Application of the age-period-cohort model for estimating the incidence rate ratio of diabetes among the elderly

Introduction

In Epidemiological studies the onset of diabetes is measured through odds ratio and incidence rate. Incidence Rate (IR) measures the frequency of new cases in a population over a specific time and restricted to single population forms a limitation for comparison. To overcome this limitation Incidence Rate Ratio (IRR) is calculated. Incidence rate ratio is a measure that compares the incidence rates of two different groups by dividing the rate of one group by the rate of another group. Thus, incidence rate ratio is a way to compare two different groups to measure the different risk patten among these groups. In a cross-sectional study data set, the initial value is taken as reference category and incidence rate ratio is calculated, Around 1.4 billion people in India estimated with diabetes adding to global burden (Sekher TV et al. 2025, Tandon et al. 2018) [15,16] . As on 2019, India is the second largest country where in 77 million people with diabetes affected and estimated to increase (Ravi Prakash Jha, et al. 2021). According to the 2019 National Diabetes and Diabetic Retinopathy Survey, prevalence was found to be 11.8% in people over the age of 50 years. The prevalence was similar in both male (12%) and female (11.7%) populations. (Tandon N, Anjana RM, Mohan V, et al.) [16] . It is an important phenomenon to know the incidence of diabetes and its variation over period.

In case of diabetes, the rates depend on current age of individual (Age), duration of disease (Period), the birth cohort (Cohort) and also by sex. Age-Period-Cohort (APC) is important because individual chronological age (current age) brings biological changes, the period (period of exposure to disease) effects changes in treatment that could affect all ages and the Cohort (birth cohort in this study) associated with long-term exposure, with different generations being exposed to different risks (Robertson C, Boyle P. 1986) [11] . Keeping this concept, the incidence rate ratios (IRR) of diabetes estimated by applying Age-Period-Cohort model using grouped cross-sectional data from Longitudinal Ageing Survey of India (LASI) and IRR estimated by age-periodcohort and by Total, Male, Female sample.

Age-Period-Cohort Model

Initially, greater emphasis was placed on cohort studies and Ryder (1965) [14] focused on birth cohorts derived social change, thereby contributing to the theoretical concept of APC analysis. APC model has since been effectively used in Cancer registers and registries for other diseases (Eva Negri et al., (1990) [4] , Clayton, D, Schiffler, S (1987) [3] and webbased tool for age-cohort-period analysis have been developed for Cancer incidence and Mortality rates. Researchers identified that APC model suffers from an identifiability problem i.e., that each temporal dimension (Age, Period, Cohort) is a linear function of the others (Glenn 2005) [5] . Mason et al. (1973) [7] suggested a common solution to overcome this problem by imposing constraints that set certain APC parameters equal. Thus, each age, period and cohort group entered into a regression model as a dummy variable, with the assumption that two categories are constrained to be equal (Carstensen, 2007) [2] . To overcome the identifiability problem Yang, Fu and Land (2004) [19] fitted age-period-cohort terms as factors, and a constraint that sets two of the categories from different components equal to one another is applied in the regression. This methodology discussed as hierarchical APC (HAPC) model. Further for the repeated cross-sectional study, Yang, Fu and Land (2004) [19] developed HAPC-GLMM frame work to overcome the classical identification conundrum. They analysed age-by-cohort and cross level interactions with STATA commends such as apc_ic and apc_cglim. As extended research Yang, & Land, (2006, 2013) [17,18] developed mixed model for cross-sectional survey data and with empirical applications. Applicability of HAPC model by Yang & Land (2013) [18] has been argued by Bell and Jones (2015) suggested modified version by applying BHPS data. Carstensen, (2007) [2] identified the problem of identifiability which is raised due to age, period and cohort (Cohort=Period-Age), suggested APC model for continues variables through the use of spline functions and incorporated his method for age-period-cohort model in the Epi package for R. Carstensen (2007) [2] introduced a unified Lexis-diagram framework for modeling rates by APC, and analysed by generalized linear model (GLM) with a Poisson family, a log link and log person-risk-time. For the Carstensen (2007) [2] APC model Rutherford et al., (2010) developed STATA command apcfit with person-risk-time. Researchers, Glenn (2005) [5] , Carstensen, (2007) [2] , argued that there is no satisfactory solution to APC collinearity because there is underlying collinearity between A-P-C. However, with necessary assumptions, models have been developed for continuous and grouped data set. Keeping above concepts in view by using APC model, Incidence rate ratio of Diabetes among elderly for total, male, female data set has been estimated.

Data and Methodology

The Longitudinal Ageing Study in India (LASI) is a multidisciplinary, internationally harmonized panel study (IIPS 2020) with a national representative sample of India's population aged 45 and older. The main objective of LASI Wave I (2016-2018) is to provide evidence base on selfreported health and wellbeing of elderly population in India, collected information about demographics, household economic status and health status of elderly. The data covered 30 major states and 6 union territories of India. Further the survey collected a wide range of data on various aspects of health and well-being, including self-reported health status, chronic conditions such as diabetes, heart, lungs, Asthama etc. And disabilities, healthcare utilization and other background information. One can access the data and LASI India Report from IIPS website (IIPS 2020) [6] . Out of total sample of 72250 older adults aged 45 and above in the LASI study, data from 65531 elderly population data have been considered for the present research. Among that sample of 64397 elderly between 45 to 85 years of age considered for analysis. To estimate the incidence rate ratio using the APC model, data were further extracted from individuals who reported their "age at last birthday," "ever diagnosed with diabetes (Yes/No)," "year of diabetes diagnosis," "birth year," and sex. Age at last birthday was taken as Age (A), birth year as Cohort (C), and the difference between the year of diagnosis and the survey year was considered as Period (P) representing exposure time to diabetes. Respondents reporting "ever diagnosed with diabetes=Yes" were classified as cases, while the total respondents (Yes + No) were treated as the population exposed to the risk of diabetes. (1995-2004, 2005-2014, 2015-2020) and C into three groups (1949-1958, 1959-1968, 1969-1979). From this coded data for each A, P, C corresponding CASES and POP, data matrix has been prepared. Sample data matrix for N=5 is listed in Table 1. By using the reconstructed data in suitable matrix format (Table 1), a GLM fitted by using the STATA 16.1 version. Core STATA commends given in Table 2

Results

The Incidence rate ratio has been derived by APC model for the data described in Table 3 for total sample, male, female separately and given in Table 4, Table 5.

Total sample APC: IRR estimates

Age effect shows the IRR for the total study sample rises (Table 3) with age up to 75-79 having peak IRR 1.6-2.6, then declined slightly for 80-85. This reveals that increasing diabetes incidence with age in the elderly having a peak in the oldest-old group in India (Table 3). Period effect also shows statistically significant increase for the periods 1995 onwards having peak in the recent periods 2005-2014 (IRR=2017, p<.01). Further Cohort effect found to be weak or no significant effect in the cohort group for this cross-sectional data. That suggest after adjusting for the age & period, birth cohort don't show higher IRR than base cohort but not statistically significant (Table 3). Graphical presentation with 95% confident intervals in Figure 1 shows predicted incidence rate peak in middle age group and declines later. Period wise the Upper and lower limits of confidence interval found to be close to estimated Incidence rates and peat at middle period. Cohort wise the IRR steep declining trend observed with wider CI (Fig 1).

Sex wise APC: IRR estimates Male Elderly

Table 4 shows IRR of diabetes among elderly males reveals that the rates are >1 but not significant when compared with 45-49 age group except in the age group 60-64 years. Period effects are large but somewhat smaller than females (e.g., 2005-2014 IRR ~15.6 men vs ~27.8 women). Cohort effects for men show some negative pattern (IRRs <1 for later cohorts) and are sometimes statistically significant-suggests men of later cohorts may have lower incidence relative to baseline after adjustment (or sample artifact), (Figure 2).

Female Elderly: IRR estimates

Analysis in Table 5 show that IRR increases Period wise larger than men, particularly strong for 2005-2014 and 2015-2020. Age effects among women are consistently higher than men (women show stronger age gradient), (Figure 3). This points to stronger secular rise in diabetes detection/incidence among older women in recent periods (or cohort-specific care effects).

Negative Binomial Model: (NBM):

The negative binomial model has been fitted by STATA 16.1 and the commands are listed in box 1. The IRR by age, period and cohort for total, male and female are given in Table 6. The Negative Binomial model for total sample data (Table 6) indicates that age effects show a decreasing incidence rate ratio (IRR) (50-54, IRR= 1.28, NS) with increasing age (55-59, IRR=.4148, p<.001), unlike the Poisson model pattern, suggesting that over dispersion adjustment changes agerelated associations considerably. Period effects (IRR=4.80, p<.001) remain strong and highly significant, with the diabetes incidence increasing substantially across later periods-particularly during 2005-2014 (IRR=19.57) and 2015-2020 (IRR=13.34), confirming rising diabetes burden over recent years.

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