Population hazard of a survival mixture ( $\overline{h} (t) = \frac{\sum _{i} F _{i} ( t ) h _{i} ( t )}{\sum _{i} F _{i} ( t )}$ )

For an equally weighted mixture of individual survivor functions

F_{i}

, the population hazard function is

\overline{h} (t) = \sum_{i} F_{i} (t) h_{i} (t) / \sum_{i} F_{i} (t)

. Thus the weights are the surviving proportions, and the cumulative hazard function is

- lo g (n^{- 1} \sum_{i} e^{- H_{i} (t)})

, not generally the mean individual integrated hazard.

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Decreasing population hazard under constant individual hazards ( $\overline{h}^{'} (t) = - Var (Λ ∣ T > t) \leq 0$ )

For a nonnegative random constant individual hazard

Λ

, write

M_{k} (t) = E [Λ^{k} e^{- Λ t}]

. The population survivor function is

M_{0}

and its hazard is

M_{1} / M_{0}

. Under the integrability needed to differentiate,

M_{k}^{'} = - M_{k + 1}

, so

\overline{h}^{'} = \frac{- M _{2} M _{0} + M _{1}^{2}}{M _{0}^{2}} = - Var (Λ ∣ T > t) .

The conditional distribution is weighted by

e^{- Λ t}

. This proves decreasing population hazard without any decrease in individual hazards and explains survival selection in a heterogeneous population.

Survival selection in a heterogeneous population ( $w_{i} (t) = \frac{p _{i} S _{i} ( t )}{\sum _{j} p _{j} S _{j} ( t )}$ )

In a population mixture with initial component probabilities

p_{i}

, conditioning on survival to time

t

changes them to the displayed probabilities by Bayes' theorem. Components with larger survivor functions receive greater relative weight among survivors. The population hazard function is the weighted mean of the component hazards with these time-dependent weights. This selection can change the population hazard even when all component hazards are constant, as shown by the hazard derivative for a mixture of exponential distributions.

Hazard derivative for a mixture of exponential distributions ( $h^{'} (t) = - Var (Λ ∣ T > t)$ )

For a finite mixture with positive weights

p_{i}

λ_{i} > 0

, the survivor function is

S (t) = \sum_{i} p_{i} e^{- λ_{i} t}

and the hazard function is the surviving-population mean of the component rates. Write

w_{i} (t) = p_{i} e^{- λ_{i} t} / S (t)

. Differentiation gives

w_{i}^{'} = w_{i} (h - λ_{i})

and therefore

h^{'} = h^{2} - \sum_{i} w_{i} λ_{i}^{2} = - Var (Λ ∣ T > t)

. The hazard is decreasing, strictly so when distinct rates have positive weights. For finitely many rates it tends to the smallest rate with positive weight. This is survival selection even though each individual component has a constant hazard.

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