Socio-demographic and clinical variables with their censoring status

The summary statistics of predictor variables in the data were displayed in (Table 1). Of a total of 334 COVID-19 patients, 258(77.2%) got the event of the first recovery and the rest 76(22.8%) were censored. The majority of the patients (51.5%) were males, 47.9% of the patients were unknowing where the diseases hold them, and also 34.7% of the patients have related co-morbid illnesses. The mean and SD (standard deviation) of age at the start of the treatment were 45.22 and 12.15 respectively. The rest variables were described in the same way.

Table 1 Summary statistics for independent variables included in the study

Also as indicated in (Table 2) the median recovery time was 16 days.

Table 2 Median recovery timeNon-parametric analysis for survival dataKaplan- meier survival curves

In the underneath KM (Kaplan- Meier) plot, recovery time is being measured in person-days. The overall Kaplan–Meier survival curve starts at zero and then the curve drops down until the follow-up recovery time happened at 30 person-days. It indicated that as follow-up time increases the curve decreased rapidly i.e. most patients recovered from COVID-19 as follow-up time increased continuously up to 30 person-days (Fig. 1A).

Fig. 1

Kaplan- Meier survival estimate curves

The plot in (Fig. 1B) suggested that the length of recovery time for male patients was greater than for female patients. That means male patients had taken a long time to recover as compared to female patients. (Fig. 1C) indicates that those COVID-19-positive patients who had one or more COVID-19 symptoms are less likely to recover from COVID-19 than those COVID-19-positive patients who hadn’t symptoms. In contrast, those COVID-19-positive patients who hadn’t related co-morbidity are more likely to recover early from COVID-19 than those who had (Fig. 1D). The remaining covariates could be presented and interpreted in the same way.

Log-rank test for each categorical variable

Log-rank test was computed to check the significant difference among the categories of categorical variables. The null hypothesis said that there is no significant difference between the survival experiences of different groups of categorical variables. Table 3 displayed the log-rank test of each categorical variable and reveals that there was a significant difference in recovery rate among males and females, having related comorbidity and no related comorbidity, presence, and absence of respiratory disease, having symptoms and not having symptoms, and patients having respiratory symptoms and not having respiratory symptoms. However, there was no difference in recovery rate between the presence and absence of asthma.

Table 3 Log-rank tests for categorical variablesCox proportional hazards modelVariable selection and Cox PH assumption

The study used a purposeful variable selection method to determine the variables to be included in the survival model. By purposeful variable selection method first test the significances of each predictor variable at a 25% level of significance, then by only the variables significant at this level; we could fit the multivariate Cox proportional hazard model. Consequently, the candidate variables for building a multivariable Cox model were the sex of the patient, age, co-morbidity, presence of respiratory disease, Severity of COVID-19, presence of symptoms, presence of asthma, and presence of respiratory symptoms were candidate variables for multivariable model building.

The proportional hazards assumption asserts that the hazard ratios are constant over time. That means the risk of failure must be the same no matter how long subjects have been followed. To test Cox proportional hazard assumption, a GLOBAL test was used.

From Table 4, the p-values of all covariates are greater than 5%, indicating that the correlation between Schoenfeld residuals and survival time is not significant; this implies that all the covariates satisfy the proportionality assumption at a 0.05 level of significance, and also the p-value of the GLOBAL test (0.326) is not significant. This indicates that the PH assumption for the Cox model is not violated.

Table 4 Cox proportional hazard assumption testMultivariable analysis for Cox Proportional Hazard (Cox-PH) model

Since the proportional hazard assumption was not violated, the data were analyzed based on Cox proportional hazard model. All of the parameter estimates were estimated by taking the other predictor variables into account using a 95% confidence interval for the hazard ratios of the statistically significant risk factors of COVID-19 which do not include 1 (the null value). In contrast, the 95% confidence intervals for the non-significant risk factors include the null value. Table 5 displayed the result of the multivariable analysis of the Cox proportional hazard model; sex of the patient, presence of co-morbidity, presence of respiratory disease, presence of asthma, presence of symptoms, and presence of respiratory symptoms were significantly associated with time to recovery from COVID-19 at a 5% level of significance.

Table 5 The multivariate Cox proportional hazards model analysis

The overall incidence rate was 13.79 per 100 (95% CI: 10.04, 18.95) person-days observations.

According to the result in (Table 5) the rate of achieving recovery for female patients was 4.9251 times higher than for male patients (AHR = 4.9201, 95% CI: 2.1660, 11.1718). That means the time needed to reach recovery for female patients was significantly shorter compared to male patients. The rate of achieving recovery for patients having severe COVID-19 cases was lower by 21.24% compared to patients having mild COVID-19 cases (AHR = 0.7876, 95% CI:0.7090, 0.8748). This means the time needed to reach recovery for patients having severe COVID-19 cases was longer.

Similarly, the rate of achieving recovery for patients who have COVID-19 symptoms was 71.86% times lower than for patients who don’t show COVID-19 symptoms (AHR = 0.2814, 95% CI: 0.1340, 0.5914). That means the time needed to reach recovery for patients who had COVID-19 symptoms take longer time to recover than patients who don’t show symptoms of COVID-19. Also, the rate of achieving recovery for patients who have other co-morbid cases in addition to COVID-19 was 83.73% lower as compared to patients with no co-morbid illness (AHR = 0.1627, 95% CI: 0.1396, 0.1897). That means the time needed to rich recovery for patients with the presence of co-morbid illness takes a longer time to recover as compared to patients with the absence of co-morbid illness.

The rate of achieving recovery for patients who have asthma was 49.45% lower as compared to patients who haven’t asthma (AHR = 0.5415, 95% CI: 0.3830, 0.7657). That means the time needed to attain recovery for patients who have asthma was longer. Also, the rate of achieving recovery from COVID-19 for patients who have the presence of respiratory disease was 76.99% times lower as compared to patients who haven’t a respiratory disease (AHR = 0.2301, 95% CI: 0.0792, 0.6607). That means the time needed to reach recovery for patients who have any respiratory disease was longer. The rate of achieving recovery for patients having oxygen saturation of above 90 was 3.237 times higher than for patients having less than or equal to 89 (AHR = 3.237, 95% CI: 2.161,4.848). Finally, for a unit increase in age, the rate of achieving recovery from COVID-19 for patients was decreased by 1.60% (AHR = 0.9840, 95% CI:0.971, 0.9973).

Model diagnostics

Once the model is fitted, the next step is to verify the entire necessary model assumptions are valid in the selected model. To check these model assumptions, we often make use of standard types of residual plots to validate the assumptions behind the Cox PH model.

From (Fig. 2), the diagnostic based on Cox-Snell residuals with the 95% point-wise CI for the Kaplan–Meier estimate of the Cox-Snell residuals along the red line. The survival function of the unit exponential distribution indicates that the survival function of the standard exponential distribution lies within the 95% CI of the Kaplan–Meier estimate. This indicates the survival process model fits the data well. Also (Fig. 3) showed that the Dfbeta residual plots were randomly distributed and a loess-smoothed curve does not exhibit more departure from the horizontal line or the origin. All the above two residual plots indicated that the selected survival model (Cox PH model) fits the data well.

Fig. 2

Cox-snell residual plots for time to recovery for COVID-19 patients

Fig. 3

Residual plots for significant predictors