dr laura bonnett department of biostatistics. understanding survival analysis
TRANSCRIPT
Dr Laura BonnettDepartment of Biostatistics
UNDERSTANDING SURVIVAL ANALYSIS
OUTLINE
• Survival analysis• Time to event data• Censoring• Kaplan-Meier curves• Log rank tests• Cox model
• Prognostic & predictive models
TIME-TO-EVENT DATA
The event might be:● discharge from hospital● weaning of a breast-fed infant● recurrence of tumour● remission of a disease etc.
The time starting point might be:● time of diagnosis● time of surgery● time of entrance into the study etc.
FOR FRED…
Event: next seizure
Starting point: time of randomisation (to treatment or no treatment)
CENSORING
Event is often not observed on all subjects:Drop-out
End of study
Individuals for whom the event is not observed are called censored
KAPLAN-MEIER CURVES
KAPLAN-MEIER CURVES
LOG-RANK TEST
p<0·0001
Years since randomisation
Cu
mu
lativ
e p
rob
abi
lity
of s
eiz
ure
(s)
HAZARD RATIO
• Hazard ratio (HR) is a measure of the relative survival in two groups• Ratio of the hazard for one group compared to
another
• Hazard is the chance that at any given moment, the event will occur, given that it hasn’t already done so.
• Confidence interval for the hazard ratio:
• Accuracy
• Significance
• Hazard ratios are similar to relative risks and
odds ratios
HAZARD RATIO
HRRR OR
MODELLING SURVIVAL
Time-to-event
Gender
Drug group
Age
• Estimate effect sizes for each risk factor, and whether these are significantly large
COX REGRESSION MODELLING
The hazard is modelled with the equation:
kk xbxbxbthth ...exp)()( 22110
Risk Factors (Covariates)
Parameters to be estimated
– related to effect sizes
Underlying hazard
INTERPRETATION
E.g. Risk of seizure for a person on Treatment (x1 = 1) compared to Control (x1 = 0), assuming they are alike for all other covariates (x2, x3, etc.).
thth)(bthth 0010 10exp
- Hazard rate in Control group at time t:
- Hazard ratio is:
)exp()(
)exp()(1
0
10 bth
bthHR
)(bth)(bthth 1010 exp1exp
- Hazard rate in treatment group at time t:
INTERPRETATION FOR BINARY VARIABLE
If b is the regression coefficient of a binary
variable, x
exp(b) = hazard ratio for x = 1 relative to x = 0
HR > 1: x = 1 has increased hazard relative to x = 0
HR < 1: x = 1 has decreased hazard relative to x = 0
HR= 1: x has no effect on survival
INTERPRETATION FOR BINARY VARIABLE
E.g. Immediate vs. delayed treatment decision
exp(b) = hazard ratio for immediate relative to delayed
HR > 1: immediate has increased hazard relative to delayed
HR < 1: immediate has decreased hazard relative to delayed
HR= 1: treatment decision has no effect on risk of seizure
INTERPRETATION FOR CONTINUOUS VARIABLE
A continuous variable x can be any value
exp(b) = hazard ratio for x = k+1 relative to x = k
i.e. as x increases by 1 unit, the hazard is multiplied by exp(b)
HR > 1: as x increases, the hazard increases
HR < 1: as x increases, the hazard decreases
HR = 1: x has no effect on survival
INTERPRETATION FOR CONTINUOUS VARIABLE
E.g. Age (in years)
exp(b) = hazard ratio for Age = k+1 relative to Age = k
HR > 1: as age increases, the chance of seizure increases
HR < 1: as age increases, the chance of a seizure decreases
HR= 1: age has no effect on the chance of a seizure
INTERPRETATION FOR CATEGORICAL VARIABLE
A categorical variable, x, can take one of several valuesTo obtain HRs, ‘dummy (binary) variables’ must be created e.g.
Interpretation is then as for binary variables
Dummy Variable 1 Dummy Variable 2
Baseline Category 0 0
Alternative Category 1 1 0
Alternative Category 2 0 1
INTERPRETATION FOR CATEGORICAL VARIABLE
E.g. EEG Results (normal, abnormal, not done)
Dummy Variable 1 Dummy Variable 2
Normal 0 0
Abnormal 1 0
Not done 0 1
Dummy Variable 1
HR > 1: abnormal results has increased hazard relative to normal results
HR < 1: abnormal results has decreased hazard relative to normal results
HR= 1: EEG result has no effect on survival
Dummy Variable 2
HR > 1: not done results has increased hazard relative to normal results
HR < 1: not results has decreased hazard relative to normal results
HR= 1: EEG result has no effect on survival
INTERPRETATION FOR CATEGORICAL VARIABLE
E.g. EEG Results (normal, abnormal, not done)
Dummy Variable 1 Dummy Variable 2
Normal 0 0
Abnormal 1 0
Not done 0 1
BACK TO FRED…
Remember, log-rank p<0.0001
Cox model (univariate):• Variable: treatment decision• Outcome: time to 1st seizure after randomisation
Variable HR (95% CI)
Treatment decision (Baseline: immediate)
1.4 (1.2, 1.7)
ASSUMPTIONS OF THE COX MODEL
Hazard for an individual in one group is proportional to the
hazard for an individual in another group for all time t.
Detected from Kaplan-Meier plots that either cross, or
diverge then converge again:
0.0
0.5
1.0
t
Su
rviv
al p
rob
abili
ty
BACK TO FRED…
Immediate antiepileptic drug treatment reduces the occurrence of seizures in the next 1-2 years, but does not affect long-term remission in individuals with single or infrequent seizures.
PROGNOSTIC & PREDICTIVE MODELS
PROGNOSTIC vs. PREDICTIVE FACTORS
Prognostic
“A situation or condition, or a characteristic of a patient, that can be used to estimate the chance of recovery from a disease or the chance of the disease recurring (coming back). “
Predictive
“A condition or finding that can be used to help predict whether a person’s cancer will respond to a specific treatment. Predictive factor may also describe something that increases a person’s risk of developing a condition or disease.”
PROGNOSTIC QUESTION
Given I have had a seizure, what is the chance I will have another?
PROGNOSTIC MODELLING
PREDICTIVE QUESTION
Given I have had a seizure, will I respond to CBZ?
PREDICTIVE MODELLING
PROGNOSTIC QUESTION
Given Fred has had a 1st seizure, how long must he refrain from driving until his risk of a seizure is less
than 20%?
PREDICTIVE QUESTION
Given Fred has had a 1st seizure, does he
have refractory epilepsy?
IN CONCLUSION…
• Survival analysis• Time to event data• Censoring• Kaplan-Meier curves• Log rank tests• Cox model
• Prognostic & predictive models
ACKNOWLEDGEMENTS
