There are six important demographic processes modelled in this study, where individuals go through the changes in Ageing, Mortality, Fertility, Household formation/Marriage and Migration.
Age and sex are the fundamental factors that contribute to demographic processes (Rowland, 2003). Transition probabilities for each of the model processes vary by geographical area as a result of the differences in social, economic and environmental profile of area populations. Marital status is important in Household Formation/Marriage and Fertility processes. Therefore this model assumes that all the demographic processes modelled are functions of age, sex/marital status (in Fertility only marital status is used, due to the female-led approach) and location. Probabilities of even occurrence are computed for population groups classified by age, sex, marital status and geographical locations from suitable census or survey data, correctly tabulated. The probabilities are used to determine whether an event occurs to an individual in the model population. Therefore a change/an event occurs to an individual in a demographic process if
) , , , ( ) 1 , 0 ( P k2a s l k1 Ran ≤ (Equation 4.1)
where Ran is a random number between 0 and 1, a is the age, s is the sex and is the location of the individual and l P(k2a, s,l,k1) is the probability
of the event occurrence for an individual of age a, sex s at location l with current characteristic of to change to a characteristic . For instance: in mortality, might be “alive” and will be “dead”; in fertility, might be “no child born alive during the year” and will be “child(ren) born alive during the year”.
1 k k2 1 k k2 k1 2 k 99
Transition probabilities for each of the events within the six demographic processes are applied at discrete one year intervals. According to the nature of specific demographic processes, extra factors will be introduced in calculating the probabilities. For instance, the Fertility process considers women by age, marital status and location. Marital status is used here as patterns of births within marriages differ hugely from births outside marriages.
Monte Carlo simulation is adopted in this MSM, because it is very useful when it is infeasible or impossible to compute an exact result with a deterministic algorithm. Monte Carlo simulation gets its name from the resort on the southern coast of France where gamblers bet at games of chance of a random number at each draw. Monte Carlo simulation uses repeated computation of random or pseudo-random numbers. Monte Carlo simulation converts uncertainties about the relationship between input variables and output variables of a model into conditional probabilities. By randomly selecting values from inputs repeatedly, it recalculates and brings out the probability of the outputs. Apart from the Ageing process, the rest five demographic processes are modelled through Monte Carlo simulation where age, sex and location specific probabilities are applied to determine whether or not an event of demographic change will happen. For example, if you want to find out if a male candidate aged 85 living in Aireborough is going to survive this year, you need to compare the mortality probability for men aged 85 in Aireborough (0.104505) to a generated random number between 0 and 1 (0.563542) as described in Equation 4.1. As the random number is bigger than the probability, it can then be decided that this person survives this year and proceeds to determine whether other demographic changes happen to him this year. Each determination requires a new random number and a corresponding event probability for a person with certain attributes in that year. All probabilities are updated annually and any change in any factor will result in the change of the probability and in turn the simulation process and result.
The general microsimulation process in this model uses the Monte Carlo method to simulate the changes within each demographic process. Monte Carlo simulation can introduce a degree of randomness and often results from different runs vary. Various what-if scenarios and sensitivity analysis have been used to help reduce variances in results from different runs. To facilitate the estimates of the absolute responses; i.e., prediction or interpolation from the observed responses for the scenarios that have already been simulated, variance reduction techniques are needed. Most used techniques include common random numbers, antithetic variates, control variates, conditioning, stratified sampling and importance sampling (Law, 2007; Rubinstein and Kroese, 2008). The common random number method (sometimes called correlated sampling, matched streams or matched pairs) is intuitively preferred, because it reduces variability based on the principle that a random number to be generated using the same draw from a random number stream for all configurations. Due to time scale and the resource limitations, this project uses a simple method that is based on the same principle. All probabilities used in this model are of 6 decimals to minimise the variance and the same draw from the number stream generated using Math.random() method from Java, employed in all runs. The current control has provided reasonable variance reduction for the model results. However, given more time and resources, a more sophisticated method can be developed to better facilitate such purpose. The general process is illustrated in Figure 4.2.
When a candidate for a certain transition enters the simulation, a process will ask the question whether this transition happens or not through a Monte Carlo simulation on the basis of a specific probability. If the answer is affirmative, then it carries on processing the changes and updating the individual’s status and household status (e.g. minus one household member in case of death), before move on to process the next event in the same fashion. If the answer is negative, then it goes directly to update the individual’s status and household status before move to the next event.
Candidate enters simulation Event Process (Monte Carlo simulation)
Figure 4.2 The general simulation procedure