Dr Laura Bonnett Department 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 diseaseetc. 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 Cumulative probability of seizure(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

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: Risk Factors (Covariates) Parameters to be estimated – related to effect sizes Underlying hazard

INTERPRETATION E.g. Risk of seizure for a person on Treatment (x 1 = 1) compared to Control (x 1 = 0), assuming they are alike for all other covariates (x 2, x 3, etc.). - Hazard rate in Control group at time t: - Hazard ratio is: - 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 values To obtain HRs, ‘dummy (binary) variables’ must be created e.g. Interpretation is then as for binary variables Dummy Variable 1 Dummy Variable 2 Baseline Category00 Alternative Category 110 Alternative Category 201

INTERPRETATION FOR CATEGORICAL VARIABLE E.g. EEG Results (normal, abnormal, not done) Dummy Variable 1 Dummy Variable 2 Normal00 Abnormal10 Not done01 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 Normal00 Abnormal10 Not done01

BACK TO FRED… Remember, log-rank p< Cox model (univariate): Variable: treatment decision Outcome: time to 1 st 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 1 st 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 1 st 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

Thank you!