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 Quantifying previous SARS-CoV-2 infection through mixture modelling of antibody levels - Nature Communications Methods .Data sources . The blood samples were collected in studies of Kenyan blood donors4 , 25, healthcare workers26, truck drivers/assistants27and pregnant women28. Most surveys were done shortly before or during the early stages of the second wave of the epidemic in Kenya (Supplementary Fig. 4). The protocols for these studies were approved by the Scientific and Ethics Review Unit (SERU) of the Kenya Medical Research Institute. The blood donors and health care workers provided written informed consent, and the truck drivers provided verbal consent. Surveillance of antenatal care attendees was conducted at the request of the Kenya Ministry of Health and consent was obtained from participating health facilities and the respective Counties. The surveillance involved analysis of anonymised residual blood volumes of samples collected in antenatal care clinics. Approval to publish the results of the antenatal care surveillance was explicitly requested from and granted by Kenyatta National Hospital, University of Nairobi Ethics Review Committee (Protocol P327/06/2020) and the Kilifi County health management rapid response team and SERU.Enzyme-linked immunosorbent assay (ELISA) . Across all serosurveys, the samples were tested for anti-SARS-CoV-2 IgG antibodies using an adaptation of the Krammer ELISA for whole length spike antigen29. Ratios of optical densities (OD) relative to a negative control were used to quantify the antibody concentrations. The assay was originally validated using 910 pre-COVID-19 serum samples collected in 2018, all of which were collected from adults and children from the Coast region of the country, and samples from 174 PCR-positive Kenyan adults, which were collected from patients admitted to Kenyatta National Hospital in Nairobi and their contacts (14 pre-symptomatic, 55 symptomatic, 92 asymptomatic and 13 unknown). For the samples obtained from PCR-positive individuals, the median time between the PCR test and blood sample collection was 21 days (IQR: 15, 34). The validation was based on a threshold OD ratio of 2, and yielded sensitivity and specificity estimates of 92.7% and 99.0% respectively. In a WHO-sponsored international standardisation study, the performance of the assay was found to be consistent with that of other assays30.Statistical analysis . Both the threshold-based analysis and the mixture model analysis were done using the Rstan package (version 2.21.2) in R version 4.0.431 , 32.Sensitivity and specificity adjusted threshold analysis . We incorporated information on the sensitivity and specificity of the threshold by simultaneously modelling the serosurvey data and validation data. Specifically, we modelled counts of (i) the number, $$y,$$of survey samples above the threshold OD ratio, (ii) the number, $$x$$, of PCR-positive samples above the threshold and (iii) the number, $$z,$$of pre-COVID-19 samples below the threshold. In the model, the observed?proportion of survey samples?above the threshold, $${p}_{{obs}}$$, is a function of the proportion previously infected, $$p$$, and the sensitivity and specificity of the threshold.Model: $$y\sim {{\mbox{Binomial}}}({p}_{{obs}},N)$$ $$x\sim {{\mbox{Binomial}}}({{{{{\rm{se}}}}}},{N}_{{{{{{\rm{se}}}}}}})$$ $$z\sim {{\mbox{Binomial}}}({{{{{\rm{sp}}}}}},{N}_{{{{{{\rm{sp}}}}}}})$$ $${p}_{{obs}}={{{{{\rm{se}}}}}}\times p+\left(1-{{{{{\rm{sp}}}}}}\right)\times \left(1-p\right).$$ Priors: $$p\sim \mbox{Uniform}(0,1)$$ $${{{{{\rm{se}}}}}}\sim {{\mbox{Uniform}}}({{{{\mathrm{0,1}}}}})$$ $${{{{{\rm{sp}}}}}}\sim {{\mbox{Uniform}}}({{{{\mathrm{0,1}}}}}).$$ Mixture model . We fitted a two-component mixture model where individuals are classified according to whether or not they have experienced SARS-CoV-2 infection. We assumed that log 2 OD ratios follow a skew normal among previously infected individuals (parameters: location = $$\xi$$, scale = $$\omega$$and skew = $$\alpha$$) and a normal distribution among previously uninfected individuals (parameters: mean = $$\theta$$, standard deviation = $$\nu$$).Model: $$\left(1-p\right)\times {{\mbox{Normal}}}(\theta ,\nu)+p\times {{\mbox{Skew}}}\, \mbox{Normal}(\xi ,\omega ,\alpha ),$$ where $$p$$= proportion previously infected.To make it easier to interpret the model parameters, we reparameterised the model in terms of the difference,?δ, between the means of the two distributions: $$\xi =\theta +\delta -\omega \sqrt{2/\pi }\frac{\alpha }{\sqrt{1+{\alpha }^{2}}}.$$ Since mixture models can be difficult to fit when there is overlap of the component distributions, we used several constraints to facilitate parameter estimation. First, we fixed the standard deviation in the uninfected, $$\nu ,$$to be equal to the standard deviation in the pre-COVID-19 samples. Second, we used an informative prior for $$\delta$$to constrain the difference in means—the prior puts 5% probability on the difference exceeding the difference between the mean in symptomatic PCR-positive cases (mean log 2 OD ratios = 3.43) and the mean in pre-COVID-19 samples (mean log 2 OD ratios = ?0.17). The prior for $$\delta$$was also designed to avoid label switching by ensuring $$\delta \, > \, 0.$$Finally, we used an informative prior for $$\alpha$$to rule out strong skew in either direction.Priors: $$\theta \sim {{\mbox{Normal}}}(0,10)$$ $$\delta \sim {{{\mbox{Normal}}}}^{+}(0,1.83)$$ $${\ln}\,\omega \sim {{\mbox{Normal}}}(0,10)$$ $$\alpha \sim {{\mbox{Normal}}}(0,1)$$ $$p\sim {{\mbox{Uniform}}}(0,1).$$ In addition to estimating the model parameters, we estimated the sensitivity and specificity at various threshold values. The sensitivity corresponds to the proportion above the threshold in the skew normal distribution and the specificity corresponds to the proportion below the threshold in the normal distribution. Both quantities were estimated using the sample of parameter values drawn from the posterior distribution.An alternative specification of the mixture model . As a sensitivity analysis, we fitted an alternative model where we assumed that the distribution among previously infected individuals follows a mixture distribution with mixing parameter $$q$$.Model: $$(1-p)\,\times \, {{\mbox{Normal}}}({\theta }_{1},{\nu }_{1})+p[q\times {{\mbox{Normal}}}({\theta }_{2},{\nu }_{2})+(1-q)\times {{\mbox{Normal}}}({\theta }_{3},{\nu }_{3})].$$ As with the skew-normal model, the model was reparameterised in terms of the difference in mean between the positive and negative component, that is we defined $${\theta }_{1}=q{\theta }_{2}+\left(1-q\right){\theta }_{3}-\delta$$, where $$\delta$$represents the difference between the means.The priors for $${\nu }_{i}$$ $$(i={{{{\mathrm{2,3}}}}})$$?were chosen to ensure that these standard deviations are of similar magnitude to the standard deviation observed in PCR-positive individuals (SD log 2 OD ratios = 1.32 = exp(0.28)) and we used the constraint $${\theta }_{3} \, > \, {\theta }_{2}$$?to avoid the problem of label switching and ensure the identifiability of these parameters.Priors: $$\delta \sim {{{\mbox{Normal}}}}^{+}(0,1.83)$$ $${\theta }_{i}\sim {{\mbox{Normal}}}(0,10)\,\,\,\,\, i=2,3 \,\,\,\,\,{\theta }_{3} \, > \, {\theta }_{2}$$ $${{{{{\rm{ln}}}}}}\,{\nu }_{i}\sim {{{{{\rm{Normal}}}}}}(0.28,0.2)\,\,\,\,\, i={{{{\mathrm{2,3}}}}}$$ $$p\sim {{\mbox{Uniform}}}(0,1)$$ $$q\sim {{\mbox{Uniform}}}(0,1).$$ Reporting summary . Further information on research design is available in the? Nature Research Reporting Summary linked to this article. . From：
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