The technique known as Negative Binomial sampling holds significant importance in the context of biological sample collection. If the proportion of individuals possessing a specific character trait is denoted as p, and sampling continues until a predetermined number, such as x individuals, is observed, the distribution of the sampled individuals follows a negative binomial distribution. Combining Negative Binomial sampling with group testing provides an attractive approach for delivering early estimates during the screening process.

This model operates under the assumption that the number of pools having a trait of interest is pre-established, and the testing process continues until the desired number of positive pools is identified.

The adaptive scheme involved testing groups in stages and adjusting the group sizes from one stage to the next. The group size used at a stage depended entirely on the outcome of the preceding stages. This implies that k’s were determined sequentially as the experiment progressed. The value of k₁ is determined by optimizing the variance of the estimator obtained from the non-adaptive Negative Binomial group testing scheme, which serves as prior information. In Stage One, the number of positive pools to be observed X₁ is fixed, and T₁ the number of groups to be tested before observing X₁ positive pools follows a Negative Binomial distribution.

In Stage Two, the group size k₂ is constructed by minimizing the variance of the estimator obtained in Stage One.

k 2 = arg min k [ var ( p ^ 1 ) ] p = p ^ 1 (1)

where k₂ was the value which minimizes the variance of p ^ 1 which is the estimator obtained in stage 1. The goal is to select the group size that minimizes the variance of the estimator from Stage One. This approach is crucial for enhancing the precision and accuracy of the overall estimation procedure. By minimizing the variance, the estimation process aims to achieve a more reliable and stable outcome, contributing to the effectiveness of the adaptive estimation model.

The number of positive pools to be observed in Stage Two X₂ is fixed, and the number of groups to be tested T₂ to achieve X₂ positive pools follows a Negative Binomial distribution, which depends on the output of Stage One. This adaptive model ensures an efficient allocation of resources and provides a robust strategy for large-scale screening and identification of positive groups in group testing scenarios.

The derivation presented focused on two models; the usual Negative Binomial model and the proposed two-stage adaptive Negative Binomial group testing model for estimating the prevalence of a rare trait.

Using the usual non-adaptive Negative Binomial to get p ^ N as in Katholi (2006). T follows a Negative Binomial with parameter x and π .

f ( t / p ) = ( t − 1 x − 1 ) [ π ( p ) ] x [ 1 − π ( p ) ] t − x (2)

The Likelihood function of Equation (2) is expressed as;

L ( p / t , x ) ∝ [ π ( p ) ] x [ 1 − π ( p ) ] t − x (3)

The log Likelihood function to base 10 is given as;

log L ( p / t , x ) ∝ x log [ π ( p ) ] + ( t − x ) log [ 1 − π ( p ) ] (4)

The maximum likelihood estimator of the non-adaptive model is obtained as the solution to

∂ log L ( . ) ∂ ( p ) = 0 (5)

Which is equivalent to;

x π ( p ) − t − x 1 − π ( p ) = 0 (6)

where π = 1 − ( 1 − p ) k which is the probability of obtaining a positive group. Equation (6) yields the results obtained by Katholi (2006) as;

p ^ N = 1 − ( 1 − x t ) 1 k (7)

Finding the variance of p ^ N the Cramer Rao Lower Bound was used where the Fisher’s information of the likelihood function was utilized;

Varianceof   p ^ N = 1 I ( p ^ N ) (8)

where the Fisher’s information is given as;

I ( p ^ N ) = − E [ ∂ 2 ∂ p 2 log L ( . ) ] − 1 (9)

The second derivative of the log likelihood function is given as;

= − t ( π ′ ( p ) ) 2 π ( p ) ( 1 − π ( p ) ) (10)

It is worth noting that π ( p ) = 1 − ( 1 − p ) k and π ′ ( p ) = k ( 1 − p ) k − 1 . Substituting in Equation (10) and taking expectation gives;

= t k 2 ( 1 − p ) 2 k − 2 1 − ( 1 − p ) k ( 1 − p ) k (11)

The E ( T ) = x π . Thus, taking the expectation and the inverse of Equation (11) will yield

Var ( p ^ 0 ) = 1 − ( 1 − p ) k t k 2 ( 1 − p ) k − 2 (12)

The proposed two-stage adaptive model aimed at optimizing resource allocation and estimation efficiency. The study considered two sets of desired positive groups, X₁ and X₂. The first stage estimator was based on the Negative Binomial distribution with a prior derived from the non-adaptive estimator. In the second stage, the study introduces the two-stage adaptive estimator.

Stage two proceeds by testing groups of size k₂ and T₂ is the number of the groups to be tested to obtain X₂ positive groups. Thus, T₂ is conditioned on T₁. This follows that T₂ has a negative binomial distribution. Specifically,

T 2 / T 1 ~ NegativeBinomial ( X 2 , π 2 / 1 = 1 − ( 1 − p ) k 2 ( t 1 ) )

Equation (6) gives the joint distribution of T₁ and T₂ as;

f ( T 2 , T 1 ) = f ( T 2 / T 1 ) × f ( T 1 ) = NegativeBinomial ( X 1 , π = 1 − ( 1 − p ) k 1 )     × NegativeBinomial ( X 2 , π 2 / 1 = 1 − ( 1 − p ) k 2 ( t 1 ) ) (13)

The joint distribution of T₁ and T₂ was used to derive the final two-stage adaptive estimator p ^ 2 and is given as;

f ( t 1 , t 2 ) = ( t 1 − 1 x 1 − 1 ) [ 1 − ( 1 − p ) k 1 ] x 1 ( 1 − p ) k 1 ( t 1 − x 1 )   × ( t 2 − 1 x 2 − 1 ) [ 1 − ( 1 − p ) k 2 ] x 2 ( 1 − p ) k 2 ( t 2 − x 2 ) (14)

( t 1 − 1 x 1 − 1 ) and ( t 2 − 1 x 2 − 1 ) are constants of proportionality thus we replace with ∝ giving;

f ( t 1 , t 2 ) ∝ [ 1 − ( 1 − p ) k 1 ] x 1 ( 1 − p ) k 1 ( t 1 − x 1 ) × [ 1 − ( 1 − p ) k 2 ] x 2 ( 1 − p ) k 2 ( t 2 − x 2 ) (15)

The log likelihood function to base 10 for the joint distribution of T₁ and T₂ was obtained as.

ln L ∝ x 1 ln [ 1 − ( 1 − p ) k 1 ] + k 1 ( t 1 − x 1 ) ln ( 1 − p )     + x 2 [ 1 − ( 1 − p ) k 2 ] + k 2 ( t 2 − x 2 ) ln ( 1 − p ) (16)

The adaptive estimator is obtained by solving Equation (17) iteratively since the solution equated to zero was not in a closed form thus was not tractable.

∂ log L ∂ p ∝ x 1 π 1 π ′ 1 − t 1 − x 1 1 − π 1 π ′ 1 + x 2 π 2 π ′ 2 − t 2 − x 2 1 − π 2 π ′ 2 (17)

where π 1 = 1 − ( 1 − p ) k 1 and π 2 = 1 − ( 1 − p ) k 2 . Also π ′ 1 = k 1 ( 1 − p ) k 1 − 1 and π ′ 2 = k 2 ( 1 − p ) k 2 − 1 .

The variance of the adaptive estimator was derived using Fisher’s information. Where the Fisher’s information is given as;

I = − E [ ∂ 2 ∂ p 2 log L ( . ) ] (18)

Thus, the variance of the adaptive estimator was derived and obtained as;

Var ( p ^ A ) = 1 ∑ i = 1 2 x i k i 2 ( 1 − p ) 2 k i − 2 π i 2 ( 1 − π i ) (19)

This variance was used to construct the Wald confidence interval as;

p ^ A ± Z α 2 Var ( p ^ A ) (20)

Data was simulated using an algorithm which mimics the Negative Binomial process as illustrated in Figure 1.

Steps for Simulation

Step 1: Specify p, k and X then set x = 0.

Step 2: Generate k Bernoulli random variables.

Set Y = (Y₁, …, Y_k).

Step 3: If the sum of y_is is greater than 0. A success is considered. If the sum of y_is is less than the group is considered negative.

Step 4: If the success was recorded, repeat the loop if x is not equal to X. If x = X, the procedure stops.

Step 5: Report T and calculate p estimate.

The Negative Binomial sampling method used in group testing experiments involves a random number of trials, while the required positive pools and group size are fixed. The number of tests needed to obtain the required positive groups depends on various variables, so it’s important to understand how changing the values of p and k affects the number of testing trials required.

As the probability of success increases in the negative binomial group testing model, the number of trials required to obtain the desired number of positive groups generally decreases as illustrated in Figure 2. Examining the plots, for a fixed value of k, it can be observed that as p increases, the number of trials decreases. This trend holds true across different values of k. This behavior is expected because a higher probability of success implies a greater likelihood of encountering positive groups during testing. Therefore, fewer trials are needed to reach the desired number of positive groups when the probability of success is higher. It is worth noting that increasing the group size from 5 to 100 in the negative binomial group testing model typically leads to a reduction in the number of trials required to achieve the desired number of positive groups as well.

The results of the maximum likelihood estimator and its properties including the variance, bias, and mean squared error for the adaptive group testing model are presented in Table 1. The results are organized based on different group sizes and true probabilities while the number of predetermined desired positive groups set at X = 30 as set by [15] .

A scrutiny of Table 1 shows that the estimated probabilities tend to increase as the probability increases, although the increase is generally small. It is important to note that the MLE values of the adaptive model exhibits monotonic behavior as the model dynamically adjusts the group size based on stage one’s outcomes which lead to more consistent and accurate estimations. The MLE generally increases as p increases for all values of k as found by [16] . The variance of the estimated probabilities remains relatively small across different values of p. The bias of the estimation shows negative values indicating a slight underestimation of the true population probability. However, the bias remains relatively small across all values of p. The MSE combines the variance and bias to provide an overall measure of estimation accuracy. The bias remains relatively small and consistent across different values of p indicating the effectiveness of the adaptive group testing model in reducing bias.

The results presented in this section examine the relationship between the adaptive maximum likelihood estimates and the true probability for different values of group size. The results are represented in four graphs for k = 5, 20, 50, 100. These findings highlighted that the adaptive nature of the model in adjusting the estimations was based on observed outcomes and the varying performance of the adaptive approach across different group sizes.

The relationship was further investigated by plotting the values p ^ against p while varying the waiting parameter X for different values of group size k.

Figure 3 illustrates the relationship between the Maximum Likelihood Estimation values and the true probability for different combinations of X in the adaptive approach. The MLE values generally increase as p increases, as well as with larger values of X and k. However, the rate and pattern of this increase vary depending on the specific combination of X and k. For X = 30, the MLE graph

shows a relatively steep curve, indicating that small changes in p lead to noticeable changes in the estimated probabilities of success. For X = 20, the MLE values also increase as p increases, but the curve is relatively flat, suggesting a less pronounced change in the MLE values as p varies. The adaptive MLE in this scenario exhibits a slower rate of increase compared to when X = 30, implying a less sensitive response to changes in p.

Similar patterns are observed for the other combinations of X and k. The adaptive approach tends to provide more conservative estimates with low MLE values. The MLE values increase with increasing p, but the specific patterns and sensitivities depend on the combination of X and k. The adaptive approach consistently provides more conservative estimates with lower MLE values, indicating a cautious approach in estimating the prevalence of the rare trait.

We examined the relationship between the variance of the estimated proportion p ^ and the true proportion p in the context of the two-stage adaptive negative binomial model. Figure 4 provides insights into this relationship for different combinations of X and k.

Figure 4 presents the interplay between X, k, and the variance of p ^ in the two-stage adaptive negative binomial model. They highlight the impact of the true proportion p, the desired number of positive groups X, and the group size k on the variability of the estimates. As the true proportion p increases, the variance of the estimated proportion p ^ also tends to increase, although the magnitude of increase varies across different X and k values. This suggests that higher probabilities of success lead to greater variability in the estimates, highlighting the increased uncertainty associated with higher p values. Comparing different X values, we find that as x increases, the variance of p ^ generally tends to increase. This implies that aiming for a higher number of positive groups introduces more variability into the estimates. Analyzing the effect of k, we notice

that for a fixed x value, as k increases, the variance of p ^ tends to decrease. This indicates that larger group sizes result in more precise estimates and lower variability. Larger group sizes provide more information, reducing the sampling error and enhancing the precision of the estimates. The insights obtained from this analysis can inform the selection of appropriate values for X and k to optimize the accuracy and reliability of the model in estimating the proportion of successes in a population.