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Transition Probabilities of Disablement for Malaysian Population in Need of Long-Term Care

This research aims to estimate the transition probabilities of lives becoming disabled and the extent to which they are disabled using the approach of functional Markov model. The transition probability from one disabled state to a more or less severely disabled state is best estimated using longitudinal data in which the change in the health status of each respondent can be monitored over one or more years. Such data are limited in Malaysia, typically covering only a smaller area of the nation. The functional Markov model overcomes such data limitations, using cross-sectional data which measures the disability status of individuals only at one point in time and build a functional form for the transition probabilities in a multiple state model. Results suggested that multiple state model's prevalence rates replicated the Malaysian prevalence rates quite well, indicating that the parameters of the probability of deterioration had been estimated accurately with sum squared of errors less than 5% for almost all age groups and disability levels. Furthermore, severely disabled elderlies, especially among the oldest age group, have the highest probability to die compared to less severely disabled elderlies.

that the initial prevalence rates are replicated over one year.
Leung (2004) adopted the Rickayzen & Walsh (2002) model to project the likely number of people requiring long term care using Australia data and proposed some modifications.
Instead of using age 20 as the beginning age, his model covers all ages to allow disabilities that may have arisen at earlier ages. Leung (2004) also proposed different functions for mortality and disability improvement using a method from the Continuous Mortality Investigation Bureau (1999).
Also, the application was extended to estimate associated long-term care costs. Hariyanto et al. (2013) also adopted the same model framework to estimate the transition probabilities across levels of disability for Australia. As opposed to Leung (2004), who used 1998data, Hariyanto et al. (2013 used more recent 2003 data. Interestingly, the model made use of two consecutive available disability prevalence rates in 1998 and 2003 to obtain the prevalence rates in every year from 1999 to . Bueno (2013 showed that the same model applies to a developing country, Brazil. However, several formula and parameters were adjusted to accommodate Brazilian disability data. Research from Omar, Shair & Asmuni (2020) proposed the cancer stage transition model for lung cancer patients in Malaysia using the same approach. However, the estimated rates were in terms of overall and did not differ by age groups and gender.
This research is the first to estimate the transitions probabilities of disablement for the Malaysian population according to different disability states, age and gender. We adopt the functional Markov Chain framework from Rickayzen & Walsh (2002) and Leung (2004). This paper extends the models, for the case of Malaysian disability in which the parameter values will be estimated and reflect Malaysian disability experience. We propose four disability levels, including able, mild, moderate and severe, to estimate the transition disablement. This paper is organised as follows. Section II describes the age and gender-specific disability prevalence data, while Section III explains the extension of the functional Markov model to suit Malaysian disability data. Section IV provides results and discussions of study. Finally, Section V concludes and provides possible future directions.

II. DATA
Malaysian disability data can be obtained from the National Health Morbidity Survey (NHMS), which is conducted once in every four years. Unfortunately, at the time we conducted this research, data according to disability levels by age groups were not available from the survey report. These age-specific data were necessary to estimate the transition probabilities across disability levels by ages.
To overcome the limitations of the NHMS data, we used the WHS (2002) There are nine main sets of questions; however, we gather data from which refers to the individual's health state description. Five possible answers for every question were set as 1 is very good, 2 is good, 3 is moderate, 4 is bad, and 5 is very bad. We recode the answer to represent disability levels such that 1 is "able", 2 is "mild", 3 is "moderate" and as data for 5 are very small, we combine these with 4 and recode as "severe".
To estimate the prevalence rates of disability for Malaysia, we collected data primarily from the following three questions of WHS (2002) and disability level and then divided by the total exposure to estimate the prevalence rates per 1,000. The prevalence rates of the three questions were then averaged to obtain the prevalence rates for Malaysia by age and severity level, as shown in Table 1 and Table 2 for males and females, respectively. The estimated WHS (2002) prevalence rates from Table 1 and Table 2 show that the disability rates are varied by age, disability level and gender. The healthy (able) rates decrease as age increases, and the disability (mild, moderates and severe) prevalence rates generally increase as age increases.
These trends appear in both males and females. The increase in disability prevalence rates by age for both males and females are considerable for the severe group such that it increases more than triples between age 55 and 75 years old.
A comparison between gender shows that females' prevalence rates for mild, moderate and severe categories are higher than that of males for age 50 years and above, but lower for ages below 50. This indicates that Malaysian elderly females suffered from disabilities, in higher percentages, compared to elderly males. The result is consistent with disability patterns of elderly people in Bangladesh (Tareque et al., 2017).

III. METHODS
The estimation of transition probabilities between several disability states followed the multiple state modelling framework from Rickayzen & Walsh (2002). Some formulae and parameters were adjusted accordingly to Malaysian data.
The model was applied to the Malaysian prevalence rate of disability according to age, gender and disability levels.
We proposed four disability states, namely able, mild, moderate and severe, and one absorbing state, death, in the model. Each disability state was denoted by : = 0, 1, 2, 3 where 0 represented the able or healthy state, 1 was mild, 2 was moderate, and 3 was severe.
There were three types of transition probabilities included in the model, including death probabilities, deterioration probabilities (probabilities of moving to any worse disability state) and improvement probabilities (probabilities of recovering to a less disabled state or able state). Similar to Rickayzen & Walsh (2002), Leung (2004) and Hariyanto et al.
(2013), the following assumptions were adopted in the model: i. deterioration to any worse disability state was allowed over a year; ii. improvement was only allowed by one category; iii.
only one transition was possible over one year. See Figure 1 for the proposed model of disability transition states for the Malaysian population.
The parameters of the equation for deterioration probabilities were estimated through the fitting process. In contrast, death and improvement probabilities were determined separately and included as an input in the fitting process. The deterioration probabilities were found by assuming a functional form for the probabilities in the multiple state model. The parameters were estimated such that the model's prevalence rates replicate the WHS estimated prevalence rates reported in Table 1 and Table 2.
We assume stationary population characteristics when deriving transition probabilities.  (2004), the observed prevalence rates were replicated using the optimisation procedure available in Excel's `Solver' function so that the sum of the squared difference of the calculated WHS prevalence rates and the model's prevalence rates were minimised. The mathematical formulae used to estimate the probability of death, probability of deterioration and probability of improvement are discussed in the following sub-sections.

A. The Probability of Death
The probability of death can be divided into two categories, overall mortality and additional mortality. The overall mortality assumes that people who are in the same age group and gender will experience the same mortality experience, regardless of their level of disability. In contrast, the additional mortality imposes an extra rate on those who are in the severe disability category ( = 3). This translates to the mortality rates of disabled people get higher as disability status becomes more severe. The probability of death is given as follows;

B. The Probability of Deterioration
The estimation of deterioration probabilities includes two types of deterioration. First, a healthy person becomes disabled, denoted as ( , 0, ) and second, a disabled person deteriorates to a more severely disabled state, denoted as ( , , ) . The formula for the first, which is the probability a healthy person ( = 0) aged deteriorates to any disability category j (j = 1,2,3), comprises two components, expressed as below; ( , 0, ) = ( ) × ( , ) The first component, ( ), represents the probability of a healthy person aged becoming disabled for the first time, which can be estimated as follows.
For Malaysian data, we apply the same formula for both males and females. The parameter is the limit of probability of becoming disabled at young ages and is the limit of probability of becoming disabled that would apply at extremely high ages. Parameters and determine how rapidly the probabilities change between the two extreme values, and . Age identifies the kink in the new disabled rates after which disability rates increase dramatically.
The second component, ( , ) , is the probability that a healthy person aged deteriorates to a particular disabled state given that he becomes disabled over the year.
The formula for ( , ) is; where ( ) = + 1 − 1 + − The second type of deterioration probability, after that discussed in equation (3), is ( , , ), which is the probability that a disabled person in disability state becomes more severely disabled---moves to state , > as follows; ( , , ) = ( , 0, ) × Equation (8) shows that the probability of a person at disability level deteriorating to level j ( > ) is times the probability of that person from the healthy state deteriorates to disability level .

D. Projection Method
To begin the one-year projections, the initial number of populations in each category is estimated such that the male and female disability prevalence rates of the model year 2002 are multiplied by the population number from DoSM by agegroup, males and females of the same year. Although the model prevalence rates are not fully consistent with the data prevalence rates, the differences are assumed to be minimal.
Below we outline the method used to project the age-sexspecific population by disability levels. Define  We can see from the Table 4 and Table 5 that the percentage errors for males and females appear relatively small, less than 5%, across almost all ages and disability states, except for the oldest age group among females. Higher errors can only be seen for females aged 70 years old and above. This result is consistent with Leung (2004) who found poor fit for age 85 and above for Australian data. ( , ) are shown in Table 6 and  We provide the estimated Malaysian disability transition rates for Malaysian elderly males and females as in Table 8 and Table 9. Although transition rates were calculated from age 18 years old for both males and females, we provide transition rates for the selected elderly group aged 60 years and above only.
We can see from the   (Lynn & Adamson 2003). Thus, more studies are critically needed to prevent and alleviate disability severity, particularly to keep physically and mentally active when getting older or after retirement. More works are needed particularly to project the number of people who likely need long term care and its related costs.

VI. ACKNOWLEDGEMENTS
The author is thankful for research funds received from the

VII. REFERENCES
Bueno, LPV 2013, 'Brazilian long-term care plan: an evaluation based on the Singaporean model', MSc. Thesis,