Measures of Association

Robert HeimerYale University School of Public Health

April 2009

Overview for Today

• Purpose of measuring associations

• The 2x2 table

• Ratio comparisons (relative measures)– Relative risk, risk/rate ratio, odds ratio

• Difference comparison (absolute measures)– Risk/rate difference, population risk/rate difference, attributable

proportion among exposed and total pop.

• Formula, examples, interpretations

• Put it all together with an example

Why Estimate Comparisons?

• Quantitative comparison is an essential element of epidemiology to identify disease determinants.

• Summarize relationship between exposure and disease by comparing at least two frequencies in a single summary estimate.

• Overall rate of disease in an exposed group says nothing about whether exposure is a risk factor for or causes a disease. – This can only be evaluated by comparing disease occurrence in

an exposed group to another group that is usually not exposed. – The latter group is usually called the comparison or reference

group.

Two Main Options for Comparison

Calculate ratio of two measures of disease frequency 1. Relative comparison2. Strength of relationship between exposure and

outcome3. Magnitude of association between exposure and

outcome4. How much more likely one group is to develop

outcome

Calculate difference between two measures of disease frequency

1. Absolute comparison (attributable)2. Public health impact of exposure3. How much greater is the frequency of outcome in one group

Comparing Ratios

The “2-by-2” Table

Yes No Total

Yes a b a+b

No c d c+d

Total a+c b+d a+b+c+d

Outcome

Exposure

Prevalence or cumulative incidence= a/(a+b) among exposed

= c/(c+d) among unexposed= (a+c)/(a+b+c+d) among total population

Example #1

Yes No Total

Yes 23 304 327

No 133 2816 2949

Total 156 3120 3276

MI

OC use

5-year CI of myocardial infarction= 23/327 = 7.0% among OC users

= 133/2949 = 4.5% among non-OC users

Data from a fixed cohort study of oral contraceptive use (OC) and myocardial infarction (MI) in pre-menopausal women followed for 5 years (adapted from Rosenberg et al AJE 1980).

The “2-by-2” Table for Person-Time Data

Yes No Total

Yes a - PTexp

No c - PTunexp

Total a+c - Total PT

Outcome

Exposure

Incidence rates= a/(PTexp) among exposed

= c/(PTunexp) among unexposed

= (a+c)/(Total PT) among total population

Example #2 -- Using Person-Time Data

Yes No Total

Yes 30 --- 54,308 PY

No 60 --- 51,477 PY

Total 90 --- 105,786 PY

CHD

PH use

Incidence rate of CHD= 30/54,306 PY = 55.2/100,000 PY among PH users= 60/51,477 PY = 116.6/100,000 PY among non-PH users

Data from a dynamic cohort study of postmenopausal hormone use (PH use) and coronary heart disease (CHD) among women (Stampfer et al NEJM 1985)

Relative Comparisons

• Based on ratio of 2 measures of frequency• Often referred to as “relative risk”• Dimensionless and ranges 0 - infinity

Prevalence ratio (PR)= Pexp / Punexp = [a / (a+b)] / [c / (c+d)]

Cumulative incidence ratio (CIR)= CIexp / CIunexp = [a / (a+b)] / [c / (c+d)]

Incidence rate ratio (IRR)= IRexp / IRunexp = [a / PTexp] / [c / PTunexp ]

Interpretations of Risk Ratios

• Gives information on the relative effect of the exposure on the disease.

• Tells you how many times higher or lower the disease risk is among the exposed as compared to the unexposed.

• Therefore, commonly used in etiologic research as a measure of risk

• Note:– Some people consider CIR and IRR better measures of “relative

risk” than PR because CIR and IRR include measures of time

Example #1 (continued)

Yes No Total

Yes 23 304 327

No 133 2816 2949

Total 156 3120 3276

MI

OC use

5-yr CIR of MI among OC users compared to non OC users:5-yr CIR= CIe/CIu = [a/(a+b)]/[c/(c+d)] = (23/327)/(133/2949) = 1.6

Interpreted as relative risk:1. Women who used OCs had 1.6 times the risk of having MI over 5

years compared to non-OC users (1.6-fold increased risk).2. There is a 60% increase risk of MI among OC users over a 5-yr

period compared to non-users (60% more likely to have MI).

Example #2 (continued)

Yes No Total

Yes 30 --- 54,308 PY

No 60 --- 51,477 PY

Total 90 --- 105,786 PY

CHD

PH use

Incidence rate ratio of CHD among PH users compared to non-PH users:IRR = IRe/IRu = [a/PTe]/[c/PTu] = 0.5

Interpreted as a relative risk: • Women who used PH had 0.5 times, or half, the risk of developing

CHD compared to non-users.

Interpretation of Relative Risk• RR=1

– Risk in exposed = risk in non-exposed– No association

• RR>1– Risk in exposed > risk in non-exposed– Positive association, factor is associated with disease– Larger RR stronger association

• RR<1– Risk in exposed < risk in non-exposed– Negative association, factor is “protective”

• RR for exposed = 1/RR for non-exposed– Need to pay attention to referent group– Risk for whom compared to whom?

Example #1: Extension to 2 x 2 Table

Yes No Total

< 1 year 4 31 35

1-4 years 5 107 112

5-9 years 7 127 134

10+ years 7 39 46

Never 133 2816 2949

Total 156 3120 3276

MI

OC use

Any Two Groups Can Be ComparedApplying Formulas on Appropriate Cells

Yes No Total

< 1 year 4 31 35

1-4 years 5 107 112

5-9 years 7 127 134

10+ years 7 39 46

Never 133 2816 2949

Total 156 3120 3276

The 5-yr CIR (“relative risk”) of MI among users of OC for greater than 10 years compared to never users is (7/46) / (133/2949) = 3.4.

The 5-yr CIR (“relative risk”) of MI among users of OC for greater than 10 years compared to users of 5-9 years is (7/49) / (7/134) = 2.7

Case-Control Studies

• Participants are selected for study participation on the basis of pre-existing disease status

• Cannot estimate prevalence, CI, or IR– Do not know the population at risk

• Cannot use previous formulas

• Relative risk can be estimated by odds ratio (OR)– ratio of odds of exposure among cases to odds of exposure

among controls

OR = (a/c)/(b/d) = ad/bc

Disease

Yes

NoExposure

Exposure

Yes

Yes

No

No

Case-control: Looking Back

Example #3: Odds Ratio Example adapted from Kehrberg et al 1981 AJE

Cases Controls Total

Rely 15 14 (29)

Other 9 45 (54)

Total 24 59 (83)

Toxic shock syndrome

• OR = (15*45)/(14*9) = 5.4• Interpretations:

– The odds of using Rely tampons among women with TSS were 5.4 times higher than the odds of using Rely tampons among those without TSS (technical)

– Women who used Rely tampons were 5.4 times more likely to develop toxic shock syndrome than women who used other brands (loosely).

Tampon brand used during

month of illness

Notes on Odds Ratios

• Can be thought of as “exposure odds ratio”

• These data provide an estimate of risk in some situations.

• They do not allow one to estimate incidence of TSS in the population of women at risk.

• To do this, one would need to know the number of all cases of TSS among women (numerator) and the number of all women who are at risk (denominator).

• Usually, denominator data are not available and numerator data may be incomplete in case-control studies.

Odds Ratio Approximates Relative Risk

• When disease is rare– Proportion of cases in exposed and unexposed groups is low– a<<b, so a+b ≈ b and c<<d, so c+d ≈ d– RR = a/(a+b)/c/(c+d) ≈ a/b / c/d = ad/bc

• If disease is not rare:– When cases are newly diagnosed– When prevalent cases are excluded, makingit more like a

cohort/incidence study

Comparing Differences

Difference (Absolute) Comparisons

• Based on difference between 2 measures of frequency

• Comparing disease occurrence among the exposed with the disease occurrence among the unexposed comparison group by subtracting one from the other.

• Gives information on:– the absolute effect of exposure on disease occurrence– the excess disease risk, or disease burden, in the exposed

group compared to the unexposed group– the public health impact of an exposure, that is, how much

disease would be prevented if the exposure were removed

• Note: this assumes that the exposure causes the disease

Risk Difference

Risk difference (RD) = Rexp – Runexp

For Prev*: RD = Pexp - Punexp = a / (a+b) – c / (c+d)

For CI: RD = CIexp - CIunexp = a / (a+b) – c / (c+d)

For IR: RD = IRexp - IRunexp = a / PTexp – c / PTunexp

• * Some epidemiologists hesitate to use “risk” when referring to prevalence estimates

• More precisely called prevalence difference, cumulative incidence difference, and incidence rate difference

• Also called attributable risk, rate difference, attributable rate– Note: “attributable” implies causality– RD = 0 when there is no association between exposure and disease

• What we can hope to accomplish in reducing risk of disease among exposed if exposure were eliminated

Example #4(adapted from Boice 1977 JNCI)

IRD = 6.7 / 10,000 PY• Interpretation:

– Broad: there are 6.7 excess cases of breast cancer for every 10,000 PY among those exposed to radiation compared to those not exposed.

– Narrow: Eliminating this radiation would prevent 6.7 cases of breast cancer for every 10,000 PY (“attribution”).

Breast cancer cases PY Rates per

10,000 PY

Radiation exposure 41 28,010 14.6

No radiation exposure 15 19,017 7.9

Total 56 47,027 11.9

Example #5: Comparison of RR and RD

Lung Cancer Coronary Heart Disease

Cigarette Smoker 140 669

Non Smoker 10 413RR 14.0 1.6RD 130/100,000/YR 256/100,000/YR

Annual Mortality Rate Per 100,000

Conclusion: Cigarette smoking is a much stronger risk factor for lung cancer but (assuming smoking is causally related to both diseases) the elimination of cigarette smoking would prevent far more deaths from coronary heart disease. Why is this so?

Death from CHD is much more common.

In Other Words

• Relative risk is a measure of strength of association between exposure and disease and is useful in analytical studies– risk

• Relative difference is a measure of how much disease incidence is attributable to exposure, and is useful in assessing exposure’s public health importance – burden

Population Risk Difference (PRD)

• Measures excess disease occurrence among the total population that is associated with the exposure.

• Helps to evaluate which exposures are most relevant to the health of a target population.– Describes impact of exposure on total population– Number of cases that would be eliminated in total

population if exposure was removed (assuming causality)• Depends on prevalence of exposure in population

– For ex, PRD will be low if exposure is rare, even if RR is high

• Also called population attributable risk

Calculating Population Risk Difference

• Two formulas for PRD:

PRD = Rt - Ru where Rt = risk total population and

Ru = risk among unexposed

PRD = (RD)*(PPexp) where PPexp is % of population that is exposed

• Note (as always) that “risk” may refer to IR or CI (more accurately) or prevalence (less accurately)

Example #4 Revisited

PRD = 11.9 – 7.9 = 4• Interpretation:

– 4 excess breast cancer cases for every 10,000 PY of observation can be attributed to radiation exposure.

– If radiation causes breast cancer, then 4 cases of breast cancer for every 10,000 person-years of observation could be prevented if the radiation exposure were removed.

Breast cancer cases PY Rates/10,000 PY

Radiation exposure 41 28,010 14.6

No radiation exposure 15 19,017 7.9

Total 56 47,027 11.9

Attributable Proportion among Exposed

• APe describes the proportion of disease among exposed that is due (attributable) to exposure or that would be prevented if exposure were eliminated

• Risk in non-exposed group can be considered “background” incidence that would occur regardless of exposure

• Interpretation may assume causal relationship

APe = [(Re – Ru)/Re] * 100= RD / Re * 100

Re = “risk” (IR, CI, P) among exposedRu = “risk” (IR, CI, P) among unexposed

• Also called etiologic fraction, attributable risk percent, attributable risk among exposed

Alternative formulaAgain, we are interested in the difference between the risk

in the exposed and risk in the unexposed:

Divide numerator and denominator by Ru

APe = [(Re – Ru)/Re] * 100

APe = [(RR-1)/RR] * 100

(Different formulas may be helpful depending on what information you have.)

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Note on alternative formula:

• Especially useful for case-control data– You cannot estimate risk directly– Can use OR to estimate RR under certain conditions.– In which case:

APe = [(OR-1)/OR] * 100

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Example(continued)

APe = [(14.6 – 7.9)/14.6] * 100 = 46%

• Interpretation: 46% of cases of breast cancer among those exposed to radiation may be attributed to radiation exposure and could be eliminated if exposure were removed.

Breast cancer cases PY Rates/10,000 PY

Radiation exposure 41 28,010 14.6

No radiation exposure 15 19,017 7.9

Total 56 47,027 11.9

Attributable proportion among total population

• APt describes the proportion of disease among total population that would be eliminated if exposure were eliminated

• What percent of disease in total population is due to exposure• Useful for setting priorities for public health action

– Elimination of exposure lead to what impact on population?

• Assumes causal relationship

APt = [(Rt – Ru)/Rt] * 100

= PRD / Rt * 100

Rt = “risk” (IR, CI, P) among total population

Ru = “risk” (IR, CI, P) among unexposed

• Also known as population attributable risk percent

Alternative formulas

APt = [1 - (Ru / Rt)] * 100

APt = [(Pe)(RR-1)]/[(Pe)(RR-1)+1] * 100

For case-control studies:

APt = [(Pe)(OR-1)]/[(Pe)(OR-1)+1] * 100

where Pe = proportion of exposed controls

(Convince yourself with some algebra if you like.)

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Example(continued)

APt = [(11.9 – 7.9)/11.9] * 100 = 34%

• Interpretation:– 34% of breast cancer cases in total study population may be attributed

to radiation exposure and could be eliminated if exposure were removed.

Breast cancer cases PY Rates/10,000 PY

Radiation exposure 41 28,010 14.6

No radiation exposure 15 19,017 7.9

Total 56 47,027 11.9

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Putting It All Together

The 58th annual convention of the American Legion was held in Philadelphia from July 21 until July 24, 1976. People at the convention included American Legion delegates, their families, and other legionnaires who were not official delegates. Between July 20th and August 30th, some of those who had been present became ill with a type of pneumonia subsequently named Legionnaire's Disease. No one attending the convention developed the disease after August 30th.

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Exercise to Practice Measures of Comparison

Yes No Total

Delegate 125 1724 1849

Non-Delegate

3 759 762Convention Status

Developed Legionnaire’s Disease

Below are the numbers of delegates and non-delegates who developed Legionnaire's Disease during the period July 20 to August 30 (41 day period).

What is the risk of Legionnaire’s disease among delegates and non-delegates?

• First: what type of measure is this?

• Cumulative incidence rate among delegates:

• Cumulative incidence rate among non-delegates:

What is the relative risk of Legionnaire’s disease among delegates compared to non-delegates?

• First: what type of measure is this?

• RR =

• Interpretation:

What is the risk difference for Legionnaire’s disease among delegates vs. non-delegates?

• First, what type of measure is this?

• RD =

• Interpretation:

What is the attributable proportion of disease among delegates?

• APe =

• Interpretation:

Further Analysis of Convention Delegates

Yes No Total

Hotel A 62 628 690

Other Hotel 63 1,098 1,161

Developed Legionnaire’s Disease

Cumulative incidence among Hotel A residents =

Cumulative incidence among other hotel residents =

CIR =

Interpretation:

Key points

• Relative and absolute measures of comparison tell us different things– Relative risk is a measure of strength of association; often of

interest to epidemiologists who do etiologic research– Absolute risk then becomes important (assuming causality) to

public health planners, policy makers, etc. for estimating public health impact of exposures on communities

• Measures go by different names and multiple formulas are sometimes available. It is important to– know what you are interesting in measuring,– know what data are available and in what form, and– if data are not available, know how to collect appropriate data.