Statistical analysis176Categorical variables were presented as n (%). We used the Wilcoxon Mann Whitney test,  
177Student t-test, χ2 test, or Fisher’s exact test to compare differences between groupsof patients  
178where appropriate.We performed multiple correspondence analysis (MCA) to investigate the  
179associations between clinical data, biologicaldata, radiological data, and the treatment  
180received.In order to control for selection bias in comparing mortality between treatment  
181groups, we used a propensity score weighting approach. The propensity score was calculated  
182using a logistic regression with sex, age groups, NEWS-2score, comorbidities and in-hospital  
183treatment(s) (HCQ, AZ, Zinc and/or corticosteroids whenappropriate) as covariates. The  
184predicted probabilities from the propensity-score model were then used to calculate the  
185stabilised inverse-probability-weighting weights (28). The association between treatment  
186groups and mortality was then assessed using a weighted multivariable Cox models. Cox 1
87models were adjusted on the following variables: sex, age groups, NEWS-2score,  
188comorbidities and in-hospital treatment (HCQ, AZ, Zinc and/or corticosteroids where  
189appropriate). Adjusted hazard ratios with 95% confidence intervals were calculated from the  
190Cox regression coefficient estimates.Sensitivity analyses wereperformed by assessing  
191whether observed effects were reproducible and consistent across subgroups according to age  
192class, sex, comorbidities, disease severity, co-medications, and reasons for non-treatment. A
193two-sided α value of less than 0.05 was considered to be statistically significant. Analyses  
194were carried out using SAS 9.4 statistical software (SAS Institute, Cary, NC).

An improvement to the inverse probability of treatment weighting is the use of stabilized weights (SW). Stabilized weights have been proposed in modeling time-varying treatment status in reducing selection bias in observational studies [16, 17]. The purpose of using SW in these studies is to reduce the weights of either those treated subjects with low propensity scores or those untreated subjects with high propensity scores.

Discussion:
 
In this paper we demonstrate several advantages of SWs over IPTWs in analyzing data obtained from observational studies. First, using SWs can reduce the weights of either those treated subjects with low propensity scores or those untreated subjects with high propensity scores in the pseudo data sets. Our serum potassium monitoring example showed that IPTWs have larger standard deviation and wider range than SWs (Table 4). Thus results using SWs are robust even with few observations with extreme IPTWs. Second, unlike variance estimators, no additional steps are needed when SWs are used because the SW approach provides appropriate variance estimates and confidence intervals of treatment effect from conventional regression models for fitting the outcome variables. Third, computer programming is simple for one to use SWs to obtain the effect of treatment effects and confidence intervals as compared to the bootstrap approach. One only needs to calculate the weights differently. Fourth, in our simulation studies and example, outcome variables are dichotomous. Unlike those developed estimators, the SW approach is applicable to outcome variables (e.g., dichotomous, continuous and count data) that have a finite distribution. Our simulation results also show that SW is a reasonable alternative to the robust variance estimator and has the advantage of reducing influential weights.
 
The limitation of the SW approach is the uncertainty of the influence of continuous confounders when their association with the probability of being treated and outcome is very strong. As shown in simulation studies, the sample size in some of the pseudo data sets can be different from the original data set when the confounding effect is strong. However, it is uncommon as our simulation results showed that the mean sample size approximated the original sample size. It is recommended that one always examine the difference between sample sizes in the original cohort and the pseudo cohort. When there is evidence that the sample size of the pseudo data is different from that of the original dataset, one can use the robust variance estimator with IPTWs although this latter method can produce slightly larger standard errors.
 
We conclude that our theoretical, simulation results and the real data example demonstrate that the use of the stabilized weights in the pseudo data preserves the sample size close to the original data. In addition, we conclude that use of stabilized weights produces the appropriate estimation of the variance of the main effect and maintains an appropriate type I error rate. Stabilized weights may be a useful tool to balance confounders between groups in observational studies.