Intermediate methods in observational epidemiology

2008

Instructor: Moyses Szklo

Measures of Disease Frequency

MEASURES OF RISK

• Absolute measures of event (including disease) frequency:

– Incidence and Incidence Odds– Prevalence and Prevalence Odds

What is "incidence"?Two major ways to define incidence

• Cumulative incidence (probability)SURVIVAL ANALYSIS (Unit of analysis:

individual)

• Rate or DensityANALYSIS BASED ON PERSON-TIME (Unit

of analysis: time)

• OBJECTIVE OF SURVIVAL ANALYSIS:To compare the “cumulative incidence” of an

event (or the proportion surviving event-free) in exposed and unexposed (characteristic present or absent) while adjusting for time to event (follow-up time)

• BASIS FOR THE ANALYSIS• NUMBER of EVENTS• TIME of occurrence

Time

Su

rviv

al1.0

Need to precisely define:• “EVENT” (failure):

– Death– Disease (diagnosis, start of symptoms, relapse)– Quit smoking– Menopause

• “TIME”:– Time from recruitment into the study– Time from employment– Time from diagnosis (prognostic studies)– Time from infection– Calendar time– Age

– Example:• Follow up of 6 patients (2 yrs)

– 3 Deaths – 2 censored (lost) before 2 years– 1 survived 2 years

Question: What is the Cumulative Incidence (or the Cumulative Survival) up to 2 years?

Death

Censored observation (lost to follow-up, withdrawal)

( ) Number of months to follow-up

Jan1999

Jan2000

Jan2001

1

3

2

4

5

6

(24)

(6)

(18)

(15)

(13)

(3)

Person ID

Crude Survival:3/6= 50%

Change time scale to “follow-up” time:

Person ID

0 1 2

1

3

2

4

5

6

(24)

(6)

(18)

(15)

(13)

(3)

Follow-up time (years)

One solution:

• Actuarial life tableAssume that censored observations over the period contribute one-half the persons at risk in the denominator (censored observations occur uniformly throughout follow-up interval).

ID

0 1 2

1

32

456

(24)

(6)(18)

(15)(13)

(3)

Follow-up time (years)

60.05

3

2216

32

yrsq

It can be also calculated for years 1 and 2 separately: Year 1: S(Y1)= [1 - {1 ÷ [6 – ½(1)]}= 0.82Year 2: S(Y2)= [1 – {2 ÷ [4 – ½(1)]}= 0.43S(2yrs)= 0.82 × 0.43= 0.35

40.01)2( 2 yrsqyrsS

10082

43

Year 1 Year 2

Cumulative Survival

Follow-up time

KAPLAN-MEIER METHODE.L. Kaplan and P. Meier, 1958*

Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event time

Step 1: Sort the survival times from shortest to longest

*Kaplan EL, Meier P.Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81.

Person ID

0 1 2

1

3

2

4

56

(24)

(6)

(18)

(15)

(3)

Follow-up time (years)

(13)

KAPLAN-MEIER METHODE.L. Kaplan and P. Meier, 1958*

Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event time

Step 1: Sort the survival times from shortest to longest

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (13)

6 (3)

Follow-up time (years)

*Kaplan EL, Meier P.Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81.

Step 2: For each time of occurrence of an event, compute the conditional survival

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (13)

6 (3)

Follow-up time (years)

When the first event occurs (3 months after beginning of follow-up), there are 6 persons at risk. One dies at that point; 5 of the 6 survive beyond that point. Thus:

• Incidence of event at exact time 3 months: 1/6• Probability of survival beyond 3 months: 5/6

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5

6 (3)

Follow-up time (years)

When the second event occurs (13 months), there are 4 persons at risk. One of them dies at that point; 3 of the 4 survive beyond that point. Thus:

• Incidence of event at exact time 13 months: 1/4

• Probability of survival beyond 13 months: ¾

(13)

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5

6 (3)

Follow-up time (years)

When the third event occurs (18 months), there are 2 persons at risk. One of them dies at that point; 1 of the 2 survive beyond that point. Thus:

• Incidence of event at exact time 18 months: 1/2• Probability of survival beyond 18 months: 1/2

(13)

Step 3: For each time of occurrence of an event, compute the cumulative survival (survival function), multiplying conditional probabilities of survival.

3 months: S(3)=5/6=0.833

12 months: S(13)=5/63/4=0.625

18 months: S(18)=5/6 3/41/2 =0.3125

CONDITIONAL PROBABILITY OF AN EVENT (or of survival)

The probability of an event (or of survival) at time t (for the individuals at risk at time t), that is, conditioned on being at risk at exact time t.

0.8330.6250.3125

Time (mo)

31318

Plotting the survival function:

0.60

0.40

0.20

0.80

Survival

2520151050

Month of follow-up

1.00

The cumulative incidence (up to 24 months): 1-0.3125 = 0.6875 (or 69%)

Si

0.833

0.625

0.3125 0.3125

0.8330.6250.3125

Time (mo)

31318

Plotting the survival function:

2520151050

Month of follow-up

0.60

0.40

0.20

0.80

Cumulative Survival1.00

0.8

0.6

0.3

CEEPlacebo

CEE

Placebo

Cumulative Hazards for Coronary Heart Disease and Stroke in the Women’s Health Initiative Randomized Controlled Trial

(The WHI Steering Committee. JAMA 2004;291:1701-1712)

EXPERIMENTAL STUDY

0.8330.6250.3125

Time (mo)

31318

Plotting the survival function:

2520151050

Month of follow-up

0.60

0.40

0.20

0.80

Cumulative Survival1.00

0.8

0.6

0.3

Cumulative Hazard

0.20

0.80

1.00

0.60

0.400.2

0.4

0.7

The cumulative incidence (hazard) at the end of 24 months: 1-0.3 = 0.7 (or 70%)

ACTUARIAL LIFE TABLE VS KAPLAN-MEIER

If N is large and/or if life-table intervals are small, results are similar

•Survival after diagnosis of Ewing’s sarcoma

ASSUMPTIONS IN KAPLAN-MEIER SURVIVAL ESTIMATES

• (If individuals are recruited over a long period of time)

No secular trends

Calendar time Follow-up time

ASSUMPTIONS IN SURVIVAL ESTIMATES(Cont’d)

• Censoring is independent of survival (uninformative censoring): Those censored at time t have the same prognosis as those remaining.

Types of censoring:• Lost to follow-up

– Migration– Refusal

• Death (from another cause)• Administrative withdrawal (study finished)

Calculation of incidenceStrategy #2

ANALYSIS BASED ON PERSON-TIME

CALCULATION OF PERSON-TIME AND INCIDENCE RATES (Unit of analysis: time)

Example 1 Observe 1st graders, total 500 hours

Observe 12 accidents

Accident rate:

hour-personper0.024500

12R

IT IS NOT KNOWN WHETHER 500 CHILDREN WERE OBSERVED FOR 1 HOUR, OR 250 CHILDREN OBSERVED FOR 2 HOURS, OR 100 CHILDREN OBSERVED FOR 5 HOURS… ETC.

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (13)

6 (3)

Follow-up time (years)

CALCULATION OF PERSON-TIME AND INCIDENCE RATES

Example 2

Person ID

No. of person-years in

Total FU1st FU year 2nd FU year

6

2

5

4

3

1

3/12=0.25

6/12=0.50

12/12=1.00

12/12=1.00

12/12=1.00

12/12=1.00

0

0

1/12=0.08

3/12=0.25

6/12=0.50

12/12=1.00

0.25

0.25

1.00

1.25

1.50

2.00

Total 4.75 1.83 6.58

Step 1: Calculate denominator, i.e. units of time (years) contributed by each individual, and total:

Step 2: Calculate rate per person-year for the total follow-up

period:

year-personper0.466.58

3R

It is also possible to calculate the incidence rates per person-year separately for shorter periods during the follow-up:

For year 1:

For year 2:

year-personper0.214.75

1R

year-personper1.09 1.83

2R

Notes:

• Rates have units (time-1). • Proportions (e.g., cumulative incidence) are unitless.• As velocity, rate is an instantaneous concept. The

choice of time unit used to express it is totally arbitrary. E.g.:

0.024 per person-hour = 0.576 per person-day = 210.2 per person-year

0.46 per person-year = 4.6 per person-decade

Person No. Year 1 Year 2 Total

1 1/12= 0.08 (D) 0 0.08

2 2/12= 0.17 (C) 0 0.17

3 3/12= 0.25 (C) 0 0.25

4 4/12= 0.33 (C) 0 0.33

5 5/12= 0.42 (C) 0 0.42

6 6/12= 0.50 (D) 0 0.50

7 7/12= 0.58 (C) 0 0.58

8 8/12= 0.67 (C) 0 0.67

9 9/12= 0.75 (C) 0 0.75

10 10/12= 0.83 (C) 0 0.83

11 11/12= 0.92 (C) 0 0.92

12 12/12= 1.00 (D) 0 1.00

13 12/12= 1.00 (C) 1/12= 0.08 (C) 1.08

14 12/12 = 1.00 (C) 2/12= 0.17 (C) 1.17

15 12/12 = 1.00 (C) 3/12= 0.25 (D) 1.25

16 12/12 = 1.00 4/12= 0.33 (C) 1.33

17 12/12 = 1.00 5/12= 0.42 (C) 1.42

18 12/12 = 1.00 6/12= 0.50 (C) 1.50

19 12/12 = 1.00 7/12= 0.58 (C) 1.58

20 12/12 = 1.00 8/12= 0.67 (C) 1.67

21 12/12 = 1.00 9/12= 0.75 (D) 1.75

22 12/12 = 1.00 10/12= 0.83 (C) 1.83

23 12/12 = 1.00 11/12= 0.92 (C) 1.92

24 12/12 = 1.00 12/12= 1.00 (C) 2.0

Total 18.5 6.5 25.0

Death rate per person-time (person-year)5 deaths/25.0 person-years= 0.20 or 20 deaths per 100 person-years

Death rate per average population, estimated at mid-point of follow-upMid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5

Death rate= 5/12.5 per 2 years= 0.40Average annual death rate= 0.40/2= 0.20 or 20/100 population

No. of person-years of follow-up

D, deathsC, censored

Person No. Year 1 Year 2 Total

1 1/12= 0.08 (D) 0 0.08

2 2/12= 0.17 (C) 0 0.17

3 3/12= 0.25 (C) 0 0.25

4 4/12= 0.33 (C) 0 0.33

5 5/12= 0.42 (C) 0 0.42

6 6/12= 0.50 (D) 0 0.50

7 7/12= 0.58 (C) 0 0.58

8 8/12= 0.67 (C) 0 0.67

9 9/12= 0.75 (C) 0 0.75

10 10/12= 0.83 (C) 0 0.83

11 11/12= 0.92 (C) 0 0.92

12 12/12= 1.00 (D) 0 1.00

13 12/12= 1.00 (C) 1/12= 0.08 (C) 1.08

14 12/12 = 1.00 (C) 2/12= 0.17 (C) 1.17

15 12/12 = 1.00 (C) 3/12= 0.25 (D) 1.25

16 12/12 = 1.00 4/12= 0.33 (C) 1.33

17 12/12 = 1.00 5/12= 0.42 (C) 1.42

18 12/12 = 1.00 6/12= 0.50 (C) 1.50

19 12/12 = 1.00 7/12= 0.58 (C) 1.58

20 12/12 = 1.00 8/12= 0.67 (C) 1.67

21 12/12 = 1.00 9/12= 0.75 (D) 1.75

22 12/12 = 1.00 10/12= 0.83 (C) 1.83

23 12/12 = 1.00 11/12= 0.92 (C) 1.92

24 12/12 = 1.00 12/12= 1.00 (C) 2.0

Total 18.5 6.5 25.0

Death rate per person-time (person-year)5 deaths/25.0 person-years= 0.20 or 20 deaths/100 person-years

Death rate per average population, estimated at mid-point of follow-upMid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5

Death rate= 5/12.5 per 2 years= 0.40Average annual death rate= 0.40/2= 0.20 or 20/100 population

No. of person-years of follow-up

D, deathsC, censored

No of person tim eEven ts D

Popu la tion N T im e N

Even ts DPopu la tion N

T im e N

.( )

( ) ( )

( )( )

( )

Notes: Rates have an undesirable statistical property• Rates can be more than 1.0 (100%):

– 1 person dies exactly after 6 months:• No. of person-years: 1 x 0.5 years= 0.5 person-years

Rate per PY per PYs 10 5

2 0 2 0 0 1 0 0.

.

Use of person-time to account for changes in exposure status (Time-dependent exposures)

Example: Adjusting for age, are women after menopause at a higher risk for myocardial infarction?

123456

Number of PY in each group

ID 1 2 3 4 5 6 7 8 9 10No. PY

PRE menoNo. PY

POST meno

C

C

: Myocardial Infarction; C: censored observation.

Rates per person-year:Pre-menopausal = 1/17 = 0.06 (6 per 100 py)Post-menopausal = 2/18 = 0.11 (11 per 100 py)

Rate ratio = 0.11/0.06 = 1.85

3 40 56 00 15 53 317 18

Year of follow-up

Note: Event is assigned to exposure status when it occurs

ASSUMPTIONS IN PERSON-TIME ESTIMATES

Risk is constant within each interval for which person-time units are estimated (no cumulative effect):– N individuals followed for t time t individuals

followed for N time– However, are 10 smokers followed for 1 year

comparable to 1 smoker followed for 10 years (both: 10 person-years)

• No secular trends (if individuals are recruited over a relatively long time interval)

• Losses are independent from survival

Rate for 1st Year= 0.21/PY

Rate for 2nd Year= 1.09/ PY

Total for 2 years = 0.46/PY

ASSUMPTIONS IN PERSON-TIME ESTIMATES

Risk is constant within each interval/period for which person-time units are estimated (no cumulative effect):– N individuals followed for t time t individuals

followed for N time– However, are 10 smokers followed for 1 year

comparable to 1 smoker followed for 10 years (both: 10 person-years)

• No secular trends (if individuals are recruited over a relatively long time interval)

• Losses are independent of survival

Method Estimate Value

Life-table

Life-table

Kaplan-Meier

q (2 years)

q(Y1) × q(Y2)

q (2 years)

0.60

0.65

0.64

Person-year

Midpoint (median) population

Rate (yearly) 0.46/py

0.43 per year

SUMMARY OF ESTIMATES

POINT REVALENCE

Point Prevalence“The number of affected persons present at the population at a specific time divided by the number of persons in the population at that time”Gordis, 2000, p.33

Relation with incidence --- Usual formula:

Point Prevalence = Incidence x Duration* P = I x D

* Average duration (survival) after disease onset.

Prevalence

1 P revalence Incidence D uration

True formula:

ODDS

OddsThe ratio of the probabilities of an event to that of the non-event.

Prob1-

ProbOdds

Example: The probability of an event (e.g., death, disease, recovery, etc.) is 0.20, and thus the odds is:

That is, for every person with the event, there are 4 persons without the event.

0.25) (or 41:0.80

0.20

0.201-

0.20Odds