Measures of association

Measures of association

Intermediate methods in observational epidemiology

2008

Measures of Association

1) Measures of association based on ratios– Cohort studies

• Relative risk (RR)

• Odds ratio (OR)

– Case control studies• OR of exposure and OR of disease

• OR when the controls are a sample of the total population

– Prevalence ratio (or Prevalence OR) as an estimate of the RR

2) Measures of association based on absolute differences: attributable risk

Cohort studies

Myocardial infarctionSevereSystolic

HTN

Number

Present Absent Probability (q) Probability oddsdis

Yes 10000 180 9820 0.0180 0.01833

No 10000 30 9970 0.0030 0.00301

Hypothetical cohort study of the one-year incidence (q) of acute myocardial infarction for individuals with severe systolic hypertension (HTN, ≥180 mm Hg) or normal systolic blood pressure (<120 mm Hg).

09.600301.0

01833.0

997030

9820180

0030.00.10030.0

0180.00.10180.0

1

1OR dis

qqqq

00600300

01800

1000030

10000180

...

RR

The OR can also be calculated from the “cross-products ratio” if the table is organized exactly as above :

bc

ad

dcba

dcddccbabbaa

dccdccbaabaa

qqqq

1

1

1

1OR disease

09.6309820

9970180OR disease


HTN

Number


Yes 10000 180 (a) 9820 (b) 0.0180 0.01833

No 10000 30 (c) 9970 (d) 0.0030 0.00301

When (and only when) the OR is used to estimate the RR, there is a “built-in” bias:

09.6018.01

003.010.6OR dis


HTN

Number


Yes 10000 180 (a) 9820 (b) 0.0180 0.01833

No 10000 30 (c) 9970 (d) 0.0030 0.00301

RR=6.0

OR=6.09

Example:

q

q

q

q

q

q

q

q

qqqq

1

11

11

1OR

“bias”RR

IN GENERAL:

• The OR is always further away from 1.0 than the RR.

• The higher the incidence, the higher the discrepancy.

Relationship between RR and OR

… when probability of the event (q) is low:

or, in other words, (1-q) 1, and thus, the “built-in bias” term,

and OR RR.

qq

q

1

0969820

997006

01801

0030106 .

.

..

.

..OR


HTN

Number

Present Absent

Yes 10000 180 9820

No 10000 30 9970

Example:

096

997030

9820180

.OR 006

1000030

10000180

.RR

11

1 0

qq

.

Relationship between RR and OR

… when probability of the event (q) is high:

818640

94006

3601

060106 .

.

..

.

..OR

Recurrent MISevereSystolic

HTN

Number

Present Absent

Yes 10000 3600 6400

No 10000 600 9400

818

9400600

64003600

.OR 006

10000600

100003600

.RR

Example:Cohort study of the one-year recurrence of acute myocardial infarction (MI) among MI survivors with severe systolic hypertension (HTN, ≥180 mm Hg) and normal systolic blood pressure (<120 mm Hg).

q

0.36

0.06

OR vs. RR: Advantages

• OR can be estimated from logistic regression.

• OR can be estimated from a case-control study

Case-control studiesA) Odds ratio of exposure and odds ratio of disease

Retrospective (case-control) studies can estimate the OR of disease because:

ORexposure = ORdisease

Hypothetical cohort study of the one-year incidence of acute myocardial infarction for individuals with severe systolic hypertension (HTN, 180 mm Hg) and normal systolic blood pressure (<120 mm Hg).


HTN

Number

Present Absent

Yes 10000 180 9820

No 10000 30 9970

096

997030

9820180

.Odds

OddsOR

exp-non dis

exp dis

dis

Hypothetical case-control study assuming that all members of the cohort (cases and non cases) were identified

Severe Syst HTN Cases Controls

Yes 180 9820

No 30 9970

096

9970982030

180

.Odds

OddsOR

cases-non exp

cases exp

exp

same

Because ORexp = ORdis, interpretation of the OR is always “prospective”.

Calculation of the Odds Ratios: Example of Use of Salicylates and Reye’s Syndrome

14027Total

871No(26/1) ÷ (53/87) =

43.0

5326Yes

Odds RatiosControlsCases

Preferred Interpretation: Children using salicylates have an odds (≈risk) of Reye’s syndrome 43 times higher than that of non-users.

(Hurwitz et al, 1987, cited by Lilienfeld & Stolley, 1994)

Past use of salicylates

Another interpretation (less useful): Odds of past salicylate use is 43 times greater in cases than in controls.

It is not necessary that the sampling fraction be the same in both cases and controls. For example, a majority of cases (e.g., 90%) and a small sample of controls (e.g., 20%) could be chosen (assume no random variability).

(As cases are less frequent, the sampling fraction for cases is usually greater than that for controls).

Severe Syst HTN Cases Controls

Yes 162 1964

No 27 1994

Toal 210 x 0.9 = 189 19790 x 0.2 = 3958

09.6

19941964

27162

ORexp

expexp

cntlsin

casesin

Odds

Odds

In a retrospective (case-control) study, an unbiased sample of the cases and controls yields an unbiased OR

Myocardial infarction Severe Systolic

HTN

Number

Present Absent

Yes 10000 180 9820

No 10000 30 9970

OROdds

O ddsd is

d is

d is un

ex p

ex p

.

1 8 0

9 8 2 03 0

9 9 7 0

6 0 9

Cohort study:

Case-control studiesB) OR when controls are a sample of the total population

In a case-control study, when the control group is a sample of the total population (rather than only of the non-cases), the odds ratio of exposure is an unbiased estimate of the RELATIVE RISK

Risk factor CASES NON-CASES TOTALPOPULATION

Present a b a+b

Absent c d c+d

dbca

cases-non exp

cases expexp Odds

OddsOR

RR

Odds

OddsOR

population exp

cases exp

exp

dccbaa

dcbaca

006

10000600

100003600

.RR

Example:Hypothetical cohort study of the one-year recurrence of acute myocardial infarction (MI) among MI survivors with severe systolic hypertension (HTN, ≥180 mm Hg) or normal systolic blood pressure (<120 mm Hg).


HTNPresent Absent

Totalpopulation

Yes 3600 6400 10000

No 600 9400 10000

006

10000600

100003600

.RR

Example:Hypothetical cohort study of the one-year recurrence of acute myocardial infarction (MI) among MI survivors with severe systolic hypertension (HTN, 180+ mm Hg) or normal systolic blood pressure (<120 mm Hg).


HTNPresent Absent

Totalpopulation

Yes 3600 6400 10000

No 600 9400 10000

• Using a traditional case-control strategy, cases of recurrent MI can be compared to non-cases, i.e., individuals without recurrent MI:

818

94006400600

3600

.ORexp

006

10000600

100003600

.RR

Example:Hypothetical cohort study of the one-year recurrence of acute myocardial infarction (MI) among MI survivors with severe systolic hypertension (HTN, 180+ mm Hg) or normal systolic blood pressure (<120 mm Hg).


HTNPresent Absent

Totalpopulation

Yes 3600 6400 10000

No 600 9400 10000

• Using a traditional case-control strategy, cases of recurrent MI are compared to non-cases, i.e., individuals without recurrent MI:

disOR 81.8

94006400600

3600

OR exp

• Using a case-cohort strategy, the controls are formed by the total population:

RR.ORexp

006

10000600

100003600

1000010000

6003600

Recurrent MI Severe Systolic

HTN Present Absent

Total population

Yes 3600 6400 10 000

No 600 9400 10 000

OR RRex p .

7 2 0

1 2 01 0 0 0

1 0 0 0

6 0

Thus… RR= unbiased exposure odds estimate in cases divided by unbiased exposure odds estimate in the total population.

Note that it is not necessary to have a total group of cases and non-cases or the total population to assess an association in a case-control study. What is needed is a sample estimate of cases and either non-cases (to obtain the odds ratio of disease) or the total population (to obtain the relative risk). Example: samples of 20% cases and 10% total population:

To summarize, in a case-control study:

What is the controlgroup? What is calculated? To obtain ...

Sample ofNON-CASES ORDisease

Sample of theTOTAL POPULATION RR

cases-non exp

cases exp

exp Odds

OddsOR

pop total exp

cases exp

exp Odds

OddsOR

How to calculate the OR when there are more than two exposure categories

Example:

Univariate analysis of the relationship between parity and eclampsia.*

* Abi-Said et al: Am J Epidemiol 1995;142:437-41.

1

2.9

7.5

0

1

2

3

4

5

6

7

8

2+ 1 Nulliparous

Number of pregnancies

OR

Parity Cases Controls OR2 or more 11 401 21 27Nulliparous 68 33

1.0 (Reference)(21/11)÷(27/40)=2.9(68/11)÷(33/40)=7.5

How to calculate the OR when there are more than two exposure categories

Parity Cases Controls OR2 or more 11 40 1.01 21 27 2.9Nulliparous 68 33 7.5

Example:

Univariate analysis of the relationship between parity and eclampsia.*

* Abi-Said et al: Am J Epidemiol 1995;142:437-41.

1

2.9

7.5

1

10

2+ 1 Nulliparous

Number of pregnancies

OR 0001.0,215.2921 ptrend linear for

Correct display:

Logscale

Baseline is 1.0

A note on the use of estimates from a cross-sectional study (prevalence ratio, OR) to estimate the RR

I

I

P

P

However, if exposure is also associated with shorter survival (D+ < D-), D+/D- <1 the prevalence ratio will underestimate the RR.

D

D

I

I

P-1PP-1

P

If this ratio= 1.0

I

I

P

P

D

D

I

I

P

PIf the prevalence is low (~≤5%)

Example? Smoking and emphysema

Duration (prognosis) of the disease after onset is independent of exposure (similar in exposed and unexposed)...

Prevalence Odds=

Measures of association based on absolute differences(absolute measures of “effect”)

• Attributable risk in the exposed:

The excess risk (e.g., incidence) among individuals exposed to a certain risk factor that can be attributed to the risk factor per se:

1000/10100010

100020qqARexp

Or, expressed as a proportion(e.g., percentage):

%5010020/1000

10/1000-20/1000100

q

qq%ARexp

In

cide

nce

(per

100

0)Unexposed Exposed

ARexp

%501002.0

1.0-2.0100

RR

1-RR%ARexp

Alternative formula for the %ARexp:

10/1000

20/1000

• Population attributable risk: The excess risk in the population that can be attributed to a given risk factor. Usually expressed as a percentage:

The Pop AR will depend not only on the RR, but also on the prevalence of the risk factor (pe).

100q

qq%PopAR

pop

popexp

10011)(RRp

1)(RRp%PopAR

e

e

exp

Levin’s formula

(Levin: Acta Un Intern Cancer 1953;9:531-41)

Inci

denc

e (p

er 1

000)

Unexposed Exposed

ARexp

Population

Low exposure prevalence

Pop AR

Inci

denc

e (p

er 1

000)

Unexposed Population Exposed

Pop AR

ARexp

High exposure prevalence

Chu SP et al. Risk factors for proximal humerus fracture. Am J Epi 2004; 160:360-367

Cases: 448 incident cases identified at Kaiser Permanente. 45+ yrs old, identified through radiology reports and outpatient records, confirmed by radiography, bone scan or MRI. Pathologic fractures excluded (e.g., metastatic cancer).

Controls: 2,023 controls sampled from Kaiser Permanente membership (random sample).

Dietary Calcium (mg/day) Odds Ratios (95% CI)

Highest quartile (≥970) 1.0 (reference)

Third quartile (771-969) 1.36 (0.96, 1.91)

Second quartile (496-770) 1.11 (0.81, 1.52)

Lowest quartile (≤495) 1.54 (1.14, 2.07)

Interpretation: If those exposed to values in the lowest quartile had been exposed to other values, their odds (risk) would have been 35% lower.

Percent ARexposed%35100

54.1

154.1100

OR

1-OR100

RR

1-RR

~

Percent Population AR

p RR

p RR

p OR

p ORex p

ex p

ex p

ex p

( )

( )

( )

( )

. ( . )

. ( . ). .

1

1 11 0 0

1

1 11 0 0

0 2 5 1 5 4 1

0 2 5 1 5 4 1 11 0 0 11 9 %~

RR estimate ~ 1.54Pexp ~ 0.25

10011)(RRP

1)(RRP

exp

exp

Levin’s formula for the Percent ARpopulation

Interpretation: The exposure to the lowest quartile is responsible for about 12% of the total incidence of humerus fracture in the Kaiser permanente population

What is the %AR in those exposed to the lowest quartile?

What is the Percent AR in the total population due to exposure in the lowest quartile?

More or less 1.0

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Measures of association