Data source and study design

The secondary data was collected from the medical chart of visceral leishmaniasis with HIV co-infected patients from January 2015 to January 2021 at Metema Hospital using the ethical clearance approval letter obtained from the University of Gondar, College of Natural and Computational Science ethical approval committee (reference number: CNCS /10/15/4/2021) and a cross-sectional study design was conducted.

Setting

The research was conducted at Metema Hospital. Metema was a town in north western Ethiopia, on the border with Sudan. This town was located in the North West Gondar Administrative Zone, Amara region, \(897\ \text\) North of Addis Ababa and \(197\ \text\) from the ancient city of Gondar.

Study populationInclusion criteria

The studies included only visceral leishmaniasis with HIV co-infected patients until death/censor with visceral leishmaniasis with HIV co-infection or cure with visceral leishmaniasis.

Exclusion criteria

All patients with incomplete data, during treatment, were excluded.

Sample size

A total of 153 visceral leishmaniases with HIV co-infection patients fulfilling the inclusion criteria in Metema hospital was used starting from January 2015 to January 2021.

Dependent variables

The response or outcome variable was the survival time of Visceral leishmaniasis with HIV co-infection patients measured in days. The outcome variable was coded as 0 for censored and 1 for death.

Independent variables

The socio-demographic factors related to the response variables were Age of patients in year, Sex of patients, Place of residence, Occupation and A Clinical variables were Baseline Weight(kg), Haemoglobin level (g/dl), Spleen size (cm), Baseline CD4+ cell count (cells/µL), Initial VL treatment regimen, treatment of VL-HIV Co-infection, history of tuberculosis, treatment of VL-HIV Co-infection, Comorbidity(opportunistic infections), Baseline Body mass index (kg/m2), Nutritional Status, Temperature (0 C) and Bleeding.

Statistical analysisSurvival data analysis

Survival model fitting to make an inference by non-parametric Model, semi-parametric Cox proportional hazard survival models. All inferences were conducted at 5% significance level data entered by SPSS and analysed by using STATA version 14 and R4.2.1 statistical software packages. The outcome variable of time until an incident occurs was a concern of the statistical technique of survival analysis for data analysis [26].

Nonparametric models

We use non-parametric methods, such as the Life-Table, the Nelson-Aalen, or the popular Kaplan–Meier estimate, to estimate (?) as well as h(t). Kaplan-Meier survival analysis was used as the nonparametric approach to event history analysis [27], and a long-rank test to compare the survival difference between two or more groups [28].

Semi-parametric cox proportional hazard model

Cox proportional hazards (PH) model was one of the mathematical models designed for the analysis of time until an event or time between events. It shows the hazard at time t of an individual given the covariates. The hazard at the time was a product of baseline hazard function h0(t) which was only a function of time and exponential to the linear sum of βixi which is a function of time-independent covariates [29, 30].

The Cox Proportional Hazard model is given by;

$$\text\left(\text, \textrm,\right)=}_\left(\textrm\right)\text\left(_=1}^}}_}}_}\right)$$

(1)

Where \(\text\left(\text, \text,\right)\) was the hazard function at a time for a subject with covariate values X1, X2, X3… Xn and the estimated coefficients of the covariates of β1,β2,… βn. h0(t) was the baseline hazard function, which was the hazard function for an individual for which all the variables included in the model are zero, X= (X1, X2, X3……………. Xn) was the value of the vectors of the explanatory/predictor variables for a particular individual, \((1,2,\dots \text)\) is a vector of the estimated coefficients of explanatory/predictor variables.

The Cox PH model’s exponential portion ensures that the fitted model will always give a non-negative hazard and by definition, a hazard function is between zero and plus infinity i.e. \(0\le \text\left(\text,\text,\right)\le \), then the hazard ratio for the two groups is defined as:

$$\text\text=\frac(\text/\text=1)}(\text/\text=0)}=}^}$$

(2)

When HR = 1, it implies that the individuals in the two categories are at the same risk of getting the event, when HR > 1, it implies that the individuals in the first category (X = 1) are at a high risk of getting the event and if HR < 1, the individuals in the second category (X = 0) are at a high risk of getting the event.

Parameter estimation in cox-PH model

The Cox model likelihood function was called a “partial” likelihood function rather than a complete likelihood function. The fitted proportional hazard regression model was interpreted based on the hazard function (\(^}}.\))\(\widehat\) was the maximum partial likelihood estimator of β. The (1-α) 100% confidence interval for the estimated parameter is given as \(\widehat}}_}}\text\;\text\left(\widehat}\right)\).

Method of variable selection

Model building starts from a single covariate analysis as suggested by Collett [31], who recommended the approach of first doing a single covariate analysis to “screen” out potentially significant variables for consideration in the multi-covariate model to identify the importance of each predictor.