![2. Actuarial (life table)
¤ R (tj-1, tj) = CI (tj-1, tj) = Ij /[N’
0j – (Wj/2)] =
# new cases during interval j
# of disease free subjects at risk at the beginning of interval j
adjusted for withdrawals during that interval
¤ tj-1, tj is a shorter time interval; calculate risks over shorter time
intervals & then accumulate them
¤ Cumulative risk = R (t0, tj) = 1 - ∏ [S (tj-1, tj)] = 1 – ∏ [1-R( tj-1, tj)]
17](https://image.slidesharecdn.com/2014labslidesmeasuresofdiseasefinal4-160503180835/85/2014-lab-slides-measures-of-disease_final-4-17-320.jpg)
![Example: Estimate CI with density method (1)
¤ Estimate 5-year risk assuming constant ID:
1-exp[-0.192*5]=0.618
¤ Be careful! The rate differs greatly from interval to interval
26
Kleinbaum
Table 6.2](https://image.slidesharecdn.com/2014labslidesmeasuresofdiseasefinal4-160503180835/85/2014-lab-slides-measures-of-disease_final-4-26-320.jpg)
![Example: Estimate CI with density method (2)
27
¤ CI(t0,tj) = 1- ∏ [1-Cij]= 1- (0.913*0.889*0.641*0.513*1)=0.733
¤ CI(t0,tj) = 1- e(-(0.091*1+ 0.118*1+ 0.444*1+ 0.667*1+ 0.000*1)=0.733
Interval
Survival
.913
.889
.641
.513
1
Kleinbaum
Table 6.2](https://image.slidesharecdn.com/2014labslidesmeasuresofdiseasefinal4-160503180835/85/2014-lab-slides-measures-of-disease_final-4-27-320.jpg)
2014 lab slides measures of disease_final (4)
- 1. Measures of Disease Discussion section September 12th, 2014 PH 250B 1
- 2. Big Picture ¤ Epidemiology (description, prediction, etiology/ causation) relies on precise measurement of outcomes ¤ To precisely measure outcome we must ¤ Define the outcome ¤ Specify whether & when outcome occurred ¤ Induction period, latent period ¤ Specify amount of time at risk for disease ¤ Determine among what population (at risk, candidate, closed/fixed, open, steady state) 2
- 3. 3 Categories of Measures ¤ Proportion ¤ Numerator is included in the denominator: (a/a+b) ¤ Ratio ¤ Numerator is distinct from the denominator : (c/d) ¤ Rate ¤ Change in disease status per unit time: # disease events/period of time 3
- 4. Types of Populations/Cohorts ¤ Closed or Fixed ¤ Can only lose members due to death or disease of interest ¤ A closed population/cohort with constant incidence rate (ID) declines in size exponentially ¤ Open ¤ Members can be added (birth, migrate in) and removed (death, migrate out) ¤ Steady state ¤ Can occur only in an open population/cohort ¤ Number of people entering is balanced by the number of people exiting within levels of age, sex, etc. 4
- 5. Measures of disease ¤ Prevalence = proportion of current cases of disease in a population at a single point (or period) in time ¤ Incidence = frequency of development of new cases of disease in a population over a defined period of time 5
- 6. Prevalence (a proportion) ¤ Prevalence = # existing cases of disease # of persons in the population of interest ¤ Two types ¤ Point Prevalence: cases at a given point in time ¤ Period Prevalence: cases during a given time period ¤ Useful for resource planning if prevalence is constant 6
- 7. Risk vs. Rate RISK RATE Defn. Probability that an individual will develop outcome over given time period, conditional on that person not dying from other causes during time period. The frequency of occurrence of new cases of disease during person-time of observation in a population at risk of developing disease Measure Cumulative incidence Hazard (instantaneous) Incidence density (average) Range 0 to 1 0 to infinity Time Refers to time period (but not used in calculations) 1/time (time-1); cases/person- time (time used in calculations) Refers to Individual (but popln. used to estimate) Population (no individual level interpretation) 7
- 8. Incidence density ¤ ID = # new cases total person-time at risk ¤ Used for populations, not individuals ¤ Interpretation of ID = 35/1000 person-years ¤ The rate of outcome in the population is 35/1000 person-years 8
- 9. Calculating person-time ¤ When you have individual-level (detailed) data: ¤ 1. If exact time contribution of each individual is known ¤ PT = ¤ Sum all contributions of disease-free time ¤ 2. If exact time contributions unknown but you have interval specific data ¤ PTj = (N’ 0j– Wj/2) Δtj ¤ N’ 0j = Number of individuals at beginning of each interval ¤ Δtj = duration of follow up in each interval ¤ Wj = # of withdrawals in each interval ¤ (Note: Variants of this formula also subtract (Ij/2) in each interval) 9 !ti i=1 N ' "
- 10. Calculating person-time ¤ When you have group level data ¤ 3. If population is in steady state à ¤ PT = N’ Δt ¤ N’= stable, disease free population size ¤ Δt = duration of follow up ¤ 4. If population is not in steady state à ¤ PT = N’ 1/2 Δt ¤ N’ 1/2 = mid-interval disease-free population size ¤ Δt = duration of follow up 10
- 11. 11 In class exercise: estimate ID ¤ Study population observed monthly for 5 months ¤ What is the person-time contributed by this population? ¤ What is the incidence density? C = censored D = died/ developed disease 0 1 2 3 4 5 1 2 3 4 Follow-up Time (months) Person# C C D
- 12. In class exercise solution j N’0j Wj ΔTj PT 1 4 1 1 (4-(1/2))*1=3.5 2 3 1 1 (3-(1/2))*1=2.5 3 1 0 1 (1-(0/2))*1=1 4 1 0 1 (1-(0/2))*1=1 5 1 0 1 (1-(0/2))*1=1 Total PT = 9 12 Use the following formula because exact time contributions are unknown and we have interval specific data PTj = (N’ 0j– Wj/2) Δtj ID=1/9 (0.111) cases per person-year
- 13. Cumulative Incidence ¤ CI= # new cases of disease # of persons at risk (at beginning of time period) ¤ Always defined over a time period ¤ Can be used to predict individual’s risk ¤ Interpretation of CI = 0.35 in a 5-year period ¤ The risk of developing the outcome in the population at risk at baseline over a 5-year period was 0.35. 13
- 14. Cumulative Incidence ¤ 4 Ways to Calculate it: ¤ Simple Cumulative ¤ Actuarial ¤ Kaplan-Meier ¤ Density Method ¤ Know the assumptions needed to use each one & how to apply them ¤ Why different approaches? 14
- 15. 1. Simple cumulative ¤ R (t0, tj) = CI (t0, tj) = I/N’ 0j = # new cases # disease free subjects at risk at t0 ¤ Assumptions: ¤ Closed population ¤ No withdrawals/loss to follow up ¤ No competing risks ¤ Best for: short time frames (e.g., outbreaks) ¤ Can’t be used when duration of follow-up varies 15
- 16. 16 In class exercise: Simple CI ¤ Study population observed monthly for 5 months ¤ What is the simple CI? C = censored D = died/ developed disease 0 1 2 3 4 5 1 2 3 4 Follow-up Time (months) Person# D
- 17. 2. Actuarial (life table) ¤ R (tj-1, tj) = CI (tj-1, tj) = Ij /[N’ 0j – (Wj/2)] = # new cases during interval j # of disease free subjects at risk at the beginning of interval j adjusted for withdrawals during that interval ¤ tj-1, tj is a shorter time interval; calculate risks over shorter time intervals & then accumulate them ¤ Cumulative risk = R (t0, tj) = 1 - ∏ [S (tj-1, tj)] = 1 – ∏ [1-R( tj-1, tj)] 17
- 18. Assumptions/benefits of actuarial method ¤ Assumptions ¤ Withdrawals occur halfway through observation period on average ¤ Independence of censoring and survival ¤ Lack of secular trends during the study period ¤ Benefits ¤ Allows for censoring 18
- 19. 19Szklo
- 20. 2. Actuarial (life table) ¤ Calculate & interpret interval-specific risks ¤ Calculate & interpret interval-specific survival ¤ Calculate & interpret the 3-year cumulative risk ¤ R(t0, t3)= 1- (0.780) (0.833) (0.892) = 0.420 ¤ Calculate & interpret the 3-year cumulative survival ¤ S (t0, t3)= 1- R(t0, t3)= 1-0.420 = 0.580 20 Year Time period Popln at risk I W Interval risk (Rj) Interval survival (1-Rj) Cumul. risk Cumul. survival 1 0,1 1000 214 55 214/(1000-55/2)= 0.220 1-0.22= 0.780 0.220 0.780 2 1,2 731 117 63 117/(731-63/2) = 0.167 0.833 0.350 0.650 3 2,3 551 57 47 57/(551-47/2)= 0.108 0.892 0.420 0.580
- 21. 3. Kaplan Meier ¤ Rj = CIj = Ij/Nj = # new cases # of individuals still at risk at time j ¤ Calculate risk at time each disease event occurs ¤ Accumulate interval-specific risks (similar to actuarial method) ¤ Need very detailed data 21
- 22. Assumptions/benefits of Kaplan-Meier Method ¤ Assumptions: ¤ Independence of censoring and survival ¤ Lack of secular trends during the study period ¤ Benefits: ¤ Calculate event probability at the time it occurs 22
- 23. 3. Kaplan Meier 23 Timej Nj Ij Interval risk (Rj) Interval survival (1-Rj) Cumul. risk Cumul. survival 1 4 1 1/4=0.25 0.75 0.25 0.75 5 2 1 1/2=0.5 0.5 0.625 0.375 1 D 2 C 3 4 D 1 2 3 4 5 6 Follow-up Time (Months) PersonNo.
- 24. ¤ Risk (CI) and rate (ID) are mathematically related ¤ CI ≈ 1-e(-ID*Δt) ¤ Rare Disease Approximation: ¤ When ID*Δt is very small (<10%) the CI≈ID*Δt ¤ 3 assumptions ¤ Closed population ¤ No competing risk ¤ Each age-specific rate (IDj) is constant over that interval Relationship between risk and rate 24
- 25. 4. Density method ¤ Estimate risk (CI) using observed incidence rates (ID) in each time interval ¤ Interval risk = R (tj-1, tj) = CI (tj-1, tj) = ¤ Cumulative risk = R (t0, tj) = 25 1!e ! (IDj"tj ) j # 1!e !IDj"tj
- 26. Example: Estimate CI with density method (1) ¤ Estimate 5-year risk assuming constant ID: 1-exp[-0.192*5]=0.618 ¤ Be careful! The rate differs greatly from interval to interval 26 Kleinbaum Table 6.2
- 27. Example: Estimate CI with density method (2) 27 ¤ CI(t0,tj) = 1- ∏ [1-Cij]= 1- (0.913*0.889*0.641*0.513*1)=0.733 ¤ CI(t0,tj) = 1- e(-(0.091*1+ 0.118*1+ 0.444*1+ 0.667*1+ 0.000*1)=0.733 Interval Survival .913 .889 .641 .513 1 Kleinbaum Table 6.2
- 28. Assumptions/benefits of density method ¤ Assumptions ¤ Within each interval, rate is constant ¤ Closed cohort ¤ No competing risks ¤ Independence of censoring and survival ¤ Lack of secular trends during the study period ¤ Benefits ¤ Can be used to extrapolate CI to intervals beyond the follow-up time 28
- 29. Relationship between prevalence and incidence ¤ In a steady-state population: ¤ P/(1-P) = Incidence (ID) x duration (D) a. If prevalence is low/disease rare: ¤ P ≈ ID x D b. If prevalence is not low/disease not rare: ¤ P = (ID x D)/(ID x D+1) ¤ So, if disease not rare, then prevalence odds may be preferred 29
- 30. Standardization of rates ¤ Types of rates ¤ Crude ¤ Specific (e.g., age-specific) ¤ Standardized/adjusted (e.g., age-adjusted) ¤ Why adjust? ¤ Comparing two or more crude rates between populations can be misleading because populations may also differ with respect to characteristics (e.g., age) that affect the rate of disease ¤ Example: Crude mortality higher in Florida than other states 30
- 31. Direct Age Adjustment Total expected outcomes Age-adjusted rate = ---------------------------------- Total standard population Age-specific rates come from your study population Age-specific population sizes (i.e., the weights) come from the standard population 31
- 32. Age-specific rates come from the standard population Age-specific population sizes (also called weights) come from your study population observed outcomes (O) SMR = ------------------------------------ x 100% expected outcomes (E) Standardized mortality ratio (SMR) = ratio of the observed number of outcomes in the study population to the expected number of outcomes if the study population had the same age-specific rates as the standard population. Indirect Age Adjustment 32
- 33. Standardization and the counterfactual ¤ What does the standard population represent? ¤ Direct: a counterfactual estimate of the age structure ¤ Answers question: what would the rate of disease look like in my study population if, counter to fact, it had the same age structure as the standard population ¤ Indirect: a counterfactual estimate of the age- specific rates ¤ Answers question: what would the rate of disease look like in my study population if, counter to fact, it had the same age-specific rates as the standard population 33
- 34. Standardization and the counterfactual ¤ What does this imply about your choice of standard population?? ¤ Your age-adjusted rate or SMR depends on your choice of standard population ¤ Changing the standard population will change your answer! ¤ Example ¤ SMR comparing rate of accidents among U.S. construction workers to the U.S. standard population = 167% ¤ SMR comparing rate of accidents among U.S. construction workers to construction workers in China = 76% 34
- 35. Key points about indirect standardization ¤ Why use indirect? ¤ When we do not have stratum-specific rates, or if stratum specific rates are based on cells with small numbers. ¤ SMR can be >100% or <100% depending on standard population (illustrated in previous slide) ¤ Caution: do not compare two SMRs 35
- 36. Don’t compare 2 SMRs! ¤ Both study groups have identical age-specific rates ¤ How do they compare to the standard? Community A Community B Age (yrs) N Deaths Rate N Deaths Rate Standard Pop. Rates <40 100 10 10% 500 50 10% 12% 40+ 500 100 20% 100 20 20% 50% Total 600 110 18.3% 600 70 11.7% 36
- 37. Don’t compare 2 SMRs Expected # of deaths obtained by applying the reference rates to Communities A & B Age (yrs) Community A Community B < 40 0.12 * 100= 12 0.12 * 500 = 60 40+ 0.5 * 500 = 250 0.5 * 100 =50 Total # expected 262 110 Total # observed 110 70 SMR (observed/expected) 0.42 0.64 The SMRs are different even though both populations had the same rates of disease in each age stratum! 37
- 38. Don’t compare 2 SMRs Another way to think about it: ¤ SMRA = obsA expA ¤ SMRB = obsB expB ¤ Directly adjusted rateA = expA standard ¤ Adjusted rateB = expB standard Different denominator: can’t compare Same denominator: can compare
- 39. Age effect ¤ Definition: ¤ Variation in health status arising from social or biological consequences of aging ¤ What to look for: ¤ Rate (of disease) changes with age ¤ Irrespective of birth cohort and calendar time 39 Source: Szklo
- 40. Period effect ¤ Definition: ¤ Variation in health status arising from changes in environment during time period ¤ What to look for: ¤ Change in rate (of disease) affecting an entire population at some point in time ¤ Irrespective of age and birth cohort 40 Source: Szklo
- 41. Cohort effect ¤ Definition: ¤ Variation in health status arising from exposures that vary by cohort ¤ What to look for: ¤ Change in the rate (of disease) according to year of birth ¤ Irrespective of age and calendar time 41 Source: Szklo
- 42. Graphing Time Scales 42 Source: Szklo
- 43. Specific measures of disease ¤ Proportionate mortality ¤ Proportionate mortality ratio ¤ Case fatality rate ¤ Death to case ratio ¤ Infant mortality rate ¤ Neonatal mortality rate ¤ Postneonatal mortality rate ¤ Maternal mortality ratio (also called maternal mortality rate) ¤ Crude birth rate ¤ General fertility rate ¤ Years of potential life lost (YPLL) 43
- 45. People moving through time E A C D Infection with HPV Cervical dysplasia detectable by pap smear (“early diagnosis”) Symptoms, e.g. bleeding (“usual diagnosis”) Death Disease detected B Changes to cervical cells Induction period Latent period 45 Lead time
- 46. Application: Hepatitis E in rural Bangladesh 46
- 47. Context – why do this study? ¤ Hepatitis E is an “emerging” pathogen ¤ Endemic in South Asia ¤ Poor understanding of patterns of infection in subpopulations ¤ Low incidence in children under 15 years ¤ High mortality rate among pregnant woman ¤ Poor understanding of how the body develops an immune response to Hep E ¤ Few longitudinal studies of Hep E 47
- 48. Study objective and design ¤ Objective: Calculate age-specific population incidence rates of HEV infection and disease under endemic, non- outbreak conditions ¤ Design: Follow randomly selected cohort of rural Bangladeshis for 18 months 48
- 49. Matlab treatment center, 1964 49
- 50. Matlab study area ¤ ADD MAP 50
- 51. Data collection ¤ Random selection from the Matlab cohort ¤ Entire cohort was enumerated in 2003 census ¤ 1,300 households randomly selected ¤ Baseline survey 2003-2004 ¤ Test for antibodies to HEV (n=1,134) ¤ 12-month & 18-month follow-up ¤ Blood sample ¤ Test for antibodies ¤ Seronegative: Titers < 20 WR-U/mL ¤ Seropositive (“seroconverted”): Titers ≥20 WR-U/mL ¤ Questionnaire – exposures, morbidity 51
- 52. Key Findings ¤ Baseline seroprevalence of HEV: 22.5% ¤ Overall incidence density : 64 per 1,000 person-years ¤ Expected lower incidence since HEV has typically been reported to be sporadic in Bangladesh 52
- 53. Linking blood test and questionnaire 53
- 54. In class example 1, extra J N’0j Ij Wj Interval Risk Interval Survival Cumulative Risk Cumulative Survival 1 4 0 1 0.00 1.0 0 1 2 3 1 1 0.40 0.6 0.4 0.6 3 1 0 0 0.00 1.0 0.4 0.6 4 1 0 0 0.00 1.0 0.4 0.6 5 1 0 0 0.00 1.0 0.4 0.6 54