1. ph250b.14 measures of disease part 1

Measures of Disease Learning Objectives
Measures of Disease: Learning Objectives
1. Understand different types of populations as conceptualized in epidemiology and the relevance
of population types to measures of disease
2. Understand concept of disease occurrence in time
a. Understand and be able to define concepts of disease occurrence in time at a population level
(age, period, cohort effects)
b. Understand and be able to define concepts of disease occurrence in time at an individual level
(i.e., latent period, lead time), and their implications for measuring disease at the population level
3. Understand and be able to define and contrast prevalence and incidence
4. Understand and be able to define and contrast risks and rates
5. Calculate and interpret prevalence (this includes knowing the formula)
6. Understand, define, calculate, and interpret cumulative incidence (this includes knowing the
formulas)
a. Know different methods for calculating cumulative incidence and the assumptions and purposes
of each
7. Understand, define, calculate, and interpret incidence density (knowing the formulae)
a. Understand and calculate person-time
8. Define and interpret a hazard rate
9. Understand and be able to convert between prevalence, cumulative incidence and incidence
density (this includes knowing the formulas)
10. Direct and indirect standardization
` a. Perform both and understand when each is appropriate (know the formulae)
b. Know what data are required for each

Measures of disease
Module 2
PH250B Epidemiologic Methods II

Measures of disease outline
– Big picture
– Illustration/discussion of measuring disease in time
– Populations
– Time scales affecting disease in populations
– Epidemiologic measures
• Basic concepts
• Measuring diseases
• Prevalence
• Incidence density (incidence rate)
• Cumulative incidence (risk)
• Relations among measures
– Standardization
– Summary
– Appendix: specific measures of disease

Big picture
• In epidemiology, one of our major goals is to
measure occurrence of disease
– Tool for surveillance (the “distribution” of disease;
descriptive epidemiology)
– Tool for etiologic/risk factor research (the
“determinants” of disease; analytical epidemiology)

Big picture
• Critical part of etiologic research
– We compare measurements of disease between
groups of people (e.g., exposed and unexposed)
because we are interested in associations between
exposures and outcomes and, ultimately, effects
(causal) of exposures on outcomes

Big picture
• Reminder: we compare disease occurrence
between groups of people that have different
exposures because we do not observe the
counterfactual outcomes for each person in the
population
– Comparisons of disease occurrence covered in next
module – measures of association
• Key step in etiologic research process is
accurate measurement of disease occurrence

Disease in time
• For measuring disease occurrence in a given population
there are two important components
– Measuring the disease outcome
– Measuring and accounting for the time over which disease
occurs
• Rothman: “disease occurrence in a population reflects
two aspects of individual experiences: the amount of
time the individual is at risk in the population, and
whether the individual actually has the focal event (e.g.,
gets disease) during that time.”

Disease in time
Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)

Disease in time
Sum of time all
members of the
population are
observed is
called person-
time
People move
through time in
a study, with
and without
disease
People are
observed for
differing
amounts of
time

Disease in time
How many
people were in
our study?
How many got
disease?
So we could
say 6/10 got
disease over 2
years

Disease in time
• Any concerns about this measure?

Disease in time
• How might you account for the time contributed
by each person?

Disease in time
• If interested in a rate of disease
– # Disease/person-time
• If interested in a risk of disease, but want to account for
different times people were observed
– ?

Cohort from Fig 2-2 by 2-month intervals
Interval Time
period
Number
in at risk
population
Number
developing
disease
Number
withdrew
Proportion of at-
risk population
developing disease
j (tj-1
, tj
) N’0j
Ij
Wj
Rj
= Ij
/ (N’0j
-(Wj
/2))
2 (0, 2) 10 1 1 1/(10-(1/2)) = 0.11
4 (2, 4) 10 - 1 - 1 =
8
1 0 1/8 = 0.125
6 (4, 6) 8 - 1 - 0 =
7
0 0 0/7 = 0
Adapted from Kleinbaum, Table 6.1

Populations
• Populations in epidemiology
– Group of people for whom we are interested in the
occurrence of disease or the effect of an exposure on
disease
– Defined by: geography, occupation, demographic
characteristics (age, race/ethnicity, gender), time etc.

Populations
• Populations in epidemiology
– Examples:
• Residents of NYC on 9/11/2001
• Women of childbearing age in Alameda County 1980-2000
• Live singleton births in Bangladesh in 2005

Populations
• Total population
– Includes everyone in a particular population
• Candidate or “at risk” population
– People in the total population who could get the
disease/condition of interest
– Excludes those who have the disease or who are
immune (or do not have the necessary organ or
physiological function, etc.)

Populations
• Candidate or “at risk” population
– Example: candidate population for pregnancy
excludes men, currently pregnant women, women
with hysterectomy, and older women

Populations
• Closed or fixed populations
– Membership is permanent and defined by some life
event
– Add no new members and only lose members to
death
– The size of the population will eventually reach 0
because everyone ultimately dies

Populations
• Closed or fixed populations
– Examples: being born in 1975, serving in Iraq or
Afghanistan

Populations
• Open populations
– Gain members over time through immigration or birth
– Lose members through emigration or death
– Sometimes called dynamic, but a misnomer b/c both
open and closed populations are changing
– If membership can be lost due to events other than
death, then the population is open

Populations
• Open populations
– Examples: most populations such as cities, states,
hospital populations, etc.

Populations
• Steady state populations – a type of open
population
– When the number of persons entering the population
is balanced by the number exiting over a period of
time
– Example: a city where the number of people moving
out or dying is approximately equal to the number
moving in or being born between over a given time
interval
– Example: population of women in the maternity ward
at Alta Bates hospital

Populations
• Distinctions can depend on measurement of
time or disease
– Example: a population that starts a new drug could
be considered closed if only the population starting at
a particular time is included but if new users of the
drug are allowed to enter the population it could be
considered open

Populations
• Relevance
– Population properties are important to consider in
study planning
• Example: when studying a particular outcome (e.g.,
pregnancy) need to make sure you study a population “at-
risk” of that outcome (e.g., women of certain ages)
• Example: should define your study population so that you
can address your study question in that population (e.g.,
differences in PTSD between OEF/OIF Veterans and
civilians vs differences in PTSD among OEF/OIF Veterans)

Populations
• Example: studying exposure to a fixed event (e.g., hurricane
Katrina) population of interest is fixed/closed and a study
would need to be designed to capture that population
appropriately
• Example: a population of interest may be open (e.g., tourists
visiting a given city) and a study would need to be designed
to capture that population appropriately
– Important to consider in calculation and interpretation
of measures of disease (more later in relations
between measures)

Time scales
• Disease occurrence at a population level
affected by different time scales
– age, period and cohort effects

Time scales
• People are conceived, born, and then move
through time until death with a variety of health
states and events along the way
• Time (and exposures in time) can affect
health/disease in three main ways
– Age effects: biological age of individuals
– Period effects: calendar time
– Cohort effects: year of birth

Time scales
• Age effect
• Definition: variation in health status arising from
social or biological consequences of aging
• What it looks like when graphed
– Rate (of disease) changes with age
– Irrespective of birth cohort and calendar time

Time scales
• Age effect
• Example: rate of heart disease increases with
age regardless of whether you examine a
population born in 1900 or 1950; at the age of
50 the rate of heart disease is higher than at
age 30

Time scales
• Age effect - depiction

Time scales
• Period effect
• Definition: Variation in health status arising from
changes in physical, ecological, or social
environment during a time period
• What it looks like
– Change in rate (of disease) affecting an entire
population at some point in time
– Irrespective of age and birth cohort

Time scales
• Period effect
• Example: DDT spraying in 1950s led to
increased risk of certain cancers for anyone
living in affected areas in the 1950s, regardless
of how old they were or when they were born

Time scales
• Period effect - depiction

Time scales
• Cohort effect
• Definition: Variation in health status arising from
exposures that vary by cohort
• What it looks like:
– Change in the rate (of disease) according to
membership in some cohort
– Birth cohort is established by year of birth
• Note that one can examine cohorts defined by any life event
which places a person permanently in a group
– Irrespective of age and calendar time

Time scales
• Cohort effect
• Example: Women exposed to DES in utero
have increased risk of vaginal and cervical
cancer at all ages and over all time periods

Time scales
• Cohort effect - depiction

Time scales
A real example
• Peptic ulcer mortality
(Susser 1982,
reprinted 2001)
– Cohort effects – for
those born after 1900
age-specific mortality
from peptic ulcer was
continually declining

Measures – basic concepts
• Proportion
• Numerator is included in the denominator (a/
(a+b))
• Range: 0 to 1 (or 0% to 100%)
• Example: number of students with tattoos/total
number of students in class (number with
tattoos + number without tattoos)

• Ratio
• Numerator is NOT included in the denominator
(a/b)
• Range: 0-infinity
• Example: number with tattoos/number without
tattoos (odds of tattoo)
• Example: number of hospital beds/number of
patients
• In epidemiology, you will see ratios of
probabilities, rates, and odds (to be elaborated
later)

• Odds
– A ratio with wide application in epidemiology (more in
measures of association, study designs, analysis of
epidemiologic data)
• Odds of disease: number with disease/number
without disease

• Rate
• Time is in the denominator
• Range: 0-infinity
• Examples: cases of flu/month, miles per hour
• Dimension is always 1/time or time-1

Measures – measuring diseases
• Disease process and measuring disease
– Induction period = time from causal action to
biological onset
– Latent period = time from biological onset to disease
detection
Biologic onset Detectable by
screening
Symptoms
develop
DeathCausal action

– Timing of disease process may differ between
individuals
– Timing of detection may differ between individuals
screening
Symptoms
develop
DeathCausal action

– Timing of disease process may differ between
individuals
screening
Symptoms
develop
DeathCausal action
Biologic onset
Detectable by
screening
Symptoms
develop
Death
Causal action
A
B
JC: mention length bias

– Timing of detection may differ between individuals
screening
Symptoms
develop
DeathCausal action
screening
Symptoms
develop
DeathCausal action
A
B
JC: mention lead time bias

• Defining disease outcome for a study
– Have to consider underlying disease process and
potential variations in that process
– Have to consider how disease is being detected
• This will influence what your measure of
disease is capturing

• Example: prevalence of cancer (proportion with
disease at a particular time) will miss cases of
aggressive cancers

Epidemiologic measures
• Prevalence vs. incidence
– Prevalence = proportion of the population with a
disease
– Incidence = frequency of development of new cases
of disease in a population
• New case is usually the first occurrence of a
disease for a non-diseased person

• Risk vs. rate
• Risk = the probability of developing disease over a
specified time period
– Population measure that is often interpreted at the individual
level
– Must specify the time period for the risk to be meaningfully
interpreted (X-year risk)
– Example: 10 year risk of mortality among men diagnosed with
prostate cancer is 0.1 or 1/10 men diagnosed with prostate
cancer die within 10 years

• Risk vs. rate
• Rate (average) = average change in disease status per
unit of time over a time period relative to the size of the
candidate population (incidence density)
• Example: There are 78 new cases of lyme disease per
100,000 population per year in CT (estimated in 2008)
• Interpreted at population level
• A rate, so time is in the denominator

• Risk vs. rate
• Rate (instantaneous) = the instantaneous potential for
change in disease status per unit of time at time t
relative to the size of the candidate (i.e., disease-free)
population at time t (hazard)
• The instantaneous rate (hazard) of lyme disease on
August 31, 2008 in CT is ?
– Instantaneous rates cannot be directly calculated from
epidemiologic data because they are defined for an infinitely
small time interval
– We can estimate average rates for smaller time intervals when
we have sufficient data

Measures - prevalence
Prevalence
• Proportion of existing disease in the total population,
without regard to when cases developed
• Numerator: number of existing cases of disease in the
population
• Denominator: number of all persons in the population of
interest
• A proportion
• Range is 0-1 - dimensionless
• Prevalence odds = prevalence of outcome/prevalence
of no outcome = P/(1-P)

Two types of prevalence measures:
• Point prevalence: the proportion of subjects who have
disease at a specified point in time
– Example: proportion of population that is HIV positive on July 1,
2010

Two types of prevalence measures:
• Period prevalence: the proportion of subjects in a
population who have disease during a certain period of
time
– Uncommon - used when exact time of onset difficult to
determine
– Example: proportion of population with an episode of
depression over the past 12 months

Uses and limitations of prevalence
• A disease that has high incidence but is rapidly fatal or
quickly cured would have low prevalence
• An exposure that increases survival with the disease will
increase prevalence
• Useful for resource planning
• Can estimate the rate under certain conditions (more to
come)

Uses and limitations of prevalence
• In measuring congenital anomalies we use prevalence
out of necessity (many incident cases are lost, as are
others in the denominator)
– Cannot measure the population at-risk (conceptions)
or person-time contributed by the population, so we
necessarily take a point prevalence—the point being
birth

Side note: Szklo and “prevalence rate”
• Szklo uses the term “prevalence rate” for prevalence
• Although you will see this in other places in the literature
as well, you should not use this term
• Use the term prevalence
• Prevalence is not a rate and thus the term “prevalence
rate” is incorrect and potentially confusing

Measures - incidence
Incidence time
• Not sufficient to just record proportion of population
affected by disease
• Necessary to account for the time elapsed before
disease occurs and the period of time during which the
disease events take place

A
B
P(DA
) = 0.6
P(DB
) = 0.6

Incidence time
• Incidence time is time from referent or zero time (e.g.,
birth, start of treatment or exposure, start of
measurement period) until the time at which the
outcome event occurs
• Also called event time, failure time, occurrence time

Incidence time
• “Censoring” occurs if the time of event is not known
because something happens before the outcome occurs
– Examples: lost to follow-up, death, surgery to make outcome
impossible like hysterectomy, end of measurement period
• Average incidence time = average time until an event
occurs

Measures – incidence density
Incidence density (ID) - aka incidence rate (IR)
• The rate of occurrence of new cases of disease during
person-time of observation in a population at risk of
developing disease
• Numerator: number of new cases of disease
– Only count cases in the numerator that are contributing to
person-time in the denominator
• Denominator: person-time of observation in population
at risk
– Only count contributions to the denominator that could yield
cases for the numerator
• A rate
• Units are “inverse time” (1/time, time-1
)
• Range is 0-infinity

Incidence density
• What is “person-time”?
• Person-time at risk: length of time for each individual
that they are in the population at risk
– Sum over population is total person time at risk
• When a person is no longer “at risk” they cease
contributing to person-time, this includes when they get
the outcome of interest
• One person year could be 2 people x 6 months each, 1
person x 12 months, 3 people x 4 months, etc.
• Helps account for censoring and different observation
periods

“Figure 2 suggests
that ID may be
viewed as the
concentration or
'density' of new case
occurrences
in a sea of
population time. The
more dots per unit
area under the
curve, the greater is
the ID.”
Morgenstern et al. 1980

Person-time calculations for individual level data
1) If exact time contribution of each individual is known:
– Sum the disease-free observation time

Person-time calculations for individual level data
2) If data on each individual is collected at regular intervals:
– Estimate the disease-free observation time in each
interval
– Note: variants of this formula also subtract Ij
/2 from N’0j

Person-time estimation from group level data
1) If the population is in steady state can estimate based on
population size (N’) and duration of follow-up (Δt)
2) If the population is not in steady state can estimate
based on mid-interval population (N’1/2
) and duration of
follow-up (Δt)
– Note: mid-interval population size can be estimated as: (Nt0
+
Nt1
)/2

Uses and limitations of incidence density
• Appropriate for fixed or dynamic populations; does not
assume that everyone is followed for specified time
period
• Does not distinguish between people who do not
contribute to disease incidence because they were not
in the study population long enough for disease to
develop and those who do not contribute because they
never got the disease (relates to next point)

Uses and limitations of incidence density
• 100 person-years could come from following 100 people
for one year or two people for 50 years – no way to tell
the difference without knowing the incidence time
– Have to consider whether study design allowed appropriate time
to elapse to plausibly consider an exposure disease relation
– Disease process is important to consider in developing
appropriate study design and disease measures
– Example: disease free cohort of 50 exposed and 50 unexposed
followed for 1 year might not allow sufficient time to elapse for
exposure to cause disease

• In class exercise
– Study population observed monthly for 6 months
– What is the person-time contributed by this
population?
– What is the incidence density?

Measures – incidence
Hazard rate
• The instantaneous potential for change in disease
status per unit of time at time t relative to the size of the
candidate (i.e., disease-free) population at time t
• Instantaneous rate in contrast to incidence density
which is an average rate
• Cannot be directly calculated because it is defined for
an infinitely small time interval
• Hazard function over time can be estimated using
modeling techniques (more in the analyzing
epidemiologic data section)

Hazard rate

Survival function

Measures – cumulative incidence
Cumulative incidence (CI) – aka risk, incidence
proportion (IP – Rothman)
• The proportion of a closed population at risk that
becomes diseased within a given period of time
• Numerator: number of new cases of a disease or a
condition (Rothman calls this A)
• Denominator: number of persons in population at risk
(Rothman calls this N)
• A proportion
• Range is 0-1 – dimensionless

Cumulative incidence
• Calculated for a fixed time period
– Only interpretable with information on time period over which it
was measured
• Population measure that translates most readily to
individual
– Interpreted as capturing individual risk of disease
• Different methods for calculating
– Variations depending on how time at risk is handled
– Option for calculating from rate measure

• Different methods for calculating
– Simple cumulative
– Actuarial
– Kaplan-Meier
– Density

• Subscript notation
– R(t0,tj)
– risk of disease over the time interval t0
(baseline) to tj (time j)
– R(tj-1,tj)
– risk of disease over the time interval tj-1
(time before time j) to tj (time j)

– N’0
– number at risk of disease at t0 (baseline)
– N’0j
– number at risk of disease at the beginning of
interval j

– Ij
– incident cases during the interval j
– Wj
– withdrawals during the interval j

Simple cumulative method:
R(t0,tj)
= CI(t0,tj)
= I
N'0
• Risk calculated across entire study period assuming all
study participants followed for the entire study period, or
until disease onset
– Assumes no death from competing causes, no withdrawals
• Only appropriate for short time frame

Simple cumulative method:
• Example: incidence of a foodborne illness if all those
potentially exposed are identified

Actuarial method:
R(tj-1, tj)
= CI(tj-1, tj)
=____Ij
____
N'0j
- Wj
/2
• Risk calculated accounting for fact that some
observations will be censored or will withdraw
• Assume withdrawals occur halfway through each
observation period on average
• Can be calculated over an entire study period
– R(t0,tj)
= CI(t0, tj)
= I/(N’0
-W/2)
• Typically calculated over shorter time frames and risks
accumulated

Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)

• Where to start – set up table with time intervals
• Fill incident disease cases and withdrawals into appropriate
intervals
• Fill in population at risk
Actuarial Method

• Calculate interval risk
• R(tj-1, tj)
= Ij
/(N’0j
-(Wj
/2))
• R(0,2)
=1/(10-(1/2)) = 0.11
Actuarial Method

• Calculate interval survival
• S(tj-1,tj)
= 1-R(tj-1,tj)
Actuarial Method

• Calculate cumulative risk – example of time 0 to 10
• R(t0, tj)
= 1 - Π (1 – R(tj-1,tj)
) = 1 - Π (S(tj-1,tj)
)
• R(0, 10)
= 1 – (0.89 x 0.88 x 1.0 x 1.0 x 0.85) = 0.34
Actuarial Method

• Calculate cumulative survival
• S(t0,tj)
= 1-R(t0,tj)
Actuarial Method

• Intuition for why R(t0, tj)
= 1 - Π (Sj
) using
conditional probabilities
• Example of 5 time intervals:
– Π (Sj
) = P(S1)*P(S2|S1)*P(S3|S2)*P(S4|S3)*P(S5|S4)
= P(S5)
– Product first two terms: P(S2|S1)*P(S1) = P(S2)
– Multiplying conditional probabilities gives you
unconditional probability of surviving up to any given
time point
– the value (1 - survival) up to (or at) a given time point
is then the probability of not surviving up to that time
point

• Exercise for home (discuss in lab)
– Study population observed monthly for 6 months
– Calculate the cumulative incidence of disease from
month 0 to 6

1. ph250b.14 measures of disease part 1

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1. ph250b.14 measures of disease part 1