Adjusting for time of infection or positive test when estimating the risk of a post-infection outcome in an epidemic

Abstract

When comparing the risk of a post-infection binary outcome, for example, hospitalisation, for two variants of an infectious pathogen, it is important to adjust for calendar time of infection. Typically, the infection time is unknown and positive test time used as a proxy for it. Positive test time may also be used when assessing how risk of the outcome changes over calendar time. We show that if time from infection to positive test is correlated with the outcome, the risk conditional on positive test time is a function of the trajectory of infection incidence. Hence, a risk ratio adjusted for positive test time can be quite different from the risk ratio adjusted for infection time. We propose a simple sensitivity analysis that indicates how risk ratios adjusted for positive test time and infection time may differ. This involves adjusting for a shifted positive test time, shifted to make the difference between it and infection time uncorrelated with the outcome. We illustrate this method by reanalysing published results on the relative risk of hospitalisation following infection with the Alpha versus pre-existing variants of SARS-CoV-2. Results indicate the relative risk adjusted for infection time may be lower than that adjusted for positive test time.

Keywords: COVID-19; epidemic phase bias; selection bias.

Figure 1.

Illustration of Example 1. Crosses show numbers of infections. Black and white circles represent cases with delays of 1 and 2 days, respectively. In left-hand graph, infection incidence is increasing. A total of 100 individuals are infected on day $t-2$ , of whom half (i.e. 50) test positive with a delay of 2 days on day $t$ . In addition, 150 individuals are infected on day $t-1$ , of whom half (i.e. 75) test positive with a delay of 1 day on day $t$ . So, $75/(50+75)=60$ % of the cases who test positive on day $t$ have a delay of 1 day. In right-hand graph, incidence is decreasing. A total of 150 individuals are infected on day $t-2$ , of whom 75 test positive with a delay of 2 days on day $t$ . In addition, 100 individuals are infected on day $t-1$ , of whom 50 test positive with a delay of 1 day on day $t$ . So, $50/(50+75)=40$ % of the cases who test positive on day $t$ have a delay of 1 day.

Figure 2.

Hospitalisation risk conditional on positive test time (solid black line) when risk conditional on infection time is 0.05 (green line). Incidence of infection is shown (dotted line). Time from infection to positive test is assumed to have a gamma distribution with mean 4 and variance 8 for the ultimately hospitalised individuals and a gamma distribution with mean 7 and variance 14 for the ultimately non-hospitalised individuals.

Figure 3.

Distributions of time from infection to positive test. Solid black line is distribution for ultimately non-hospitalised individuals. Dotted line is same distribution shifted by three days. Red line is distribution for ultimately hospitalised individuals.

Figure 4.

Summary of implementation of proposed sensitivity analysis.