Epidemiological Measures

• Measures of disease frequency- prevalence and incidence

• Measures of risk- RR, OR, ARR, NNH/NNT,

etc…

Outline and Objectives1. Understand the Concepts of Uncertainty, Probability and Odds

2. Measures of Disease Frequency- Prevalence- Incidence

Cumulative incidenceIncidence density (the concept of person-time)

3. Relationship between Incidence, Duration & Prevalence

4. Risk Estimates (and their uses)

We will cover these conceptually and how to calculate them.

Measuring Disease and Defining Risks

• Clinicians are required to know or make estimates of many things:– The occurrence of disease in a population– The “risk” of developing a disease or an

outcome (prognosis) – The risks and benefits of a proposed treatment

• This requires an understanding of:– Measures of disease frequency

• Proportions and odds• Prevalence and incidence rates• Risk (relative and absolute)

Uncertainty• Medicine (or almost anything) isn’t an exact

science, uncertainty is ever present.

• Uncertainty can be expressed either:

– Qualitatively using terms like ‘probable’, ‘possible’, ‘unlikely’

• Study: Doctors asked to assign prob. to commonly used words:

– ‘Consistent with’ ranged from 0.18-0.98– ‘Unlikely’ ranged from 0.01 to 0.93

– Quantitatively using probabilities (p) • Advantage: explicit interpretation, exactness • Disadvantage: may force one to be more exact than is

justified!

Probability vs. Odds (review)

• Odds for disease =

e.g. In North Carolina one estimate of the pre-term birth probability is .1136. Thus the odds is for pre-term birth is

Relationship between Probability and Odds

• Probability and odds are more alike the lower the probability of the outcome, p = P (disease)

Probability Odds0.80 40.67 20.60 1.50.50 1.00.40 0.670.33 0.50.25 0.330.20 0.250.10 0.110.05 0.0530.01 0.0101

• p = Odds/(1 + Odds)• Odds = p/(1 – p)

Example: If odds for disease = 2.00

p = 2/(1 + 2) p = 2/3 = 0.67If probability of disease =

0.67Odds= 0.67/(1 - 0.67)Odds= 0.67/0.33 = 2

Disease PrevalenceDef’n: the proportion of a defined group or population that has a

clinical condition or outcome at a given point in time.

– Prevalence = Number of cases observed at time t__ Total number of individuals at time t

The prevalence ranges from 0 to 1 (i.e. it’s a proportion), but usually referred to as a rate and is often expressed as a %.

– Example:

• Of 100 patients hospitalized with stroke, 18 had ICH• The our estimate of the prevalence of ICH among

hospitalized stroke patients = 18%

The prevalence rate answers the question: – “what fraction of the group is affected at this moment in time?”

Incidence Rates• A special type of proportion that includes a

specific time period and population-at-risk

• Numerator = the number of newly affected individuals occurring over a specified time period.

• Denominator = the population-at-risk over the same time period

• There are two types of incidence rates: incidence density rate (IDR) or cumulative incidence rate (CIR).

Incidence Density Rate (ID)Def’n: the speed at which a defined at-risk group or population

develops a new clinical condition or outcome over a given time period.

IDR or ID = Number of newly diseased individuals or cases_______ Sum of time periods for all disease-free individuals-at-risk

– denominator is "person-time“, typically person-years.

– a measure of the instantaneous force or speed of disease

– IDR or ID ranges from 0 to infinity (i.e. it is not a proportion!)

– dimension = per unit time or the reciprocal of time (time-1), typically per year or per month.

Incidence Density Rate is generally just referred to as the Incidence Rate. It is sometimes denoted by the Greek letter ().

Person-Time Data• A person-time data arises when we follow

individuals over time.

• The most common person-time unit is person-years.

• One person-year is accumulated by following one person for an entire year.

• If we follow an individual for five years, then that person accumulates or provides 5 person-years of information.

Person-Time Data• The sum of the disease-free time experience for individuals at risk in

the population.

• 100 people followed for 6-months have same person-time experience as 50 people followed for one year.

• 100 x 0.5 = 50 person-years• 50 x 1.0 = 50 person-years

• How to calculate? (add up disease-free time)• 100 subjects followed for 6-months• Suppose 1 new case develops on day 1 of each successive month

(i.e., 2 thru 6): • Person time is the sum of disease free-time for each month (1 thru 6)

– = 100 + 99 + 98 + 97 + 96 + 95 = 585 months• ID = 5/585 person-months or 8.54 per 1,000 person months

• Person time can be measured with whatever scale that makes the most sense i.e., person-days, person-weeks, person-months, person-years (PY).

Incidence Density Rate (IDR)Example:• Approximately 100,000 women in the Nurses’ Health

Study, ages 30–64, were followed for 1,140,172 person-years from 1976 to 1990, during which time 2,214 new cases of breast cancer occurred.

• The incidence density or incidence rate for breast cancer for this population is then estimated to be:

l = 2,214/1,140,172 = 0.00194 events per person-year or l = 194 events per 100,000 (105) person-years.

Incidence Density Rate (IDR)Recall: A measure of the “speed” that disease is occurring

– IDR answers the question: “At what rate are new cases of disease occurring in the population?”

• Common Incidence Rates– Mortality Rate (used in Vital Statistics)

• Lung CA mortality rate = 50 per 100,000 PY • Breast CA mortality rate = 15 per 100,000 PY

– Disease Incidence Rates• IDR of neonatal diarrhea = 280 per 1,000 child weeks

– Cohort specific Incidence Rates• Calculated for specific sub-sets defined by age, gender or race

– Black Men: Lung CA incidence rate = 122 per 100,000 PY – White Females: Lung CA incidence rate = 43 per 100,000 PY

Cumulative Incidence Rates (CIR)Def’n: the proportion of a defined at-risk group or population that

develops a new clinical condition or outcome over a given time period.

CIR= Number of disease cases for a specific time period Total # of population-at-risk for same time period

– Measures the proportion of at-risk individuals who develop a condition or outcome over a specified time period

– Ranges from 0 to 1 (so it’s a proportion), but it is called a rate because it includes time period and population-at-risk

– Must be accompanied by a specified time period to be interpretablebecause the CIR must increase with time.

e.g.

7-day CIR of stroke following Transient Ischemic Attack (TIA) = 5% 90-day CIR of stroke following Transient Ischemic Attack (TIA) = 10%

Cumulative Incidence of GI side effects for Rofecoxib (VIOXX) vs. Naproxen - The VIGOR Trial (Bombardier et al., NEJM 2000)

Cumulative Incidence Rates (CIR)• The Cumulative Incidence (CIR or CI(t)) can also be

thought of the probability of developing a disease over a time period (t).

• if , the incidence rate, is l<.10.

Example: Approximately 100,000 women in the Nurses’ Health Study, ages 30–64, were followed for 1,140,172 person-years from 1976 to 1990, during which time 2,214 new cases of breast cancer occurred. We saw the incidence rate

Thus the cumulative incidence for years is given by, or 966 cases per 100,000 women followed for five years. Another way to interpret this quantity is as the probability of developing breast cancer within the next 5-years for woman in this population is .00966 or .966% chance.

Diagram by Mathew J. Reeves, Dept. of Epidemiology, Mich State Univ.

Concept of the Prevalence “Pool”

New cases(Incidence)

Deathrate

Recoveryrate

Concept of the Prevalence “Pool”

New cases occurring between 10/1/90 – 9/30/91 = 4 casesTotal population at midpoint = 20 – 2 = 18ID = 4/18 = .222 or approx. 22 cases per 100 population

Another Incidence Rate ExampleScenario: Investigators enrolled 2,100 women in a study and followed them annually for four years to determine the incidence rate of heart disease. After one year, none had a new diagnosis of heart disease, however 100 had been lost to follow-up. After two years, one had a new diagnosis of heart disease, and another 99 had been lost to follow-up. After three years, another seven had a new diagnosis of heart disease, and 793 had been lost to follow-up. After four years, 8 had a new diagnosis of heart disease, and 392 were lost to follow-up.

Since we don’t know exactly when the individuals lost to follow-up left the study or when the cases were diagnosed with heart disease we will assume they remained disease-free for half of the year. Making this assumption we calculate the following:

Total number of cases = 0 + 1 + 7 + 8 = 16 cases of heart disease.Person-years = ???? If no one had dropped then we would have had 2,100 X 4 = 8,400 person-years (PY) for the denominator. However, the losses at follow-up need to be taken into account.

Another Incidence Rate ExampleScenario: Investigators enrolled 2,100 women in a study and followed them annually for four years to determine the incidence rate of heart disease. After one year, none had a new diagnosis of heart disease, however 100 had been lost to follow-up. After two years, one had a new diagnosis of heart disease, and another 99 had been lost to follow-up. After three years, another seven had a new diagnosis of heart disease, and 793 had been lost to follow-up. After four years, 8 had a new diagnosis of heart disease, and 392 were lost to follow-up.

Total number of cases = 0 + 1 + 7 + 8 = 16 cases of heart disease.Person-years = Thus the incidence rate is or 2.5 cases per 1,000 person-years.

Prevalence and IncidencePrevalence is a function of: the incidence of the condition, and the average duration of the condition. The duration is influenced in turn by the recovery rate and mortality rate.

Prevalence Incidence Duration

This relationship explains why…- Arthritis is common (“prevalent”) in the

elderly- Rabies is rare- Influenza is only common during epidemics

Other related measuresMortality Frequency MeasuresMeasure Numerator Denominator 10n

Crude death rate Total number deaths during year

Total population as of July 1st

Typically 1,000

Specific death rate

Total number of deaths in a specific subgroup

Total population in specific subgroup as of July 1st

Typically 1,000

Maternal mortality rate

Deaths from all puerperal causes during a year

Total live births during the year (preferred would be # of women pregnant during the year)

Typically 1,000 (3rd world) to 100,000 (U.S.)

Infant mortality rate

Number of deaths under 1 year of age during a year

Total number of live births during the year

Typically 1,000

Neonatal mortality rate

Number of deaths under 28 days of age during a year

Total number of live births during the year

Typically 1,000

Fetal death rate Total number of fetal deaths during a year

Total deliveries during the year

Typically 1,000

Fetal death ratio Total number of fetal deaths during a year

Total number of live births during the year

Typically 100 or 1,000

Perinatal mortality rate

Total number of fetal deaths of 28 weeks or more + infant deaths under 7 days

Total number of fetal deaths of 28 weeks or more + number of live births

Typically 1,000

Proportional mortality ratio

Number of deaths in a particular subgroup

Total number of deaths 100

Cause-of-death ratio

Number of deaths due to a specific cause during year

Total number of deaths due to all causes during the year

100

Other related measuresMeasures of Fertility

Measures of Morbidity (community’s status in terms of disease)

Measure Numerator Denominator 10n

Crude birth rate Total number of live births during a year

Total population as of July 1st

Typically 1,000

General fertility rate

Total number of live births during a year

Total number of women of childbearing age (e.g. 15-44 or 15-49)

Typically 1,000

Age-specific fertility rate

Number of births to women of a certain age or age range in a year

Total number of women of a certain age or age range

Typically 1,000

Measure Numerator Denominator 10n

Incidence rate Total number of new cases of a specific disease during a year (or some other time unit)

Total population as of July 1st

Depends on rarity of disease (100-100,000)

Prevalence Total number of cases, new or old, existing at a point in time

Total population at that same point in time

Depends on rarity of disease (100-100,000)

Case-fatality ratio Total number of deaths due to a disease

Total number of cases due to the disease

100

Immaturity ratio Number of live births under 2,500 grams during a year

Total number of live births during the year

100

Secondary attack rate

Number of additional cases among contacts to primary case within max incubation period

Total number of susceptible contacts

100

Quantifying Risk (or Benefit)Presentation and Interpretation of Information on Risk

• Information on the effect of a potential risk factor or beneficial treatment can be presented in several different ways:

– Relative Risk (RR)– Odds Ratio (OR)– Absolute Risk Reduction (ARR)– Number Needed to Harm/Treat (NNH and NNT)– Attributable Risk (AR)– Others?

• The way risk information is presented and interpreted can have a profound effect on clinical decisions (both on part of patients and doctors).

Quantifying Risk (or Benefit)Presentation and Interpretation of Information on Risk

We might also think of these as measures of association. Measures of association quantify the potential relationship between “exposure” and “disease” among two groups.

The two main measures are Relative Risk (RR) and Odds Ratio (OR) as defined in the prerequisite courses.

However, with the new concepts of incidence rate, cumulative incidence, prevalence, and other measures we might think of other ratios of interest.

Relative Risk, Risk Ratio, or Rate Ratio (RR)

The general definition of RR is

Here we could be taking the ratio of two prevalence measures (i.e. proportions) or two rate measures (e.g. incidence rates, person-time rates, mortality rates, fertility rates, or morbidity rates).

A RR = 1.0 the risk is the same for both groups, if RR > 1.0 the risk is greater for group in numerator and if RR < 1.0 it indicates decreased risk for group in numerator.

Examples of RR

Examples of RR

Examples of RR

These three examples were taken from:

This publication is linked under Research Papers (Deppa) on D2L. It would be a good idea to read through this entire document and do the exercises!

Odds Ratio (OR)• In some cases due to the nature of the

study design it is not possible to compute measures of disease prevalence or incidence, and thus the RR is not computable.

e.g. case-control study

• The interpretations of factors or effects in a logistic regression analysis are done using Odds Ratios (OR).

Odds Ratio (OR) Example: Age at First Pregnancy and Cervical

Cancer A case-control study was conducted to determine whether there was increased risk of cervical cancer amongst women who had their first child before age 25. A sample of 49 women with cervical cancer was taken of which 42 had their first child before the age of 25. From a sample of 317 “similar” women without cervical cancer it was found that 203 of them had their first child before age 25.

Q: Do these data suggest that having a child at or before age 25 increases risk of cervical cancer?

The Odds for an event A are defined as

Odds for A = _______

Odds Ratio (OR)

P(A)1 – P(A)For example suppose we roll a single

die the odds for a 3 are:Odds for 3 = P(3)/(1 – P(3)) = (1/6)/(1 – (1/6)) = (1/6)/(5/6) = 1/5 Interpretation: We expect one 3 for every five rolls that don’t result in a 3.(Odds for a 3 are 1:5 and odds against a 3 are 5:1)

Odds Ratio (OR)The Odds Ratio (OR) for a disease associated with a

risk factor is ratio of the odds for disease for those with risk factor and the odds for disease for those without the risk factor

OR = _________________________

P(Disease|Risk Factor)1 – P(Disease|Risk

Factor)

_____________________

P(Disease|No Risk Factor)1 – P(Disease|No Risk

Factor)

_______________________

The Odds Ratio gives us the multiplicative increase in odds associated with having the “risk factor”.

Odds for disease amongst those with risk factor present

Odds for disease amongst those without the risk factor.

Odds Ratio (OR)Age at 1st Pregnancy Case Contro

l

Row Totals

Age < 25

42 203 245

Age > 25

7 114 121

ColumnTotals 49 317 n = 366

Cervical Cancer

a) Why can’t we calculate P(Cervical Cancer | Age < 25)?Because the number of women with disease (49 cases) was fixed in advance and therefore NOT RANDOM !

Odds Ratio (OR)Age at 1st Pregnancy Case Contro

l

Row Totals

Age < 25

42 203 245

Age > 25

7 114 121

ColumnTotals 49 317 n = 366

Cervical Cancer

b) What is P(risk factor|disease status) for each group?P(Age < 25|Case) = 42/49 = .857 or 85.7%P(Age < 25|Control) = 203/317 = .640 or 64.0%

Odds Ratio (OR)Age at 1st Pregnancy Cas

eContr

ol

Row Totals

Age < 25

42 203 245

Age > 25

7 114 121

ColumnTotals 49 317 n = 366

Cervical Cancer

c) What are the odds for the risk factor amongst the cases? Amongst the controls?Odds for risk factor cases = .857/(1-.857) = 5.99Odds for risk factor controls = .64/(1- .64) = 1.78

Odds Ratio (OR)Age at 1st Pregnancy Case Contro

l

Row Totals

Age < 25

42 203 245

Age > 25

7 114 121

ColumnTotals 49 317 n = 366

Cervical Cancer

d) What is the odds ratio for the risk factor associated with being a case?

Odds Ratio (OR) = 5.99/1.78 = 3.37, the odds for having 1st child on or before age 25 are 3.37 times higher for women who currently have cervical cancer versus those that do not have cervical cancer.

Odds Ratio (OR)Odds RatioThe ratio of dark to light shading is 3.37 times larger for the cervical cancer (case) group than it is for the women without cervical cancer (control) group.

e) Even though it is inappropriate to do so calculate P(disease|risk status).

P(case|Age<25) = 42/245 = .171 or 17.1%P(case|Age>25) = 7/121 = .058 or 5.8%f) Now calculate the odds for disease given

the risk factor statusOdds for Disease for 1st Preg. Age < 25 = .171/(1 - .171) = .207Odds for Disease for 1st Preg. Age > 25 = .058/(1 - .058) = .061

Odds Ratio (OR)

g) Finally calculate the odds ratio for diseaseassociated with 1st pregnancy age < 25 years of age.

Odds Ratio = .207/.061 = 3.37

This is exactly the same as the odds ratio for having the risk factor (Age < 25) associated with being in the cervical cancer group!!!!

Odds Ratio (OR)

g) Finally calculate the odds ratio for diseaseassociated with 1st pregnancy age < 25 years of age.

Odds Ratio (OR) = .207/.061 = 3.37

Odds Ratio (OR)

Final Conclusion: Women who have their first child at or before age 25 have 3.37 times the odds of developing cervical cancer when compared to women who had their first child after the age of 25.

Odds Ratio (OR)

Risk FactorStatus

Case Control

Risk Factor Present

a bRisk Factor Absent c d

Disease Status

OR = _____a x db x c

Much easier computational formula!!!

Relative Risk (RR) and Odd’s Ratio (OR)

When the disease is fairly rare, i.e. P(disease) < .10 or 10%, then one can show that the odds ratio and relative risk are similar.

OR is approximately equal to RR when P(disease) < .10 or less than a 10% chance.

In these cases we can use the phrase:“… times more likely” when interpreting the

OR, just as we would for the RR.

Relative Risk (RR) and Odds Ratio (OR)

Age at 1st Pregnancy Case Contro

l

Row Totals

Age < 25

a42

b203 245

Age > 25

c7

d114 121

ColumnTotals 49 317 n = 366

OR = (42 X 114)/(7 X 203) = 3.37 Because less than 10% of the population of women would develop cervical cancer we can say women who have their first child at or before age 25 are 3.37 times more likely to develop cervical cancer than women who have their first child after age 25.

Relative Risk (RR) and Odds Ratio (OR)

The most commonly cited advantage of the RR over the OR is that the former is the more natural interpretation. The relative risk comes closer to what most people think of when they compare the relative likelihood of events.

e.g. Suppose there are two groups, one with a 25% chance of mortality and the other with a 50% chance of mortality. Most people would say that the latter group has it twice as bad. But the odds ratio is 3.00, which seems too big!

RR = .50/.25 = 2.00

OR = P(death|risk)/P(survive|risk)_______ P(death|no risk)/P(survive|no risk)

= .50/(1 - .50) = 3.00 .25/(1 - .25)

Relative Risk (RR) and Odds Ratio (OR)

Even more extreme examples are possible. A change from 25% to 75% mortality associated with having the risk factor represents a relative risk of 3.00, but an odds ratio of 9.00. A change from 10% to 90% mortality represents a relative risk of 9.00 but an odds ratio of 81.00!

RR = .90 /.10 = 9.00

OR = P(death|risk)/P(survive|risk)_____ P(death|no risk)/P(survive|no risk)

= .90/(1 - .90) = 81.00 !! .10/(1 - .10)

Relative Risk (RR) and Odds Ratio (OR)

• Any study of risk generally benefits from adjustments for potential confounding factors which is typically done using logistic regression in the study of disease.

• We have seen that OR’s arise as part of the interpretation of the results from a logistic regression analysis.

• Despite their pitfalls OR’s are really the only option when case-control studies are used.

Absolute Risk Reduction (ARR)• Def’n: The difference in absolute risk (or probability of

events) between the two groups. (e.g. exposed vs. unexposed or treatment vs. control)

ARR = Risk1 – Risk2 (need to careful with units!)

• A simple and direct measure of the impact of risk exposure or treatment.

• Also called the risk difference (RD) or attributable risk (AR).

• The ARR depends on the background baseline risk which can vary markedly from one population to another,

e.g. 78% vs. 75% compared to 6% vs. 3%.

Attributable Proportion or Attributable Risk Percent

The attributable proportion or attributable risk percent is calculated as follows:

It represents the expected reduction in disease if the exposure were removed or never existed.

Attributable Proportion or Attributable Risk Percent

Example: Lung Cancer and Smoking 1-14 cigs/day

Number Need to Harm (NNH)

The number needed to harm (NNH) is an indicates how many patients need to be exposed to a risk factor over a specific period to cause harm in one patient that would not otherwise have been harmed.  It is defined as the inverse of the ARR,

Intuitively, the lower the number needed to harm, the worse the risk factor.

Number Needed to Harm (NNH)Example: Smoking and Low Birth WeightP(Low Birth|Smoker) = .1479P(Low Birth|Non-smoker) = .0839

Thus it would take approximately 16 additional mothers to smoke during pregnancy to produce one more infant in North Carolina born with a low birth weight (< 2,500 g). For every 16 women who smoke during pregnancy we expect one more infant to be born with a low birth weight.

Treatment Efficacy or Treatment Effectiveness

The treatment efficacy or treatment effectiveness is calculated as follows:

It represents the percentage reduction in risk of “disease” among treated persons relative to untreated persons.

Treatment Efficacy or Treatment Effectiveness

Suppose that 11.8% of those vaccinated for the H1N1 flu got it anyway vs. 42.9% for those who were not vaccinated. The estimated treatment efficacy of the vaccination is:

Thus the vaccinated group experienced 72% fewer cases than they would have if they had not been vaccinated.

Number Needed to Treat (NNT)The number needed to treat (NNT) is an epidemiological measure used in assessing the effectiveness of a health-care intervention, typically a treatment with medication. The NNT is the average number of patients who need to be treated to prevent one additional bad outcome (i.e. the number of patients that need to be treated for one to benefit compared with a control in a clinical trial). It is defined as the inverse of the ARR,

Intuitively the lower the NNT, the better the treatment.

Number Needed to Treat (NNT)

Suppose that 11.8% of those vaccinated for the H1N1 flu got it anyway vs. 42.9% for those who were not vaccinated. The estimated NNT associated with the vaccination is:

Thus for approximately every 3 people vaccinated for the flu we expect to see one fewer flu case.

Some ReferencesHere are some references:1) Principles of Epidemiology (CDC Publication – free)

2) Biostatistics: A Foundation for the Health Sciences by Wayne Daniel (Vital Statistics - Ch. 14)