measures of disease association measuring occurrence of new outcome events can be an aim by itself,...

Measures of Disease Association

• Measuring occurrence of new outcome events can be an aim by itself, but usually we want to look at the relationship between an exposure (risk factor, predictor) and the outcome

• The type of measure showing an association between an exposure and an outcome event is linked to the study design

Main points to be covered

• Measures of association compare measures of disease between levels of a predictor variable

• Prevalence ratio versus risk ratio

• Probability and odds

• The 2 X 2 table

• Properties of the odds ratio

• Absolute risk versus relative risk

• Disease incidence and risk in a cohort study

Cross-Sectional Study Design: A Prevalent Sample

Measures of Association in a Cross-Sectional Study

• Simplest case is to have a dichotomous outcome and dichotomous exposure variable

• Everyone in the sample is classified as diseased or not and having the exposure or not, making a 2 x 2 table

• The proportions with disease are compared among those with and without the exposure

• NB: Exposure=risk factor=predictor

2 x 2 table for association of disease and exposureDisease

Yes NoE

xpos

ure

Yes

No

a b

c d

a + b

c + d

a + c b + d N = a+b+c+d

Note: data may not always come to you arranged as above.STATA puts exposure across the top, disease on the side.

Prevalence ratio of disease in exposed and unexposedDisease

Yes No

Exp

osur

e

Yes

No

a b

c d

a + b

c + d

c

a

PR =

Prevalence Ratio

• Text refers to Point Prevalence Rate Ratio in setting of cross-sectional studies

• We like to keep the concepts of rate and prevalence separate, and so prefer to use prevalence ratio

Exposed Unexposed | Total---------------------------------------------------

Cases | 14 388 | 402Noncases | 17 248 | 265---------------------------------------------------

Total | 31 636 | 667 | |

Risk | .4516129 .6100629 | .6026987

Point estimate [95% Conf. Interval] ---------------------------------------------

Risk ratio .7402727 | .4997794 1.096491 ----------------------------------------------- chi2(1) = 3.10 Pr>chi2 = 0.0783

Prevalence ratio (STATA output)

STATA calls it a risk ratio by default

Prevalence ratio of disease in exposed and unexposedDisease

Yes No

Exp

osur

e

Yes

No

a b

c d

a + b

c + d

c

a

PR =

So a/a+b and c/c+d = probabilities of diseaseand PR is ratio of two probabilities

Probability and Odds

• Odds another way to express probability of an event

• Odds = # events # non-events

• Probability = # events # events + # non-events

= # events # subjects


• Probability = # events

# subjects

• Odds = # events

# subjects = probability

# non-events (1 – probability)

# subjects

• Odds = p / (1 - p)

[ratio of two probabilities]


• If event occurs 1 of 5 times, probability = 0.2.

• Out of the 5 times, 1 time will be the event and 4 times will be the non-event, odds = 0.25

• To calculate probability given the odds:

probability = odds / 1+ odds

Odds versus Probability• Less intuitive than probability (probably wouldn’t

say “my odds of dying are 1/4”)

• No less legitimate mathematically, just not so easily understood

• Used in epidemiology because the measure of association available in case-control design is the odds ratio

• Also important because the log odds of the outcome is given by the coefficient of a predictor in a logistic regression

Odds ratio• As odds are just an alternative way of

expressing the probability of an outcome, odds ratio (OR), is an alternative to the ratio of two probabilities (prevalence or risk ratios)

• Odds ratio = ratio of two odds

Probability and odds in a 2 x 2 table

DiseaseYes No

Exp

osur

e

Yes

No

2 3

1 4 5

5

103 7

What is p of diseasein exposed?

What are odds ofdisease in exposed?

And the same forthe un-exposed?

Probability and odds ratios in a 2 x 2 table

DiseaseYes No

Exp

osur

e

Yes

No

2 3

1 4 5

5

103 7

PR = 2/5 1/5= 2

0R = 2/3 1/4= 2.67

Odds ratio of disease in exposed and unexposed

DiseaseYes No

Exp

osur

e

Yes

No

a b

c d

a + b

c + d

c

a

OR =

aa + b

1 -

c

c + d1 -

Formula of p / 1-p in exposed / p / 1-p in unexposed

Odds ratio of disease in exposed and unexposed

a + b

c + dc

a

OR =

aa + b

1 -

cc + d

1 -

=

aa + b ba + b cc + d dc + d

a b c d

= =adbc

Important Property of Odds Ratio #1

• The odds ratio of disease in the exposed and unexposed equals the odds ratio of exposure in the diseased and the not diseased– Important in case-control design

Odds ratio of exposure in diseased and not diseased

DiseaseYes No

Exp

osur

e

Yes

No

a b

c d

a + c

b + d

b

a

OR =

aa + c

1 -

b

b + d1 -

OR for disease = OR for exposure

a + c

b + db

a

ORexp =

aa + c

1 -

bb + d

1 -

=

aa + c ca + c bb + d db + d

a c b d

= =adbc

Important characteristic of odds ratio

Measures of Association Using Disease Incidence

• With cross-sectional data we can calculate a ratio of the probability or of the odds of prevalent disease in two groups, but we cannot measure incidence

• A cohort study allows us to calculate the incidence of disease in two groups

Measuring Association in a CohortFollowing two groups by exposure status within a cohort:Equivalent to following two cohorts defined by exposure

Analysis of Disease Incidence in a Cohort

• Measure occurrence of new disease separately in a sub-cohort of exposed and a sub-cohort of unexposed individuals

• Compare incidence in each sub-cohort

• How compare incidence in the sub-cohorts?

Relative Risk vs. Relative Rate• Risk is based on proportion of persons with

disease = cumulative incidence

• Risk ratio = ratio of 2 cumulative incidence estimates = relative risk

• Rate is based on events per person-time = incidence rate

• Rate ratio = ratio of 2 incidence rates = relative rate

• We prefer risk ratio, rate ratio, odds ratio

A Note on RR or “Relative Risk”

• Relative risk or RR is very common in the literature, but may represent a risk ratio, a rate ratio, a prevalence ratio, or even an odds ratio

• We will try to be explicit about the measure and distinguish the different types of ratios

• There can be substantial difference in the association of a risk factor with prevalent versus incident disease

Difference vs. Ratio Measures

• Two basic ways to compare measures:– difference: subtract one from the other– ratio: form a ratio of one over the other

• Can take the difference of either an incidence or a prevalence measure but rare with prevalence

• Example using incidence: cumulative incidence 26% in exposed and 15% in unexposed,– risk difference = 26% - 15% = 11% – risk ratio = 0.26 / 0.15 = 1.7

Summary of Measures of Association

Ratio Difference

Cross-sectional prevalence ratio prevalence difference

odds ratio odds difference

Cohort risk ratio risk difference

rate ratio rate difference

odds ratio odds difference

Why use difference vs. ratio?

• Risk difference gives an absolute measure of the association between exposure and disease occurrence– public health implication is clearer with absolute

measure: how much disease might eliminating the exposure prevent?

• Risk ratio gives a relative measure– relative measure gives better sense of strength of an

association between exposure and disease for inferences about causes of disease

Relative Measures and Strength of Association with a Risk Factor

• In practice many risk factors have a relative measure (prevalence, risk, rate, or odds ratio) in the range of 2 to 5

• Some very strong risk factors may have a relative measure in the range of 10 or more – Asbestos and lung cancer

• Relative measures < 2.0 may still be valid but are more likely to be the result of bias– Second-hand smoke relative risk < 1.5

Example of Absolute vs. Relative Measure of Risk

TB recurrence

No TB recurrence

Total

Treated

> 6 mos 14 986 1000

Treated

< 3 mos 40 960 1000

Risk ratio = 0.04/0.014 = 2.9

Risk difference = 0.04 – 0.014 = 2.6%

If incidence is very low, relative measurecan be large but difference measure small

Reciprocal of Absolute Difference ( 1/difference)

• Number needed to treat to prevent one case of disease

• Number needed to treat to harm one person• Number needed to protect from exposure to

prevent one case of disease• TB rifampin example: 1/0.026 = 38.5,

means that you have to treat 38.5 persons for 6 mos vs. 3 mos. to prevent one case of TB recurrence

Table 2. Survival and Functional Outcomes from the Two Study Phases

Study Phase

Return of Spontaneous Circulation

Risk

Difference

(95% CI) p-value

Rapid Defibrillation

(N=1391) 12.9%

-- --

Advanced

Life Support

(N=4247) 18.0% 5.1% (3.0-7.2) <0.001

Stiel et al., NEJM, 2004

Example of study reporting risk difference

Risk difference = 0.051; number needed to treat = 1/0.051 = 20

Risk Ratio Diarrheal Disease

Yes No Total

Ate potato salad 54 16 70

Did not eat potato salad

2 26 28

Total 56 42 98

Probability of disease, ate salad = 54/70 = 0.77

Probability of disease, no salad = 2/28 = 0.07

Risk ratio = 0.77/0.07 = 11 Illustrates risk ratio in cohort with complete follow-up

Risk Ratio in a Cohort with Censoring

Choose a time point for comparing two cumulative incidences:At 6 years, % dead in low CD4 group = 0.70 and in high CD4group = 0.26. Risk ratio at 6 years = 0.70/0.26 = 2.69

Comparing two K-M Curves

Risk ratio would be different for different follow-up Times. Entire curves are compared using log rank test(or other similar tests).

OR compared to Risk Ratio

0 1 ∞Stronger effect

OR Risk Ratio

Stronger effect

Risk Ratio OR

If Risk Ratio = 1.0, OR = 1.0;otherwise OR farther from 1.0

Risk ratio and Odds ratio

If Risk Ratio > 1, then OR farther from 1 than Risk Ratio:

RR = 0.4 = 2 0.2

OR = 0.4

0.6 = 0.67 = 2.7 0.2 0.25 0.8


If Risk Ratio < 1, then OR farther from 1 than RR:

RR = 0.2 = 0.67 0.3

OR = 0.2

0.8 = 0.25 = 0.58 0.3 0.43 0.7

Important property of odds ratio #2

• OR approximates Risk Ratio only if disease incidence is low in both the exposed and the unexposed group


If risk of disease is low in both exposed and unexposed, RR and OR approximately equal.

Text example: incidence of MI risk in high bp group is 0.018 and in low bp group is 0.003:

Risk Ratio = 0.018/0.003 = 6.0

OR = 0.01833/0.00301 = 6.09

Risk ratio and Odds ratioIf risk of disease is high in either or both exposed and unexposed, Risk Ratio and OR differ

Example, if risk in exposed is 0.6and 0.1 in unexposed: RR = 0.6/0.1 = 6.0

OR = 0.6/0.4 / 0.1/0.9 = 13.5

OR approximates Risk Ratio only if incidence is low in both exposed and unexposed group

“Bias” in OR as estimate of RR

• Text refers to “bias” in OR as estimate of RR (OR = RR x (1-incid.unexp)/(1-incid.exp))– not “bias” in usual sense because both OR and

RR are mathematically valid and use the same numbers

• Simply that OR cannot be thought of as a surrogate for the RR unless incidence is low


• Unlike Risk Ratio, OR is symmetrical:

OR of event = 1 / OR of non-event

Symmetry of odds ratio versus non-symmetry of risk ratio

OR of non-event is 1/OR of eventRR of non-event = 1/RR of eventExample: If cum. inc. in exp. = 0.25 andcum. inc. in unexp. = 0.07, thenRR (event) = 0.25 / 0.07 = 3.6 RR (non-event) = 0.75 / 0.93 = 0.8 Not reciprocal: 1/3.6 = 0.28 = 0.8

Symmetry of ORExample continued:OR(event) = 0.25 (1- 0.25) = 4.4 0.07 (1- 0.07)OR(non-event) = 0.07 (1- 0.07) = 0.23 0.25 (1- 0.25)Reciprocal: 1/4.46 = 0.23


• Coefficient of a predictor variable in logistic regression is the log odds of the outcome (e to the power of the coefficient = OR)

– Logistic regression is the method of multivariable analysis used most often in cross-sectional and case-control studies

3 Useful Properties of Odds Ratios

• Odds ratio of disease equals odds ratio of exposure– Important in case-control studies

• Odds ratio of non-event is the reciprocal of the odds ratio of the event (symmetrical)

• Regression coefficient in logistic regression equals the log of the odds ratio

Summary points

• Cross-sectional study gives a prevalence ratio

• Risk ratio should refer to incident disease

• Relative ratios show strength of association

• Risk difference gives absolute difference indicating number to treat/prevent exposure

• Properties of the OR important in case-control studies– OR for disease = OR for exposure– Logistic regression coefficient gives OR

measures of disease association measuring occurrence of new outcome events can be an aim by itself,...

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