Assessing Survival:

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Statistics for Health Research
Assessing Survival Cox Proportional Hazards
Model
Peter T. Donnan Professor of Epidemiology and
Biostatistics
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Objectives of Workshop

• Understand the general form of Cox PH model
• Understand the need for adjusted Hazard Ratios
  (HR)
• Implement the Cox model in SPSS
• Understand and interpret the output from SPSS

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Modelling Detecting signal from background noise
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Survival Regression Models
Expressed in terms of the hazard function
formally defined as The instantaneous risk of
event (mortality) in next time interval t,
conditional on having survived to start of the
interval t
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What is hazard?

• Hazard rate is an instantaneous rate of events as
  a function of
• time

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Plot of hazard

• Note that the hazard changes over time denoted
  by h(t)

h(t)
time
Old age
Birth
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Survival Regression Models
The Cox model expresses the relationship between
the hazard and a set of variables or covariates
These could be arm of trial, age, gender, social
deprivation, Dukes stage, co-morbidity, etc.
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How is the relationship formulated?
Simplest equation is
h is the hazard K is a constant e.g. 0.3 per
Person-year
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How is the relationship formulated?
Next Simplest is linear equation
h is the outcome a is the intercept ß is the
slope related to x the explanatory variable
and e is the error term or noise
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Linear model of hazard
Hazard
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Cox Proportional Hazards Model (1972)
h0 is the baseline hazard r ( ß, x) function
reflects how the hazard function changes (ß)
according to differences in subjects
characteristics (x)
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Exponential model of hazard
Hazard
Age in years
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What is Hazard Ratio?

• Hazard Ratio (HR) is ratio of hazards in two
  groups
• e.g. men vs women, new drug vs. BSC
• N.B. It is the improvement in one group over the
  other in terms of rate at which events will occur
  from a particular time point to another time point

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What is Hazard Ratio?

• Hazard Ratio (HR) is ratio of hazards in two
  groups and remains constant over time (n.b.
  survival curve widens)

Survival
Time
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Interpretation of HR comparing two groups

• HR 1 Do NOT reject null hypothesis (i.e. no
  difference)
• HR lt 1 Reduction in Hazard relative to
  comparator (e.g. HR 0.6 is 40 reduction)
• HR gt 1 Increase in Hazard relative to
  comparator (e.g. HR 1.7 is 70 increase)

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Cox Proportional Hazards Model Hazard Ratio
Consider hazard ratio for men vs. women, then -
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Cox Proportional Hazards Model Hazard Ratio
If coding for gender is x1 (men) and x0 (women)
then
where ß is the regression coefficient for gender
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Hazard ratios in SPSS
SPSS gives hazard ratios for a binary factor
coded (0,1) automatically from exponentiation of
regression coefficients (95 CI are also given as
an option) Note that the HR is labelled as EXP(B)
in the output
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Fitting Gender in Cox Model in SPSS
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Output from Cox Model in SPSS
p-value
Standard error
Degrees of freedom
Variable in model
HR for men vs. women
Regression Coefficient
Test Statistic ( ß/se(ß) )2
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Logrank Test Null Hypothesis

• The Null hypothesis for the logrank test
• Hazard Rate group A
• Hazard Rate for group B
• HR OA / EA 1
• OB / EB

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Wald Test Null Hypothesis

• The Null hypothesis for the Wald test
• Hazard Ratio 1
• Equivalent to regression coefficient ß0
• Note that if the 95 CI for the HR includes 1
  then the null hypothesis cannot be rejected

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Hazard ratios for categorical factors in SPSS

• Enter factor as before
• Click on categorical and choose the reference
  category (usually first or last)
• E.g. Dukes staging may choose Stage A as the
  reference category
• HRs are now given in output for survival in each
  category relative to Stage A
• Hence there will be n-1 HRs for n categories

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Fitting a categorical variable Dukes Staging
Reference category

B vs. A
C vs. A
D vs. A
UK vs. A
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One Solution to Confounding
Use multiple Cox regression with both predictor
and confounder as explanatory variables i.e fit
x1 is Dukes Stage and x2 is Age
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Fitting a multiple regression Dukes Staging and
Age
Age adjusted for Dukes Stage
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Interpretation of the Hazard Ratio
For a continuous variable such as age, HR
represents the incremental increase in hazard per
unit increase in age i.e HR1.024, increase 2.4
for a one year increase in age
For a categorical variable the HR represents the
incremental increase in hazard in one category
relative to the reference category i.e. HR 6.66
for Stage D compared with A represents a 6.7 fold
increase in hazard
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First steps in modelling

• What hypotheses are you testing?
• If main exposure variable, enter first and
  assess confounders one at a time
• Assess each variable on statistical significance
  and clinical importance.
• It is acceptable to have an important variable
  without statistical significance

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Summary

• The Cox Proportional Hazards model is the most
  used analytical tool in survival research
• It is easily fitted in SPSS
• Model assessment requires some thought
• Next step is to consider how to select multiple
  factors for the best model

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Check assumption of proportional hazards (PH)

• Proportional hazards assumes that the ratio of
  hazard in one group to another remains the same
  throughout the follow-up period
• For example, that the HR for men vs. women is
  constant over time
• Simplest method is to check for parallel lines in
  the Log (-Log) plot of survival

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Check assumption of proportional hazards for each
factor. Log minus log plot of survival should
give parallel lines if PH holds
Hint Within Cox model select factor as
CATEGORICAL and in PLOTS select log minus log
function for separate lines of factor
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Check assumption of proportional hazards for each
factor. Log minus log plot of survival should
give parallel lines if PH holds
Hint Within Cox model select factor as
CATEGORICAL and in PLOTS select log minus log
function for separate lines of factor
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Proportional hazards holds for Dukes Staging
Categorical Variable Codings(b)
Frequency (1) (2) (3) (4) dukes(a) 0A 18 1 0 0
0 1B 107 0 1 0 0 2C 188 0 0 1 0
3D 123 0 0 0 1 9UK 40 0 0 0 0 a Indicator
Parameter Coding b Category variable dukes
(Dukes Staging)
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Proportional hazards holds for Dukes Staging
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Summary

• Selection of factors for Multiple Cox regression
  models requires some judgement
• Automatic procedures are available but treat
  results with caution
• They are easily fitted in SPSS
• Check proportional hazards assumption
• Parsimonious models are better

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Practical

• Read in Colorectal.sav and try to fit a multiple
  proportional hazards model
• Check proportional hazards assumption