Introduction to medical survival analysis by HC12100415177

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									Introduction to medical survival
            analysis



  John Pearson
  Biostatistics consultant
  University of Otago Canterbury
  7 October 2008

                                   1
Objectives
•   Describe survival data
•   Define survival analysis terms
•   Compare survival of groups
•   Describe study design

Acknowledgement:
Thanks to Colm Fahy for providing the
example data.


                                        2
Omissions
• Not covered:
  – most methodology issues
  – mathematical justification
• See
  – Collett: Modelling Survival Data in Medical
    Research
  – Hosmer & Lemeshow: Applied Survival
    Analysis
  – Many other good texts.

                                                  3
Example: Metastatic Parotid SCC
• Disease risk factors:
  – >50 yo
  – Male
  – Exposure to sun
  – Caucasian ancestry
• 61 patients operated on since 1990
• Audit done 1/6/8
• 14 patients died from SCCMP, 20 died
  from other causes, 1 couldn’t be found
                                           4
Example: Patient data
     OpDate      Died           Status   Preserved   RadioTx   ICOMP
       7/05/2002                ALIVE    PARTIAL     YES       N
      15/11/2007                ALIVE    NO          YES       N
      12/10/2007    1/03/2008   DOC      YES         YES       N
      17/04/1992    1/08/1993   DOD      YES         YES       Y
       7/10/1996    1/04/1997   DOC      NO          YES       N
       1/05/1991                LOST     YES         YES       N
      12/03/2003    1/05/2005   DOC      YES         YES       Y




Only 7 patients shown.
Dates have been confidentialized.




                                                                       5
Example: Patient data
              Parotidectomy patient medical records
Patient




                  Alive
          7       Dead OC
                  Dead OD
          6

          5

          4

          3

          2




                                                                                      audit
          1                                           ? Lost to follow up


               1990              1995                  2000                 2005   6/2008




                                                                                              6
Example: Patient data
              Parotidectomy patient medical records
Patient




                  Alive                                                                       ?
          7       Dead OC
                  Dead OD
          6

          5

          4                                                                                   ?

          3

          2




                                                                                      audit
          1                                           ? Lost to follow up


               1990              1995                  2000                 2005   6/2008




                                                                                                  7
Example: Survival Data
              Parotidectomy patient survival data
Patient




                                                                  Alive
          7                                                       Dead OC
                                                                  Dead OD
          6

          5

          4

          3

          2

          1                                                            ?


                0                       5           10     15
                                                         Years post operation




                                                                                8
Example: Survival Data
               Parotidectomy patient survival data
 Patient




                                                                    Alive
           7                                                        Dead OC
                                                                    Dead OD
           6

           5

           4

           3

           2

           1                                                             ?


                 0                       5           10      15
                                                           Years post operation

Date formats and manipulation can cause headaches.
Check what happens when your software subtracts dates to get survival time.
                                                                                  9
Example: Survival Data
              Parotidectomy patient survival data
Patient




                                                                            Alive
          7                                         censored                Dead OC
                                                                            Dead OD
          6

          5

          4                                         censored
          3

          2

          1
                                                    Missing data                 ?


                0                       5                   10       15
                                                                   Years post operation




                                                                                          10
Example: Survival Data
              Parotidectomy patient survival data
Patient




                                                                            Alive
          7                                         censored                Dead OC
                                                                            Dead OD
          6                                         censored
          5                                         censored
          4                                         censored
          3                                         censored
          2

          1
                                                    Missing data                 ?


                0                       5                   10       15
                                                                   Years post operation




                                                                                          11
                   Data
Example: Survivaldata is explicitly addressed by survival
          Censored
                                      analysis, using simple linear regression is not
                                      recommended.
               Parotidectomy patient survival data
                                      Options:
 Patient




                                      1. SPSS censored                          Alive
           7                                                                    Dead OC
                                      2. SAS                                    Dead OD
           6                          3. R         censored
           5                          4. Other software
                                                   censored
           4

           3                                 censored
           2

           1
                                             Missing data                             ?


                 0                  5                  10                 15
                                                                        Years post operation




                                                                                               12
Example: Survival Data
                                       Missing data can have a large effect on results,
                                       requires careful management.
              Parotidectomy patient survival data
                                       Options:
Patient




                                                                             Alive
          7                            1. Omit censored                      Dead OC
                                       2. Impute  censored
                                                                             Dead OD
          6
                                       3. Model
          5                                 censored
          4

          3                                 censored
          2

          1
                                            Missing data                            ?


                0                  5                 10                 15
                                                                      Years post operation




                                                                                             13
What is survival analysis
• Time to event data
  – Continuous
  – Right skewed, ≥0, not normal
  – Censored
  – Analyse risk (hazard function)
• Examples
  – Time to death
  – Time to onset/relapse of disease
  – Length of stay in hospital
                                       14
What is survival analysis
• Time to event data                   Post operative survival




                        Patients
  – Continuous
  – Right skewed, ≥0, not15normal
  – Censored
                          10
  – Analyse risk (hazard function)
• Examples                         5

  – Time to death
  – Time to onset/relapse 0of disease
                             0   2  4                            6   8   10
  – Length of stay in hospital                                                Years




                                                                                15
Censoring
• Right censoring
• Left censoring
• Interval censoring




Censoring is also categorised by
1. Fixed study length
2. Fixed number of events
3. Random entry to study
                                   16
Censoring
• Right censoring
     – observed survival time is less than actual
     – Study ends before event
               Parotidectomy patient medical records

• Left Alive
       censoring
 Patient




    7                                                                                          ?
       Dead OC
• Interval censoring
    6
       Dead OD


           5

           4                                                                                   ?

           3

           2




                                                                                       audit
           1                                           ? Lost to follow up


                1990              1995                  2000                 2005   6/2008         17
Censoring
• Right censoring
• Left censoring
  – Time to relapse
    Surgery       Recurrence

      0       t       3 month exam
  – Time to event is less than observed t < 3
• Interval censoring



                                                18
Censoring
• Right censoring
• Left censoring
• Interval censoring
  – Time to relapse
    Surgery     Free of disease      Recurrence

      0               3 month exam   t   6 month exam
  – 3<t<6



                                                        19
Censoring
Independent censoring
Survival time is independent of censoring
process.

A censored patient is representative of those at
risk at censoring time.

The methods described here assume
independent censoring
                                            20
Censoring
Independent censoring
Survival time is independent of censoring
process.

Informative censoring
Patients removed from study if condition
deteriorates.


                                            21
Censoring example
How are the SCCMP patients censored?




                                       22
Censoring example
How are the SCCMP patients censored?
• Enter study on surgery date
• Last known status is at audit

Random right censoring.




                                       23
Survival function
The survival function S(t) is the probability of
surviving longer than time t.

                   S(t) = P(T>t)

Where T is the survival time.

       Number of patients surviving longer than t
S(t) 
              total number of patients

                                                    24
Hazard function
The hazard function λ(t) is the                     f(t)
probability of dying “at” time t.            (t) 
                                                    S(t)
Also called the instantaneous failure
rate and force of mortality.


Usually plotted is the cumulative
hazard function, that is the            (t)   log S (t )
accumulated hazard until time t.


                                                          25
Survival function
For censored data the survival function can only
be estimated.
                    Parotidectomy patient survival data
      Patient




                                                                         Alive
                7                                                        Dead OC
                                                                         Dead OD
                6

                5

                4

                3

                2

                1

                     0.0       0.5       1.0       1.5    2.0        2.5       3.0
                                                                Years post operation



                                                                                       26
Survival function
Life table estimates
                                                 All causes mortality

                        100
    Percent surviving




                        80

                        60                                                              NZ
                                                                                        Australia
                        40                                                              Chad

                        20

                         0
                              0   10   20   30   40    50    60    70   80   90   100
                                                      Age



WHO, StatsNZ
                                                                                                    27
Survival function
Kaplan Meier estimates
     Months    n   d   (n-d)/n     S(t)
 1      2.2   57   1    0.982    0.982
 2     6.12   51   1    0.980    0.963
 3    10.32   46   1    0.978    0.942
 4    10.78   45   1    0.978    0.921
 5    10.88   44   1    0.977      0.9
 6    13.08   41   1    0.976    0.878
 7    13.35   39   1    0.974    0.856
 8    16.11   37   1    0.973    0.833
 9     26.2   34   1    0.971    0.808
10    29.42   31   1    0.968    0.782
11    37.48   26   1    0.962    0.752
12    45.86   23   1    0.957    0.719
13    59.08   19   1    0.947    0.682
14    65.33   14   1    0.929    0.633
                                          28
      1. Order data by time to event
Survival function
      (death)                                   2. Number at risk of
                                                event is number
                                                surviving less number
Kaplan Meier estimates                          censored.

     Months     n      d     (n-d)/n     S(t)
 1      2.2    57      1      0.982    0.982
                                                 3. Estimate of
 2     6.12    51      1      0.980    0.963
                                                 probability of surviving
 3    10.32    46      1      0.978    0.942     to next event
 4    10.78    45      1      0.978    0.921
 5    10.88    44      1      0.977      0.9
 6    13.08    41      1      0.976    0.878
 7    13.35    39      1      0.974    0.856
 8    16.11    37      1      0.973    0.833
                                                   4. Multiply probabilities
 9     26.2    34      1      0.971    0.808
                                                   to estimate survival
10    29.42    31      1      0.968    0.782
11    37.48    26      1      0.962    0.752
12    45.86    23      1      0.957    0.719
13    59.08    19      1      0.947    0.682
14    65.33    14      1      0.929    0.633
                                                                        29
Kaplan Meier plot
                               Kaplan Meier estimate
 Estimated survivor function




                     1.0



                     0.8



                     0.6



                     0.4



                     0.2



                     0.0
                               0           20          40   60   80   100   120
                                                                                  Months


                                                                                           30
Kaplan Meier plot                                                      SCCMP
                                 Kaplan Meier estimate
   Estimated survivor function




                       1.0



                       0.8



                       0.6



                       0.4

                                                              Standard errors and 95%
                       0.2                                    CI’s calculated by most
                                                              software      (SPSS, R,
                                                              SAS)
                       0.0
                                 0           20          40   60        80       100    120


Usually use Greenwood’s or Tsiatis’ formula, software dependent.                              31
Cumulative Hazard                                           SCCMP
                     Cumulative Hazard Function
 Cumulative hazard




              0.4




              0.3




              0.2




              0.1




              0.0
                     0          20           40   60   80    100   120
                                                                         Months



                                                                                  32
Summary statistics
1. Median survival: time when S(t) = 0.5
    • Must have enough data
2. Mean survival: area under the survival
   curve
3. 5 year survival is survival rate at 5 years




                                                 33
Kaplan Meier estimate
KM and lifetables are non-parametric
methods: no assumptions are made about
the distribution on the survival times.

Typical distributions are exponential and
Weibull. More powerful but can be sensitive
to getting the distribution right.



                                          34
Disease specific survival
                               SCCMP survival
 Estimated survivor function




                     1.0
                                                                 Disease specific
                                                                 All causes
                     0.8



                     0.6



                     0.4



                     0.2



                     0.0
                               0         20     40   60   80   100         120
                                                                                 Months


                                                                                          35
Comparing 2 groups
Log rank test
• Computed in SPSS, SAS, R
• Most popular
  – (Bland Altman BMJ 2004;328:1073 (1 May)
• Limitations
  – No estimate of size
  – Unlikely to detect a difference when risk is not
    consistent


                                                   36
Immuno compromised
                               SCCMP survival: Immuno Compromised
 Estimated survivor function




                     1.0



                     0.8
                                                                                No

                     0.6



                     0.4



                     0.2



                     0.0                         Yes
                               0     20     40         60   80      100   120       140
                                                                                Months
                                                                                     37
Immuno compromised
                               SCCMP survival: Immuno Compromised
 Estimated survivor function




                     1.0



                     0.8
                                                                                                    No

                     0.6                                     Case Processing Summary

                                                                                         Censored
                                            ICOMP       Total N    N of Events         N       Percent
                     0.4
                                            N                 53            9            44      83.0%
                                            Y                  7            5             2      28.6%
                                            Overall           60           14            46      76.7%
                     0.2



                     0.0                              Yes
                               0     20      40             60       80          100          120       140
                                                                                                    Months
                                                                                                         38
Immuno compromised
                               SCCMP survival: Immuno Compromised
 Estimated survivor function




                     1.0



                     0.8
                                                                                                                 No

                                                           Means and Medians for Surv iv al Time
                     0.6
                                                                         a
                                                                   Mean                        Median


                     0.4                         ICOMP      Estimate    Std. Error    Estimate      Std. Error
                                                 N           101.048        7.616             .              .
                                                 Y            22.978        7.653       16.110          3.293
                                                 Overall      91.761        7.842             .              .
                     0.2
                                                   a. Estimation is limited to the largest survival time if it
                                                      is censored.

                     0.0                             Yes
                               0     20     40               60              80          100            120          140
                                                                                                                 Months
                                                                                                                      39
Immuno compromised
                               SCCMP survival: Immuno Compromised
 Estimated survivor function




                     1.0



                     0.8
                                                                                                              No

                     0.6
                                                                Ov erall Comparisons

                                                                 Chi-Square              df          Sig.
                     0.4                   Log Rank (Mantel-Cox)     19.579                   1        .000
                                          Test of equality of survival distributions for the different levels of
                                          ICOMP.
                     0.2



                     0.0                             Yes
                               0     20      40            60            80           100           120           140
                                                                                                              Months
                                                                                                                   40
Age group
                               SCCMP survival: Age group
 Estimated survivor function




                     1.0



                     0.8


                                                                         <75
                     0.6                                                             75+



                     0.4 Call:
                               survdiff(formula = Surv(mths,Status == "DOD") ~ ICOMP)

                     0.2                  N Observed Expected (O-E)^2/E (O-E)^2/V
                               Age75=<75 24        7     5.63     0.332     0.557
                               Age75=75+ 36        7     8.37     0.224     0.557
                     0.0

                                         0.6
                               0 Chisq= 20                                p=
                                               on401 degrees of freedom,100 0.455
                                                         60      80            120       140
                                                                                     Months
                                                                                           41
Facial Nerve
                               SCCMP survival: Facial Nerve Preserved
 Estimated survivor function




                     1.0



                     0.8                                                               YES


                                                                 PARTIAL
                     0.6



                                                                            NO
                     0.4



                     0.2


                                   Log rank p value: 0.09
                     0.0
                               0        20        40        60    80       100   120       140
                                                                                       Months
                                                                                             42
Multiple independent variables
Cox proportional hazards model
• Most common model
• Linear model for the log of the hazard ratio

            h1 (t )    B1Z1  B2 Z 2 
                    e
            h0 (t )

• Baseline hazard unspecified


                                                 43
SCCMP example
CPH model:
Survival ~ Preserved + Age + ICOMP

Preserved and ICOMP categorical
Age continuous

Plot survival for patients with each of
/Y/N/partial nerve preservation adjusted for
age and immuno compromised status

                                               44
SCCMP example - SPSS
Analyze > Survival > Cox Regression

COXREG
 Months /STATUS=Status('DEAD')
 /PATTERN BY Preserved
 /CONTRAST (Preserved)=Indicator /CONTRAST (ICOMP)=Indicator(1)
 /METHOD=ENTER Preserved Age ICOMP
 /PLOT SURVIVAL
 /SAVE=PRESID XBETA
 /PRINT=CI(95) CORR SUMMARY BASELINE
 /CRITERIA=PIN(.05) POUT(.10) ITERATE(20) .




                                                                  45
SCCMP example - SPSS
                              Variables in the Equation

                                                                   95.0% CI for Exp(B)
              B        SE       Wald      df    Sig.      Exp(B)    Lower      Upper
 Preserved                       8.493     2     .014
   No        2.535     .871      8.470     1     .004     12.617     2.288    69.564
   Partial   2.091    1.110      3.549     1     .060      8.093      .919    71.279
 ICOMP       3.588     .918     15.274     1     .000     36.166     5.981   218.676
 Age          -.011    .028       .149     1     .700       .989      .936     1.046




Patients with their facial nerve preserved have 12.6 times
less hazard ratio, (95% CI 2-70) .

Preserving the facial nerve significantly reduces patients
risk, (p value <0.001 CPH model).

                                                                                         46
SCCMP CPH model
                              SCCMP survival: Facial nerve preserved
Estimated survivor function




                    1.0

                                                                                          YES

                    0.8



                    0.6

                                                                                          PARTIAL
                    0.4
                                                                                          NO

                    0.2


                                  Adjusted for age and immuno compromised patients
                    0.0
                              0        10       20      30       40       50         60        70
                                                                                                Months
                                                                                                     47
Next Steps:
• Check proportional hazards assumption
  – Residual plots for groups
• Time dependent covariates
• More complex models

• we also didn’t do power calculations



                                          48
Summary
• Survival analysis accounts for censoring in
  time to event data
• Log rank test: difference in survival
  between 2 groups
• Cox proportional hazard model
• More complex/powerful models available
• SPSS, R, SAS, Stata


                                            49

								
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