Multiple imputations of missing data in meta-analysis of diagnostic accuracy

$$$$

Set-up and notation

Suppose there are $m$ studies to evaluate the diagnostic accuracy of some test. For the $i$th study, let

• ${\text{TP}}_i$: number of true positives
• ${\text{FN}}_i$: number of false negatives
• ${\text{FP}}_i$: number of false positives
• ${\text{TN}}_i$: number of true negatives

These are the data we wish to collect for each $i=1,\dots ,m$. Write $N_i={\text{TP}}_i+{\text{FN}}_i$, the number of cases; ${\overline{N}}_i={\text{FP}}_i+{\text{TN}}_i$, the number of non-cases; and $n_i=N_i+{\overline{N}}_i$, the total sample size.

Let $p_i$ denote the sensitivity, $q_i$ the specificity, and ${\pi }_i$ the prevalence of cases in the $i$th study. If there is no missing data, we can easily estimate them by ${\hat{p}}_i={\text{TP}}_i/N_i$, ${\hat{q}}_i={\text{TN}}_i/{\overline{N}}_i$, and ${\hat{\pi }}_i=N_i/n_i$, respectively. We can also feed the entire data $({\text{TP}}_i,{\text{FN}}_i,{\text{FP}}_i,{\text{TN}}_i)$ $(i=1,\dots ,m)$ into any standard software, e.g., mada (Sousa-Pinto 2022), for meta-analysis.

Methods

A motivating example

Consider the scenario where there are no non-cases in some study $i$, so ${\text{FN}}_i$, ${\text{TN}}_i=$NA. An example:

	TP	FN	FP	TN
Ahn	19	0	NA	NA
Bock	5	5	2	17
Chung	5	0	48	114
Cordoba	21	1	NA	NA
Espinosa	5	0	NA	NA
Haliloglu	3	0	1	73
Langer	57	17	NA	NA
Liberman	6	0	NA	NA
Nishanova	26	4	NA	NA
Obenauer	2	2	0	23
Oh	9	0	NA	NA
Qian	50	1	63	92
Reyes	32	1	NA	NA
Robbins	4	0	17	101
Taskin	47	0	NA	NA
Taylor	21	0	NA	NA
Yang	21	0	NA	NA
Myers	33	0	NA	NA
Taron	19	0	NA	NA
Son	6	0	6	23
Jafari	31	0	NA	NA
Wang	110	18	NA	NA

We aim to fill in the missing values fof ${\text{FN}}_i$ and ${\text{TN}}_i$ by multiple imputations (MI).

References

Sousa-Pinto, Philipp Doebler with contributions from Bernardo. 2022. “Mada: Meta-Analysis of Diagnostic Accuracy.” https://CRAN.R-project.org/package=mada.