Descriptive summaries
The data for this study was collected from 409 patients who received treatments for HF at least once at Jimma University Medical Center in Jimma, Ethiopia, between first January 2016, and first January 2019. The minimum and maximum event time observed from HF patient's follow-up were 6 and 36 months respectively. Among those HF patients, about 59.90% were censored (right censored), and the remaining 40.10% have died. Fifty percent of HF patients who received treatment survived 31 months or above (Table
4).
Almost half, 52.81%, of the HF patients were female and the remaining were male during the follow-up study. However, the survival time of male patients seems lower. The majority of HF patients, approximately 64.79%, live in rural areas, with the remainder living in urban areas. The survival time of HF patients seems less as they get older. About 20.05%, 22.25%, 23.72%, 25.43%, and 8.55% of HF patients were ischemic heart disease, rheumatic valvular heart diseases, cardiomyopathy heart disease, hypertensive heart disease, and other diseases respectively.
By observing the smoking status of HF patients, most HF patients were 74.82% non-smokers and the death proportion seems highest for those HF patients who were smokers, which was 54.88% compared to non-smokers which were 45.12%. About 64.55% of HF patients were not alcohol users and 35.45% were alcohol users.
Moreover, about 19.08% of HF patients are treated in the hospital with a combination of two or more treatments, and 19.32% of HF patients take digoxin. In addition, the remaining 24.2%, 25.18%, and 11.49% of HF patients were treated with spironolactone, atorvastatin, and other treatments respectively. About 58.19%, 13.69%, and 28.12% of HF patients were non-diabetic, type I diabetes mellitus, and type II diabetes mellitus respectively.
By observing the chronic kidney disease of HF patients, about 30.32% and 69.68% were HF patients with chronic kidney disease and without chronic kidney disease respectively, in which HF patients with chronic kidney disease seem to have lower survival time. Most HF patients have no hypertension, 60.64%, and the remaining have hypertension.
Looking at the stage at which the HF patients go to the hospital for treatment, about 36.92%, 28.61%, 19.07%, and 15.4% were in stage IV, in stage III, in stage II, and in stage I respectively. Most of, about 54.87% death, HF patients go for treatment in the hospital at a later stage and their survival time seems low at this stage (Table
5).
Bayesian survival analysis
As it can be shown in Table
1, the assumption of the Cox-PH model was not valid for the HF data set; in this case, parametric AFT models were used for the HF data set. For the HF data set, the time
\(t_{i}\) where i = 1, 2,….,409 of HF patients. Given that
\(\beta = (\beta_{0} , \beta_{1} , \ldots .,\beta_{p} )^{^{\prime}}\) is the vector of coefficients of the covariates considered for analysis
\(\beta_{0}\) is the intercept and p the number of covariates (p = 13), we assume that all these coefficients have a normal prior with mean 0 and variance of 1000. We assume that a gamma prior was applied to the Weibull, Log-normal, and Log-logistic distributions with shape parameter 1 and inverse scale parameter 0.001 [
6]. Table
2 below, shows the analysis of the HF data set for model comparison using the INLA method. To compare the efficiency of these various models, DIC and WAIC were used, and the one with the smallest value and the best fit was chosen. Accordingly, the Bayesian lognormal AFT model (DIC = 1297.84; WAIC = 1297.47) was found to be the best for survival time of HF patients data-set from the given alternative because the bold values are smallest.
Table 2
The comparisons of Bayesian AFT model using INLA methods
Exponential | 12.24 | 1400.62 | 1522.88 |
Log-Normal | 17.06 | 1297.84 | 1297.47 |
Weibull | 11.63 | 1389.20 | 1383.43 |
Log-logistic | 7.59 | 1326.88 | 1326.39 |
Table
3 shows the final results for the Bayesian log-normal AFT model, and as this result shows, the survival time of HF patients is statistically significantly affected by age, chronic kidney disease, diabetes mellitus, etiology of HF, hypertension, anemia, smoking, and stages of HF.
Table 3
Indicating the results for Bayesian log-normal AFT model using INLA method
| Intercept | 4.953 | 0.221 | 4.945 | [4.541, 5.409]* | 4.929 | 0 |
Age | ≤ 49–65 | Ref | | | | | |
49–65 | −0.258 | 0.115 | −0.256 | [−0.488, −0.036]* | −0.253 | 0 |
≥ 65 | −0.336 | 0.110 | −0.335 | [−0.557, −0.125]* | −0.331 | 0 |
Hypertension | No | Ref | | | | | |
Yes | −0.301 | 0.076 | −0.300 | [−0.452, −0.153]* | −0.299 | 0 |
CKD | No | Ref | | | | | |
Yes | −0.389 | 0.075 | −0.388 | [−0.537, −0.244]* | −0.387 | 0 |
Etiology of Heart failure | IHD | Ref | | | | | |
RVHD | −0.302 | 0.116 | −0.302 | [−0.533, −0.076]* | −0.300 | 0 |
Cardiomyopathy | −0.158 | 0.113 | −0.158 | [−0.382, 0.063] | −0.157 | 0 |
HDD | −0.258 | 0.115 | −0.257 | [−0.486, −0.035]* | −0.255 | 0 |
Others | −0.381 | 0.160 | −0.381 | [−0.693, −0.066]* | −0.382 | 0 |
Smoking cigarette | No | Ref | | | | | |
Yes | −0.156 | 0.073 | −0.155 | [−0.300, −0.014]* | −0.154 | 0 |
Stages of Heart failure | I | Ref | | | | | |
II | −0.400 | 0.190 | −0.397 | [−0.782, −0.038]* | −0.389 | 0 |
III | −0.423 | 0.176 | −0.419 | [−0.781, −0.090]* | −0.410 | 0 |
IV | −0.506 | 0.173 | −0.501 | [−0.857, −0.180]* | −0.492 | 0 |
Diabetes mellitus | Not | Ref | | | | | |
Type I | −0.232 | 0.10 | 0.231 | [−0.431, −0.036]* | −0.230 | 0 |
Type II | −0.422 | 0.086 | −0.421 | [−0.593, −0.255]* | −0.419 | 0 |
Anemia | No | Ref | | | | | |
Yes | −0.154 | 0.072 | −0.153 | [−0.298, −0.013]* | −0.152 | 0 |
Tau parameter | For log-normal | 4.30 | 0.497 | 4.28 | [3.38, 5.33]* | 4.24 | – |
From Table
3, the final model was interpreted using acceleration factor, 95% credible interval of Bayesian accelerated failure time estimated values. The estimated acceleration factor is defined as
\(\gamma = \left[ {exp\left( {\hat{\beta }} \right)} \right] = \left[ {exp\left( {posterior \, mean} \right)} \right].\)
Under the Bayesian log-normal AFT model, keeping the effect of other factors constant, the estimated acceleration factor for the age group of HF patients were 49 to 65 and greater than or equal to 65 years old is estimated to be 0.7726 with [95% CrI 0.6138, 0.9646] and 0.7146 with [95% CrI 0.5729, 0.9875] respectively. Thus, the expected survival time of HF patients decreased by 22.74% and 28.54% for HF patients aged group 49 to 65 and 65 or above 65 years older respectively as compared to HF patients aged group 49 or below 49 years (Reference).
The 95% CI for acceleration factor for both age groups did not include one, implying that both age groups have a significant effect on HF patients' survival time. Looking for chronic kidney disease and controlling for other factors, the estimated acceleration factor of HF patients with chronic kidney disease is 0.6777 with [95% CrI 0.5844, 0.7835], implying that the expected survival time decreases by 32.23% compared to HF patients without chronic kidney disease. The 95% CrI for acceleration factor of HF patients with chronic kidney disease did not include one, implying that HF patients with chronic kidney disease have a significant effect on HF patients' survival time.
By observing hypertension, keeping the effect of other factors constant, the estimated acceleration factor for HF patients with hypertension is estimated to be 0.74 with [95% CrI 0.5844, 0.7834] in which the expected survival time is a 26% decrease as compared to HF patients without hypertension (Reference). The 95% credible interval for acceleration factor of HF patients with hypertension did not include one which implies that HF patients with hypertension have a significant (in the Bayesian sense) effect on the survival time of HF patients.
On the other hand, keeping the effect of other factors constant, the estimated acceleration factor for HF patients who were smoking cigarettes is estimated to be 0.8555 with [95% CrI 0.7408, 0.986]. The 95% credible interval for the acceleration factor of HF patients who were smoking cigarettes did not include one. Thus, HF patients who smoked cigarettes had a significant effect on patient survival time, and the expected survival time of HF patients who smoked cigarettes was 14.45% shorter than that of HF patients who did not smoke cigarettes.
Regarding the etiologies of HF, keeping the effect of other factors constant, the estimated acceleration factor for etiologies of HF were rheumatic valvular heart disease, hypertensive heart disease, and other heart diseases are estimated to be 0.7393 with [95% CrI 0.5868, 0.9268], 0.772 with [95% CrI 0.615, 0.965] and 0.683 with [95% CrI 0.5, 0.936] respectively. Thus, the expected survival time of HF patients decreased by 27.07% were rheumatic valvular heart disease, 22.8% were hypertensive heart disease and 31.7% were other heart diseases as compared to ischemic heart disease of HF patients. The 95% CrI for acceleration factor of HF patients for etiology of HF were rheumatic valvular heart disease, hypertensive heart disease, and other heart disease did not include one, implying that HF patient with rheumatic valvular heart disease, hypertensive heart disease, and other heart disease has a significant effect on HF patients' survival time, whereas cardiomyopathy does not affect.
Moreover, for diabetes mellitus, keeping the effect of other factors constant, the estimated acceleration factor for HF patients with type I diabetic and type II diabetic is estimated to be 0.793 with [95% CrI 0.649, 0.964] and 0.655 with [95% CrI 0.552, 0.774] respectively. Thus, the expected survival time of HF patients decreases by 20.7% for type I diabetics and 34.5% for type II diabetics as compared to HF patient’s non-diabetic (Reference). The 95% credible interval for acceleration factor of HF patients with both types of diabetes did not include one, implying that HF patients with both types of diabetes have a significant effect on HF patients' survival time.
Looking for anemia while controlling for other factors, the estimated acceleration factor of HF patients with anemia is 0.857 with [95% CrI 0.742, 0.987], implying that the expected survival time decreases by 14.3% compared to HF patients without anemia. The 95% CrI for acceleration factor in HF patients with anemia did not include one, implying that HF patients with anemia have a significant effect on HF patients' survival time.
Finally, observing stages of HF, keeping the effect of other factors constant, the estimated acceleration factor for stage II, III and IV of HF patients is estimated to be 0.67 with [95% CrI 0.457, 0.962], 0.655 with [95% CrI 0.457, 0.913] and 0.602 with [95% CrI 0.424, 0.835] respectively. Thus, the expected survival time of HF patients decreases by 33%, 34.5%, and 39.8% for stage II, III, and IV of HF patients respectively as compared to stage I. The 95% credible interval for acceleration factor of HF patients with stage II, III, and IV did not include one, indicating that stage II, III, and IV have a significant effect on heart failure patients' survival time.
From Table
3, the Kullback–Leibler divergence values for all significant parameters in the Bayesian log-normal AFT model were 0, and thus, small values indicate that the posterior distribution was well approximated by a normal distribution. The most efficient algorithm was a simplified Laplace approximation, which improved efficiency and resulted in faster computation speed.