Group testing started or originated with [1] during World War II as a frugal method of testing blood specimens of army inductees in order to detect the presence of infection. Group testing has been applied in many areas: plant disease assessment [2] , fisheries [3] , quality control, drug discovery and transmission of viruses by insect vectors [4] . It has also been used to screen the population for the presence of HIV/Aids antibody [5] . [1] proposed that, rather than testing each blood specimen individually, portions each of k specimens can be pooled and the pooled specimen can be tested. If the pooled specimen is free of infection, all k inductees are passed with no further tests, otherwise the remaining portions of each of the blood specimens are to be tested individually. If the prevalence of infection is low, the expected number of tests per inductee and thus the expected cost per inductee, would be reduced. [1] assumed that tests were perfect, that is, a negative reading indicated the group contained no defective item and a positive reading indicated the presence of at least one defective item.

Group tests save resources since many units are tested without testing them individually. One of their advantages as a method of estimation is they are time efficient. In most of the studies carried out, the experimental unit/group is a litter. It has been observed that there is a tendency of littermates to respond more alike than animals from different litters, the “litter effect”. This litter effect is also known as the extra-dispersion (over/under- dispersion) or the intra-litter correlation. These litters may be of equal or unequal sizes. The concern here is with methods of establishing the confidence interval for the proportion of defective units p in unequal groups with adjustment for overdispersion using bootstrap technique.

Overdispersion is the phenomenon of having greater variability than predicted by the random component of the model; this is common in the modeling of binomial distribution for group testing [6] . Overdispersion is a very common feature in applied data analysis because naturally populations are frequently heterogeneous contrary to the assumptions implicit within widely used simple parametric models. The over-dispersion parameter is common in proportions in biology, toxicology, medicine, genetics and other similar fields and is important in making inference regarding the regression parameters on the mean [7] .

Maximum likelihood estimation gives a unified approach to estimation, which is well-defined, in binomial distribution and many other problems. The maximum likelihood as an estimator has been studied and seconded as an approach for using the proposed extended Beta-Binomial (BB) model to analyze over/under-dispersed proportions [8] . They estimated this parameter using maximum likelihood estimation (MLE). This concurred with the observation that the MLEs may be biased when the sample size n or the total Fisher information is small [9] . This bias is usually ignored in practice, the justification being that it is small compared with the standard errors.

Studies on point estimation in terms of bias and efficiency and the test for presence of overdispersion for both counts and proportions data have been done [10] . Interval estimation methods in which groups were of unequal sizes were carried out using interval based on the MLE, the Logit and CLL [11] . The quasi-likelihood approach to correct for overdispersion is used by [6] , he examines interval estimation methods based on functions of the MLE, the Logit and CLL functions.

First, positive groups, are simulated from the number of groups, with their respective sizes for. This is followed by bootstrapping Y_i’s and then computing the estimate of p. Coverage probabilities of the three interval methods for the estimate of p for 500 bootstrap simulations are obtained, and the program is run repetitively in order to see how the three method’s coverage probabilities vary.

The nominal coverage probability is set at 0.95.

This is then compared to estimating p from the same data that is the Y_i’s without bootstrapping and the program run repetitively so as to calculate the coverage probability.

The number of bootstraps B’s were varied for the same combination of p, and. Coverage probabilities are generated. This was to assess the performance of the coverage probabilities as the number of bootstrap simulations increase for each method.

Next p was varied for the same N_i, k_i and the same number of bootstrap simulations (500), and coverage probabilities are obtained.

Lastly, interval widths for the data are calculated for bootstrapping technique for the three interval methods and compared to those intervals got from each of the three methods without bootstrapping. This is in order to investigate whether bootstrapping gives more precise intervals.

Group testing was primarily used to inspect individual members from a large population [1] . Therefore probability used in this study ranged from 0.01 to 0.2. From Table 1 it was seen that the CLL and the Logit gave more conservative coverage probabilities consistently for different bootstrap simulations for p = 0.02. All the three methods differ only slightly with CLL and Logit sometimes taking the same values of coverage probability. The coverage probabilities oscillate about the nominal level with Wald being less conservative than CLL and Logit. This was then compared to coverage probabilities of the three methods for the same data combination without applying bootstrapping technique. Similarly CLL outperformed the Wald interval method and at times the Logit Interval method as seen in Table 2. It is clearly seen that without bootstrapping technique, CIs generated have lower coverage probabilities than CIs generated after applying bootstrapping.

In Table 3 and Figure 1, the number of bootstrap simulations was varied. It was seen that as the number of

bootstrap simulations increased, all the three methods became more precise, that is, their coverage probabilities hovered slightly above and below the nominal coverage probability.

Group testing works best when the prevalence rate or the proportion is sufficiently small. According to Table 4 and Figure 2, we see that as the proportion increases to 0.1 the coverage probability becomes very low for all the three interval methods. Between 0.03 and 0.08, the coverage probabilities for the three methods are conservative.

From Table 5 and Table 6, it is seen that the interval widths for both CLL and Logit are not differing a great deal for the same bootstrap simulations. CLL has shorter widths than logit. However, interval widths for the Wald for the three bootstrap simulations are considerably wider than those of the other two interval methods. When interval widths are compared without bootstrap simulations to those interval widths with bootstrap simulations we note that they are wider.

According to Table 4 and Figure 2, all the three methods perform well as the number of bootstrap simulations increases, with the coverage probabilities of the CLL and Logit being oftentimes greater than the nominal coverage probability.

Bootstrap methods produce good confidence intervals by an order of magnitude upon the accuracy of standard intervals [19] . In this paper we implement bootstrapping to the three interval methods: Wald, CLL and Logit. The comparison results in this study show that bootstrapping produces more precise confidence intervals than standard intervals for all the three procedures. Likewise the coverage probabilities from the three methods are consistent with the CLL being most conservative. Wald is more liberal. As the number of bootstrap simulations increases the more conservative the coverage becomes.

Olivia WanjeriMwangi,AliIslam,OrawoLuke, (2015) Bootstrap Confidence Intervals for Bootstrap Confidence Intervals for Adjusted for Overdispersion. Open Journal of Statistics,05,502-510. doi: 10.4236/ojs.2015.56052