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Rate versus Risk

Rate versus Risk. Two basic measures of the occurrence of new events (disease) Cumulative incidence=Risk=Probability Incidence rate=Rate=events per time units Last week we discussed the concept of cumulative incidence

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Rate versus Risk

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  1. Rate versus Risk • Two basic measures of the occurrence of new events (disease) • Cumulative incidence=Risk=Probability • Incidence rate=Rate=events per time units • Last week we discussed the concept of cumulative incidence • Commonly calculated by the Kaplan-Meier method when different follow-up times exist • Incidence rate of disease is somewhat less intuitive but is the more fundamental measure

  2. Main Points to be Covered • Cumulative incidence and person-time incidence rate: different but related • Hazard • Calculating a person-time incidence rate • Uses of person-time incidence rates • STATA commands for rates • Assumptions of survival and person-time analyses

  3. 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

  4. 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) = incidencerate

  5. 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

  6. 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

  7. 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 20 deaths in 200 persons followed for 50 years gives the same incidence rate as 20 deaths in 10,000 persons followed 1 year • The rate is constant for the time period during which it is calculated • Rates calculated over long time periods may be less meaningful

  8. Understanding the Difference between a Rate and Cumulative Incidence • Rate can be thought of as how likely an event is to happen at any moment in time • Cumulative incidence is the result of applying that rate to a defined population for a specified period of time • A rate is calculated by using data from a time period, but the rate is assumed constant during that period (i.e., at any moment in time during the period the rate is the same)

  9. Illustration of Rate versus Cumulative Incidence • The mortality rate in the U.S. population in 2001 was 855 per 100,000 person-years (or 0.855 per 100 person-years) • If everyone alive at the beginning of the period were followed for 5 years, the cumulative incidence of death (if the rate held constant) would be 4.2% at 5 years; at 10 years it would be 8.2%.

  10. 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 and 1 - F(t) = cumulative survival; e= 2.71828;  = rate; t = time units

  11. Constant Rate Increasing Rate

  12. Effect of high and low constant incidence rates on cumulative incidence

  13. Hazard • Hazard is an instantaneous incidence rate • h(t) = P(event in interval between t and [t+∆t] | alive at t) ∆t • Hazard function: • Shape of the relationship between time and hazard. • Constant rate, or constantly increasing rate, shown in previous slides are particular examples • Can take on any shape

  14. Hazard function for mortality in general population Years

  15. Note on Person-Time Rates • Person-time concept may seem unfamiliar because often described as “annual rate” or “annual rate per 100,000 persons” or “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” • Better to make person-time explicit: “incidence among children was 1.13 per 100,000 person-years”

  16. How to Calculate a Person-Time Rate: Obtaining the Denominator • Method 1: If have exact entry, censoring, and event times for each person, can sum person-time for each person for denominator • Method 2: If no individual data but have the time interval and average population size, can take their product as denominator • Some datasets may only have the average population size at risk

  17. c

  18. Rate: 6/9.583 = 0.626 per person-year = 62.6 per 100 person-years

  19. Method 2: Using average number of persons at risk during time interval 10 persons at baseline; 1 person at end of 2 years (6 deaths + 3 censored before 2 years = 9 losses) Formula: Average number of persons at risk = N baseline + N end / 2 = 11 / 2 = 5.5 Rate = 6/5.5 over 2 years = 0.545 per person-year or 54.5 per 100 person-years

  20. 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)

  21. Waiting Time Property of Incidence Rates • Waiting time to an event is reciprocal of the incidence rate (1/rate) • Eg, if rate 300 per 100 person-years, reciprocal is 1 (300/100 person-years) = (1/3) person-year • Average waiting time between events is 0.33 person-year = 4 person-months

  22. Why Use Incidence Rates? • To calculate incidence from population-based disease registries

  23. (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?

  24. Large Population Person-Time Rates “Since the production of stable rates for cancers at most individual sites requires a population of at least one million subjects, the logistic and financial problems of attempting to maintain a constant surveillance system [of everyone in the 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. To get an incidence 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).

  25. 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

  26. Potential Weakness of Using Census Data • Calculating rates from census population data is very useful but caution is required as a full census is only done every 10 years • Interim estimates of population change are made by the Census but over 10 years denominators may become inaccurate

  27. Invasive Breast Cancer Rates for Marin County versus Other California, 1995-2000 Rates per 100,000 person-years *Excluding 5 Bay Area Counties

  28. Census Denominators for Incidence Rates are Estimates 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.

  29. Why Use Incidence Rates? • To calculate incidence from population-based disease registries • To compare disease incidence in a cohort (individual-level data) with rate from the general population OR to compare incidences between 2 or more general populations

  30. (2) Comparing a rate from a cohort to the rate in the general population • A cohort study of petroleum refinery workers followed 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?

  31. Example of Using Person-Time Rates for Cohort Analysis • Cohort of petrochemical workers • 6,588 white 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

  32. Cohort of petrochemical workers • Could calculate KM estimate of cumulative incidence (for 36 years of follow-up), but what is the comparison group? • Using the person-time rate, the observed rate can be compared to the rate that would be expected if the person-time rate from a reference population (eg, U.S. population) is applied to the cohort

  33. Standardized Mortality Ratio • If U.S. death rates for age-sex-race-calendar period groups applied to the cohort, 924 deaths were expected in the cohort versus the 765 observed. • Ratio of 765 observed/924 expected = 0.83. This is called a Standardized Mortality Ratio (SMR).

  34. Obtaining an expected rate for comparison

  35. Cause Specific SMR’s Austin SG, et al., J Occupat Med, 1983

  36. Example of using both cumulative incidence and incidence rates in the same analysis for different purposes End stage renal disease: Cumulative incidence (survival) within cohorts defined by age at diagnosis Ratios of mortality incidence rates in renal disease children compared with national child mortality rates McDonald et al., NEJM 2004

  37. Another example of SMR: Is mortality higher after a fracture? Bluic et al. JAMA 2009

  38. (2b) Comparing hip fracture incidence in different populations Per 100,000 person-years e Standardized to 1990 non-Hispanic white US population

  39. Why Use Incidence Rates? • To calculate incidence from population-based disease registries • To compare disease incidence in a cohort with a rate from the general population OR to compare incidence in 2 or more populations • To compare incidence from a time-varying exposure in persons while exposed and unexposed

  40. (3) To compare incidence from a time-varying exposure in persons while exposed and unexposed • Research question: In a Medicaid database is there an association between use of non-aspirin non-steroidal anti-inflammatory drugs (NSAID) and coronary artery disease (CAD)? • How would you study the relationship between NSAID use and CAD?

  41. 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 a way to deal with risk factors that change over time

  42. Analysis of changing exposure and disease incidence • Tennessee Medicaid data base, 1987-1998: are NSAIDs associated with CAD risk? • Same person could both use and not use NSAIDs at different times over the 11 years • Can’t do cumulative incidence because would have to define groups by baseline characteristics without accounting for changes in subsequent behavior Ray, Lancet, 2002

  43. Analysis of changing exposure with person-time rates • Person-time totaled for using and not using NSAIDs; MI or CAD death outcome • 181,441 periods of “new” NSAIDS use in 128,002 individuals; 181,441 periods of non-use in 134,642 individuals (matched by age, sex, and calendar date) • A person can contribute to the denominator both for use and non-use but only after a 365 day “wash out” period between use and non-use

  44. Analysis of changing exposure with person-time rates • Rate ratio = 1.01 • Concluded no evidence that NSAIDS reduced risk of CHD events Ray, Lancet, 2002

  45. Calculating Rates in STATA Declare data set survival data: . stset timevar, fail(failvar) .strate gives person-years rate .strate groupvar gives rates within groups Example: Biliary cirrhosis time to death data .use biliary cirrhosis data, clear .stset time, fail(d) .strate D Y Rate Lower Upper 96 747.04 0.1285 0.1052 0.1570 .strate treat Treat D Y Rate Lower Upper Placebo 49 355.0 0.138 0.104 0.183 Active 47 392.0 0.120 0.090 0.160

  46. Hazard function in Stata • sts graph, hazard K-M survival curve for same data Incidence rate (from previous slide) = 0.13 deaths per person-year 10 yr cum incidence = 0.2375

  47. Immediate Commands in STATA • STATA has an option to use it like a calculator for • various computations without using a data set. • Called immediate commands. • Example, to calculate the confidence interval • around a person-time rate: • . cii #person-time units #events, poisson • E.g. 6 events occur in 10 person-years of follow-up: • . cii 10 6, poisson • 95% CI = 0.220 – 1.306

  48. Incidence rate Cumulative incidence

  49. Survival changing over calendar time

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