Statistics 262: IntermediateBiostatistics

Modelling Hazard: Parametric Models

Jonathan Taylor & Kristin Cobb

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Today’s class

Hazard rates (yes, again).

Parametric models: accelerated failuremodels.

Likelihood.

Diagnostics.

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Hazard Rates & Densities

f(t)dt = P (t ≤ T < t + dt) .

h(t)dt = P(t ≤ T < t + dt

∣∣T > t)

=f(t)dt

S(t)

Both model instantaneous probability offailure at time t, but hazard is conditional onsurviving up to time t.

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Hazard Rates & Densities

Conceptually, it can be easier to modelhazard than density.

Hazard incorporates “what-ifs”: “If you surviveup to 100 days, then the probability you willfail within a day is ' h(100).”

Density does not: “The probability you will failon the 101-st day is ' f(100).”

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Weibull

0.0 0.5 1.0 1.5 2.0

01

23

45

Time

Haz

ard

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Log-logistic

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

Time

Haz

ard

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Log-normal

0 2 4 6 8 10

0.0

0.5

1.0

1.5

Time

Haz

ard

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Accelerated failure model

Log-linear model Wi = log Zi, V ar(Wi) = σ2:

log Ti = β0 +

p∑

j=1

Xijβj + Wi

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Accelerated failure model

Survival function:

S(t∣∣β,X

)= SZ

(exp

(β0 +

p∑

j=1

Xijβj

)t

).

Hazard rate:

h(t∣∣β,X

)= exp

(β0 +

p∑

j=1

βjXj

)

× hZ

(exp

(β0 +

p∑

j=1

βjXj

)t

)

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Accelerated failure model

“Accelerated” refers to the fact that increasein linear predictor “accelerates” your positionalong the hazard curve.

All variables have the same shape of hazardfunction, hZ.

Covariates act multiplicatively in front ofhazard and on the time scale.

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Likelihood of model

L(β, σ|(ti, δi), 1 ≤ i ≤ n)

∝

n∏

i=1

(1 − F (ti|β, σ))1−δif(ti|β, σ)δi.

Only difference with this and Kaplan-Meier isthat F is a parametric model so likelihood isactually a function of parameters β, σ.

Maximization is carried out numerically.

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Interpretation of coefficients

Coefficient βj affects median (or any otherquantile of) survival time.

Specifically,

eβj =t50(X1, . . . , Xj + 1, . . . , Xp)

t50(X1, . . . , Xj, . . . , Xp)

= TR((X1, . . . , Xj + 1, . . . , Xp),

(X1, . . . , Xj + 1, . . . , Xp))

Another reason why it is called “accelerated”failure: i.e. for every cigarette you smoke,your median survival time goes down by afactor eβcigarette.

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Fisher information

Unlike linear models there is no explicitformula for variance / covariance ofparameter estimates (β̂, σ̂).

V̂ ar(σ̂, β̂) =

(−

∂2 log L

∂σ∂β

)−1 ∣∣∣∣(β̂,σ̂)

.

Generic way to estimate variance/covarianceof MLE estimates. Obeys “delta rule.”

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Example: variance of sample mean

If σ2 is known

− log L(µ|Z1, . . . , Zn) ∝1

2σ2

(‖Z‖2 − 2nZµ + nµ2

).

Second derivative gives:

−∂2

∂µ2log L(µ|Z1, . . . , Zn) =

n

σ2.

“Inverse” yields

V̂ ar(Z) =σ̂2

n.

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Residual diagnostics: Cox-Snell

Like regression models, to assess goodnessof fit, we use residuals.

Parametric model imposes a (cumulative)hazard function H(t,X, β). Leads toCox-Snell residuals (similar to “QQplot”)

ri = H(ti, Xi, β̂).

If model is correct, the collection (ri, δi)1≤i≤n

should be like a sample from a censoredExp(1) sample. Plot of ri vs. Nelson-Aalenestimator should be linear with slope 1.

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Residual diagnostics: martingale

Mi = δi − ri.

Martingale residuals can be used to seerelation between j-th covariate and hazard byplotting (Xij , ri) and using “lowess” or othersmoother.

Residuals are motivated the Cox model, anddefinitions are basically identical. We’ll revisitthem later.

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Example: HMO-HIV+

We will look at many models:Exponential: only drug.Weibull: only drug.Weibull: drug + age.Lognormal: drug + age.Loglogistic: drug + age.

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Exponential vs. Weibull

Exponential is special case of Weibull withα = 1.

Can test if exponential is appropriate bytesting α = 1 or not.

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Cox-snell: exponential

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

1.0

2.0

3.0

Model hazard

Est

imat

ed h

azar

d

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Cox-snell: Weibull

0 1 2 3 4 5

01

23

Model hazard

Est

imat

ed h

azar

d

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Martingale: Weibull

20 25 30 35 40 45 50 55

−4

−3

−2

−1

01

Age

Mar

tinga

le r

esid

ual

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