# Logistic Regression _ Survival Analysis by huanghengdong

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```									       Logistic Regression & Survival Analysis

Analysis of binary outcome & time to event data
Larry Holmes, Jr
Joabyer Hossain

Stats Research, Lecture 7                              November 13, 2008
Presentation Objectives

   At the end of this presentation, participants should be able to :
   Rationale for logistic regression, conduct and interpretation of result
   Survival analysis
– Measure Time and Events
– Understand Truncation and Censoring
– Understand Survival and Hazard Functions
– Define Competing Risks
– Understand Models and Hypothesis Testing
 Log rank

 Kaplan- Meier survival curve & estimates

 Cox Proportional Hazards Model (semi-parametric model)
What is Logistic Regression?

– Logistic regression is often used
because the relationship between
the DV (a discrete variable) and a
predictor is non-linear
 Blood glucose level and diabetes
mellitus
 Hypertension and LDL level
Logistic Regression
In logistic regression:
 Outcome variable is binary
 Purpose of the analysis is to assess the
effects of multiple explanatory variables,
which can be numeric and/or categorical, on
the outcome variable.
Requirements for Logistic Regression

The Following need to be specified:
1) An outcome variable with two possible categorical
outcomes (1=success; 0=failure).
2) Estimating the probability P of the outcome variable.
3) Linking the outcome variable to the explanatory
variables.
4) Estimating the coefficients of the regression equation, as
well as their confidence intervals.
5) Testing the goodness of fit of the regression model.
Measuring the Probability of Outcome

The probability of the outcome is measured
by the odds of occurrence of an event.
If P is the probability of an event, then (1-P) is
the probability of it not occurring.
Odds of success = P / 1-P
P
1 P
The logistic function
The logistic function

u
e
Yi 
1 e u

   Where Y-hat is the estimated probability
that the ith case is in a category and u is the
regular linear regression equation:

u  A  B1 X 1  B2 X 2              BK X K
Logistic function

For a response variable y with p(y=1)= P and p(y=0) = 1- P

1.0

0.8
e  x
Probability
of disease

P( y x ) 
0.6              1  e  x

0.4

0.2

0.0
x
The logistic function

   Change in probability is not constant
(linear) with constant changes in X
   This means that the probability of a
success (Y = 1) given the predictor
variable (X) is a non-linear function,
specifically a logistic function
The logistic function

   It is not obvious how the regression
coefficients for X are related to changes in
the dependent variable (Y) when the
model is written this way
   Change in Y(in probability units)|X
depends on value of X. Look at S-shaped
function
The Logistic Regression

The joint effects of all explanatory variables put together on
the odds is
Odds = P/1-P = e α + β1X1 + β2X2 + …+βpXp

Taking the logarithms of both sides
Log{P/1-P} = log α+β1X1+β2X2+…+βpXp
Logit P = α+β1X1+β2X2+..+βpXp

The coefficients β1, β2, βp are such that the sums of the
squared distance between the observed and predicted
values (i.e. regression line) are smallest.
The Logistic Regression

Logit p = α + β1X1 +β2X2 + .. + βpXp
α represents the overall disease risk
β1 represents the fraction by which the disease risk is
altered by a unit change in X1
β2 is the fraction by which the disease risk is altered
by a unit change in X2
……. and so on.
What changes is the log odds. The odds themselves
are changed by eβ
If β = 1.6 the odds are e1.6 = 4.95
Logistic Regression-Demo

   MS-Excel: No default functions
   SPSS: Analyze > Regression > Binary Logistic > Select
Dependent variable: > Select independent variable
(covariate)
Logistic Regression SPSS output
Dependent Variable Encoding

Original Value    Internal Value
0                                   0
1                                   1

Categorical Variables Codings

Parameter
coding

Frequency                (1)
2                      30                 .000

Classification Table(a,b)

Predicted

pc
Percentage
Observed                                                 0                1               Correct
Step 0   pc                            0                               0                  30                 .0
1                               0                  30              100.0
Overall Percentage                                                                                50.0
a Constant is included in the model.
b The cut value is .500

Variables in the Equation

B              S.E.           Wald           df                  Sig.       Exp(B)
Step 0   Constant                 .000          .258           .000               1              1.000       1.000

Variables not in the Equation

Score           df                  Sig.
Step 0   Variables                Shades(1)                  17.067                   1            .000
Overall Statistics                                   17.067                  1            .000
Logistic Regression SPSS output
Omnibus Tests of Model Coefficients

Chi-square        df            Sig.
Step 1   Step             17.985                 1      .000
Block            17.985                 1      .000
Model            17.985                 1      .000

Model Summary

-2 Log       Cox & Snell   Nagelkerke R
Step      likelihood      R Square       Square
1           65.193(a)           .259            .345
a Estimation terminated at iteration number 4 because parameter estimates changed by less than .001.

Classification Table(a)

Predicted

pc
Percentage
Observed                                               0               1            Correct
Step 1   pc                        0                                23               7             76.7
1                                 7              23             76.7
Overall Percentage                                                                        76.7
a The cut value is .500

Variables in the Equation

B             S.E.        Wald               df              Sig.      Exp(B)
Step     Shades(1)          -2.379           .610      15.189                  1            .000          .093
1(a)     Constant           1.190            .432           7.594              1            .006         3.286
a Variable(s) entered on step 1: Shades.
Regression vs. Survival Analysis

Technique     Predictor         Outcome        Censoring
Variables         Variable       permitted?
Linear     Categorical or        Normally          No
Regression  continuous           distributed

Logistic   Categorical or Binary (except in        No
continuous     polytomous log.
Regression                   regression)

Survival         Time and         Binary           Yes
Analyses       categorical or
continuous
Regression vs. Survival Analysis

Technique    Mathematical            Yields
model
Linear         Y=B1X + Bo         Linear changes
Regression       (linear)

Logistic     Ln(P/1-P)=B1X+Bo      Odds ratios
(sigmoidal prob.)
Regression
Survival           h(t) =          Hazard rates
Analyses     ho(t)exp(B1X+Bo)
What is survival analysis?

   Model time to failure or time to event
– Unlike linear regression, survival analysis has a dichotomous
(binary) outcome
– Unlike logistic regression, survival analysis analyzes the time
to an event
   Why is that important?
 Able to account for censoring
 Can compare survival between 2+ groups
 Assess relationship between covariates and survival
time
Survival Analysis

 Survival analysis deals with making inference about
EVENT RATES
 Rate at t = Rate among those at risk at t
 Deals with Median survival (50%) .
 Not Mean survival (need everyone to have an event)
…..Why?
 Survival vs. time-to-event
 Outcome variable = event time
 Examples of events:
– Death, infection, MI,prostate cancer death, hospitalization
– Recurrence of cancer after treatment
Types of censoring

 Subject does not
experience event of
interest
 Incomplete follow-up
– Lost to follow-up
– Withdraws from study
– Dies (if not being studied)
   Left or right censored
Survival Function

 S(t) = P[ T ≥ t ] = 1 – P[ T < t ]
 Plot: Y axis = % alive, X axis = time
 Proportion of population still without the
event by time t
Survival Curve
1.0
0.2 0.4 0.6 0.8               Survival Curve
Proportion Alive
0.0

0   1   2    3    4     5    6     7   8   9
Months since surgery
Hazard Function

   Also termed incidence rate, instantaneous risk,
force of mortality
   λ(t)
   Event rate at t among those at risk for an event
   Key function
   Estimated in a straightforward way
– Censored
– Truncated
Time to Cardiovascular Adverse Event in VIGOR Trial
Hazard Function

 Event = death, scale = months since Tx
 “λ(t) = 1% at t = 12 months”
 “At 1 year, patients are dying at a rate of
1% per month”
 “At 1 year the chance of dying in the
following month is 1%”
Relationship between survivor function and
hazard function
 Survivor function, S(t) defines the probability of
surviving longer than time t
– this is what the Kaplan-Meier curves show.
– Hazard function is the derivative of the survivor
function over time h(t)=dS(t)/dt
   instantaneous risk of event at time t (conditional failure
rate)
   Survivor and hazard functions can be converted
into each other
Use of survival analysis: clinical trial

 Accrual into the study over 2 years
 Data analysis at year 3
 Reasons for exiting a study
– Died
– Alive at study end
– Withdrawal for non-study related reasons
(LTFU)
– Died from other causes
Kaplan-Meier

 One way to estimate survival
 Nice, simple, can compute by hand
 Cannot evaluate covariates like Cox model
 No sensible interpretation for competing
risks
Kaplan-Meier estimate

   Multiply together a series of conditional probabilities

Time ti         # at risk     # events                 ˆ
S
0             20            0                                 1.00
5             20            2              [1-(2/20)]*1.00=0.90
6             18            0              [1-(0/18)]*0.90=0.90
10            15            1              [1-(1/15)]*0.90=0.84
13            14            2              (1-(2/14)]*0.84=0.72
Proportion Surviving (95% Confidence)
0.6    0.7     0.8   0.9    1.0

10 0
5
Survival Time
Kaplan-Meier Curve

15
20
Kaplan Meier Curve
Limit of Kaplan-Meier curves

   What happens when you have several covariates that you
believe contribute to survival?
   Example
– Smoking, hyperlipidemia, diabetes, hypertension, contribute to time
to myocardial infarct
   Can use stratified K-M curves – for 2 or maybe 3 covariates
   Need another approach – multivariate Cox proportional
hazards model is most common -- for many covariates
– (think multivariate regression or logistic regression rather than a
Student’s t-test or the odds ratio from a 2 x 2 table)
Multivariable method: Cox proportional
hazards

 Needed to assess effect of multiple covariates
on survival
 Cox-proportional hazards is the most
commonly used multivariable survival
method
Cox proportional hazard model

   Works with hazard model

   Conveniently separates baseline hazard function from
covariates
– Baseline hazard function over time
  h(t) = ho(t)exp(B1X+Bo)
– Covariates are time independent
– B1 is used to calculate the hazard ratio, which is similar to the relative
risk
   Semi-parametric
Cox Proportional Hazards Model

 Add covariates to the model
 Change in a prognostic factor →
proportional change in the hazard (on the
log scale)
 Can test the effect of the prognostic factor
as in linear regression - H0: β=0
Limitations of Cox PH model

   Does not accommodate variables that change
over time
– Most variables (e.g. gender, ethnicity, or congenital
condition) are constant
   If necessary, one can program time-dependent variables
   When might you want this?
   Baseline hazard function, ho(t), is never specified
– You can estimate ho(t) accurately if you need to
estimate S(t).
Summary
   Survival analyses quantifies time to a single,
dichotomous event
   Handles censored data well
   Survival and hazard can be mathematically converted to
each other
   Kaplan-Meier survival curves can be compared
statistically and graphically
   Cox proportional hazards models help distinguish
individual contributions of covariates on survival,
provided certain assumptions are met.
SPSS output of Survival functions
Survival Table

Cumulative Proportion               N of              N of
Surviving at the Time            Cumulative        Remaining
Time             Status         Estimate      Std. Error           Events            Cases
1                6.000                   1          .800           .179                    1                   4
2               14.000                   1          .600           .219                    2                   3
3               21.000                   0              .              .                   2                   2
4               44.000                   1          .300           .239                    3                   1
5               62.000                   1          .000           .000                    4                   0

Means and Medians for Survival Time
a
Mean                                                           Median
95% Confidence Interval                                         95% Confidence Interval
Estimate     Std. Error Lower Bound Upper Bound                Estimate      Std. Error Lower Bound Upper Bound
35.800       11.810        12.652         58.948               44.000        23.875          .000         90.794
a. Estimation is limited to the largest survival time if it is censored.
SPSS output of KM plot
SPSS output of cumulative hazard
SPSS output of Cox Regression

Omnibus Tests of Model Coefficientsa,b

-2 Log                 Overall (score)                       Change From Previous Step             Change From Previous Block
Likelihood    Chi-square        df             Sig.       Chi-square      df           Sig.      Chi-square      df          Sig.
6.732         .468             1            .494          .646           1          .422         .646           1         .422
a. Beginning Block Number 0, initial Log Likelihood function: -2 Log likelihood: 7.378
b. Beginning Block Number 1. Method = Enter

Variables in the Equation

B                    SE                   Wald                    df            Sig.             Exp(B)
psa                -1.393                2.305                 .365                       1           .546              .248

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