Dr Laura Bonnett Department of Biostatistics. UNDERSTANDING SURVIVAL ANALYSIS.

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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

LOG-RANK TEST

p<0·0001

Years since randomisation

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

• 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

10 bth

)(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

E.g. EEG Results (normal, abnormal, not done)

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

E.g. EEG Results (normal, abnormal, not done)

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:

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

L.J.BONNETT@LIV.AC.UK

Thank you!

Dr Laura Bonnett Department of Biostatistics. UNDERSTANDING SURVIVAL ANALYSIS.

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