Why ADL thresholds matter
A long-term-care contract may use the insured’s ability to perform activities of daily living (ADLs) as one trigger for benefits. The model must therefore connect the clinical assessment to the states used to represent an insured’s health. A threshold that requires assistance with two ADLs does not identify the same group as a threshold requiring assistance with three. That difference can change who is classified as eligible, where entry into eligibility occurs in the transition graph, and the transition rate observed between model states.
For modelling, the ADL rule needs to be stated precisely. The count must refer to a specified set of activities and a specified assessment standard—for example, whether a person needs another person’s assistance to perform an activity. This tutorial treats the count as already determined under the hypothetical contract’s assessment rules. It does not address other possible triggers, such as cognitive impairment.
From ADL counts to model states
If the contract assesses six ADLs, the exact count can take values \(0,1,\ldots,6\). A model may retain all seven counts as distinct states, or group several counts into broader care states. For example, with a threshold of two, counts 0 and 1 are ineligible, while counts 2 through 6 are eligible. The groups are a mapping of exact states; they do not change the underlying assessment.
A threshold is not itself a transition assumption. If an insured moves from one ADL count to another, the threshold determines whether that movement crosses the eligibility boundary. With a threshold of two, a movement from count 1 to count 2 enters the eligible group. With a threshold of three, the same movement remains within the ineligible group; entry occurs at count 2 to count 3. If recovery is possible, a movement back below the threshold exits the eligible group. The model must include such movements if they are permitted by its transition structure.
The count can also conceal clinically relevant differences. Two people may each need help with two ADLs, but need help with different activities. A count-based model treats them as the same exact state unless the state definition includes more detail. That simplification is appropriate only when the model’s purpose and assumptions support it. The assessment rule and the state definition should be documented together so that eligibility is reproducible.
Worked Example: Locating the Eligibility Boundary
A hypothetical policy assesses six ADLs and defines eligibility as needing assistance with at least two. An insured’s recorded counts at successive assessment dates are 1, 2, 4, 3, and 1. Identify the eligibility status at each date, the first entry into eligibility, and any exit from eligibility.
Method. Apply the rule \(N(t)\geq 2\) separately to each recorded count. Do not treat a higher count as a different trigger once the threshold has been met.
At count 1, the insured is ineligible because \(1<2\). At count 2, the insured is eligible because \(2\geq2\). At count 4, the insured remains eligible; at count 3, the insured remains eligible; and at the final count 1, the insured is ineligible again.
Thus, the first entry occurs when the count moves from 1 to 2. The exit occurs when the count later moves from 3 to 1. The assessment sequence does not state whether the change from 3 to 1 happened in one transition or through an unobserved intermediate count; the model’s permitted transitions and assessment timing would determine that detail.
Check. The eligible dates are exactly those with counts 2, 4, and 3, all of which meet the stated threshold. The final count of 1 is below the threshold, so it is an exit, not continued eligibility.
Exact-state intensities and aggregate transitions
Suppose the exact states form a continuous-time Markov model. For distinct exact states \(i\) and \(j\), let \(q_{ij}(t)\) denote the transition intensity from \(i\) to \(j\) at time \(t\). For a small interval \(h\), the conditional probability of a transition from \(i\) to \(j\) is \(q_{ij}(t)h+o(h)\), given that the process is in state \(i\) at time \(t\). The intensity can depend on time. Death can be included as an absorbing state, and its transition intensity is handled in the same way.
To represent eligibility, group exact states into blocks. For example, under threshold \(k\), the ineligible block is \(I=\{0,\ldots,k-1\}\) and the eligible block is \(E=\{k,\ldots,6\}\). From exact state \(i\), the intensity of moving into the eligible block is the sum of intensities to each exact state in \(E\):
If the insured is known only to be in the ineligible block, the instantaneous rate of entering eligibility depends on the distribution of exact states within that block. Write \(\pi_i(t)=P(N(t)=i\mid N(t)\in I)\) for that conditional distribution. The aggregate intensity is then
This is a weighted average of exact-state transition totals. It is generally not enough to know the threshold group alone: the weights may change as the insured population moves among the exact states in that group.
For a particular population rather than strong lumpability, the aggregate rate is constant over time precisely when its weighted average remains constant over time. Constant exact-state totals with equal values within a block are one sufficient case. Fixed weights are not sufficient by themselves if the exact-state totals vary over time; their weighted average must also remain constant. Likewise, equality between exact-state totals is not sufficient for a constant aggregate rate if the common total varies with time.
Worked Example: Calculating an Aggregate Entry Intensity
A hypothetical threshold defines counts 0 and 1 as ineligible and counts 2 or more as eligible. At a particular time, the intensities from count 0 into the eligible block and from count 1 into the eligible block are respectively 0 and 0.12 per year. Among insureds currently in the ineligible block, 40% are at count 0 and 60% are at count 1. Calculate the aggregate entry intensity. Then repeat when the conditional proportions are 25% and 75%.
Method. Weight each exact-state transition total by the proportion in that exact state. The exact-state totals are \(q_{0E}=0\) and \(q_{1E}=0.12\) per year.
For the first distribution,
For the second distribution,
The exact-state totals did not change, but the aggregate intensity increased because the ineligible group now has a larger proportion at count 1, the state from which entry is possible. This calculation is an instantaneous intensity, not a one-year probability.
Check and common mistake. Both answers lie between 0 and 0.12, the smallest and largest exact-state totals. Using 0.12 as the aggregate rate would incorrectly assume that every ineligible insured is at count 1. The aggregate rate is constant over time only if the relevant weighted average stays constant; fixed exact-state totals alone do not guarantee that when the weights change.
How a threshold changes entry probabilities
An intensity is a local rate. To calculate a probability over a specified period, the model must account for the time spent in each state and for competing transitions such as death. The following hypothetical model isolates the effect of raising an ADL threshold. It assumes that the count can only deteriorate through the stated transitions, that death is absorbing, and that all intensities are constant over the one-year period. These assumptions are specific to this example, not general properties of ADL models.
Worked Example: Comparing Two Eligibility Thresholds
An insured starts at ADL count 1. The only living transitions are \(1\to2\) and \(2\to3\), each with intensity 0.20 per year. From either count 1 or count 2, death occurs with intensity 0.05 per year. Consider the probability of becoming eligible within one year under threshold 2 and under threshold 3.
Method. Treat each step as a competing-risk transition. In counts 1 and 2, the total rate of leaving the current state is \(0.20+0.05=0.25\) per year. Eligibility under threshold 2 occurs on the first living transition, \(1\to2\). Its probability by time 1 is the integral of the transition density:
Rounded to four decimals, this probability is 0.1770. Under threshold 3, the insured must make both living transitions before death and before the end of the year. If the first transition occurs at time \(s\) and the second at time \(t\), then \(0\leq s\leq t\leq1\). The joint density contribution is \(0.20e^{-0.25s}\times0.20e^{-0.25(t-s)}\). Integrating gives
Rounded to four decimals, the threshold-3 probability is 0.0170. The probability is smaller because the insured must make two transitions instead of one, and death can occur before either required transition.
Check and common mistake. The threshold-3 probability is less than the threshold-2 probability, as it must be for nested eligibility sets. Using \(1-e^{-0.20}\) for threshold 2 would ignore death as a competing transition. The correct total exit rate from count 1 is 0.25, not 0.20.
A practical modelling sequence
When an ADL threshold is translated into a model, keep the clinical rule, state map, and transition assumptions distinct. A useful sequence is:
List the ADLs being assessed and state what qualifies as needing assistance for each one.
State whether the model records the count alone or distinguishes particular patterns of limitations.
Map counts below \(k\) to the ineligible block and counts at least \(k\) to the eligible block.
Identify which exact transitions cross the boundary, then test whether the exact-state totals satisfy the aggregation condition.
A higher threshold generally delays the first possible entry into eligibility when the exact-state process is otherwise unchanged. It does not automatically imply a constant reduction in benefit incidence: that depends on the transition intensities, recovery patterns, mortality, and distribution of insureds across exact states. Similarly, two models using the same threshold may have different aggregate transition rates if their exact-state occupancy distributions differ.
Common exam traps
- Confusing a threshold with an exact state. “Eligible at two or more ADLs” is a group of exact states, not a single ADL count.
- Changing the threshold but not the boundary transitions. Entry under threshold 2 and entry under threshold 3 occur on different exact transitions.
- Treating an aggregate intensity as a probability. An intensity is a rate per unit time; a probability over a year requires the transition model over that year.
- Assuming aggregation is automatically Markov. If exact states in a group have different rates to a target group, the grouped rate may depend on the within-group distribution.
- Claiming a constant aggregate rate from equality or fixed weights alone. The weighted average of the exact-state totals must remain constant over time.
- Ignoring recovery. If transitions can reduce the number of limitations, an insured can leave eligibility by crossing below the threshold.
Check Your Understanding
For each problem, use the stated assessment rules and transition assumptions. Probabilities are reported to four decimals.
- A hypothetical contract assesses six ADLs and requires assistance with at least three for ADL eligibility. Classify counts 2, 3, and 5. (Answer: count 2 is ineligible; counts 3 and 5 are eligible.)
- For threshold 2, counts 0 and 1 are ineligible. Their intensities into the eligible block are 0.04 and 0.16 per year. The conditional proportions at counts 0 and 1 are 0.25 and 0.75. Find the aggregate entry intensity. (Answer: 0.130 per year.)
- For a block containing exact states 0 and 1, the intensities from each state to every target block are equal at each time. Does this satisfy the strong lumpability condition for the grouped process? (Answer: Yes, provided the equality holds for every target block and at every time.)
- An insured starts below a threshold and has one transition into eligibility at intensity 0.10 per year and death at intensity 0.05 per year. Both intensities are constant, and there are no other transitions before eligibility. Find the probability of entering eligibility before death within one year. (Answer: 0.0929.)
- An insured starts at count 1. The only required transitions to reach threshold 3 are \(1\to2\) and \(2\to3\), each at intensity 0.10 per year. Death has intensity 0.10 per year in both states, and all rates are constant. Find the probability of reaching count 3 within one year before death. (Answer: 0.0044.)