Encoding Care Levels and Permitted Deterioration

Why Care-Level Encoding Matters

A long-term care model must distinguish how much care a person needs from whether the person is alive. Someone moving from moderate to severe care has deteriorated, but has not died. If those outcomes are combined in the model, probabilities of death and periods of benefit eligibility can both be calculated incorrectly.

This tutorial represents care levels as separate states and encodes which transitions are permitted. The examples use a discrete-time model with annual transitions. This is a convenient way to specify and check a model’s state structure; the probabilities are hypothetical and are not industry assumptions.

Definition: A state is one of the mutually exclusive conditions an individual can occupy at a specified time. In a simple long-term care model, let $H$ denote alive and not receiving qualifying care, $M$ denote alive and receiving moderate care, $S$ denote alive and receiving severe care, and $D$ denote dead. Care levels describe different conditions while alive; $D$ is a separate absorbing state.

The care labels must be defined so that a person occupies exactly one of them at a time. A model might, for example, classify an individual as severe whenever the severe-care criteria are met, and otherwise classify the individual as moderate if the moderate-care criteria are met. The model specification must resolve any overlap between clinical criteria; otherwise, the state assignment is ambiguous.

For this tutorial, deterioration means movement toward a more intensive care level: $H$ to $M$, or $M$ to $S$. Whether a direct transition such as $H$ to $S$ is permitted is a model assumption, not something implied by the labels. If direct transitions are ruled out, their probabilities are zero. Death remains distinct from every care-level change.

Encoding Permitted Transitions

Let $p_{ij}$ be the probability that an individual in state $i$ at the start of a one-year period is in state $j$ at its end. Use the state order $H,M,S,D$, and put these probabilities in a transition matrix $P$, with the starting state in the row and the ending state in the column. Each row must sum to one. A zero in the matrix records a transition ruled out by the model for that period.

In a model that allows only gradual deterioration, permits death from each living state, and treats death as absorbing, one possible structure is:

P= \begin{pmatrix} p_{HH} & p_{HM} & 0 & p_{HD}\\ 0 & p_{MM} & p_{MS} & p_{MD}\\ 0 & 0 & p_{SS} & p_{SD}\\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad \sum_j p_{ij}=1\text{ for every state }i.

This structure is an assumption, not a universal rule. For example, if the model permits someone in $H$ to enter care directly at the severe level, the entry in row $H$, column $S$ must be modeled rather than set to zero. Conversely, a positive entry for a transition that the model says is impossible is a coding error.

The matrix also makes the separation between deterioration and death explicit. The transition $M\to S$ belongs in the severe-care column, not in the death column. The death probability from $M$ is $p_{MD}$, and the two outcomes are distinct even if both are adverse events.

Worked Example: Two-Year Outcomes With No Direct Jump to Severe Care

Suppose a hypothetical annual model has states $H,M,S,D$ and transition matrix

P= \begin{pmatrix} 0.78&0.16&0&0.06\\ 0&0.68&0.22&0.10\\ 0&0&0.72&0.28\\ 0&0&0&1 \end{pmatrix}.

An individual starts in $H$. Calculate the probabilities of occupying each state after two years. In particular, calculate the probability of death by then and the probability of being in severe care.

Method. Add the probabilities of all two-period paths that end in the requested state. Equivalently, use the initial row vector $(1,0,0,0)$ and multiply it by $P$ twice. The matrix says that a direct $H\to S$ transition is not allowed.

The only two-year path from $H$ to $H$ is $H\to H\to H$, so $P(X_2=H)=0.78(0.78)=0.6084$.

To end in $M$, the paths are $H\to H\to M$ and $H\to M\to M$: $P(X_2=M)=0.78(0.16)+0.16(0.68)=0.1248+0.1088=0.2336$.

To end in $S$, the only possible path is $H\to M\to S$: $P(X_2=S)=0.16(0.22)=0.0352$. A path $H\to H\to S$ contributes zero because $p_{HS}=0$.

Death can occur in the first year, or in the second year after being alive in $H$ or $M$ at the end of the first year. Thus $P(X_2=D)=0.06+0.78(0.06)+0.16(0.10)=0.06+0.0468+0.0160=0.1228$.

The endpoint probabilities check: $0.6084+0.2336+0.0352+0.1228=1.0000$. The probability of severe care is $0.0352$, while the probability of death is $0.1228$; neither is included in the other. The calculation avoids the common mistake of counting a move into severe care as a death.

Why the Path Calculation Works

For a discrete-time Markov model, $p_{ij}$ is the one-period probability of moving from state $i$ to state $j$, conditional on being in $i$ at the period’s start. The Markov assumption says that, given the current state, the next-state distribution does not additionally depend on the earlier path. For two periods, the law of total probability gives the endpoint probability by summing over every possible intermediate state $k$:

P(X_2=j\mid X_0=i) =\sum_k P(X_1=k\mid X_0=i)P(X_2=j\mid X_1=k) =\sum_k p_{ik}p_{kj}.

The sum includes every state, but a forbidden transition contributes zero. Applying the same reasoning repeatedly gives $P(X_n=j\mid X_0=i)=(P^n)_{ij}$ for $n$ periods when the same transition matrix applies in each period. The formula depends on the specified discrete-time Markov model; it should not be used as though annual probabilities were continuous-time transition intensities.

Worked Example: A Mixed Starting Population

A hypothetical book has this one-year transition matrix, again ordered $H,M,S,D$:

P= \begin{pmatrix} 0.80&0.15&0&0.05\\ 0&0.65&0.20&0.15\\ 0&0&0.70&0.30\\ 0&0&0&1 \end{pmatrix}.

At issue, 70% of the individuals are in $H$ and 30% are in $M$. Calculate the state probabilities after one year and after two years.

Method. Represent the starting proportions as a row vector and multiply it by the transition matrix once for each year. This preserves the distinction between the starting state and the ending state.

The starting vector is $(0.70,0.30,0,0)$. After one year, the $H$ probability is $0.70(0.80)=0.56$. The $M$ probability is $0.70(0.15)+0.30(0.65)=0.105+0.195=0.30$. The $S$ probability is $0.30(0.20)=0.06$. The $D$ probability is $0.70(0.05)+0.30(0.15)=0.035+0.045=0.08$. These sum to $0.56+0.30+0.06+0.08=1.00$.

For the second year, use the one-year results as the new starting distribution. The probability of $H$ is $0.56(0.80)=0.448$. The probability of $M$ is $0.56(0.15)+0.30(0.65)=0.084+0.195=0.279$. The probability of $S$ is $0.30(0.20)+0.06(0.70)=0.060+0.042=0.102$. The probability of $D$ includes those already dead and new deaths from each living state: $0.08+0.56(0.05)+0.30(0.15)+0.06(0.30)=0.08+0.028+0.045+0.018=0.171$.

The two-year distribution is therefore $(0.448,0.279,0.102,0.171)$, and its entries sum to $1.000$. The calculation shows why a mixed starting population must be weighted by its initial state proportions: the same transition matrix applies, but individuals in $H$ and $M$ have different one-year possibilities.

State Encoding and Benefit Eligibility

A state model is useful only if its states distinguish the conditions that matter to the contract. A hypothetical policy might pay a monthly benefit in either moderate or severe care, but at different amounts. In that case, $M$ and $S$ must remain separate states even though both qualify for payment. If they were merged, the model would lose the information needed to apply different benefit amounts or represent deterioration between the care levels.

The following example uses an end-of-year state to illustrate this point. It calculates an expected benefit amount conditional on the model’s endpoint probabilities; it does not calculate a present value or a premium.

Worked Example: Keeping Severe Care Separate From Death

A hypothetical insured starts in moderate care. The annual transition probabilities from $M$ are $0.60$ to remain in $M$, $0.25$ to move to $S$, and $0.15$ to die. From $S$, the probabilities are $0.70$ to remain in $S$ and $0.30$ to die. Death is absorbing. A benefit paid at the end of year two is $800 per month in $M$, $1,400 per month in $S$, and zero in $D$. Find the year-two state probabilities and the expected monthly benefit amount at that time.

Method. Enumerate the two-year paths from the starting state. Then multiply each state’s probability by its applicable benefit. The transition $M\to S$ is deterioration, so it remains a path to an eligible care state rather than a path to death.

The probability of being in $M$ after two years is the probability of $M\to M\to M$: $0.60(0.60)=0.36$.

To be in $S$, the paths are $M\to M\to S$ and $M\to S\to S$. Therefore, $P(X_2=S)=0.60(0.25)+0.25(0.70)=0.15+0.175=0.325$.

Death by the end of year two includes death in year one, death in year two after remaining in $M$, and death in year two after moving to $S$: $P(X_2=D)=0.15+0.60(0.15)+0.25(0.30)=0.15+0.09+0.075=0.315$. The check is $0.36+0.325+0.315=1.000$.

The expected monthly benefit amount at the end of year two is $0.36(800)+0.325(1{,}400)+0.315(0)=288+455+0=\$743$. This is an expectation across the modeled outcomes, not a benefit paid to every individual. Treating the transition $M\to S$ as death would wrongly remove the $0.325$ severe-care probability and the associated benefit.

Common Encoding Errors

The most important check is to read each transition as a movement between named conditions, not as a generic adverse event. For the first worked example, the correct two-year death probability is $0.1228$. A mistaken calculation might add the probability of moving from $M$ to $S$ in year two as though that transition were death:

\text{Incorrect: }0.06+0.78(0.06)+0.16(0.10)+0.16(0.22)=0.1580.

The extra term $0.16(0.22)=0.0352$ is the probability of the path $H\to M\to S$, not a death path. The corrected death probability is $0.06+0.0468+0.0160=0.1228$. A second common error is to include a path such as $H\to H\to S$ when the model sets $p_{HS}=0$. A structural zero must remain zero in path calculations.

Keep the state meanings explicit. Severe care is an alive care state; death is a separate state.
Check every row. Transition probabilities from a state must sum to one, including the probability of remaining in that state.
Apply structural zeros. Do not invent a direct transition that the specified model forbids.
Check endpoint probabilities. The probabilities across all states at a fixed time should sum to one.

Key takeaway: Encode each materially different care level as its own state, specify permitted deterioration transitions explicitly, and represent death separately. This structure preserves the information needed to calculate care-state probabilities and apply state-dependent benefits.

Check Your Understanding

For each problem, use the stated state order and include only permitted transitions. All figures are hypothetical.

A one-year model has states $H,M,S,D$. From $H$, the probabilities of remaining in $H$, moving to $M$, and dying are $0.75,0.20,0.05$, with no direct move to $S$. Starting in $H$, give the one-year probability of each state. (Answer: $H=0.75,\ M=0.20,\ S=0,\ D=0.05$.)
In a model, the probabilities from $M$ are $0.65$ to remain in $M$, $0.20$ to move to $S$, and $0.15$ to die. From $S$, the probabilities are $0.70$ to remain in $S$ and $0.30$ to die. Starting in $M$, calculate the probabilities of $M,S,D$ after two years. (Answer: $M=0.4225,\ S=0.2700,\ D=0.3075$.)
A population starts with proportions $0.40$ in $H$ and $0.60$ in $M$. The annual probabilities from $H$ are $0.80$ to $H$, $0.15$ to $M$, and $0.05$ to $D$; from $M$, they are $0.65$ to $M$, $0.20$ to $S$, and $0.15$ to $D$. From $S$, they are $0.70$ to $S$ and $0.30$ to $D$. Find the one-year distribution. (Answer: $H=0.3200,\ M=0.4500,\ S=0.1200,\ D=0.1100$.)
For the same transition rules as question 2, an analyst reports the two-year death probability from $M$ as $0.15+0.65(0.15)+0.20=0.4475$, counting the $M\to S$ probability as death. Give the correct death probability and identify the endpoint probability wrongly counted. (Answer: $D=0.3075$; the $0.20$ term is the first-year probability of reaching $S$, not death, and its year-two death contribution is $0.20(0.30)=0.0600$.)
Starting in $M$, use these annual probabilities: from $M$, $0.60$ to $M$, $0.25$ to $S$, and $0.15$ to $D$; from $S$, $0.70$ to $S$ and $0.30$ to $D$. At the end of year two, the monthly benefit is $900 in $M$, $1,500 in $S$, and zero in $D$. Calculate the year-two state probabilities and expected monthly benefit. (Answer: $M=0.360,\ S=0.325,\ D=0.315$; expected monthly benefit $=\$811.50$.)

What You'll Learn