Chapter 9 Duration Models

9.1 Introduction

Examples: Typical economic examples for duration responses.

  • Macroeconomics.
    \(t_i\): Duration of strike \(i\).
    Potential covariates: Macroeconomic indicators, …
  • Labor economics
    \(t_i\): Duration of unemployment of person \(i\).
    Potential covariates: Education, age, gender, …
  • Marketing.
    \(t_i\): Duration between two purchases of person \(i\).
    Potential covariates: Price, price of alternative products, …
  • Demography/population economics.
    \(t_i\): Age of woman \(i\) at birth of first child.
    Potential covariates: Education, income, …

Remarks:

  • Duration: Non-negative, continuous random variable.
  • In practice: Discrete durations in weeks/months are often treated like continuous variables.
  • A lot of terminology stems from epidemiology/biostatistics (survival analysis), and technical statistics (analysis of failure time data).
  • Related to count data.
  • Count data: Often number of events in a time interval.
  • Duration data: Often time between subsequent events.
  • Censoring: Durations are often only observed up to a certain maximal time interval. Thus, observations for which the event of interest has not occurred in the time interval are right-censored.
  • In social sciences: Event history analysis typically encompasses analysis of count data and duration data.

9.2 Basic Concepts

Duration: The random variable \(T\) measures duration until event, also known as survival time or spell length.

  • \(T > 0\),
  • \(T\) has CDF \(F(t) = P(T \le t)\),
  • \(T\) has PDF \(f(t) = F'(t)\).

Survivor function: \(S(t)\) is the proportion of events that has not yet occurred at time \(t\).

\[\begin{equation*} S(t) ~=~ P(T \ge t) ~=~ 1 ~-~ F(t). \end{equation*}\]

Hazard function: \(h(t)\) measures the risk of an event at time \(t\) given survival/duration up to time \(t\).

\[\begin{equation*} h(t) ~=~ \lim_{\delta \rightarrow 0} \frac{P(t \le T < t + \delta ~|~ T \ge t)}{\delta}. \end{equation*}\]

Also known as hazard rate, failure rate, or exit rate.

Cumulative hazard function:

\[\begin{equation*} H(t) ~=~ \int_0^t h(u) ~ du. \end{equation*}\]

Remark: \(f(t)\), \(S(t)\), \(h(t)\), and \(H(t)\) are equivalent ways of defining or characterizing a specific duration pattern uniquely.

Relationships:

\[\begin{eqnarray*} h(t) & = & \frac{f(t)}{S(t)}, \\[0.3cm] f(t) & = & F'(t) ~=~ -S'(t), \\[0.3cm] h(t) & = & - \frac{S'(t)}{S(t)} ~=~ - \frac{\partial \log S(t)}{\partial t}, \\[0.3cm] H(t) & = & - \log S(t), \\[0.3cm] S(t) & = & \exp (-H(t)). \end{eqnarray*}\]

Kaplan-Meier estimator: Nonparametric estimator of survivor function. Also called product limit estimator.

Idea: Assume \(n\) observations of survival times (durations) without censorings occurring at \(p\) distinct times

\[\begin{equation*} t_{(1)} ~<~ \dots ~ < t_{(p)} \end{equation*}\]

and let \(d_i\) be the number of deaths (events/exits/…) at \(t_{(i)}\). Then \[\begin{eqnarray*} \hat S(t) & = & 1 - \hat F(t) ~=~ \frac{n - \sum_{j = 1}^s d_j}{n} \qquad (t_{(s)} \le t < t_{(s + 1)}) \\ & = & \frac{n - d_1}{n} \cdot \frac{n - d_1 - d_2}{n - d_1} \ldots \frac{n - d_1 - \ldots - d_s}{n - d_1 - \ldots - d_{s - 1}} \\ & = & \left( 1 - \frac{d_1}{r_1} \right) \ldots \left( 1 - \frac{d_s}{r_s} \right) ~=~ \prod_{j = 1}^s \left( 1 - \frac{d_j}{r_j} \right). \end{eqnarray*}\]

Number at risk: \(r_i\), i.e., alive, just before \(t_{(i)}\).

Without censoring: \(r_{i+1} = r_i - d_i\). Number alive at \(t_{(i+1)}\) are those from \(t_{(i)}\) without the deaths from \(t_{(i)}\).

With censoring: \(r_{i+1} = r_i - d_i - c_{i + 1}\). Additionally, remove censorings in the interval (\(t_{(i-1)}, t_{(i)}\)).

Alternatively: Employ \(r_{i+1} = r_i - d_i - c_{i + 1}/2\) as a continuity-corrected version. Also known as lifetable method.

Remark: If there are censorings after the last event \(\hat S(t) > 0\) for all \(t\).

In R: Methods for survival analysis in package survival.

Basic utility:

  • Declare survival times as such using Surv(), encompassing information about censoring (if any).
  • Surv(time, time2, event, type, origin = 0) can declare left-, right-, and interval-censored data.
  • Basic usage for right-censored data has an indicator event where 0 corresponds to “alive”, and 1 to “dead” (or “with event”).

Example: Age at first birth (AFB) in US General Social Survey 2002. AFB is only known for women that have children. For women without children (but within child-bearing age) AFB is censored at their current age.

Load data and select subset:

data("GSS7402", package = "AER")
gss2002 <- subset(GSS7402,
  year == 2002 & (agefirstbirth < 40 | age < 40))

Compute censored AFB and declare censoring:

gss2002$afb <- with(gss2002, ifelse(kids > 0, agefirstbirth, age))
gss2002$afb <- with(gss2002, Surv(afb, kids > 0))

Observations now contain censoring information:

head(gss2002$afb)
## [1] 25+ 19  27  22  29  19+

Compute Kaplan-Meier estimator via survfit():

afb_km <- survfit(afb ~ 1, data = gss2002)
afb_skm <- summary(afb_km)
print(afb_skm)
## Call: survfit(formula = afb ~ 1, data = gss2002)
## 
##  time n.risk n.event survival  std.err lower 95% CI upper 95% CI
##    11   1371       1   0.9993 0.000729       0.9978       1.0000
##    13   1370       3   0.9971 0.001457       0.9942       0.9999
##    14   1367       2   0.9956 0.001783       0.9921       0.9991
##    15   1365      14   0.9854 0.003238       0.9791       0.9918
##    16   1351      40   0.9562 0.005525       0.9455       0.9671
##    17   1311      66   0.9081 0.007802       0.8929       0.9235
##    18   1245      97   0.8373 0.009967       0.8180       0.8571
##    19   1148     106   0.7600 0.011534       0.7378       0.7830
##    20   1030     122   0.6700 0.012726       0.6455       0.6954
##    21    898     123   0.5782 0.013406       0.5525       0.6051
##    22    767      97   0.5051 0.013612       0.4791       0.5325
##    23    654      72   0.4495 0.013600       0.4236       0.4770
##    24    563      61   0.4008 0.013480       0.3752       0.4281
##    25    484      60   0.3511 0.013248       0.3261       0.3781
##    26    407      45   0.3123 0.012986       0.2878       0.3388
##    27    351      45   0.2723 0.012618       0.2486       0.2981
##    28    294      37   0.2380 0.012223       0.2152       0.2632
##    29    246      32   0.2070 0.011795       0.1852       0.2315
##    30    209      38   0.1694 0.011119       0.1489       0.1926
##    31    163      18   0.1507 0.010730       0.1311       0.1733
##    32    135      19   0.1295 0.010264       0.1108       0.1512
##    33    108      17   0.1091 0.009766       0.0915       0.1300
##    34     83       7   0.0999 0.009542       0.0828       0.1205
##    35     65      13   0.0799 0.009101       0.0639       0.0999
##    36     45       7   0.0675 0.008815       0.0522       0.0872
##    37     29       7   0.0512 0.008572       0.0369       0.0711
##    38     17       3   0.0422 0.008499       0.0284       0.0626
##    39     12       2   0.0351 0.008411       0.0220       0.0562
plot(afb_km, conf.int = FALSE)

with(afb_skm, plot(n.event/n.risk ~ time, type = "s"))

Remarks:

  • \(\hat h(t) = d_s / r_s\) for \(t_{(s)} \le t < t_{(s + 1)}\) is a less reliable estimator for the hazard function. Grouping or smoothing (similar to histograms or kernel densities) would be necessary but rare in practice.
  • Parametric approach: Estimate distribution parameters (and thus the associated survivor and hazard functions).
  • Moreover: Regression models are required that capture how survivor and hazard functions change with covariates.
  • Simple groupings can be incorporated nonparametrically in survfit(y ~ group).

Example: For data exploration, one could consider:

gss2002$edcat <- cut(gss2002$education, c(0, 12, 14, 20))
afb_km2 <- survfit(afb ~ edcat, data = gss2002)
plot(afb_km2, col = c(1, 2, 4))

Observations: Survivor functions for different groups look very similar. This conveys that hazards are similar up to a factor. This is often exploited in modeling and inference.

Formally: The assumption of proportional hazards for two groups entails

\[\begin{eqnarray*} h_2(t) & = & c \cdot h_1(t), \qquad \mbox{(or equivalently)}\\ S_2(t) & = & S_1(t)^c. \end{eqnarray*}\]

Implication: Survivor functions cannot cross.

Remark: Violations in practice are more serious for lower death/exit times (which are based on more observations).

9.3 Continuous Time Duration Models

Basic model: Constant hazard rate \(h(t) = \lambda\) (independent of duration \(t\)) is the simplest conceivable model for duration data.

Formally: Exponential distribution with hazard rate \(\lambda > 0\).

\[\begin{eqnarray*} f(t) & = & \lambda ~ \exp (- \lambda ~ t),\\ S(t) & = & \exp (- \lambda ~ t), \\ h(t) & = & \lambda. \end{eqnarray*}\]

Remarks:

  • Constant hazard rate is also called lack of memory.
  • In practice nonconstant hazards (depending on \(t\)) are relevant. In economics called duration dependence.

In R: d/p/q/rexp with parameter \(\mathtt{rate} = \lambda\).

Generalization: Weibull distribution with additional shape parameter \(\alpha > 0\).

\[\begin{eqnarray*} f(t) & = & \lambda ~ \alpha ~ t^{\alpha - 1} ~ \exp (- \lambda ~ t^\alpha),\\ S(t) & = & \exp (- \lambda ~ t^\alpha), \\ h(t) & = & \lambda ~ \alpha ~ t^{\alpha - 1}. \end{eqnarray*}\]

Properties: The hazard function is always monotonic, for

  • \(\alpha > 1\) increasing,
  • \(\alpha = 1\) constant (exponential distribution),
  • \(\alpha < 1\) decreasing.

In R: d/p/q/rweibull with parameters \(\mathtt{scale} = \lambda^{-1/\alpha}\) and \(\mathtt{shape} = \alpha\). (Further parametrizations exist in the literature.)

Example: For \(\lambda = 1\) and \(\alpha = 1/4, 1, 4\).

Furthermore: Distributions with potentially non-monotonic hazard functions.

  • Lognormal distribution: \(\log(t) \sim \mathcal{N}(\mu, \sigma^2)\).
  • Loglogistic distribution.

9.4 Proportional Hazard Models

Idea: Assume proportional hazards for observations \(i = 1, \ldots, n\).

\[\begin{equation*} h_i(t) ~=~ \lambda_i \cdot h_0(t), \end{equation*}\]

where \(h_0(t)\) is called baseline hazard.

Regression: The factor \(\lambda_i\) is allowed to depend on a set of regressors \(x_i\), typically using a linear predictor and a log link.

\[\begin{equation*} \log(\lambda_i) ~=~ x_i^\top \beta. \end{equation*}\]

Thus: Baseline hazard corresponds to observation with \(x = 0\).

Parameter estimation: Two approaches for estimation of \(\beta\).

  1. parametric: \(h_0(t, \alpha)\) parametric function,
  2. semi-parametric: leave \(h_0(t)\) unspecified.

Parametric regression models: Most common models are

  • Exponential: \(h_0(t) = 1\).
  • Weibull: \(h_0(t) = \alpha ~ t^{\alpha - 1}\).

Inference: Via maximum likelihood, e.g., for exponential case.

\[\begin{eqnarray*} L(\beta) & = & \prod_{i = 1}^n \left\{\lambda_i \cdot \exp(- \lambda_i t_i)\right\}^{\delta_i} \exp(- \lambda_i t_i)^{1 - \delta_i} \\ & = & \prod_{i = 1}^n \lambda_i^{\delta_i} \exp(- \lambda_i t_i) \\ \log L(\beta) & = & \sum_{i = 1}^n \delta_i \cdot x_i^\top \beta - \sum_{i = 1}^n \exp(x_i^\top \beta) \cdot t_i, \end{eqnarray*}\]

where \(\delta_i = 1\) indicates an event and \(\delta_i = 0\) censoring.

In R: survreg() from package survival.

survreg(formula, data, subset, na.action, dist, ...)

with

  • formula: Response must be a "Surv" object from Surv().
  • dist: Distribution (implying link and variance functions etc.). Available distributions include "weibull" (default), "exponential", "logistic", "gaussian", "lognormal", "loglogistic".
  • Similar in spirit to GLMs. However, even without censoring not always special cases of exponential families.
  • Slightly different parametrization: Reports estimates for coefficients \(\tilde \beta\) and “scale” parameter \(\sigma\).
  • Corresponds to \(\beta = - \tilde \beta/\sigma\) in previous notation.
  • For Weibull: \(\alpha = 1/\sigma\).

Semiparametric regression: Cox proportional hazards model.

Idea: Maximum likelihood requires specification of the hazard, hence construct the conditional likelihood. Instead of \(P(T_i = t_i)\) use

\[\begin{equation*} P(T_i = t_i ~|~ \mbox{one event at } t_i). \end{equation*}\]

This leads to

\[\begin{equation*} \frac{h_0(t_i) \exp(x_i^\top \beta)}{\sum_{j: t_j \ge t_i} h_0(t_i) \exp(x_j^\top \beta)} ~=~ \frac{\exp(x_i^\top \beta)}{\sum_{j: t_j \ge t_i} \exp(x_j^\top \beta)} \end{equation*}\]

and thus, the conditional likelihood is

\[\begin{equation*} L(\beta) ~=~ \prod_{i = 1}^n \left(\frac{\exp(x_i^\top \beta)}{\sum_{j: t_j \ge t_i} \exp(x_j^\top \beta)}\right)^{\delta_i}. \end{equation*}\]

If there are censorings this is called {partial likelihood}.

In R: coxph() from survival.

Inference: For both parametric and semi-parametric models asymptotic normality can be shown. Hence, the usual inference (LR/Wald/score test, confidence intervals, information criteria) can be performed.

Interpretation: Similar to semi-logarithmic models. For observations \(x_1\) and \(x_2\)

\[\begin{equation*} \frac{h_1(t)}{h_2(t)} ~=~ \exp( (x_1 - x_2)^\top \beta), \end{equation*}\] independent of \(t\).

Example: Determinants of AFB. Employ model formula

afb_f <- afb ~ education + siblings + ethnicity + immigrant + lowincome16 + city16

Of increased interest: influence of education.

afb_ex <- survreg(afb_f, data = gss2002, dist = "exponential")
summary(afb_ex)
## 
## Call:
## survreg(formula = afb_f, data = gss2002, dist = "exponential")
##                  Value Std. Error     z       p
## (Intercept)     2.7068     0.1703 15.90 < 2e-16
## education       0.0468     0.0108  4.34 1.4e-05
## siblings       -0.0155     0.0103 -1.51    0.13
## ethnicitycauc   0.0456     0.0721  0.63    0.53
## immigrantyes    0.0940     0.0926  1.02    0.31
## lowincome16yes -0.0272     0.0738 -0.37    0.71
## city16yes       0.0372     0.0605  0.61    0.54
## 
## Scale fixed at 1 
## 
## Exponential distribution
## Loglik(model)= -4978   Loglik(intercept only)= -4994
##  Chisq= 31.37 on 6 degrees of freedom, p= 2.2e-05 
## Number of Newton-Raphson Iterations: 3 
## n= 1371
coeftest(afb_ex)
## 
## z test of coefficients:
## 
##                Estimate Std. Error z value Pr(>|z|)
## (Intercept)      2.7068     0.1703   15.90  < 2e-16
## education        0.0468     0.0108    4.34  1.4e-05
## siblings        -0.0155     0.0103   -1.51     0.13
## ethnicitycauc    0.0456     0.0721    0.63     0.53
## immigrantyes     0.0940     0.0926    1.02     0.31
## lowincome16yes  -0.0272     0.0738   -0.37     0.71
## city16yes        0.0372     0.0605    0.61     0.54
afb_wb <- survreg(afb_f, data = gss2002, dist = "weibull")
coeftest(afb_wb)
## 
## z test of coefficients:
## 
##                Estimate Std. Error z value Pr(>|z|)
## (Intercept)     2.89004    0.03718   77.73  < 2e-16
## education       0.02563    0.00229   11.17  < 2e-16
## siblings       -0.00733    0.00234   -3.13  0.00177
## ethnicitycauc   0.06330    0.01640    3.86  0.00011
## immigrantyes    0.03429    0.02095    1.64  0.10164
## lowincome16yes  0.00701    0.01672    0.42  0.67496
## city16yes       0.02575    0.01366    1.89  0.05941
## Log(scale)     -1.48673    0.02137  -69.56  < 2e-16

Comparison: Exponential and Weibull model can be compared using the standard likelihood methods.

lrtest(afb_ex, afb_wb)
## Likelihood ratio test
## 
## Model 1: afb ~ education + siblings + ethnicity + immigrant + lowincome16 + 
##     city16
## Model 2: afb ~ education + siblings + ethnicity + immigrant + lowincome16 + 
##     city16
##   #Df LogLik Df Chisq Pr(>Chisq)
## 1   7  -4979                    
## 2   8  -3836  1  2284     <2e-16
AIC(afb_ex, afb_wb)
##        df  AIC
## afb_ex  7 9971
## afb_wb  8 7689
afb_cph <- coxph(afb_f, data = gss2002)
coeftest(afb_cph)
## 
## z test of coefficients:
## 
##                Estimate Std. Error z value Pr(>|z|)
## education       -0.1138     0.0103  -11.04  < 2e-16
## siblings         0.0296     0.0103    2.86   0.0042
## ethnicitycauc   -0.2907     0.0730   -3.98  6.8e-05
## immigrantyes    -0.2011     0.0928   -2.17   0.0303
## lowincome16yes   0.0304     0.0742    0.41   0.6820
## city16yes       -0.0689     0.0607   -1.13   0.2564

Comparison: Parameter estimates.

cf <- cbind(
  "Exponential" = -coef(afb_ex),
  "Weibull" = -coef(afb_wb)/afb_wb$scale,
  "Cox-PH" = c(NA, coef(afb_cph))
)
cf
##                Exponential  Weibull   Cox-PH
## (Intercept)       -2.70684 -12.7815       NA
## education         -0.04679  -0.1133 -0.11380
## siblings           0.01549   0.0324  0.02961
## ethnicitycauc     -0.04560  -0.2800 -0.29069
## immigrantyes      -0.09401  -0.1517 -0.20108
## lowincome16yes     0.02725  -0.0310  0.03039
## city16yes         -0.03722  -0.1139 -0.06887

Hazard proportions:

exp(cf[-1,]) - 1
##                Exponential  Weibull   Cox-PH
## education         -0.04571 -0.10715 -0.10756
## siblings           0.01561  0.03293  0.03006
## ethnicitycauc     -0.04458 -0.24419 -0.25225
## immigrantyes      -0.08973 -0.14072 -0.18215
## lowincome16yes     0.02762 -0.03053  0.03086
## city16yes         -0.03654 -0.10765 -0.06655

Compare survivor functions at \(\bar x\):

xbar <- colMeans(model.matrix(afb_ex, data = gss2002))

Distribution parameters:

lambda_ex <- exp(c(- xbar %*% coef(afb_ex)))
lambda_wb <- exp(c(- xbar %*% coef(afb_wb)) / afb_wb$scale)
alpha_wb <- 1/afb_wb$scale

Visualization (using Kaplan-Meier at \(\bar x\) for Cox-PH):

plot(afb_km, conf.int = FALSE)
tt <- 0:400/10
lines(tt, exp(-lambda_ex * tt), col = 3)
lines(tt, exp(-lambda_wb * tt^alpha_wb), col = 4)
lines(survfit(afb_cph), col = 2)

Alternatively: Using R’s own exponential and Weibull functions requires reparametrization, i.e.

pexp(tt, rate = lambda_ex, lower.tail = FALSE)
pweibull(tt, scale = lambda_wb^(-1/alpha_wb), shape = alpha, lower.tail = FALSE)

Conclusions:

  • Exponential is clearly inappropriate.
  • Weibull and Cox-PH lead to similar results.
  • Cox-PH has least restrictive assumptions.

9.5 Outlook

Alternative models: Instead of proportional hazards (PH) models, employ accelerated failure time (AFT) models.

\[\begin{equation*} \log(t_i) ~=~ x_i^\top \beta ~+~ \varepsilon_i, \end{equation*}\]

e.g., leading to lognormal model (if \(\varepsilon\) is normal) or loglogistic model (if \(\varepsilon\) is logistic).

Failure time acceleration can be seen via reparametrization

\[\begin{equation*} \log(\psi_i \cdot t_i) ~=~ \varepsilon_i, \end{equation*}\]

with \(\psi_i = \exp(- x_i^\top \beta)\) leading to shortened/lengthened survival times for \(\psi_i > 1\) and \(\psi_i < 1\), respectively.

Weibull/exponential model is the only model that is both of AFT and PH type.

Unobserved heterogeneity: Account for via additional random effects. In duration analysis called frailty models. Example: Exponential duration with gamma frailty

\[\begin{equation*} \lambda_i ~=~ \nu \cdot \exp(x_i^\top \beta), \end{equation*}\]

where \(\nu \sim \mathit{Gamma}(\theta, \theta)\) leading to Pareto type~II distribution.

Further extensions:

  • Need to incorporate time-varying regressors \(x_{it}\).
  • Competing risks: Deaths/exits can be caused by different types of events.
    Example: Unemployment can be terminated by job or withdrawal from labor market.