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EPI 5344: Survival Analysis in Epidemiology Numerical Problems & Other Issues April 1, 2014

EPI 5344: Survival Analysis in Epidemiology Numerical Problems & Other Issues April 1, 2014. Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa. Objectives. Monotone likelihood Colinearity. Monotone Likelihood (1).

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EPI 5344: Survival Analysis in Epidemiology Numerical Problems & Other Issues April 1, 2014

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  1. EPI 5344:Survival Analysis in EpidemiologyNumerical Problems & Other IssuesApril 1, 2014 Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

  2. Objectives • Monotone likelihood • Colinearity

  3. Monotone Likelihood (1) • In most cases of Cox regression, the likelihood function looks like this: • Function has a clear maximum, giving the MLE • Curve is peaked which gives valid variance estimates MLE

  4. Monotone Likelihood (2) • In some cases, likelihood function looks like this: • L(p) is always bigger for bigger values of ‘p’ • Never reaches a maximum no MLE exists • Variance can not be estimated either Monotonic

  5. Monotone Likelihood (3) • When does this occur? • Consider a risk set. • One subject will have the highest value of the predictor variable ‘x’. • Suppose, the subject with highest value of ‘x’ is the one who has the event • Suppose this is true for all risk sets •  Monotone Likelihood • Also true if subject with event always has lowest ‘x’.

  6. Monotone Likelihood (4) • Consequences • Computer has trouble detecting this situation • Latest SAS gives a warning (no convergence). • Works hard to find a maximum (where the convergence criterion is met) • if convergence fails, an error is logged • Easy to miss • It will produce an estimate, with variance, etc. • BUT, ALL estimates will be invalid!!!!!! • Standard errors will be HUGE. • Use this as a method to detect that there is a problem.

  7. Monotone Likelihood (5) • Example • Cocaine Adoption study • 1,640 subjects • Outcome is time to adoption of cocaine • I added a small random variable to outcome time in order to avoid ‘ties’ • Define a ‘stupid’ predictor variable • Order data by time to event • For subject having last event, set x=1999 • For each earlier event, reduce x by 1

  8. Risk set at time 25.8668

  9. 10 followed by 129 ‘0’s’!!

  10. Colinearity (1) • Identical issue as with linear regression • 2 predictor variables are strongly correlated • OR, more generally, some set of predictors is linearly dependent • Shows up with invalid estimates of standard error, etc.

  11. Colinearity (2) • Data set has time-to-follow-up information • Variable ‘x1’ is defined in data set • Define a new variable x2: • x2=x1+ u • Where u is a uniform random variable ranging from 0 to 0.01 • Very small difference between x’s • r(x1,x2)≈1.00

  12. Colinearity (3) Hazard Ratios

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