitivity increases the cut point value between ¹E- and ³E-experiments increases while that between ²E- and ³E-experiments decreases. It is observed from Table 1 that as the pool size (k) increases, keeping and the same, the region in which the ³E-experiment is better than the ²E- and ¹E- shrinks. Increase in sensitivity and specificity leads to decrease of the area which ¹E- and ³E -experiments is better than ²E-experiment. In general the exact values of p where one experiment is better than the others depends on the pool size (k), sensitivity and specificity of the tests.

Table 1. Cut-point point values for various values of

For example if k = 3 and N tests are available, then the allocation that maximizes the information about is:

In general, if N tests are available, then the allocation that maximizes the information about is

Note also that the region where one experiment is better than the other depends on the unknown parameter hence adaptive rule is suggested where is estimated at each stage and the next observation is allocated depending on the relationship between the estimated and the cut-point value.

5. The Joint Model

If r, s and t are the total number of observations from ¹E-, ²E- and ³E-experiment respectively, then the joint probability density function of the random variables and from the ¹E-, ²E- and ³E-experiments respectively is a multinomial probability density function. The joint probability density function is given by the product of their respective density functions, since the random variables are assumed to be independent, therefore

(16)

The joint likelihood function of (16) is

(17)

Taking logarithm on both sides of (17) and differentiating with respect to yields

(18)

where and

Equating (18) to zero leads to

(19)

The only variable in (19) is Hence

(20)

is a function of which can be solved iteratively. The value of computed from (20) is denoted by hence the maximum likelihood estimator of p, denoted by of the joint model is The asymptotic variance of is obtained by solving where is the joint probability density function given by (16). Therefore

(21)

where

6. Estimator of Prevalence Rate, Its Variance and Confidence Interval

The maximum likelihood estimator of the prevalence rate of the joint model, the variance and 95% Wald-type confidence interval of the MLE for values of and are computed in this section.

From Tables 2 and 3 it is observed that the maximum likelihood estimators of the prevalence rate are very close to the actual value which were used to simulate the estimators. The population estimators resulting from the experiments are used to evaluate the confidence limits of the confidence interval of the simulated estimators where is the level of significance and it is noted from Tables 2 and 3 that the actual value is within the limits.

Table 2. Maximum likelihood estimator, variance and Confidence interval for different values of p for       and

Table 3. Maximum likelihood estimator, variance and Confidence interval for different values of p for       and

7. Comparison of Variances

In this section, the graphs of the variance of of ¹E-, ²E- and ³E-experiments and joint model for various values of and versus are plotted for comparison purposes.

It is observed from Figures 7 to 12 that:

i) is smaller than and for values of p close to 0,

ii) for values of p close to 1 the is smaller than and while

iii) for some values of p between 0 and 1 the is smaller than the variance of the other two models.

It is also noted that holding sensitivity and specificity constant and increasing the value of k from 2 to 10, makes the area in which the is smaller than the variance of the other models shrinks. Increasing sensitivity and specificity of the tests shrinks the area between and It is also observed from Figures 7 to 12 that the variance of the joint model is smaller than the variance of p of the ¹E-, ²E- and ³E-models. Hence the joint model is more reliable compared to the other three models.

Figure 7. A graph of as a function of with and

Figure 8. A graph of as a function of with and

Figure 9. A graph of as a function of with and

Figure 10. A graph of as a function of with and

Figure 11. A graph of as a function of with and

Figure 12. A graph of as a function of with and

8. Asymptotic Relative Efficiency (ARE)

The and are compared in this section. This is accomplished by computing asymptotic relative efficiency (ARE) values for various values of and Let and

From Tables 4 and 5 it is observed that increase in sensitivity and specificity of test leads to increase in the values of ARE. For the given values of pool size, sensitivity and specificity the highest value of ARE is 0.761. Hence the other models under consideration in the study ¹E-, ²E- and ³E-models can only be 76.1% efficiency as the joint model.

Table 4. The ARE of the joint model relative to the 1E-, 2E- and 3E-models with       and

Table 5. The ARE of the joint model relative to the 1E-, 2E- and 3E -models with

9. Conclusions

This study focused on construction of the a new model for approximating the prevalence rate of a trait in a population with imperfect tests by consecutively choosing between three experiments namely ¹E-, ²E- and ³E -experiments. The model should select the better experiment and once the better experiment is being used, the estimator should approximate the individual maximum likelihood estimator (MLE) for that experiment. From this study it is clear that the best estimators for small, medium and large values of p, respectively, are and From Tables 2 and 3, the computed values of asymptotic relative efficiency (ARE) for various values of and are less than one hence the proposed joint model for sequential choice of the best experiment for optimal estimation of a trait with misclassification is more efficient than the ¹E-, ²E- and ³E-models.

It is assumed that the samples being pooled for use in the model are independent and identically distributed Bernoulli random variables. It is also assumed that there is a laboratory test that can determine whether or not a unit or at least a unit in a pool has the characteristic of interest and that the tests are conditionally independent of each other. Sensitivity and specificity of the test kits are also assumed to be the same at each step of testing and for all samples in use in the model.

The findings of the study of a better approximation of the prevalence rate have important health implications for prevention, intervention and treatment of HIV infections in a population. HIV infected population are known to greatly increase the spread of HIV infection. Our improved estimate of the prevalence rate of HIV infection could substantially reduce the potential risks for secondary HIV transmission by HIV infected population who are unaware of HIV infections. A prompt diagnosis of HIV infection might prevent the infected population from engaging in high-risk behaviours with uninfected population and avoid new HIV infections to occur. But the population with HIV infection should take counselling, regarding risk-reduction strategies such as abstinence and safer sexual behaviours such as 100% use of condoms.

Based on the model developed a pool testing model of retesting of both positive and negative pools can be studied. A model based on cost analysis when sampling from different experiments can also be looked at when using imperfect kits.

Appendix

Appendix A: R code to Determine the Cut-Point Value of and

Appendix B: R Code to Determine the Cut-Point Value of and

Appendix C: R Code to Determine the Cut-Point Value of and