Regression Mortality rates by transplant type

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Regression Mortality rates by transplant type Powered By Docstoc
					Mortality rates by transplant type
   Proportional hazards assumption


• The hazard ratio does not vary with time

• In the previous table the hazard ratios
  (death rate ratios) remained about 2
  across the years of transplantation
Kidney transplant results using the
   Proportional Hazards Model




Hazard ratio = 2.1 (95% CI 1.6-2.6, P < 0.0005)
           Regression coefficients

• Predictors with hazard ratios less than 1 (β < 0)
  are associated with longer survival times

• Predictors with hazard ratios greater than 1
  (β > 0) are associated with shorter survival times
Kaplan-Meier curves for transplant recipients
          Potential confounders

• Living donors are more likely to be related
  and thus are closer tissue matches

• Cold ischemia time (the time spent in
  transport) is shorter for kidneys obtained
  from living donors
Adjusted survival curve
Adjusted survival curve for transplant recipients
     Mortality among patients with
       primary biliary cirrhosis


• A placebo controlled trial of D-penicillamine
  (DPCA)
Age
Results with age coded by 5 year age
      intervals (age5 = age / 5)
         Percent increase in rates


• With the age5 coefficient of 1.22 a one unit
  increase implies a 22% change in risk
  (100*(1.22-1)%).
           Histology as an ordinal variable


                  Analysis of Maximum Likelihood Estimates

Variable    DF   Parameter   Standard   Chi-Square    Pr > ChiSq   Hazard
                  Estimate     Error                                Ratio

 histol     1    0.81733     0.12383    43.5673      <.0001        2.264
Histology (4 categories)
 Model with linear and categorical terms




Detects a departure from non-linearity (not statistically
significant here (p = 0.54)
                     Treatment alone



Analysis of Maximum Likelihood Estimates
Variable DF Parameter Standard Chi-Square Pr > ChiSq Hazard Variable
            Estimate  Error                          Ratio  Label
rx       1   -0.05709 0.17916 0.1015       0.7500    0.945   rx
Are there differences in the drug effect between
     patients with and without a hepatoma?
                 Analysis of Maximum Likelihood Estimates

 Variable   DF   Parameter   Standard   Chi-Square   Pr > ChiSq   Hazard
                  Estimate     Error                               Ratio

 hepatom    1    1.05276     0.60570    3.0210       0.0822       2.866

    rx      1    -0.17849    0.33216    0.2888       0.5910       0.837

  rxhepa    1    0.09512     0.39490    0.0580       0.8097       1.100


Hazard ratio for those on drug vs placebo without
hepatoma = exp(-0.17849) = 0.837

Hazard ratio for those on drug vs placebo with hepatoma
= exp(-0.17849 + 0.09512) = 0.920
                       Bilirubin




Patients with higher bilirubin may also be more likely to
have hepatomegaly, edema, or other signs of liver damage
Treatment with DPCA (rx) and bilirubin levels as
             predictors of survival




Patients receiving DPCA have 82% the mortality rate of other patients

For each mg/dL increase in bilirubin the mortality rate increased by 16%
 Potential confounders of the bilirubin
              association

Patients with higher bilirubin may also be
 more likely to have hepatomegaly, edema,
 or spiders – other signs of liver damage
 which are correlated with elevated bilirubin
 levels but not meditors of its effects, and
 all associated with higher mortality
               The adjusted model




The bilirubin hazard ratio decreased from 1.16 to 1.12 (25%)
Survival for a patient with hepatomegaly and a
           bilirubin level of 4.5 mg/dL
 Proportional hazards: the log minus log survival curves
are parallel if the proportional hazards assumption holds



                                          Curves
                                          converge
                                          suggestion non-
                                          proportionality
                 Stratification

• Stratification adjusts for the stratification
  variable, but you do not get a regression
  coefficient for the stratification variable
• Stratification requires a reasonable
  number of outcomes in each strata
• Stratification often is a good choice in
  randomized trials to model possibly
  different hazards across multiple centers
   Model with stratification by edema


                   Analysis of Maximum Likelihood Estimates

Variable   DF   Parameter     Standard     Chi-Square    Pr > ChiSq   Hazard
                 Estimate       Error                                  Ratio

bilirub    1 0.10292 0.01591 41.8245 <.0001 1.108
hepatom    1 0.78186 0.21508 13.2143 0.0003 2.186
spiders    1 0.33529 0.19773 2.8754                      0.0899 1.398


  Note: Stratification adjusts for the stratification variable
  although no Parameter Estimate is provided
          Interactions with the stratification
              variable can be modeled

                     Analysis of Maximum Likelihood Estimates

   Variable     DF   Parameter       Standard     Chi-Square    Pr > ChiSq   Hazard
                      Estimate         Error                                  Ratio

    bilirub     1 0.10320          0.01579 42.7283 <.0001 1.109
   hepatom      1 0.89999          0.23899 14.1810 0.0002 2.460
   spiders      1 0.31302          0.19763 2.5085               0.1132 1.368
edema_hematom   1 -0.58049 0.48928 1.4076                       0.2355 0.560
 Test for proportional hazard assumption by
adding an interaction of the predictor and time

                      Analysis of Maximum Likelihood Estimates

 Variable    DF   Parameter      Standard      Chi-Square    Pr > ChiSq   Hazard
                   Estimate        Error                                   Ratio

  edema      1 5.61684         0.54151 107.5904 <.0001 275.019
edema_time   1 -1.05186 0.16808 39.1659                      <.0001 0.349
Time dependent covariate: A predictor
  whose values may vary with time

Example
• Lung transplantation in cyctic fibrosis
    Time dependent covariate
• Use a 0/1 indicator for transplantation and
  change the child’s status once the
  transplant occurs

• Create two lines in the dataset for children
  who receive the transplant (the child is
  transplant=0 before the transplant and
  transplant=1 afterwards)
    Time dependent measurements


• Use the most recent

• Or use the most recent with a lag (a gap in
  time as in using the most recent, but from
  the prior year)
    Time-dependent variables with
           randomization

Be sure the time-dependent change is not a
 consequence of the randomized
 treatment.
               SAS Enterprise

• Open the dataset PCBdata
• Select Proportional hazards under Survival
  Analysis from the task list
Select years as the survival time, status as the
  Censoring variable (censor=0), and bilirub
 hepatom spiders and edema as predictors
           Results of running regression


                   Analysis of Maximum Likelihood Estimates

Variable   DF   Parameter     Standard     Chi-Square    Pr > ChiSq   Hazard
                 Estimate       Error                                  Ratio

bilirub    1 0.11180 0.01487 56.5108 <.0001 1.118
hepatom    1 0.71813 0.21187 11.4888 0.0007 2.051
spiders    1 0.38848 0.19479 3.9775                      0.0461 1.475
edema      1 0.75447 0.22220 11.5288 0.0007 2.126
Repeat the analysis using edema as the Strata
variable instead of as an Explanatory variable
Open Life Tables from the Task list
Select years as the Survival time, status as
 the Censoring variable (censor=0), and
       edema as the Strata variable
Under Plots check log(-log(survival function))
Plot to check the proportional hazards
             assumption
Add the edema*time interaction term to
    the previous regression model
  Further evidence of non-proportional
           hazards for edema

                   Analysis of Maximum Likelihood Estimates

 Variable    DF   Parameter     Standard    Chi-Square   Pr > ChiSq   Hazard
                   Estimate       Error                                Ratio

  bilirub    1    0.06796      0.01732 15.3975 <.0001 1.070
 hepatom     1    0.81387      0.21582 14.2216 0.0002 2.257
 spiders     1    0.32513      0.20121 2.6109            0.1061 1.384
  edema      1    4.51011      0.58355 59.7343 <.0001 90.932
edema_time   1    -0.95281 0.18008 27.9948 <.0001 0.386