Some properties of maximum likelihood estimators of Weibull parameters with Type I censored data.

This thesis looks at extending previous work in the field of Type I censored reliability experiments. Due to its popularity and wide use, we use the Weibull distribution, and provide formulae on asymptotically valid variances and covariance of the maximum likelihood estimates and quantiles. We also examine the effect that sample size and censoring levels have on such properties. Theoretical results are validated with simulation studies throughout. These results are then used to obtain measures of precision in the Weibull parameter and quantile estimates given the assumption of asymptotic Normality. The suitability of using this large sample Normal theory in finite samples is consequently studied, and we provide an alternative measure of precision using relative likelihood methods. Confidence regions for each method are compared using published data. We investigate the concept of undertaking interim analysis of reliability data, where maximum likelihood estimates are calculated at successive times during an experiment, but the experiment is only stopped when adequate precision in the censored estimate is obtained. That is, when the censored estimate can provide a reliable guide to the complete estimate. Finally we summarise our results and conclusions, and some ideas for future research are discussed

Community prevalence of SARS-CoV-2 in England from April to November, 2020: results from the ONS Coronavirus Infection Survey

Background: Decisions about the continued need for control measures to contain the spread of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) rely on accurate and up-to-date information about the number of people testing positive for SARS-CoV-2 and risk factors for testing positive. Existing surveillance systems are generally not based on population samples and are not longitudinal in design. Methods: Samples were collected from individuals aged 2 years and older living in private households in England that were randomly selected from address lists and previous Office for National Statistics surveys in repeated crosssectional household surveys with additional serial sampling and longitudinal follow-up. Participants completed a questionnaire and did nose and throat self-swabs. The percentage of individuals testing positive for SARS-CoV-2 RNA was estimated over time by use of dynamic multilevel regression and poststratification, to account for potential residual non-representativeness. Potential changes in risk factors for testing positive over time were also assessed. The study is registered with the ISRCTN Registry, ISRCTN21086382. Findings: Between April 26 and Nov 1, 2020, results were available from 1 191 170 samples from 280327 individuals; 5231 samples were positive overall, from 3923 individuals. The percentage of people testing positive for SARS-CoV-2 changed substantially over time, with an initial decrease between April 26 and June 28, 2020, from 0·40% (95% credible interval 0·29–0·54) to 0·06% (0·04–0·07), followed by low levels during July and August, 2020, before substantial increases at the end of August, 2020, with percentages testing positive above 1% from the end of October, 2020. Having a patient facing role and working outside your home were important risk factors for testing positive for SARS-CoV-2 at the end of the first wave (April 26 to June 28, 2020), but not in the second wave (from the end of August to Nov 1, 2020). Age (young adults, particularly those aged 17–24 years) was an important initial driver of increased positivity rates in the second wave. For example, the estimated percentage of individuals testing positive was more than six times higher in those aged 17–24 years than in those aged 70 years or older at the end of September, 2020. A substantial proportion of infections were in individuals not reporting symptoms around their positive test (45–68%, dependent on calendar time. Interpretation: Important risk factors for testing positive for SARS-CoV-2 varied substantially between the part of the first wave that was captured by the study (April to June, 2020) and the first part of the second wave of increased positivity rates (end of August to Nov 1, 2020), and a substantial proportion of infections were in individuals not reporting symptoms, indicating that continued monitoring for SARS-CoV-2 in the community will be important for managing the COVID-19 pandemic moving forwards