Calculate maximum likelihood estimates of transition intensities and probabilities for multiple state and multiple decrement models, assuming piecewise constant transition intensities.

You watch a portfolio of disability-income policies for a year. Twenty-three active lives became disabled, nine died directly from the active state, and fourteen disabled lives died. The active cohort logged 1,840 person-years of exposure; the disabled cohort logged 312. Those five numbers, plus the assumption that intensities are constant inside an age band, are everything you need to build the MLE.

A life under observation contributes two kinds of information. While occupying state $i$, it contributes a survivorship factor $\exp(-\mu^{i\bullet} t)$, where $\mu^{i\bullet} = \sum_{j \neq i} \mu^{ij}$ is the total force out of $i$. At a transition $i \to j$, it contributes the intensity $\mu^{ij}$.

$$L = \prod_{i} \left[ \exp\!\left(-\mu^{i\bullet} v^{i}\right) \prod_{j \neq i} \left(\mu^{ij}\right)^{d^{ij}} \right]$$

Here $v^{i}$ is the total central exposure in state $i$ (sum of time all lives spent there) and $d^{ij}$ counts observed $i \to j$ transitions.

Logging converts the product into a sum, so each $(i,j)$ pair contributes an independent term you can maximize on its own.

Common mistakes

• Dividing by wrong exposure. $\hat\mu^{12}$ uses disabled exposure 312, not active 1,840. Mixing them gives 14/1840 = 0.00761, far too low.
• Forgetting censored time. A life observed for only half the band contributes 0.5 person-years, not 1.0. Skipping censoring inflates denominators.
• Using $d^{ij}/(d^{ij} + \text{still in }i)$. That is an empirical probability, not an intensity. The denominator must be exposure time, not headcount.

Bottom line

• Core MLE: $\hat\mu^{ij} = d^{ij} / v^{i}$. Transitions out of $i$ into $j$ over total waiting time in $i$.
• Independent estimation: the likelihood factors by origin-destination pair, so each $\mu^{ij}$ is maximized on its own.
• Piecewise constant means $\mu^{ij}$ is flat inside each age band; you re-estimate for each band.
• Right exposure: the denominator is time in origin state $i$ only. Time in state $j$ never enters $\hat\mu^{ik}$ for $i \neq j$.

Exam shortcut

Write down a "transitions over exposure" table before any algebra. Rows are origin states, columns are destinations, cells are $d^{ij}/v^{i}$. The diagonal stays blank. DECISION: One band asked → use $e^{-\mu^{i\bullet} h}$ and the ratio formula directly. Multiple bands asked → chain via ${}_{a+b}p^{ii} = {}_{a}p^{ii} \cdot {}_{b}p^{ii}$ with each factor band-specific. For competing decrements, $q^{(j)} = (\mu^{(j)}/\mu^{(\tau)})(1 - e^{-\mu^{(\tau)}})$. Memorize the split-then-scale pattern; it recurs everywhere intensities are constant.

The full lesson (about 1,798 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1h

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