Estimating multiplicity of infection, allele frequencies, and prevalences accounting for incomplete data.

BACKGROUND: Molecular surveillance of infectious diseases allows the monitoring of pathogens beyond the granularity of traditional epidemiological approaches and is well-established for some of the most relevant infectious diseases such as malaria. The presence of genetically distinct pathogenic variants within an infection, referred to as multiplicity of infection (MOI) or complexity of infection (COI) is common in malaria and similar infectious diseases. It is an important metric that scales with transmission intensities, potentially affects the clinical pathogenesis, and a confounding factor when monitoring the frequency and prevalence of pathogenic variants. Several statistical methods exist to estimate MOI and the frequency distribution of pathogen variants. However, a common problem is the quality of the underlying molecular data. If molecular assays fail not randomly, it is likely to underestimate MOI and the prevalence of pathogen variants. METHODS AND FINDINGS: A statistical model is introduced, which explicitly addresses data quality, by assuming a probability by which a pathogen variant remains undetected in a molecular assay. This is different from the assumption of missing at random, for which a molecular assay either performs perfectly or fails completely. The method is applicable to a single molecular marker and allows to estimate allele-frequency spectra, the distribution of MOI, and the probability of variants to remain undetected (incomplete information). Based on the statistical model, expressions for the prevalence of pathogen variants are derived and differences between frequency and prevalence are discussed. The usual desirable asymptotic properties of the maximum-likelihood estimator (MLE) are established by rewriting the model into an exponential family. The MLE has promising finite sample properties in terms of bias and variance. The covariance matrix of the estimator is close to the Cramér-Rao lower bound (inverse Fisher information). Importantly, the estimator's variance is larger than that of a similar method which disregards incomplete information, but its bias is smaller. CONCLUSIONS: Although the model introduced here has convenient properties, in terms of the mean squared error it does not outperform a simple standard method that neglects missing information. Thus, the new method is recommendable only for data sets in which the molecular assays produced poor-quality results. This will be particularly true if the model is extended to accommodate information from multiple molecular markers at the same time, and incomplete information at one or more markers leads to a strong depletion of sample size.

Bias-corrected maximum-likelihood estimation of multiplicity of infection and lineage frequencies.

BACKGROUND: The UN's Sustainable Development Goals are devoted to eradicate a range of infectious diseases to achieve global well-being. These efforts require monitoring disease transmission at a level that differentiates between pathogen variants at the genetic/molecular level. In fact, the advantages of genetic (molecular) measures like multiplicity of infection (MOI) over traditional metrics, e.g., R0, are being increasingly recognized. MOI refers to the presence of multiple pathogen variants within an infection due to multiple infective contacts. Maximum-likelihood (ML) methods have been proposed to derive MOI and pathogen-lineage frequencies from molecular data. However, these methods are biased. METHODS AND FINDINGS: Based on a single molecular marker, we derive a bias-corrected ML estimator for MOI and pathogen-lineage frequencies. We further improve these estimators by heuristical adjustments that compensate shortcomings in the derivation of the bias correction, which implicitly assumes that data lies in the interior of the observational space. The finite sample properties of the different variants of the bias-corrected estimators are investigated by a systematic simulation study. In particular, we investigate the performance of the estimator in terms of bias, variance, and robustness against model violations. The corrections successfully remove bias except for extreme parameters that likely yield uninformative data, which cannot sustain accurate parameter estimation. Heuristic adjustments further improve the bias correction, particularly for small sample sizes. The bias corrections also reduce the estimators' variances, which coincide with the Cramér-Rao lower bound. The estimators are reasonably robust against model violations. CONCLUSIONS: Applying bias corrections can substantially improve the quality of MOI estimates, particularly in areas of low as well as areas of high transmission-in both cases estimates tend to be biased. The bias-corrected estimators are (almost) unbiased and their variance coincides with the Cramér-Rao lower bound, suggesting that no further improvements are possible unless additional information is provided. Additional information can be obtained by combining data from several molecular markers, or by including information that allows stratifying the data into heterogeneous groups.