Parametric Survival Models (ch. 7)

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Parametric Survival Models (ch. 7)

• notations
• Y survival variable Ssurvival function,
  fdensity (pdf) Fcdf1-S. All these functios
  involve the unknown parameter (or vector of
  parameters) theta (??. The 2 examples well
  consider are
• exponential with
• Weibull with

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• The maximum likelihood estimator of
• and is a function of the
  observed Ys which maximizes the likelihood,
  L(???a constant multiple of the joint
  distribution of the observed data. So theta-hat
  maximizes L over all possible values of theta
• the mathematical method is discussed at the top
  of page 120 - note that usually the log of the
  likelihood is maximized

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• Ill list some of the important results given in
  this section
• define the score function as the derivative of
  the log-likelihood
• define Fishers information as
• then
• at the bottom of p. 120 and top of p.121, the
  method of actually finding maximum likelihood
  estimators is discussed and it is shown that

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• Then hypothesis tests about ? can be based on any
  of the 3 statistics below. To test use one of
  
• Wald test
• LR test
• Score test

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• Lets use the LR test to compare models (see p.
  179-180) - use the notation there
• The likelihood ratio test of the current model is
  
• The likelihood ratio test of the full model is
• Their difference (subtract the full minus the
  current likelihoods) is asymptotically chi-square
  with q d.f. and may be used to test whether the
  additional q parameters in the full model are
  zero.
• This difference is called the deviance

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• Now go to p.127, the exponential model
• Def. 7.2 Y ?(???? if the pdf of Y is
• here ? is the gamma function. Since ??1)1,, the
  exponential model is a special case of the gamma
  for ??1.
• The chi-square distribution is also a special
  case of the gamma.

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• here are some other relationships(Lemma 7.1 on
  page 128)
• now go to page 131 Maximum likelihood under Type
  I censoring . Suppose we have a survival
  variable Y exp(?), subject to Type I censoring.
  Beta is the MTTF and we may use maximum
  likelihood methods to estimate it(p.131 and 132)

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• Maximizing the likelihood equation gives the
• MLE of ? as
• so we may use this to construct a 95 CI for the
  MTTF

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• In the Weibull model we assume Y W(???? subject
  to Type I censoring. Or we may work with the logs
  of the survival data, Xloge(Y).
• The Xs have the 2-parameter (here u and b)
  extreme value distribution
• The survival function is given by
• The original Weibull parameters can be estimated
  if u and b are estimated by u-hat and b-hat

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• So, now go over example 7.4. Use SAS to read in
  the switch failure time data on p.135 (see
  website). Then get estimates of the the Weibull
  parameters for both the log-transformed and
  non-transformed data - use PROC LIFEREG - dont
  forget
• Notice that we may use the NOLOG option in the
  MODEL statement to not take logs of the data
• proc lifereg dataswitch model ucensor(0)
  /nolog distweibull title 'Modeling ulog(Y) w/
  NOLOG option'
• proc lifereg dataswitch model ycensor(0)
  /distweibull
• title 'Modeling non-transformed Y' run quit
• Return to Section 4.4 (p. 61-62) and use SAS to
  get a probability plot (formula 4.4) of this
  data

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• Read section 8.5 on Diagnostics for choosing
  between models
• Recall the empirical
• survivor plot
• Diagnostic 1 Plot the empirical survivor plots
  for each of the groups in a classification
  variable on the same plot. They should be
  location shifted versions of the baseline
  survival function when an accelerated lifetimes
  model is appropriate
• Diagnostic 2 Compare the standard deviations of
  the data in each group (or when enough data
  points permits, the s.d.s at each covariate
  level). This allows us to check the constancy of
  sigma in the AFT model.
• Diagnostic 3 Plot to see if they are
  parallel - if so, the proportional hazards is
  correct

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• Diagnostic 3 (continued) These curves will be
  straight lines when log(y) is plotted as the
  x-coordinate (rather than y) and the Weibull
  model fits the data well.
• Go over Example 8.3 and use either R or SAS (or
  both!) to reproduce the diagnostic plots for this
  data.