Multiple Summary Knowledge‐Based Disease Prevalence Estimation for Group Testing Data

ABSTRACT

Group testing is widely recognized for its ability to reduce time and cost in screening for rare diseases. However, with limited samples, the parameter estimation from group‐testing data might be inefficient. In some scenarios, the external summary statistics are available. To make full use of them, we propose a multiple‐source knowledge transfer method. The resulting prevalence estimator is consistent and more efficient than the conventional estimator based on the target group‐testing data alone. We establish the asymptotic normality of the proposed estimator. In addition, we develop a data‐driven screening procedure to identify the transferable sources. We extend this framework to Dorfman's two‐stage group testing design and show that integrating the retesting data yields efficiency gains. The asymptotic normality of the corresponding estimator is also established. Extensive simulations and a real‐data application support the theoretical results and demonstrate the favorable finite‐sample performance of the proposed method.

Abbreviations

GT

group testing

HIV

human immunodeficiency virus

NT

naive transfer

ST

screening transfer

1. Introduction

Group testing (GT) with the samples being pooled and tested as a single unit is an efficient tool to reduce time and cost in screening rare diseases [1]. It has been widely adopted in HIV surveillance [2, 3, 4], veterinary medicine [5, 6], drug development [7, 8], food safety [9, 10], and plant pathology [11]. Besides these applications, GT data have also been used to estimate rare‐disease prevalence. Tu et al. [12] and Liu et al. [13] proved the variance reduction for the maximum likelihood estimator of disease prevalence. Zhang et al. [14] incorporated individual retesting after initial pooling and achieved substantial efficiency gains.

GT typically focuses on rare diseases. With limited budgets, the number of positive pools may be too small to yield a precise prevalence estimate. When auxiliary information is available, appropriately integrating it can improve the efficiency of disease prevalence estimation [15, 16]. However, the effective use of such information is nontrivial since the auxiliary data might not be homogeneous with the current data. Directly using them could introduce bias. This motivates us to develop a transfer‐learning approach that adaptively borrows strength while guarding against nontransferable sources. Recent studies on transfer learning have been conducted in linear regression models [17], generalized linear models [18, 19] and interval‐censored survival models [20]. To the best of our knowledge, there is no work on handling GT data.

Adapting a transfer learning strategy to group testing data faces three key challenges. First, the auxiliary studies often differ from the target study in group size and assay characteristics such as sensitivity and specificity, especially in the scenario of retesting. Second, the raw data from external laboratories are typically inaccessible due to privacy or proprietary restrictions [21, 22], with only summary statistics available for analysis. Finally, for the unseen source data, it is difficult to assess the quality of data and determine whether to include it in the transfer set.

To address the above challenges, we develop a multiple‐source knowledge transfer method to improve parameter estimation using the auxiliary summary statistics. The proposed approach imposes no constraints on design‐specific features such as sensitivity, specificity, and group size. It can be broadly applicable across heterogeneous source data. By minimizing a composite loss that integrates both target data and transformed auxiliary summaries, we obtain a transfer estimator and establish its theoretical properties. We also extend this framework to the retesting setting and provide an algorithm for identifying the transferable sources. The developed R package and all reproducible code for numerical experiments are available at https://github.com/amss‐stat/GT‐Transfer/tree/main.

2. Method

2.1. Target Data and Classical Estimation

To estimate the disease prevalence $p$ of a rare disease, $n$ subjects are randomly divided into $m$ groups of size $k$. Among these, $m_1$ groups test positive, and $m_0(=m-m_1)$ groups are negative. Without loss of generality, we assume that the sensitivity ${\pi }_1(>0.5)$ and specificity ${\pi }_0(>0.5)$ are known at the group level. Then the probability of a group being reported as positive is $g={\pi }_1-({\pi }_0+{\pi }_1-1){(1-p)}^k,$ and $m_1$ follows a binomial distribution $\text{Bino}(m,g)$. The log‐likelihood function is

$${\ell }_1(g)=\log \left(\left(\genfrac{}{}{0ex}{}{m}{m_1}\right)\right)+m_1\log (g)+m_0\log (1-g).$$

The maximum likelihood estimator (MLE) of $g$ is $\tilde{g}=m_1/m,$ and the corresponding MLE of $p$ is

$$\tilde{p}=1-{\left[\frac{{\pi }_1-\min \left\{{\pi }_1,\max \{1-{\pi }_0,\tilde{g}\}\right\}}{{\pi }_0+{\pi }_1-1}\right]}^{\frac{1}{k}},$$

with its asymptotic variance given by

$$\tilde{v}=\frac{\tilde{g}(1-\tilde{g}){(1-\tilde{p})}^{2-2k}}{{({\pi }_0+{\pi }_1-1)}^2{k}^{2}m}.$$

2.2. Multiple Knowledge Transfer Method

In addition to the target data, assume that $J$ auxiliary studies are available and provide summary statistics ${\{({\tilde{p}}_j,{\tilde{v}}_j)\}}_{j\in \{1,\dots ,J\}}$, where ${\tilde{p}}_j$ represents an estimate of $p$ and ${\tilde{v}}_j$ denotes the corresponding asymptotic variance for $j=1,\dots ,J$. Let $F_g$ be the Bernoulli distribution with success probability $g$. The Kullback–Leibler divergence from $F_g$ to $F_{{\tilde{g}}_j}$ is

$$D_{\text{KL}}(F_g\Vert F_{{\tilde{g}}_j})=g\log \left(\frac{g}{{\tilde{g}}_j}\right)+(1-g)\log \left(\frac{1-g}{1-{\tilde{g}}_j}\right),$$

where ${\tilde{g}}_j={\pi }_1-({\pi }_0+{\pi }_1-1){(1-{\tilde{p}}_j)}^k$ for $j=1,\dots ,J.$ To leverage the information from auxiliary summary statistics, we define the transfer‐regularized loss function as

$${LOSS}_1(g)=-\frac{1}{m}{\ell }_1(g)+\frac{\lambda }{s}\sum _{j\in 𝕁}{D}_{\text{KL}}(F_g\Vert F_{{\tilde{g}}_j}),$$

where $𝕁\subset \{1,\dots ,J\}$ denotes the set of transferable sources with cardinality $s=|𝕁|$, and $\lambda >0$ tunes the magnitude of borrowing from the sources. If all the sources are transferable, then $𝕁=\{1,\dots ,J\}$. A data‐driven procedure for identifying $𝕁$ is presented in Section 2.3.

Remark 1

In contrast to the common practice in the transfer‐learning literature, which typically imposes penalties directly on the parameter differences between the target and auxiliary studies [17, 18, 19], our approach employs a Kullback–Leibler divergence at the distributional level. Because the KL divergence compares the Bernoulli distributions implied by the group‐testing model, it naturally accommodates heterogeneous group sizes and different assay characteristics such as sensitivity and specificity. Accordingly, ${LOSS}_1(g)$ balances the target likelihood with a Kullback–Leibler penalty towards the transferable sources.

Let $ĝ=\arg {\min }_g{LOSS}_1(g)$. The transfer‐learning estimator of $p$ is

$$\widehat{p}=1-{\left[\frac{{\pi }_1-\min \{{\pi }_1,\max (1-{\pi }_0,ĝ)\}}{{\pi }_0+{\pi }_1-1}\right]}^{1/k}.$$

Denote the true prevalence values for the target and source studies by $p^{\ast },p_1^{\ast },\dots ,p_J^{\ast }$, and the corresponding asymptotic variances for $\tilde{p},{\tilde{p}}_1,\dots ,{\tilde{p}}_J$ by $v^{\ast },v_1^{\ast },\dots ,v_J^{\ast }$. To study the theoretical properties of $\widehat{p}$, we introduce the following source‐target transferability condition on the transferable set $𝕁$.

Condition 1

There exist constants $\eta >\frac{1}{2}$ and $C_{\eta }>0$ such that $|p_j^{\ast }-p^{\ast }|<C_{\eta }{m}^{-\eta }$ for $j\in 𝕁$. The variance satisfies $s{\tau }^{\ast }=o(m)$, where ${\tau }^{\ast }={\max }_{j\in 𝕁}\{v_j^{\ast }/v^{\ast }\}$. All the source statistics ${\tilde{p}}_j$ for $j\in \mathbb{J}$ are asymptotically normal.

Condition 1 formalizes a local asymptotic source—target similarity requirement. Specifically, the bound $|p_j^{\ast }-p^{\ast }|<C_{\eta }{m}^{-\eta },\eta >\frac{1}{2}$ implies that the source—target discrepancy is asymptotically negligible relative to the estimation error of the target‐only estimator, which is of order $m^{-1/2}$. Under this local similarity regime and together with the variance condition in Condition 1, sufficiently similar auxiliary sources can improve estimation efficiency. This requirement is analogous to the source—target similarity conditions commonly assumed in transfer‐learning analyses [17, 18, 19]. Practically, Condition 1 characterizes sources whose discrepancy from the target is small relative to the target‐only estimation error. When the target sample size $m$ is small, the target‐only estimator is less precise, so moderately different auxiliary sources may still be useful. As $m$ increases and the target‐only estimator becomes more precise, this tolerance shrinks, and only more closely matched sources remain beneficial for transfer. If the discrepancy is of order $m^{-1/2}$ or larger, the theoretical gain from transfer is no longer guaranteed, and negative transfer may arise. This also motivates the screening procedure introduced below. The performance of the proposed method under different source—target discrepancy regimes relative to Condition 1 is examined in the Supporting Information. Meanwhile, the variance of the source data should be small enough to improve the estimation. We require that the sum of the variances of the source statistics tends to 0 when $m\to \infty$. Let $\lambda =C\sqrt{s/\tau }$, where $C>0$ is a constant and $\tau ={\max }_{j\in 𝕁}\{\frac{{\tilde{v}}_j}{\tilde{v}}\}>0$. Then we have the following asymptotic results for $\widehat{p}$.

Property 1

If Condition 1 holds, then $\widehat{p}-p^{\ast }=O_p\left(m^{-\eta }+\sqrt{{\tau }^{\ast }/(sm)}\right)$.

Theorem 1

Assume that Condition 1 holds. As $m\to \infty$, $(\widehat{p}-p^{\ast })/{\sigma }_1\stackrel{d}{\to }N(0,1),$ where

$$\begin{array}{cc}\hfill {\sigma }_1^2& =\frac{{(g^{\ast })}^2{(1-g^{\ast })}^2{(1-p^{\ast })}^{2-2k}}{k^2{(1+\lambda )}^2{({\pi }_0+{\pi }_1-1)}^2}\hfill \\ \hfill & \times \left[\frac{{\lambda }^2{k}^2{({\pi }_0+{\pi }_1-1)}^2}{s^2}\sum _{j\in 𝕁}\frac{{(1-p_j^{\ast })}^{2k-2}{v}_j^{\ast }}{{(g_j^{\ast })}^2{(1-g_j^{\ast })}^2}+\frac{1}{m{g}^{\ast }(1-g^{\ast })}\right],\hfill \end{array}$$ (1)

$g^{\ast }={\pi }_1-({\pi }_0+{\pi }_1-1){(1-p^{\ast })}^k$ and $g_j^{\ast }={\pi }_1-({\pi }_0+{\pi }_1-1){(1-p_j^{\ast })}^k$ for $j\in 𝕁$.

The proofs of Property 1 and Theorem 1 are given in the Supporting Information. It is easy to see that $\widehat{p}\stackrel{p}{\to }{p}^{\ast }$, $ĝ\stackrel{p}{\to }{g}^{\ast }$, ${\tilde{p}}_j\stackrel{p}{\to }{p}_j^{\ast }$, ${\tilde{g}}_j\stackrel{p}{\to }{g}_j^{\ast }$ and ${\tilde{v}}_j\stackrel{p}{\to }{v}_j^{\ast }$. We can plug them into (1) to estimate ${\sigma }_1^2$.

Remark 2

Note that $\tilde{p}-p^{\ast }=O_p\left(m^{-\frac{1}{2}}\right)$. When $\eta >\frac{1}{2}$ and ${\tau }^{\ast }=o(s)$, $|\widehat{p}-p^{\ast }|/|\tilde{p}-p^{\ast }|=o_p(1).$ This shows that $\widehat{p}$ has a faster convergence rate than $\tilde{p}$. On the other hand, let $v^{\ast }=\frac{g^{\ast }(1-g^{\ast }){(1-p^{\ast })}^{2-2k}}{{({\pi }_0+{\pi }_1-1)}^2{k}^{2}m}$. Then we have

$$\begin{array}{ll}\hfill \frac{{\sigma }_1^2}{v^{\ast }}& \le \frac{{\lambda }^2{\tau }^{\ast }{(g^{\ast })}^2{(1-g^{\ast })}^2{(1-p^{\ast })}^{2-2k}}{{(1+\lambda )}^2{s}^2}\sum _{j\in 𝕁}\frac{{(1-p_j^{\ast })}^{2k-2}}{{(g_j^{\ast })}^2{(1-g_j^{\ast })}^2}+\frac{1}{{(1+\lambda )}^2}\\ \hfill & \stackrel{p}{\to }\frac{C^{\prime }{C}^2}{{\left(1+C\sqrt{\frac{s}{{\tau }^{\ast }}}\right)}^2}+\frac{1}{{\left(1+C\sqrt{\frac{s}{{\tau }^{\ast }}}\right)}^2}\\ \hfill & \le \left(C^{\prime }+\frac{1}{C^2}\right)\frac{{\tau }^{\ast }}{s},\end{array}$$

where $C^{\prime }$ is a constant. It follows that, if ${\tau }^{\ast }=o(s)$, then $\mathrm{var}(\widehat{p})=o(\mathrm{var}(\tilde{p}))$; that is, the transfer‐based estimator $\widehat{p}$ attains a smaller asymptotic variance than the direct estimator $\tilde{p}$.

2.3. Source Screening

Negative transfer is most likely to occur when some source studies are not sufficiently close to the target, especially when their prevalences differ substantially from the target or when their summary estimators are relatively noisy. In such cases, directly borrowing from all sources may introduce bias, thereby weakening or even overturning the gain from transfer. Property 1 and Theorem 1 further show that smaller values of ${\tau }^{\ast }/s$ lead to a more favorable asymptotic variance. Motivated by this observation, we propose a two‐stage adaptive procedure to select transferable sources. First, let $\mathrm{\Phi }(·)$ and $Z_q$ be the cumulative distribution function and the $q$th quantile of the standard normal distribution, respectively. Let $b_1,b_2\in (0,1)$ be two small real numbers and $\eta \in [0.5,0.5+{\log }_{m}J-{\log }_m\tau ]$. For $j=1,\dots ,J$, we calculate $t_{1j}=m^{-\eta }+\sqrt{\tilde{v}+{\tilde{v}}_j}{Z}_{\frac{b_2}{2}+\mathrm{\Phi }\left(\frac{-m^{-\eta }}{\sqrt{\tilde{v}+{\tilde{v}}_j}}\right)}$, $t_{2j}=\sqrt{\tilde{v}+{\tilde{v}}_j}{Z}_{1-\frac{b_1}{2}}$ and

$$\begin{array}{cc}\hfill t_j& =\underset{z\in [t_{1j},t_{2j}]}{\text{arg min}}\left\{\mathrm{\Phi }\left(\frac{z-m^{-\eta }}{\sqrt{\tilde{v}+{\tilde{v}}_j}}\right)-\mathrm{\Phi }\left(\frac{-m^{-\eta }}{\sqrt{\tilde{v}+{\tilde{v}}_j}}\right)\right.\left.-\mathrm{\Phi }\left(\frac{z}{\sqrt{\tilde{v}+{\tilde{v}}_j}}\right)\right\},\hfill \end{array}$$ (2)

${𝕁}_1=\left\{j\in \{1,\dots ,J\}||\tilde{p}-{\tilde{p}}_j|<t_j\right\}$ with cardinality $s_1=|{𝕁}_1|$. Second, for $j\in {𝕁}_1$, we calculate ${\tau }_j={\tilde{v}}_j/\tilde{v}$ and rank them increasingly to get ${\tau }_{(1)}\le \dots \le {\tau }_{(s_1)}$. Let $R(j)$ be the rank of ${\tau }_j$. When ${\min }_{a=1,\dots ,s_1}\frac{{\tau }_{(a)}}{a}<1$, the transferable index of the source summary data is $𝕁=\{R^{-1}(1),\dots ,R^{-1}(r)\}$ with $r={\arg \min }_{a=1,\dots ,s_1}\frac{{\tau }_{(a)}}{a}.$ Otherwise, $𝕁=\varnothing$.

The above procedure screens the source data based on two criteria: small estimator discrepancy and small variance. The first stage screens for source—target similarity, while the second stage selects a subset with favorable variance properties. In the first stage, (2) determines a threshold for the discrepancy between the target and source summary estimators by combining the local similarity scale $m^{-\eta }$ with the aggregate variability $\tilde{v}+{\tilde{v}}_j$. A source is retained only if its discrepancy from the target falls below this threshold. Thus, the first stage serves as a data‐driven implementation of the source—target similarity requirement. It balances the risk of discarding useful sources against that of retaining nontransferable ones. Even within ${𝕁}_1$, some sources may still be unsuitable for transfer because they are too noisy. The second stage, therefore, selects a subset that balances information borrowing against the inclusion of noisier sources.

For each $j\in {𝕁}_1$, we compute ${\tau }_j={\tilde{v}}_j/\tilde{v}$ and rank these values increasingly as ${\tau }_{(1)}\le \cdots \le {\tau }_{(s_1)}$. For any subset $A\subset {𝕁}_1$ with size $|A|=a$, let ${\tau }^{\ast }(A)=\underset{j\in A}{\max }{\tau }_j$. By Property 1 and Theorem 1, the asymptotic performance of the transfer estimator depends on ${\tau }^{\ast }(A)/a$, with smaller values being more favorable. Hence, for a fixed $a$, the optimal subset consists of the $a$ sources with the smallest ${\tau }_j$, and the criterion becomes ${\tau }_{(a)}/a$. Minimizing ${\tau }_{(a)}/a$ over $a=1,\dots ,s_1$ reflects a trade‐off: retaining more sources enables greater information borrowing, but may require including noisier or less compatible sources. The rule $\underset{a=1,\dots ,s_1}{\min }{\tau }_{(a)}/a<1$ therefore serves as a simple and interpretable criterion: it indicates the existence of a subset whose asymptotic variance is smaller than that of the target‐only estimator; otherwise, we set $𝕁=\varnothing$. Empirically, as shown in Table 2 and Table 3, the performance of naive transfer may deteriorate when unsuitable sources are included in the transfer set, while the proposed screening transfer remains substantially more stable and achieves better estimation accuracy.

TABLE 2.

$\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ and $\text{MSE}(\tilde{p})⁄\text{MSE}(\stackrel{\mathrm{˘}}{p})$ for scenarios incorporating small‐sample source data without using a screening procedure. The MSE values (multiplied by $10^6$) are shown in parentheses.

		${\pi }_0$	${\pi }_1$	$p^{\ast }=0.01$	$p^{\ast }=0.02$	$p^{\ast }=0.05$
$\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$	$k=5$	0.95	0.95	3.44 ($8.92/2.60$)	3.44 ($14.1/4.11$)	3.11 ($27.0/8.68$)
			0.99	3.29 ($8.93/2.71$)	3.12 ($12.5/3.99$)	3.16 ($28.0/8.86$)
		0.99	0.95	2.48 ($4.94/1.99$)	2.68 ($10.0/3.74$)	2.88 ($24.2/8.40$)
			0.99	2.51 ($4.86/1.93$)	2.54 ($8.43/3.32$)	2.89 ($22.4/7.75$)
	$k=10$	0.95	0.95	2.26 ($3.50/1.55$)	2.20 ($5.98/2.72$)	2.38 ($16.3/6.83$)
			0.99	2.18 ($3.30/1.51$)	2.19 ($5.69/2.60$)	2.35 ($14.2/6.04$)
		0.99	0.95	1.91 ($2.40/1.26$)	1.98 ($5.13/2.60$)	2.28 ($14.7/6.45$)
			0.99	1.85 ($2.44/1.32$)	1.90 ($4.67/2.45$)	2.20 ($12.7/5.79$)
$\text{MSE}(\tilde{p})/\text{MSE}(\breve{p})$	$k=5$	0.95	0.95	5.14 ($8.92/1.74$)	4.59 ($14.1/3.08$)	3.87 ($27.0/6.98$)
			0.99	5.41 ($8.93/1.65$)	3.86 ($12.5/3.23$)	3.96 ($28.0/7.07$)
		0.99	0.95	2.56 ($4.94/1.93$)	2.91 ($10.0/3.44$)	3.10 ($24.2/7.81$)
			0.99	2.82 ($4.86/1.72$)	2.81 ($8.43/3.00$)	3.14 ($22.4/7.14$)
	$k=10$	0.95	0.95	3.00 ($3.50/1.17$)	2.71 ($5.98/2.21$)	3.48 ($16.3/4.68$)
			0.99	2.65 ($3.30/1.25$)	2.57 ($5.69/2.21$)	3.08 ($14.2/4.60$)
		0.99	0.95	1.99 ($2.40/1.21$)	2.19 ($5.13/2.34$)	2.75 ($14.7/5.34$)
			0.99	1.93 ($2.44/1.26$)	2.12 ($4.67/2.21$)	2.64 ($12.7/4.82$)

TABLE 3.

$\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ under the heterogeneous disease prevalences. The MSE values (multiplied by $10^6$) are shown in parentheses.

		$D=0.1p^{\ast }$		$D=0.8p^{\ast }$
	$k$	NT	ST	NT	ST
$p^{\ast }=0.01$	5	24.68 ($9.30/0.377$)	20.83 ($9.30/0.446$)	1.68 ($9.30/5.55$)	3.31 ($9.30/2.81$)
	10	11.43 ($3.30/0.289$)	9.87 ($3.30/0.335$)	0.79 ($3.30/4.20$)	4.09 ($3.30/0.807$)
$p^{\ast }=0.02$	5	17.51 ($14.1/0.808$)	14.63 ($14.1/0.967$)	0.69 ($14.1/20.6$)	5.59 ($14.1/2.53$)
	10	8.64 ($5.91/0.684$)	8.09 ($5.91/0.731$)	0.35 ($5.91/16.9$)	10.25 ($5.91/0.577$)
$p^{\ast }=0.05$	5	8.01 ($26.8/3.35$)	7.13 ($26.8/3.77$)	0.22 ($26.8/121$)	10.15 ($26.8/2.64$)
	10	5.03 ($15.4/3.07$)	4.36 ($15.4/3.54$)	0.14 ($15.4/108$)	8.88 ($15.4/1.74$)

3. Multiple Knowledge Transfer With Retesting Data

Dorfman's two‐stage design is widely applied in nowadays routine laboratory testing, where specimens are pooled and tested at the first stage, and individuals from positive groups are retested [1]. Let $S_p$ and $S_e$ denote the specificity and sensitivity of the assay for individual testing with $S_p>0.5$ and $S_e>0.5$. For $i\in \{0,\dots ,k\}$, denote by ${\mathrm{\Theta }}_{1i}(g)$ the probability that a group is reported positive at the first stage and that exactly $i$ individuals test positive upon retesting at the second stage. To derive ${\mathrm{\Theta }}_{1i}(g)$, we consider two situations. The first corresponds to a pool that contains no infected individuals. The second corresponds to a group that contains $a+b$ infected individuals, where $a$ infected individuals test positive at the second stage and $b$ infected individuals test negative. Then

$$\begin{array}{ll}\hfill {\mathrm{\Theta }}_{1i}(g)& =\left(\genfrac{}{}{0ex}{}{k}{i}\right)\left\{\frac{{\pi }_1-g}{{\pi }_0+{\pi }_1-1}{(S_p)}^{k-i}{(1-S_p)}^i(1-{\pi }_0)\right.\\ \hfill & +\sum _{a+b=1}^k\left(\genfrac{}{}{0ex}{}{i}{a}\right)\left(\genfrac{}{}{0ex}{}{k-i}{b}\right){\left[1-{\left(\frac{{\pi }_1-g}{{\pi }_0+{\pi }_1-1}\right)}^{\frac{1}{k}}\right]}^{a+b}\\ \hfill & \times {\left(\frac{{\pi }_1-g}{{\pi }_0+{\pi }_1-1}\right)}^{1-\frac{a+b}{k}}{(S_e)}^a{(1-S_e)}^b{(S_p)}^{k-i-b}\\ \hfill & \left.{(1-S_p)}^{i-a}{\pi }_1\right\}.\end{array}$$

Let $m_{1i}$ denote the number of positive groups that contain exactly $i$ individuals retested positive, $i\in \{0,1,\dots ,k\}$. Then $m_1={\sum }_{i=0}^k{m}_{1i}$. The log‐likelihood based on $\{m-m_1,m_{10},\dots ,m_{1k}\}$ is

$$\begin{array}{ll}\hfill {\ell }_2(g)& =\log \left(\frac{m!}{(m-m_1)!{\prod }_{i=0}^k{m}_{1i}!}\right)+(m-m_1)\log (1-g)\\ \hfill & +\sum _{i=0}^k{m}_{1i}\log \left({\mathrm{\Theta }}_{1i}(g)\right).\end{array}$$

Let $G_g$ and $G_{{\tilde{g}}_j}$ denote the multinomial distributions with cell probabilities $\left(1-g,{\mathrm{\Theta }}_{10}(g),\dots ,{\mathrm{\Theta }}_{1k}(g)\right)$ and $\left(1-{\tilde{g}}_j,{\mathrm{\Theta }}_{10}({\tilde{g}}_j),\dots ,{\mathrm{\Theta }}_{1k}({\tilde{g}}_j)\right)$, respectively. Define the transfer‐regularized loss function as

$${LOSS}_2(g)=-\frac{1}{m}{\ell }_2(g)+\frac{\lambda }{s}\sum _{j\in 𝕁}{D}_{\text{KL}}(G_g\Vert G_{{\tilde{g}}_j}),$$

where

$$\begin{array}{ll}\hfill D_{\text{KL}}(G_g\Vert G_{{\tilde{g}}_j})& =(1-g)\log \left(\frac{1-g}{1-{\tilde{g}}_j}\right)+\sum _{i=0}^k{\mathrm{\Theta }}_{1i}(g)\log \left(\frac{{\mathrm{\Theta }}_{1i}(g)}{{\mathrm{\Theta }}_{1i}({\tilde{g}}_j)}\right).\end{array}$$

Let $ğ=\arg {\min }_g{LOSS}_2(g)$. The corresponding estimator of $p$ is

$$\stackrel{\mathrm{˘}}{p}=1-{\left[\frac{{\pi }_1-\min \{{\pi }_1,\max (1-{\pi }_0,\stackrel{\mathrm{˘}}{g})\}}{{\pi }_0+{\pi }_1-1}\right]}^{1⁄k}.$$

Property 2

If Condition 1 holds, then $\stackrel{\mathrm{˘}}{p}-p^{\ast }=O_p\left(m^{-\eta }+\sqrt{{\tau }^{\ast }⁄(sm)}\right).$

Theorem 2

Assume that Condition 1 holds. As $m\to \infty$, $(\stackrel{\mathrm{˘}}{p}-p^{\ast })⁄{\sigma }_2\stackrel{d}{\to }N(0,1),$ where

$$\begin{array}{ll}\hfill {\sigma }_2^2& =\frac{{(1-p^{\ast })}^{2-2k}\mathrm{\Psi }(g^{\ast })}{{\mathrm{\Xi }}^2(g^{\ast }){(1+\lambda )}^2{k}^2{({\pi }_0+{\pi }_1-1)}^2},\\ \hfill \mathrm{\Psi }(g^{\ast })& =\frac{{\lambda }^2}{s^2}\sum _{j\in 𝕁}{\left[\frac{1}{1-g_j^{\ast }}+\sum _{i=0}^k\frac{{\mathrm{\Theta }}_{1i}^{\prime }(g_j^{\ast }){\mathrm{\Theta }}_{1i}^{\prime }(g^{\ast })}{{\mathrm{\Theta }}_{1i}(g_j^{\ast })}\right]}^2\\ & \times k^2{({\pi }_0+{\pi }_1-1)}^2{(1-p_j^{\ast })}^{2k-2}{v}_j^{\ast }\\ & +\sum _{i=0}^k\frac{{\mathrm{\Theta }}_{1i}^{\prime 2}(g^{\ast })}{m{\mathrm{\Theta }}_{1i}(g^{\ast })}+\frac{1}{m(1-g^{\ast })},\\ \hfill \mathrm{\Xi }(g^{\ast })& =\frac{1}{1-g^{\ast }}+\sum _{i=0}^k\frac{{\mathrm{\Theta }}_{1i}^{\prime 2}(g^{\ast })}{{\mathrm{\Theta }}_{1i}(g^{\ast })},\text{and}{\mathrm{\Theta }}_{1i}^{\prime }(g)=\frac{\partial }{\partial g}{\mathrm{\Theta }}_{1i}(g).\end{array}$$

The estimator $\stackrel{\mathrm{˘}}{p}$ attains the same convergence rate as $\widehat{p}$, whereas its asymptotic variance satisfies ${\sigma }_2^2<{\sigma }_1^2$ when ${\pi }_1(1-S_e)(1-S_p)-S_p{S}_e(1-{\pi }_0)<0$. This inequality demonstrates that, for a fixed number of subjects and initial pooled tests, the additional retesting stage yields a nonnegligible reduction in variance and thus improves the overall inferential efficiency. This finding is consistent with the well‐established result that retesting enhances efficiency when only the target data are available [14]. The proofs of Property 2 and Theorem 2 are given in the Supporting Information.

4. Numerical Studies

Extensive simulation studies and a real‐data analysis are conducted to assess the performance of the proposed estimators $\widehat{p}$ and $\stackrel{\mathrm{˘}}{p}$, compared with $\tilde{p}$.

4.1. Homogeneous Disease Prevalence

We consider homogeneous disease prevalences for the target and source summary data. Two situations are carried out. One corresponds to the case where all auxiliary summary statistics are generated from studies with a large number of pooled groups, whereas the other allows some auxiliary summaries to originate from large studies and others from much smaller ones. For the first situation, 500 group tests ($m=500$) are used for the target dataset. We randomly generate 5 source datasets with sample sizes $m_j\in [2000,5000]$ and group sizes $k_j\in \{5,10\}$ for $j=1,\dots ,5$. All the sources share the same prevalence as the target dataset, meaning that $p_j^{\ast }=p^{\ast }$. The sensitivity (${\pi }_{1j}$) and specificity (${\pi }_{0j}$) for the $j$th source data are drawn independently from $[0.9,0.999]$. The sensitivity ($S_e$) and specificity ($S_p$) of individual testing in target data are $S_e=S_p=0.999$. For the tuning parameter $\lambda$, we use the theory‐guided form $\lambda =C\sqrt{s/\tau }$ with $C=1$ as a default choice, and apply this specification uniformly throughout the simulation studies and the real‐data analysis. We compare the estimation performance under various testing conditions with target prevalence $p^{\ast }=0.01,0.02,0.05$, group size $k=5,10$, group testing sensitivity ${\pi }_1=0.95,0.99$ and group testing specificity ${\pi }_0=0.95,0.99$. In the source screening procedure, we set $b_1=b_2=0.01$ and $\eta =1$. Each configuration is replicated 1000 times. We report the ratios of mean squared error (MSE) as $\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ and $\text{MSE}(\tilde{p})⁄\text{MSE}(\stackrel{\mathrm{˘}}{p})$. As to the second situation, besides generating the data as above, we also generate 3 additional source datasets with smaller sample sizes in $[50,100]$.

Table 1 reports the results for the above two situations when the screening method is used. Across all the scenarios, the proposed $\widehat{p}$ and $\stackrel{\mathrm{˘}}{p}$ yield smaller MSE than $\tilde{p}$, with the ratios being greater than 7. In some cases, they can be larger than 20. Moreover, using the retesting data improves the estimation efficiency with $\text{MSE}(\tilde{p})⁄\text{MSE}(\stackrel{\mathrm{˘}}{p})$ being larger than that of $\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ under the same settings. We also find that the screening method is able to identify and exclude all small‐sample datasets in our experiment. Notably, the MSE ratios do not change after adding additional summary data (results not shown here since the ratios are the same as those in Table 1). Conversely, the introduction of small samples reduces the ratio of MSE when the screening procedure is not used, as shown in Table 2. For example, for $p^{\ast }=0.01$, $k=5$, ${\pi }_0={\pi }_1=0.95$, the MSE of $\tilde{p}$ is 13.72 times that of $\widehat{p}$, and it increases to 24.03 after retesting the positive groups. If the small‐sample sources are included without screening, the ratios will decrease to 3.44 and 5.14, respectively.

TABLE 1.

$\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ and $\text{MSE}(\tilde{p})⁄\text{MSE}(\stackrel{\mathrm{˘}}{p})$ for homogeneous disease prevalences using the screening procedure. The MSE values (multiplied by $10^6$) are shown in parentheses.

		${\pi }_0$	${\pi }_1$	$p^{\ast }=0.01$	$p^{\ast }=0.02$	$p^{\ast }=0.05$
$\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$	$k=5$	0.95	0.95	13.72 ($8.92/0.650$)	13.79 ($14.1/1.02$)	13.69 ($27.0/1.97$)
			0.99	11.67 ($8.93/0.765$)	13.05 ($12.5/0.955$)	11.01 ($28.0/2.54$)
		0.99	0.95	14.07 ($4.94/0.351$)	12.14 ($10.0/0.825$)	14.96 ($24.2/1.62$)
			0.99	10.63 ($4.86/0.457$)	10.09 ($8.43/0.836$)	15.21 ($22.4/1.47$)
	$k=10$	0.95	0.95	10.30 ($3.50/0.339$)	11.30 ($5.98/0.529$)	12.04 ($16.3/1.35$)
			0.99	9.05 ($3.30/0.364$)	10.35 ($5.69/0.550$)	11.64 ($14.2/1.22$)
		0.99	0.95	7.69 ($2.40/0.312$)	8.35 ($5.13/0.614$)	9.72 ($14.7/1.51$)
			0.99	8.10 ($2.44/0.301$)	8.15 ($4.67/0.573$)	10.51 ($12.7/1.21$)
$\text{MSE}(\tilde{p})/\text{MSE}(\breve{p})$	$k=5$	0.95	0.95	24.03 ($8.92/0.371$)	20.51 ($14.1/0.689$)	21.32 ($27.0/1.27$)
			0.99	25.35 ($8.93/0.352$)	19.87 ($12.5/0.627$)	16.28 ($28.0/1.72$)
		0.99	0.95	15.51 ($4.94/0.319$)	13.35 ($10.0/0.751$)	17.67 ($24.2/1.37$)
			0.99	12.28 ($4.86/0.395$)	11.79 ($8.43/0.715$)	16.77 ($22.4/1.34$)
	$k=10$	0.95	0.95	14.36 ($3.50/0.244$)	14.12 ($5.98/0.423$)	16.11 ($16.3/1.01$)
			0.99	12.03 ($3.30/0.274$)	12.89 ($5.69/0.442$)	14.20 ($14.2/0.999$)
		0.99	0.95	8.41 ($2.40/0.286$)	9.22 ($5.13/0.557$)	11.81 ($14.7/1.24$)
			0.99	8.67 ($2.44/0.281$)	9.30 ($4.67/0.503$)	12.79 ($12.7/0.996$)

4.2. Heterogeneous Disease Prevalences

We first generate five homogeneous source datasets by setting ${\pi }_0={\pi }_1=0.95$ under the configuration in Section 4.1. We then additionally generate three heterogeneous source datasets with the disease prevalence $p_j^{\ast }=p^{\ast }+D$, $D\in \{0.1p^{\ast },0.8p^{\ast }\}$ while keeping the other settings the same as above. Table 3 shows the MSE ratios $\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ under two transfer strategies: naive transfer (NT) and screening transfer (ST), where NT means using all source summary statistics without excluding potentially noisy sources, and ST is conducted via the source screening procedure in Section 2.3. From Table 3, we find two phenomena. (1) Directly using all source summary statistics may induce negative transfer when the difference $D$ is large; (2) Using the proposed screening procedure to filter out noisy summary statistics can enhance MSE reductions. For instance, when $p^{\ast }=0.05$, $k=5$, $D=0.8p^{\ast }$, the difference between the source and target prevalences reaches 0.04. This substantial difference results in the MSE of the NT estimator increasing by approximately 4.5‐fold compared to $\tilde{p}$, substantially deteriorating estimation accuracy. In contrast, the ST estimator maintains excellent performance, with MSE being about 1/10 of that of $\tilde{p}$.

4.3. Real‐Data Analysis

We analyze the HIV antiretroviral therapy study of Tayong et al. [4] to illustrate the practical gains of the proposed estimators. The primary aim is to estimate the post‐treatment non‐suppression rate using group testing data with individual retests for initially positive groups and two auxiliary surveillance datasets. The study assayed 4132 three‐member groups for plasma HIV RNA. All 814 positive groups were subsequently retested individually. The design yielded a sensitivity of 0.9999 and an empirical specificity of 0.8943 at the group level. Meanwhile, a study of routine antiretroviral therapy monitoring reported an estimate of 0.04667 with variance $2.04\times 10^{-6}$ [4]. Another auxiliary study reported an estimate of 0.04651 with variance $2.06\times 10^{-4}$ [23]. These two summaries constitute the source statistics. The two auxiliary studies are relevant external sources because all three studies concern HIV antiretroviral therapy monitoring and target closely related viral‐load monitoring outcomes. Similar to the simulations, we set $C=1$ to calculate the tuning parameter $\lambda$. For each of 100 random subsamples comprising 5% of the target groups, we computed three estimators: (i) the classical maximum likelihood estimator based solely on the subsample; (ii) the transfer estimator $\widehat{p}$ of Section 2.2; and (iii) the retesting transfer estimator $\stackrel{\mathrm{˘}}{p}$ of Section 3. Table 4 reports the average point estimate and its Monte Carlo variance.

TABLE 4.

The estimated mean and variance of $\tilde{p}$, $\widehat{p}$ and $\stackrel{\mathrm{˘}}{p}$.

Estimators	$\tilde{p}$	$\widehat{p}$	$\stackrel{\mathrm{˘}}{p}$
Mean	0.03550	0.04263	0.04393
Variance	$1.23\times 10^{-4}$	$7.11\times 10^{-5}$	$2.99\times 10^{-5}$

After transfer learning, the average estimate increases from 0.03550 to 0.04263 and further rises to 0.04393 if retesting information is considered. The variances of $\tilde{p}$, $\widehat{p}$ and $\stackrel{\mathrm{˘}}{p}$ are $1.23\times 10^{-4}$, $7.11\times 10^{-5}$ and $2.99\times 10^{-5}$, respectively. The use of source data and retesting information leads to a substantial reduction in estimator variance. The estimate based only on the target group testing data is noticeably lower, which may reflect the limited subsample size and the resulting instability of target‐only estimation. Information from related HIV monitoring studies supplements the limited target data. Both $\widehat{p}$ and $\stackrel{\mathrm{˘}}{p}$ are also closer to the external reference value of 0.05 reported by [24]. Overall, these results indicate that incorporating auxiliary information and retesting contributes to more stable prevalence estimation in this application.

5. Conclusion

Group testing is highly efficient for prevalence estimation, yet estimators based only on a small‐sample target study may remain statistically noisy. We propose a principled framework that borrows external summary statistics to improve estimation precision while accommodating study‐level heterogeneity. This approach facilitates information borrowing without requiring restrictive design assumptions on group sizes, assay sensitivity, and specificity. The resulting multiple knowledge transfer estimator and its retesting extension possess desirable large‐sample properties, with explicit variance characterizations established. Both simulations and the HIV application confirm substantial MSE reductions, particularly when retesting data are leveraged.

In some screening applications, covariates are available and the target of inference naturally extends from an overall prevalence to covariate‐adjusted rates [25, 26, 27]. Future work will generalize our summary‐transfer framework to this setting by allowing the disease probability to depend on covariates and by treating each auxiliary study as providing transferable regression‐type summaries, rather than a single prevalence estimate. In the current implementation, we use the theory‐guided form of the tuning parameter $\lambda =C\sqrt{s/\tau }$ with $C=1$. More data‐adaptive choices of $C$, such as those based on a plug‐in estimate of the asymptotic variance, may be explored in future work. Such extensions would broaden the scope of the method and have the potential to further improve efficiency by exploiting richer auxiliary information.

Funding

This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 12325110, 12288201) and the CAS Project for Young Scientists in Basic Research (Grant No. YSBR‐034). Ding's research was supported by the Fundamental Research Funds for the Central Universities (B240201095).

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Table S1. $\text{MSE}(\tilde{p})/\text{MSE}(\widehat{p})$ under the heterogeneous source‐prevalence settings. The MSE values are shown in parentheses.

Data Availability Statement

The data used in this study are publicly available in the electronic Supporting Information of Tayong et al. [4], specifically Supporting Information 2 for the article (DOI: https://10.1038/s41598‐025‐92709‐y).