Main Points to be Covered

• Cumulative incidence using life table method• Difference between cumulative incidence and

person time incidence rate• Uses of person-time incidence rates• Calculating a person-time incidence rate• Relation of incidence rate to cumulative

incidence• Odds versus probability

Cumulative incidence: Life table

• No exact times of events or censoring needed

• Assume events and censoring occurred uniformly during the fixed time intervals (uniformity assumption)

• Therefore assume on average each censored person at risk for half of the time period

• Subtract one-half of subjects lost during interval from denominator at interval beginning

• Calculations just like Kaplan-Meier

Calculating a Life Table

Cumulative

Int. Total D Lost N At-risk P Event Survival

0-1 184 18 7 184 – 7/2=180.5 18/180.5=0.0997 0.9003

Subtract ½ of lost during interval from denominator

1-2 159 19 9 159 – 9/2=154.5 19/154.5=0.1229 0.7896

Repeat for next interval and so forth

2-3 131 13 9 131 – 9/2=126.5 13/126.5=0.1028 0.7084

Life Table analysis of primary biliary cirrhosis data:

Interval Total Deaths Lost Survival

0 - 1 184 18 7 0.9003

1 - 2 159 19 9 0.7896

2 - 3 131 13 9 0.7084

3 - 4 109 9 12 0.6465

Kaplan-Meier analysis (only showing survival close to year integers):

Time Total Deaths Lost Survival

1.038 159 0 1 0.9011

2.034 131 1 0 0.7846

3.001 109 1 0 0.7025

4.104 88 1 0 0.6397

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Survival Curve from Life Table: Cirrhosis Data

Where a life table is more appropriate than Kaplan-Meier

• Infectious diseases studied in a cohort – Subjects seen at intervals for exam and lab tests

(several months to a year or more)– Laboratory test for antibodies indicates the occurrence

of a new infection sometime between a negative and a positive test

• Example: Date of HIV infection estimated as mid-point between last negative and first positive test– Assumes uniformity of seroconversions on average

across the intervals (the life table assumption)– Not true Kaplan-Meier knowledge of event– Life table would be an appropriate way to analyze

Two assumptions in survival analysis

• Censoring is unrelated to the probability of the outcome

• There are no temporal trends in the probability of the outcome

Long-Term Survival Data May Be Invalid Due to Temporal Trends

Analysis of data from National Cancer Institute’s Follow-up of Diagnoses 1978 –1998 (SEER program):

Overall survival cohort method = 40% at 20 years

Overall survival with period analysis allowing forTemporal trend Changes in survival in recentCalendar periods = 51% at 20 years

Brenner, The Lancet, Oct 12, 2002

Improving Cancer Survival Times by Calendar Period

Brenner, The Lancet, Oct 12, 2002

Preservation of glomerular filtration rate on dialysis when adjusted for patient dropout.

BACKGROUND: Residual renal function (RRF) plays an important role in dialysis patients. … We speculated that regardless of the patient's type of therapy, the estimate obtained for the rate of decline in glomerular filtration rate (GFR) may be biased because of informative censoring associated with patient dropout. Informative censoring occurs when patients who die or transfer to another modality very early have associated with them a lower starting GFR or a higher rate of decline of GFR than patients who either complete the study or who die or transfer much later. If patient dropout is indeed related to the rate of decline in GFR and if this relationship is ignored in the analysis, then the estimate obtained of the rate of decline in GFR may be biased. …The results show that for the CANUSA cohort, the mean initial GFR was significantly lower, and the rate of decline was significantly higher for patients who died or transferred to HD than for patients who were randomly censored or received a transplant. CONCLUSION: In any longitudinal study designed to estimate trends in an outcome measured over time, it is important that the analysis of the data takes into account any effect patient dropout may have on the estimated trend. This analysis demonstrates that among PD patients, both the starting GFR and the rate of decline in GFR are associated with patient dropout.

Misra et al., Kidney Int 2000

The Three Elements in Measures of Disease Incidence

• E = an event = a disease diagnosis or death

• N = number of at-risk persons in the population under study

• T = time period during which the events are observed

Two Measures of Incidence

• The proportion of individuals who experience the event in a defined time period (E/N during some time T) = cumulative incidence

• The number of events divided by the amount of person-time observed (E/NT) = incidence rate (not a proportion)

Person-Time Incidence Rates

• The numerator is the same as incidence based on proportion of persons = events (E)

• The denominator is the sum of the follow-up times for each individual

• The resulting ratio of E/NT is not a proportion--may be greater than 1

• Value depends on unit of time used

Incidence rate value depends on the time units used

Incidence rate of 8 cases per 100 person-years:

• 0.67 cases per 100 person-months

• 0.15 cases per 100 person-weeks

• Note: time period during which rate is measured can differ from the units used (use data from 2 years of follow-up but report a rate per person-months)

Why Use Person-Time Incidence Rates?

1. To calculate incidence from population-based disease registries

2. To compare disease incidence in a cohort with a rate from the general population

3. To compare incidence from a time-varying exposure in persons while exposed and unexposed

4. Several commonly used statistical models use rates

Person-time incidence can be calculated from individual data or from the average

population at risk

• If have exact entry, censoring, and event times for each person, can calculate person-time for each person and sum them all

• Some large datasets may only have the average population size at risk

• Total person-time for grouped data is based on the time interval x the average population at risk during the interval

Note on Person-Time Rates• Person-time concept may seem unfamiliar because

often described as annual rate per 100,000 persons (i.e., person-time denominator is not made explicit)

• Example: “The incidence of Pediatric Cardiomyopathy in two regions of the United States” (NEJM, 2003)– 467 cases of cardiomyopathy in registry of 38 centers (New

England, Southwest) 1996 - 1999 – denominator “population estimates…1990 census with an

in- and out-migration algorithm” ages 1 - 18– “overall annual incidence of 1.13 per 100,000 children” – 1.44 per 100,000 in New Eng. vs. 0.98 in Southwest

Person-time incidence based on grouped vs. individual data

• Szklo and Nieto use incidence rate when based on group data (average population at risk) and incidence density when based on individual data

• This terminology distinction is not followed by most

• Average population method assumes uniform occurrence of events and of censoring during the interval (like life table)

c

rates: year 1 = 3/7.083 = 42.4/100 person-years year 2 = 3/2.50 =120/100 person-years both yrs 6/9.583 = 62.6/100 person-years

(1) Calculating a rate from population-based registry of diagnoses

• Research question: What is the incidence rate for first diagnoses of breast cancer in Marin County and how does it compare with rates from other counties?

• Nearly all new breast cancer diagnoses are reported to the SEER cancer registry

• How to obtain a denominator for a rate?

Large Population Person-Time Rates

“Since the production of stable rates for cancers at most individual sites requires a population of at least one millionsubjects, the logistic and financial problems of attemptingto maintain a constant surveillance system [of everyone inthe population] are usually prohibitive.” Breslow and Day, Statistical Methods in Cancer Research

Solution: Do surveillance of all the cancer diagnoses and estimate the population denominator to get person-time at risk.

For an annual rate, person-time denominator by the group method requires only an estimate of the average population size during the year (=the population at mid-year).

Average Population (Group data) rates versus individual data rates

• If losses are perfectly uniform, total person-time calculation for the denominator (and thus the rate) is the same whether based on average population size or individual follow-up

• For large populations the rate will be nearly identical calculated by either method

The estimates of breast cancer incidence (number of new cancers per year) most recently reported for Marin and other areas of the country were based on 1990 census information. Data from Census 2000 have enabled researchers to recalculate rates for Marin. Preliminary results show that revised incidence rates for Marin County based on the 2000 census are substantially lower than the rates calculated using 1990 census information. The discrepancy between using the 1990 and 2000 census data is due to projected population growth differing considerably from actual population growth. Comparisons of recalculated rates to other parts of California and the country are in progress.

Census Denominators for Incidence Rates are Estimates

(2) Comparing a rate from a cohort to the rate in the general population

• A cohort study follows up subjects for mortality for 36 years and found 765 deaths.

• Research question: Was the cohort mortality incidence high, low, or just average for those calendar years?

• How would you calculate the mortality incidence in the cohort?

Example of Using Person-Time Rates for Cohort Analysis

• Cohort of petrochemical workers– 6,588 male employees of Texas plant– Mortality determined from 1941-1977– 137,745 person-years of follow-up time– 765 deaths

• Overall death rate = 765 / 137,745 person-years = 5.6 per 1000 person-years

• Question: Is this a high death rate?

Austin SG, et al., J Occupat Med, 1983

Cohort of petrochemical workers

• Could calculate KM estimate of cumulative incidence (for 36 years of follow-up), but what is the comparison group?

• If calculate a person-time rate, it can be compared to the expected death rate in the U.S. population (within age and calendar date groups)

Cohort of petrochemical workers

• Applying U.S. population rates to the cohort age groups, get an expected 924 deaths in the cohort versus the 765 observed.

• Ratio of 765 observed/924 expected = 0.83 = 83%. This is called a Standardized Mortality Ratio (SMR).

(3) Comparing rates among persons with time-varying exposure

• Research question: In a Medicaid database is there an association between use of non-aspirin non-steroidal anti-inflammatory drugs (NANSAID) and coronary artery disease (CAD)?

• How would you study the relationship between NANSAID use and CAD?

Calculating stratified person-time incidence rates in cohorts

• For persons followed in a cohort some potential risk factors may be fixed but some may be variable – gender is fixed; taking medications or getting

regular exercise are behaviors that can change over time

• Adding up person-time in an exposure category to get a denominator of time at risk is one way to deal with risk factors that change over time

Analysis of changing exposure and disease incidence

• Tennessee Medicaid data base, 1987-1998• Use of NANSAIDs could change over 11 years

of study: same person could be in both using and non-using group at different times

• Could construct some fixed classification of persons as never, sometime, and frequent users and do cumulative incidence in each group.

Analysis of changing exposure with person-time rates

• Person-time totaled for time using NANSAIDs and not using and CHD events occurring during times of use and non-use

• 181,441 person-years of use (128,002 persons)

• 181,441 person-years of non-use (134,642 persons)

• 69,314 persons contributed to the denominator for both use and non-use Ray, Lancet, 2002

Analysis of changing exposure with person-time rates

• 6362 overall CHD events in 532,634 overall person-years, rate = 11.9 per 1000 person-years

• Rate for NANSAID use = 12.02 per 1000 pers-yrs

• Rate for non use = 11.86 per 1000 pers-yrs

• Rate ratio = 1.01

• Concluded no evidence that NANSAIDS reduced risk of CHD events

Ray, Lancet, 2002

Assumption of Person-Time Incidence Estimation

• “A” time units of follow-up on “B” persons is the same as “B” time units on “A” persons

• Observing 2 deaths in 2 persons followed for 50 years gives the same incidence rate as 2 deaths in 100 persons followed 1 year

• One reason rates are usually calculated for relatively short periods of time

• In other words, assumption is that the rate is constant

Relationship between Incidence Rate and Cumulative Incidence

• A constant rate produces an exponential cumulative incidence (or survival) distribution

• If know the instantaneous incidence rate, can derive the cumulative incidence/survival function or vice-versa

where F(t) = cumulative incidence; e= 2.71828; = rate; t = time units

etF )(1

Relationship between Incidence Rate and Cumulative Incidence

• If time period is short, incidence rate and cumulative incidence will be close

• If rate is low, incidence rate and cumulative incidence will be close (unless study period is very long)

Calculating Rates in STATAA few STATA survival analysis and rate commands:

Declare data set survival data:. stset timevar, fail(failvar)

. ltable timevar, graph gives life table analysis & graph

.strate gives person-years rate

.strate groupvar gives rates within groups

Immediate Commands in STATASTATA has an option to use it like a calculator forvarious computations without using a data set.

Called immediate commands.

Example, to calculate the confidence intervalaround a person-time rate:

. cii #person-time units #events, poisson

Eg. 6 events occur in 10 person-years of follow-up. cii 10 6, poisson

95% CI = 0.220 – 1.306

Summary Points• Person-time incidence rate or density is not

the same thing as cumulative incidence and is not a proportion

• Person-time incidence rate can be calculated with individual or group data– Allows incidence estimates in large populations

that are not completely enumerated– Allows comparison with population reference

rates from other data sources– Allows accumulation of time at risk for different

exposure strata

Measures of Disease Occurrence: PrevalenceOdds versus Probability

• Odds based on probability; expresses probability (p) as ratio: odds = p / (1 - p)– odds is always > p because divided by < 1

• For example, if probability of dying = 1/5, then odds of dying = 1/5 / 4/5 = ¼

• Thinking of odds as 2 outcomes, the numerator is the # of times of one outcome and the denominator the # of times of the other

Odds versus Probability

• Less intuitive than probability (probably wouldn’t say “my odds of dying are 1/4”)

• No less legitimate mathematically, just not so easily understood

• Used in epidemiology primarily because the log odds of the outcome is given by the coefficient of a predictor in a logistic regression