Statistical analysis
comorbidity, illness
REGRESSION
Cox regression was used for the primary outcome, and for total symptom duration, adjusting for age, baseline symptom severity, prior duration of illness, and comorbidity. Linear regression was used for symptom severity, and logistic regression for re-consultation, progression of illness, and side effects, adjusting for the same baseline covariates as in the primary analysis. To aid interpretation, risk ratios were also calculated for the binary outcomes using a log-binomial model. Analysis was by intention to treat (antibiotic group as randomised in the trial sample, and by initial antibiotic treatment in the observational sample) regardless of non-adherence or protocol deviations. Multiple imputation was used as the primary analysis, agreed with the funder. Multiple imputation included all variables from the analysis model and any predictors of missingness (further details are provided in Supplementary Table S1).The plan had been to control for confounding by indication in the observational dataset by using inverse probability of treatment weighting (IPTW) using propensity scores in each of the regression models. However, the IPTW approach did not achieve good balance on the key covariates, whereas a post hoc analysis using stratification by propensity score did improve balance, and therefore was used in analysing both the observational data and the combined dataset that included both observational and trial data. Participants were stratified based on their propensity to receive antibiotics, with the aim of balancing the covariates between those who receive antibiotics and those who did not within each stratum. The stratum-specific treatment effects were then combined to obtain an overall treatment effect. The propensity scores were calculated separately for the observational and trial data, and regression analyses were then carried out on the pooled sample.