Statistical modeling and estimating number of healthy life years lost and healthy life expectancy in India, 2000–2019

Abstract Objective In this study, our objective is to propose various models to estimate healthy life year lost (HLYL) and healthy life expectancy (HLE) in India. Methods The HLYL and HLE were estimated and further these estimates were compared with the direct life table method and the World Health Organization (WHO) method. From the mortality perspective, we have developed a log‐logistic model for estimating the parameter (bx), which is characterized by HLYL. The results were compared with other models, such as the Gompertz and Weibull model. Here, we have also obtained the HLE by subtracting HLYL from the total life expectancy. Results The result shows an increasing trend of HLYL among the male, female, and the total population in India. Conclusion From the log‐logistic model, the HLYL was estimated as 8.79 years, 8.36 years, and 9.38 years for the total, male, and female populations, respectively, in India during 2019.

average number of years expected to be lived at a particular age, considering current mortality conditions.Therefore, HLE measures the number of years that a population may anticipate to live in good health assuming that they survive the current state of health and mortality circumstances.For instance, if a person has an LE of Over the years, many general health indicators have been proposed, however, relatively few have been applied.The WHO provides the most powerful health estimate as HALE.The concept of HLE was introduced in the 1960s and was developed by Sullivan. 2 is an indicator of mortality in a population, whereas HLE is an indicator of both mortality and morbidity.A demographic transition from high to low levels of mortality and fertility, along with a rise in LE has resulted in population aging.[5][6][7] The health transition of India began at a low life expectancy at birth (LEB; ie, 24.8 years).8 In the early 1980s, female LE exceeded male LE at birth.9 Gradually, the overall life expectancy in India increased and has been recorded as 70.79 years, 69.52 years, and 72.17 years for the total population, male population, and female population, respectively.HLE was recorded as 60.3, 60.3, and 60.4 years for total, male, and female populations,respectively. 3 Based on the latest Sample Registration System (SRS) abridge life table report (2015-2019), the LE for the urban and rural populations in India was recorded as 73 years and 68.3 years, respectively.10 The average LE of the global population in the year 2019 was 73.3 years.11 It is widely known from previous reports that LE at birth in India has doubled over the past 50 years.Although this accomplishment is commendable, it is important to recognize that these additional years might not necessarily be years of good health.
India continues to face challenges with infectious diseases, malnutrition, and the rapid rise of noncommunicable diseases, as well as age-related changes in physical health that lead to disability.
The aging process is a significant factor contributing to the increasing burden on social, economic, and health care systems in virtually all countries.Developing nations, in particular, experience the dual challenge of dealing with illnesses and disabilities, which poses a threat to the overall quality of life for their aging populations.With India being the most populous country in the world and experiencing a rapid growth in its older population, there is a pressing need to estimate healthy LE.HLE can be obtained by subtracting the number of HLYL from the total LE, where HLYL is the number of years that a population generally loses due to their illness by assuming they survive at a current state of health and mortality conditions.Therefore, HLYL plays a major role to understand the severity of public health issues and helps to allocate scarce resources effectively in public health planning to set priorities for prevention, and to control the GBD among the populations.
To measure the health state of a population, HALE plays an important role and was provided by the WHO.The practical approach for estimating the HLE was given by Jagger et al., 12 Romero et al., 13 estimated the HLE in Brazil by applying the Sullivan method.The estimation of LE and HLE for the Japanese population was done by Tokudome et al., 14 The interconnection between the HLYL and the Weibull shape parameter is observed by Matsushita et al., 15 and further explored by Skiadas and Skiadas. 16Skiadas et al., 17 estimates the LE and HLE for men and women in France during 1900 to 2017 and forecasted the same for the year 2060.Skiadas and Skiadas, 18 have also estimated the HLE and HALE using best fit logistic model during 1751-2016 in Sweden and found very close estimates as given by the WHO.
It is crucial to monitor the change in levels of physical and mental wellbeing in a population, as increased longevity alone holds little value without a healthy life.The data on LE and HLE allows us to recognize disparities in overall health among different groups according to age, gender, socioeconomic status, living conditions, and other factors.This information also enables us to identify and measure the impact of illness on the overall health of the population.The information on LE, HLE, and HLYL is useful in determining the allocation of resources for health promotion and in providing an improved understanding of the determinants of health.This information can be used to predict the future needs of a population, to provide information in planning of health and social services, and to identify trends and inequalities present in the population.The aim of this research is to calculate HLE and HLYL for the Indian population using the loglogistic model and also to compare the results with Gompertz and Weibull models.

| Methodology
In this study, we have applied four different methods to estimate

| Direct method
The direct methods (without using a model) for estimating HLYL were given by Skiadas and Skiadas, 16,19,20 which is based on averaging the health state of a population.The use of life table data for this purpose is advantageous because it can be applied to any population, regardless of whether or not direct data on health and diseases are available.The full life table from the Human Mortality Database (HMD) is followed by four more columns for the estimation of bx.In the first, the cumulative mortality is estimated from M x = ∑ x 0 m x .The average mortality  21 where they used the expression of the survival function as H(x) denotes the cumulative hazard function, which is equivalent to the area under the hazard function m x .The area under the hazard function was defined by taking the corresponding integration limits ranging from x (current age of an individual) to x + y x (age at death or quantity of time lived from birth to death).The calculated area will give the risk of dying at a given age x up to a particular future time y x .This approach was developed by Skiadas and Skiadas 1,16,19,20,22,23 to set a timevarying fraction b x of the form given below: Accordingly, the mortality process will have two alternatives expressed by simple Equation 16;

| Log-logistic model
The log-logistic distribution (LLD) is obtained by applying the logarithmic transformation to the logistic distribution (LD).LLD is mathematically more tractable when compared with the log-normal distribution.For this property, LLD is being used in survival data analysis. 24The LLD can also be a good replacement for the Weibull distribution.As it has a closed form of distribution function and its hazard function is quite flexible, this distribution has greater scope and may be applicable to a wide variety of problems in various areas.
Let X be a non-negative random variable that follows log-logistic distribution with two parameters and l .Then, the probability density function f(x), survival function S(x), and hazard function h(x) becomes 25 : (1)

Mx
To derive log-logistic generating function, here, we take the shape parameter as unity, that is, = 1, hence we have: The selected value for the estimation of the HLYL is provided by the parameter l .

| Weibull model
Matsushita et al., 15 had suggested the Weibull model for a life time data analysis of disease and aging.It was also established that the Weibull model can be used to estimate the HLYL due to disabilities.Skiadas and Skiadas 16 highlight the importance of introducing the Weibull shape parameter in connection to survival rates.
From the above Equation 1 16 taking b x as constant and the found that the estimator b x follows the hazard function of the Weibull distribution.
Let X be a non-negative random variable that follows the Weibull distribution with two parameters b and l .Then, the probability density function f(x), survival function S(x), and hazard function h(x) becomes 25 :

| Gompertz model
In the Gompertz model, Skiadas put b x as a multiple of a constant with age x, that is b x = ax in Equation 1, which is inter-related with hazard function of the Gompertz distribution. 26The Gompertz All the statistical analysis and plotting has been done using two statistical software programs viz., R version 3.6.2and Microsoft Excel.

| ANALYS IS AND RE SULTS
Primarily, HLYL was estimated by using the above discussed four methods viz.Direct, Gompertz, Weibull, and log-logistic for the total, male, and female populations, respectively, in India from 2000 to 2019 and presented in the Table 2.Note that the direct method given by Skiadas HLYL was estimated by using three probabilistic models viz.
Gompertz, Weibull, and log-logistic for the total, male, and female populations, respectively, in India from 2000 to 2019 and the model fit summary of these considered models is given in Table 1.
In this study, we have considered three life time models, that is, the log-logistic, the Weibull, and the Gompertz models to fit agespecific mortality data for the total Indian population and the male and female populations, respectively.The model summary is given in Table 1, which shows all the three considered models and it gives very similar values of R 2 , SSE, and standard error.For the log-logistic model, the R 2 value is found to be maximum for the total, male, and female populations as 0.947, 0.936, and 0.960, respectively.Therefore, the log-logistic model is considered to be the best fitted model with the greater accuracy.
Table 2 shows an increasing trend of HLYL for the total, male, and female populations in India in all the four discussed methods Further, HLE also been estimated viz. the direct, the Gompertz, the Weibull, and the Log-logistic models for the total, male, and female populations in India from 2000 to 2019 and presented in Table 3. LE values for the male, female, and total populations were retrieved from the complete life table given by HMD and presented in Table 3.Then, HLE is being calculated by subtracting HLYL from their corresponding LE.
Table 3 shows an increasing trend of HLE for the total, male, and female populations in India in all the four methods from 2000 to 2019.By comparing the estimated HLE with the WHO estimates it is found that the log-logistic model provides closer estimates than the other considered methods.Further, the log-logistic model shows that the HLE for the female population are comparatively higher than the male population in India during 2000 to 2019.From the loglogistic model, the current estimates of HLE for the total, male, and female populations in India becomes 60.85, 60.09, and 61.58 years, respectively.for both male and female cases.In addition to this, the log-logistic model gives more close estimates than the other considered methods or models as compared to the estimates given by the WHO.

| DISCUSS ION
In this study, we have used four different techniques viz. the direct, the Weibull, the Gompertz, and the log-logistic models to estimate HLYL and HLE.The applications of the Weibull and Gompertz models can be found in several studies. 15,19,28But the use of the log-logistic model is not in common with the others.Therefore, in this study, we have proposed and established the analytical explanation of the link between the HLYL (b x ) parameter and the hazard function of the log-logistic model.In the same way, Skiadas and Skiadas, 16 derived a relationship between the Weibull shape parameter and HLYL and applied it on the Japanese population to estimate their HLYL.They have found that the direct method and the Weibull method provides nearly the same estimates as provided by the WHO.Skiadas and Skiadas, 22 developed a health mortality approach to estimate HLYL of United States and the Japanese populations using the Weibull and Gompertz models and they have also found estimates of HLYL which are very close to the one provided by the WHO.
In our study, we have found an increasing trend of HLE and HLYL over the years irrespective of the gender.Similar findings have been years among both men and women.Further, it is also found that there is a significant difference between men and women with respect to their LE and HLE and similar findings were also found in a recent study in India by Borah. 9 In this study, they have reported that the women have a higher LE than men, which is also observed around the world. 30e coronavirus disease 2019 (COVID-19) pandemic has had a devastating impact on LE and HLYL.In 2020, LE at birth declined by an average of 1.33 years in 27 countries.This means that people born in 2020 are expected to live 1.33 years less than people born in 2019.
The pandemic has also had a disproportionate impact on the elderly, with LE at age 65 years declining by an average of 0.9 years in 27 countries. 31Brazil's life expectancy at birth declined by 1.3 years in 2020 due to COVID-19, reaching a level not seen since 2014.The LE at age 65 years also declined by 0.9 years, setting Brazil back to 2012 levels. 32t only the proposed model, that is, the log-logistic model, provides a good and close estimate of HLYL, the Gompertz and Weibull models also provide good estimates as given by the WHO.Here, in Appendix S1, we have provided the analytical explanation of the relationship between HLYL and the modified Weibull model.This explanation can be used for future study in demography and population health research.Researchers may further extend this model to estimate the HLYL due to COVID-19 in India.

| CON CLUS ION
The evidence presented indicates that the proportion of healthy Furthermore, our findings revealed that HLYL are consistently higher for women compared to men, thus compensating for the longer LE observed in women.As LE increases, there is a corresponding rise in the number of healthy years lost.It is crucial to address this issue by not only extending the lifespan but also by implementing strategies to reduce the HLYL.Moreover, it is essential to enhance the social security system, which often relies solely on LE measures, by incorporating healthy life year data into relevant plans and policies.Interestingly, although women tend to have higher healthy LE compared to their male counterparts, the gap between male and female LE at birth is larger than the gap in healthy LE.
The health care system faces a challenge in adapting its support to the growing segment of the population that surpasses the healthy LE threshold.The concepts of healthy LE and HLYL can serve as valuable indicators for health care resource planning and future policy development in India.Monitoring healthy LE could provide valuable tools for crafting health care strategies aimed at postponing morbidity and disabilities for the entire population.Given that India is a welfare state with a dynamic demographic dividend and a rapidly growing population, addressing concerns related to the aging population in a timely manner is crucial to ensure that the demographic dividend does not transform into a demographic disaster.
Calculating healthy LE at a national level could yield policy recommendations that improve health practices and enhance access to quality health care for the elderly.We hope that this paper will stimulate further research and foster meaningful dialogue on the topic of healthy LE.
80 years and a health adjusted life expectancy (HALE) of 74 years, 6 of the 80 years are effectively "lost" due to ill health.Major planning in health care is based upon objective indicators, such as mortality, morbidity, or disability statistics.The combination of mortality figures with LE, HALE and healthy life years lost (HLYL) provides meaningful health outcomes at the population level.
HLE and HLYL for the Indian population and further compared with the values provided by the WHO.The four different methods are (a) the direct method, (b) the Gompertz model, (c) the Weibull model, and (d) the log-logistic model.The detailed methodologies for estimating HLE and HLYL have been discussed below.
ls l−1 ds.Now considering the denominator of the above equation we have: Now, Further rewriting Equation (1) we get: Now, putting the value of b x in the above equation we have: Here, m x is the hazard function or generating function of LLD.
model is generally used to handle the mortality data.A convenient Gompertz model is provided by Carriere 27 as x = Bc x , where B and c are the parameters.This model provides a probability density function f(x) expressing the distribution of deaths over age for a population at a specific period of time.Then the probability density function f(x), cumulative distribution function F(x), survival function S(x), and hazard function h(x) becomes: figure shows, up to 21 years of age for both male and female subjects are having almost the same years of HLYL.But the population with 22 and above years of age are having more or less different HLYL.From the age 22-59 years, HLYL for the male population is quite higher than the female population.But for the age 60 and over, women are dominant over men in India.By looking at overall characteristics of HLYL, as shown in Figure2, it can be clearly observed that the value of HLYL increases for all three categories of populations, that is, total, male, and female populations up to 92 years of age and decreases on reaching 93 years and above.

from 2000 to 2019 .
The related figures for 2000 are b x = 6.82 with the direct estimate, b x = 5.74 with the Gompertz estimate, b x = 5.88 for the Weibull model, and b x = 7.30 via the log-logistic model.The related figures for 2019 are b x = 7.26 for direct life table estimation, b x = 6.64 for the Gompertz model, b x = 6.66 for the Weibull model, and b x = 8.79 for the log-logistic model.Note that the figure for the HLYL provided by the WHO is 10.49 years of age.By comparing the estimated HLYL with the WHO estimates it is found that the loglogistic model provides more close estimates than the other considered methods.Further, the outcomes of the log-logistic model shows that the HLYL for women are comparatively higher than their counterparts, that is, the male population in India during 2000 to 2019.From the log-logistic model, current estimates of HLYL for the male and female populations in India in 2019 becomes 8.36 and 9.35 years, respectively.

Figure 3
Figure 3 shows the graphical comparison between HLE (in the left) and HLYL (in the right) estimated by various methods for India during 2000 to 2019.The figure shows the value of LE and HLE increases over the years.However, the gap between LE and HLE slowly increases as their value increases with the years.As a result, we can see the HLYL also shows an increasing trend over the years.The values of HLE estimated by four different methods viz.the direct, the Weibull, the Gompertz, and the log-logistic models show nearly similar values.But the estimated value of HLE by the log-logistic model is closer to the estimated value given by the WHO than the value estimated by the other considered methods or models, which confirms that the log-logistic model is an appropriate model to estimate HLE for the Indian population.Similarly, for HLYL, the log-logistic model shows a closer estimate as given by the WHO than the other considered models and confirms as an Abbreviations: F, female; HLYL, healthy life years lost; M, male; T, total; WHO, World Health Organization.
appropriate model to estimate HLYL for the Indian population.Figures 4 and 5 show the graphical comparison between HLE (in the left) and HLYL (in the right) estimated by various methods for the Indian male and female populations, respectively, during 2000 to 2019.These figures also show an increasing trend of HLE and HLYL

F I G U R E 3
Comparison of life expectancy and healthy life expectancy (left).Comparison of healthy life years lost (right) in India from 2000 to 2019.HLE, healthy life expectancy; HLYL, healthy life years lost; LE, life expectancy; WHO, World Health Organization.F I G U R E 4 Comparison of life expectancy and healthy life expectancy (left).Comparison of healthy life years lost (right) among women in India from 2000 to 2019.HLE, healthy life expectancy; HLYL, healthy life years lost; LE, life expectancy; WHO, World Health Organization.observedby Lau et al.,29 an increasing trend of LE and HLE in all age groups as well as for men and women in India over the period 2007 to 2020.In their study, the largest gain in LE and HLE is observed in the age 70+ years among women and the age 65+ years among men.However, in this study, the largest gain in HLE and HLYL observed in the age 90+ years in a person's life has remained relatively stable, suggesting that the additional years gained are generally characterized by poor health.If individuals can enjoy these extra years in good health, and if they are supported by an enabling environment, their ability to engage in activities they value may not significantly differ from that of younger individuals.However, if these additional years are primarily marked by physical and mental decline, it has more negative implications for both older individuals and society as a whole.

F I G U R E 5
Comparison of life expectancy and healthy life expectancy (left).Comparison of healthy life years lost (right) among men in India from 2000 to 2019.HLE, healthy life expectancy; HLYL, healthy life years lost; LE, life expectancy; WHO, World Health Organization.

The secondary data have been collected on various columns of com- plete life table from 2000 to 2019 for India from http
Model fit summary.HLYL estimates and comparison for the total, male, and female populations in India from 2000 to 2019 HLYL for Total, Male and Female of India(2019)