A joint penalized spline smoothing model for the number of positive and negative COVID-19 tests

One of the key tools to understand and reduce the spread of the SARS-CoV-2 virus is testing. The total number of tests, the number of positive tests, the number of negative tests, and the positivity rate are interconnected indicators and vary with time. To better understand the relationship between these indicators, against the background of an evolving pandemic, the association between the number of positive tests and the number of negative tests is studied using a joint modeling approach. All countries in the European Union, Switzerland, the United Kingdom, and Norway are included in the analysis. We propose a joint penalized spline model in which the penalized spline is reparameterized as a linear mixed model. The model allows for flexible trajectories by smoothing the country-specific deviations from the overall penalized spline and accounts for heteroscedasticity by allowing the autocorrelation parameters and residual variances to vary among countries. The association between the number of positive tests and the number of negative tests is derived from the joint distribution for the random intercepts and slopes. The correlation between the random intercepts and the correlation between the random slopes were both positive. This suggests that, when countries increase their testing capacity, both the number of positive tests and negative tests will increase. A significant correlation was found between the random intercepts, but the correlation between the random slopes was not significant due to a wide credible interval.