Midterm Exam

• Yes, I know the first poll had a bug• Second poll is up and running:

– www.tinyurl.com/epiexam2

• I will accept responses until Monday Morning and will announce the results in Monday’s class

• There will have to be an overwhelming majority for me to change the exam date

HSS4303B – Intro to EpidemiologyJan 28, 2010 - Kaplan-Meier Survival Curves

Abstracts

• Don’t forget…. Due by midnight tonight• Email assignments to hss4303@gmail.com

(already received 6 as of 2:AM)• Include name and student number in subject

heading• Do not CC me or Erin!!!!!(!!!!)

Have you ever wished to have some of your work published but never had an opportunity?

Well now’s your chance!

Submit your work to the IJHS today!

Details: www.IJHS.ca

Inquiries: IJHS@hssuottawa.ca

Deadline for submissions: February 20th, 2010

5

Early example of survival analysis, 1669

Christiaan Huygens' 1669 curve showing how many out of 100 people survive until 86 years.From: Howard Wainer STATISTICAL GRAPHICS: Mapping the Pathways of Science. Annual Review of Psychology. Vol. 52: 305-335.

6

Early example of survival analysis

What was a person’s chance of surviving past 20? Past 36?

This is survival analysis! We are trying to estimate this curve—only the outcome can be any binary event, not just death.

7

Probabilities

P(T>76)=.01

P(T>36) = .16

P(T>20) ~ 0.32, etc.

8

Retrospective cohort study:From December 2003 BMJ:

Aspirin, ibuprofen, and mortality after myocardial infarction: retrospective cohort study

Curits et al. BMJ 2003;327:1322-1323.

9

Survival Analysis: Terms

• Time-to-event: The time from entry into a study until a subject has a particular outcome

• Censoring: Subjects are said to be censored if they are lost to follow up or drop out of the study, or if the study ends before they die or have an outcome of interest. They are counted as alive or disease-free for the time they were enrolled in the study. – If dropout is related to both outcome and treatment,

dropouts may bias the results

What is Survival Time?• Survival time refers to a variable

which measures the time from a particular starting time (e.g., time initiated the treatment) to a particular endpoint of interest (e.g., attaining certain functional abilities)

• It is important to note that for some subjects in the study a complete survival time may not be available due to censoring

Censored DataSome patients may still be alive or in remission at the end of the study period

The exact survival times of these subjects are unknown

These are called censored observation or censored times and can also occur when individuals are lost to follow-up after a period of study

Right Censoring

• Pretending that all subjects began at the same time

13

Choice of time of origin. Note varying start times.

14

Count every subject’s time since their baseline data collection.

Right-censoring!

Kaplan-Meier Survival Curve (K-M)

• K-M curves represent the proportion of the study population still surviving (or free of disease or some other outcome) at successive times

• as the number of subjects in each intervention group decreases over time, the curves are more precise in the earlier periods (left hand side of the survival curves) than later periods (right hand side of the survival curves)

18

Introduction to Kaplan-Meier

Non-parametric estimate of the survival function:

Simply, the empirical probability of surviving past certain times in the sample (taking into account censoring).

19

Introduction to Kaplan-Meier

• Non-parametric estimate of the survival function.

• Commonly used to describe survivorship of study population/s.

• Commonly used to compare two study populations.

• Intuitive graphical presentation.

Treatment #1

Treatment #2

Beginning of study End of study Time in months

Subject B

Subject A

Subject C

Subject D

Subject E

Survival Data (right-censored)

1. subject E dies at 4 months

X

100%

Time in months

Corresponding Kaplan-Meier Curve

Probability of surviving to 4 months is 100% = 5/5

Fraction surviving this death = 4/5

Subject E dies at 4 months

Beginning of study End of study Time in months

Subject B

Subject A

Subject C

Subject D

Subject E

Survival Data 2. subject A drops out after 6 months

1. subject E dies at 4 months

X

3. subject C dies at 7 monthsX

100%

Time in months

Corresponding Kaplan-Meier Curve

subject C dies at 7 months

Fraction surviving this death = 2/3

Beginning of study End of study Time in months

Subject B

Subject A

Subject C

Subject D

Subject E

Survival Data 2. subject A drops out after 6 months

4. Subjects B and D survive for the whole year-long study period

1. subject E dies at 4 months

X

3. subject C dies at 7 monthsX

26

100%

Time in months

Corresponding Kaplan-Meier CurveP(surviving intervals 1 and 2)=P(surviving interval 1)*P(surviving interval 2)

Product limit estimate of survival = P(surviving interval 1/at-risk up to failure 1) * P(surviving interval 2/at-risk up to failure 2) = 4/5 * 2/3= .5333

27

The product limit estimate• The probability of surviving in the entire year, taking

into account censoring• = (4/5) (2/3) = 53%

• NOTE: 40% (2/5) because the one drop-out survived at least a portion of the year.

• AND <60% (3/5) because we don’t know if the one drop-out would have survived until the end of the year.

Comparing 2 groups

KM Curves comparing Thiotepa to Placebo

S(t

)

Months0 20 40 60

0.00

0.25

0.50

0.75

1.00

Thiotepa

Control

30

What happens to our precision as the experiment lingers on in time?

• Survival estimates can be unreliable toward the end of a study when there are small numbers of subjects at risk of having an event.

31

Limitations of Kaplan-Meier• Mainly descriptive• Doesn’t control for covariates• Requires categorical predictors• Can’t accommodate time-dependent variables

Example of K-M Curve

• In 1982, 38 infertile women underwent a special laparascopic technique, and attempted to get pregnant

• They were each followed for 24 weeks to see who would conceive and when

What is the outcome variable?

Is this population likely to be right-censored?

33

(This is how it’s going to look)

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

Months to conception or censoring in 38 sub-fertile women after laparoscopy

1 21 31 41 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

1 21 31 41 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

Months to conception or censoring in 38 sub-fertile women after laparoscopy

36

Corresponding Kaplan-Meier Curve

6 women conceived in 1st month (1st menstrual cycle). Therefore, 32/38 “survived” pregnancy-free past 1 month.

37

Corresponding Kaplan-Meier Curve

S(t=1) = 32/38 = 84.2%

S(t) represents estimated survival probability: P(T>t)Here P(T>1).

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

1 2.11 31 41 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

Important detail of how the data were coded:Censoring at t=2 indicates survival PAST the 2nd cycle (i.e., we know the

woman “survived” her 2nd cycle pregnancy-free).

Thus, for calculating KM estimator at 2 months, this person should still be included in the risk set.

Think of it as 2+ months, e.g., 2.1 months.

Months to conception or censoring in 38 sub-fertile women after laparoscopy

39

Corresponding Kaplan-Meier Curve

40

Corresponding Kaplan-Meier Curve5 women conceive in 2nd month.

The risk set at event time 2 included 32 women.

Therefore, 27/32=84.4% “survived” event time 2 pregnancy-free.

S(t=2) = ( 84.2%)*(84.4%)=71.1%

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

1 2.11 3.11 41 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

Risk set at 3 months includes 26 women

Months to conception or censoring in 38 sub-fertile women after laparoscopy

42

Corresponding Kaplan-Meier Curve

43

Corresponding Kaplan-Meier Curve

S(t=3) = ( 84.2%)*(84.4%)*(88.5%)=62.8%

3 women conceive in the 3rd month.

The risk set at event time 3 included 26 women.

23/26=88.5% “survived” event time 3 pregnancy-free.

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

1 21 3.11 41 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

Risk set at 4 months includes 22 women

Months to conception or censoring in 38 sub-fertile women after laparoscopy

45

Corresponding Kaplan-Meier Curve

46

Corresponding Kaplan-Meier Curve

S(t=4) = ( 84.2%)*(84.4%)*(88.5%)*(86.4%)=54.2%

3 women conceive in the 4th month, and 1 was censored between months 3 and 4.

The risk set at event time 4 included 22 women.

19/22=86.4% “survived” event time 4 pregnancy-free.

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

1 21 31 4.11 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

Risk set at 6 months includes 18 women

Months to conception or censoring in 38 sub-fertile women after laparoscopy

48

Corresponding Kaplan-Meier Curve

49

Corresponding Kaplan-Meier Curve

S(t=6) = (54.2%)*(88.8%)=42.9%

2 women conceive in the 6th month of the study, and one was censored between months 4 and 6.

The risk set at event time 5 included 18 women.

16/18=88.8% “survived” event time 5 pregnancy-free.

50

Skipping ahead to the 9th and final event time (months=16)…

S(t=13) 22%(“eyeball” approximation)

Data from: Luthra P, Bland JM, Stanton SL. Incidence of pregnancy after laparoscopy and hydrotubation. BMJ 1982; 284: 1013-1014

1 21 31 41 71 71 82 82 92 92 92 113 243 243  4  4  4  6  6  9  9  9  10  13  16  

Conceived (event) Did not conceive (censored)

2 remaining at 16 months (9th event time)

Months to conception or censoring in 38 sub-fertile women after laparoscopy

52

Skipping ahead to the 9th and final event time (months=16)…

S(t=16) =( 22%)*(2/3)=15%

Tail here just represents that the final 2 women did not conceive (cannot make many inferences from the end of a KM curve)!

Observed survival: rationale for life tables

Table 6-1. Hypothetical Study of Treatment Results in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to Follow-up)

    Number Alive on Anniversary of Treatment

Year of Treatment

No. of Patients Treated

2001 2002 2003 2004 2005

2000 84 44 21 13 10 8

2001 62   31 14 10 6

2002 93     50 20 13

2003 60       29 16

2004 76         43

Observed survival: rationale for life tables

Table 6-2. Rearrangement of Data in Table 5-1, Showing Survival Tabulated by Years Since Enrollment in Treatment (None Lost to Follow-up)

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Observed survival: one yearTable 6-3. Analysis of Survival in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to

Follow-up): I

P1 = Probability of surviving the 1st year

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Totals 375 197        

     

=those alive / risk set = 197/375 = 0.53

Observed survival: two years

Table 6-4. Analysis of Survival in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to Follow-up): II

P2 = Probability of surviving the 2nd year

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Totals   197 71      

   

=those alive at end of 2nd year / those at risk (risk set)

=71 / (197-43) = 71/154 = 0.46

Observed survival: three yearsTable 6-5. Analysis of Survival in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to

Follow-up): III

P3 = Probability of surviving the 3rd year

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Totals     71 36    

   

=36/(71-16) = 0.66

Observed survival: four yearsTable 6-6. Analysis of Survival in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to

Follow-up): IV

P4 = Probability of surviving the 4th year

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Totals       36 16  

   

=16/(36-13) = 0.70

Observed survival: five years

Table 6-7. Analysis of Survival in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to Follow-up): V

P5 = Probability of surviving the 5th year

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Totals         16 8

   

=8/(16-6) = 0.80

So…

• What is the probability of surviving all 5 years?

Probability for each year of survivalTable 6-8. Probability of Survival for Each Year of the Study

P1 = Probability of surviving the 1st year

P2 = Probability of surviving the 2nd year given survival to the end of the 1st year

P3 = Probability of surviving the 3rd year given survival to the end of the 2nd year

P4 = Probability of surviving the 4th year given survival to the end of the 3rd year

P5 = Probability of surviving the 5th year given survival to the end of the 4th year

Cumulative probabilities of survival

Table 6-9. Cumulative Probabilities of Surviving Different Lengths of Time

Probability of surviving 1 year = P1 =.525 =52.5%

Probability of surviving 2 years = P1 ×P2 =.525 ×.461 =.242 =24.2%

Probability of surviving 3 years = P1 ×P2 ×P3 =.525 ×.461 ×.655 =.159 =15.9%

Probability of surviving 4 years = P1 ×P2 ×P3 ×P4 =.525 ×.461 ×.655 ×.696 =.110 =11.0%

Probability of surviving 5 years = P1 ×P2 ×P3 ×P4 ×P5 =.525 ×.461 ×.655 ×.696 ×.800 =.088 =8.8%

Survival curve for five-year survival

Life table5. Life tables

– Also know as mortality tables show the effect of mortality on life expectancy

– They are created by different agencies for different purposes (insurance companies, government agencies, vital statistics, etc.)

– Different approaches are also used depending on the answer that is sought from the life table

• Probability method• Kaplan-Meier method

– Allow calculation of survival– Allow consideration to loss to follow-up or withdrawals– Allow calculation of life expectancy

Observed survival: one yearTable 6-3. Analysis of Survival in Patients Treated from 2000 to 2004 and Followed to 2005 (None Lost to

Follow-up): I

    Number Alive at End of Year

Year of Treatment

No. of Patients Treated

1st Year

2nd Year

3rd Year

4th Year

5th Year

2000 84 44 21 13 10 8

2001 62 31 14 10 6  

2002 93 50 20 13    

2003 60 29 16      

2004 76 43        

Totals 375 197  71 36  16  8 

     

Life table calculations

Table 6-10. Rearrangement of Data in Standard Format for Life Table Calculations

(1) Interval Since Beginning Treatment

(2) Alive at Beginning of Interval

(3) Died During Interval

(4) Withdrew During Interval

x lX dX wX

1st year 375 178 0

2nd year 197 83 43

3rd year 71 19 16

4th year 36 7 13

5th year 16 2 6

We will not complete the life table calculations today. But review the basic logic for Monday.

However……

Here are your homework assignments….

Patient

Fifteen patients infected with dumbass disease are followed for 51 weeks. The table below indicates at which week each patient died of the disease.

Construct a K-M curve for this datasetQ.1

Patient

Q.2

MMMFFMFFFMMMFFM

SEX

Now, construct 2 K-M curves on the same graph for Men and Women.

Can you work out a system?