{'fraction_non_alphanumeric': 0.02592852137351086, 'fraction_numerical': 0.01051156271899089, 'mean_word_length': 4.798026697620429, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Examining the Effect of Reduction of Predictors Affecting the Survival Time of HIV/ AIDS Patients using a Multiple Correlation/ Association Technique

The main objective of this paper is to reduce the number of predictors using multiple correlation/ association while estimating the survival of HIV/AIDS. MANOVA is used to test the association when we have two or more continuous predictors and two or more categorical predictors. Also, log-linear models are used to test the joint independence of two or more predictors. The survival times are estimated using AFTM. It is observed that the estimated survival times are not affected by the reduction of predictors from 11 to 4. Also, the estimates so obtained are more efficient as they have reduced standard error. Results of the proposed reduction method were also compared with the results of the two existing variable reduction methods, viz, LASSO and Net- Elastic method.


Introduction
Human Immunodeficiency Virus (HIV) is a virus that attacks and destroys the infection-fighting CD4 cells of the body's immune system. Due to continuous loss of CD4 cells, it becomes very difficult for the immune system of the body to fight infections. As a result, the immune system of the patient damages progressively. Due to the progressively damaged immune system, the infected person becomes immunosuppressed and is, therefore, vulnerable to other opportunistic infections, especially tuberculosis. 1 There is another advanced and symptomatic form of HIV, known as, Acquired Immunodeficiency Syndrome (AIDS). In the modern world, this epidemic is considered as one of the most destructive health crises of modern times. This epidemic is destroying families and communities around the world, causing huge socio-economic burdens. It is assumed that worldwide 36 million persons are infected from HIV and this disease has caused 1.2 million deaths globally. 2 In India only, 2.5 million people were estimated to have been suffering from HIV till 2014. 3 This virus can be transmitted through many ways which includes transmission of blood, semen, genital fluids, or breast milk of an infected person. Among all the modes of transmission, most common ways through which HIV is spread are unprotected sex or sharing drug injection equipment with an infected person. Many tremendous researches have been conducted in the field of HIV/AIDS, but there is currently no cure for this infection. There are, however, steps that can be taken to delay the onset of full blown AIDS and to reduce its progression. The most promising advance has been the advent of potent combination of therapy, the Anti-retroviral therapy (ART) in 1996. The ART can prolong the life of the infected patient by slowing down the wasting period as it boosts the CD4 count in the immune system. Markus Abigo Erango. (2017) predicted and compared the survival time of HIV/AIDS patients in three hospitals in Ethiopia. Three parametric accelerated failure time distributions: lognormal, log logistic and Weibull are used to analyze, predict and compare survival probabilities of the patients. The results indicated that the empirical hazard rates of the three data sets reveal maximal peaks. 8 Jaya Chakravarty et al. (2013) assessed the factors determining survival of patients on ART under routine programme conditions in an ART centre in north India five years after its inception. The study findings revealed poor survival in the first six months of therapy especially in those with severe immunosuppression. This emphasizes the need for early enrolment into the programme. The high LFU occurring early after initiation of therapy suggests the urgent need to build an efficient patient retrieval system in the programme. 9 Derek Ngbandor Nawumbeni et al. Colins Kingoum Nubed (2016) aimed at assessing the KAPs of senior secondary school students in Fako Division, Cameroon, on HIV/AIDS. This was a cross-sectional study carried out on 464 students aged 13-25 years, selected by systematic quota random sampling from some secondary schools in Fako, from April to June 2014, to evaluate their KAPs regarding HIV/AIDS. 12 Hanhui Ye et al., (2016) conducted an analysis to determine the correlation between AIDS restriction and metabolic pathway gene expression. The results showed that HIV-1 postentry cellular viral cofactors from AIDS restriction genes are coexpressed in human transcriptome microarray datasets. 13 Grover and Swain (2013) estimated the survival of HIV/ AIDS patients who were undergoing Antiretroviral Therapy in an ART centre, Delhi. Non Parametric Maximum Likelihood Estimation NPMLE (E-M) for interval censoring and KM survival plot for left, right and mid-point imputation had been used to estimate the survival of these patients. 14 Richard Hafner et. al (1999) studied the relationship between Mycobacterium avium complex (MAC) infection of blood and bone marrow in human immunodeficiency virus-infected patients before and during treatment. 15 Sergio Rueda (2016) conducted a systematic review and series of meta-analyses on the association between HIV related stigma and health among people living with HIV. They conducted a structured search on 6 electronic databases for journal articles reporting associations between HIV-related stigma and health related outcomes published between 1996 and 2013. 16 However, none of these studies aimed at finding out the association among three or more predictors at a single time. In this paper, we have proposed a multi correlation/ association screening procedure for reducing the number of predictors to be included in the model without affecting the estimated survival time and also resulting in lower Standard error. So, this research paper aims at: We have used SPSS and R software to fit the models and to compute the associations/correlations among predictors.

Methods used
Suppose that n number of HIV/AIDS patients are under ART, Let Ti be the survival time of i th patient with survival function S (t). Then assuming a linear relationship between log (T i ) and "p" predictors, we have Where, associated with a single predictor; we can replace those predictors with a single predictor with which they are associated. For example, in the model, (1), X 2 and X 3 may be significantly correlated/ associated with, X 1 , then we can replace X 2 and X 3 in the model with X 1 and it will fulfill the need of all the three variables. So, if we suppose out of these "p" explanatory, variables, "r 1 " predictor are associated with a single predictor, then, instead of using these "r 1 " predictors, we can use only that predictor with which they are associated. There by, reducing the number of predictors in the model to "p-r". The modified model will then be given by: where the notations have their usual meaning.
To test the independence among two or more continuous predictors and two or more categorical predictors, MANOVA is used and then on the basis of p-value, we rejected or accepted null hypothesis that these variables are jointly independent or not.
Also, Log-linear models for joint independence are used to test the independence when we have all the predictors as categorical predictors.

Model Comparison
These two models can be compared on the basis of their AIC values where AIC is given by: Where LL = Log-likelihood of the model, a = number of parameters of the assumed probability distribution (for example; a = 2 for Log-Logistic AFT model as there are two parameters involved) and c, the number of coefficients (excluding constant) in the final model. The model with smaller value of AIC can be considered as a better model compared to other models under consideration.
All these models are compared using two statistical criterion (likelihood ratio test and AIC). The nested AFT models can be compared using the likelihood ratio (LR) test. The exponential model, the Weibull model and the log-normal model are nested within the gamma model (Table 1).
Since the AIC for logistic model is least among all the fitted models. Also, according to the LR test, the logistic model fits better (Table1). So, we fit the survival data using Logistic AFT model and results are shown in the table 3 below: Then, we have tried to determine the pairs of predictors which are associated/ correlated with each other. We have used One-way Anova to test the dependence between a categorical and numerical variable whereas Chi-Squared test is used to the test the correlation/ association between two categorical variables. The pairs of predictors which are found to be significantly correlated/ associated are listed below in the Table 4 below: After finding these pairs of associated/ correlated predictors, we have tried to find the joint association among the predictors which are associated/ correlated with a single predictor. For eg, since Sex, alcohol, occupation and Spouse status of the patients are found to associated/ correlated with Age, we have tried to find that whether these 4 predictors are jointly associated with Age or not. We have used MANOVA to test the joint independence among two or more continuous and two or more categorical predictors and Log-linear models are used to test the joint independence among categorical predictors only. So, it can be seen that p-values for each group is less than 0.05, we can reject the null hypothesis that these predictors are not significantly jointly independent. As a result, we can jointly replace dependent predictors with the independent predictor. Then the survival times of the patients are estimated using AFTM with only 4 selected predictors. Again, for this model, Logistic model is found to be appropriate AFT model according to LR test and AIC values. Results are presented below in Table 6. Since Logistic AFTM has the minimum AIC, therefore, it is considered to be the best fit model. Then we tried to find the multiple/ associated predictors using MANOVA and Log-Linear models. Finally, only 4 predictors are selected. The