1、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 Density ANALYSIS BA
2、SED 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
3、NUMBER of EVENTS TIME of occurrence,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 A
4、ge,Example: Follow up of 6 patients (2 yrs) 3 Deaths 2 censored (lost) before 2 years 1 survived 2 yearsQuestion: 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,Jan 1999,Jan 200
5、0,Jan 2001,1,3,2,4,5,6,(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),(13),Follow-up time (years),One solution:,Actuarial life tableAssume that censored observations over the period contribute one-half the person
6、s at risk in the denominator (censored observations occur uniformly throughout follow-up interval).,ID,0,1,2,1,3,2,4,5,6,(24),(6),(18),(15),(13),(3),Follow-up time (years),It can be also calculated for years 1 and 2 separately: Year 1: S(Y1)= 1 - 1 6 (1)= 0.82 Year 2: S(Y2)= 1 2 4 (1)= 0.43 S(2yrs)=
7、 0.82 0.43= 0.35,KAPLAN-MEIER METHOD E.L. Kaplan and P. Meier, 1958*,Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event timeStep 1: Sort the survival times from shortest to longest,*Kaplan EL, Meier P.Nonparametric estimation from incomplete
8、 observations. J Am Stat Assoc 1958;53:457-81.,Person ID,0,1,2,1,3,2,4,5,6,(24),Follow-up time (years),(13),KAPLAN-MEIER METHOD E.L. Kaplan and P. Meier, 1958*,Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event timeStep 1: Sort the survival
9、times from shortest to longest,Person ID,0,1,2,4,5,(13),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,0,1,2,4,5,Follow-up ti
10、me (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,0,1,2,4,2,(6),5,6,(3)
11、,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: ,Person ID,0,1,2,4,2,(6),(15),5,6,(3),F
12、ollow-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 occurren
13、ce of an event, compute the cumulative survival (survival function), multiplying conditional probabilities of survival.3 months: S(3)=5/6=0.83312 months: S(13)=5/63/4=0.62518 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 survi
14、val) at time t (for the individuals at risk at time t), that is, conditioned on being at risk at exact time t.,Plotting the survival function:,The cumulative incidence (up to 24 months): 1-0.3125 = 0.6875 (or 69%),Si,Plotting the survival function:,Month of follow-up,Cumulative Hazards for Coronary
15、Heart Disease and Stroke in the Womens Health Initiative Randomized Controlled Trial,(The WHI Steering Committee. JAMA 2004;291:1701-1712),EXPERIMENTAL STUDY,Plotting the survival function:,Month of follow-up,The cumulative incidence (hazard) at the end of 24 months: 1-0.3 = 0.7 (or 70%),ACTUARIAL L
16、IFE TABLE VS KAPLAN-MEIERIf N is large and/or if life-table intervals are small, results are similar,Survival after diagnosis of Ewings 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,ASSUM
17、PTIONS IN SURVIVAL ESTIMATES,(Contd)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),Calcul
18、ation of incidence Strategy #2 ANALYSIS BASED ON PERSON-TIME,CALCULATION OF PERSON-TIME AND INCIDENCE RATES (Unit of analysis: time)Example 1 Observe 1st graders, total 500 hoursObserve 12 accidentsAccident rate:,IT IS NOT KNOWN WHETHER 500 CHILDREN WERE OBSERVED FOR 1 HOUR, OR 250 CHILDREN OBSERVED
19、 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 RATESExample 2,Step 1: Calculate denominator, i.e. units of time (years) contributed by each individual, and total:,Step 2:
20、Calculate rate per person-year for the total follow-up period:,It is also possible to calculate theincidence rates per person-year separately for shorter periods during the follow-up:For year 1:For year 2:,Notes:,Rates have units (time-1). Proportions (e.g., cumulative incidence) are unitless. As ve
21、locity, 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-year0.46 per person-year = 4.6 per person-decade,Death rate per person-time (person-year) 5 deaths/25.0 person-years= 0.20
22、or 20 deaths per 100 person-years,Death rate per average population, estimated at mid-point of follow-up Mid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5,Death rate= 5/12.5 per 2 years= 0.40 Average annual death rate= 0.40/2= 0.20 or 20/100 population,No. of pe
23、rson-years of follow-up,D, deaths C, censored,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-up Mid-point (median) population (When calculating yearly rate in Vital Statistics
24、) = 12.5,Death rate= 5/12.5 per 2 years= 0.40 Average annual death rate= 0.40/2= 0.20 or 20/100 population,No. of person-years of follow-up,D, deaths C, censored,Notes: Rates have an undesirable statistical property,Rates can be more than 1.0 (100%): 1 person dies exactly after 6 months: No. of pers
25、on-years: 1 x 0.5 years= 0.5 person-years,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?,Rates per person-year:Pre-menopausal = 1/17 = 0.06 (6 per 100 py)Post-me
26、nopausal = 2/18 = 0.11 (11 per 100 py)Rate ratio = 0.11/0.06 = 1.85,3,4,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 individua
27、ls 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 fo
28、r 1st Year= 0.21/PYRate for 2nd Year= 1.09/ PYTotal 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, a
29、re 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,SUMMARY OF ESTIMATES,POINT REVALENCE,Point Prevalence “The number of affecte
30、d 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.,True formula:,ODDS,Odds The ratio of the probabilities of an event to that of the non-event.,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.,
