Measuring Associations Between Measuring Associations Between Exposure and OutcomesExposure and Outcomes

Chapter 3, Szklo and Nieto

Measures of Association can be Measures of Association can be based on:based on:

Absolute differences Between Groups (e.g., disease risk among exposed – disease risk among unexposed)

Relative differences or ratios Between Groups (e.g., disease risk ratio or relative risk: disease risk in exposed/disease risk in unexposed)

Absolute differencesAbsolute differences

Public Health activitiesPreventive activitiesMeasure of association when the outcome

of interest is continuous Examples: PAR , Mean Differences

Relative differences or ratiosRelative differences or ratios

For discrete variableTo assess causal associationsExamples: Relative Risk/Rate,

Relative odds

Types of VariablesTypes of Variables

Discrete/categorical– Dichotomous, binary

• Absolute Difference?• Relative Difference

Continuous– Difference between

means

Cohort StudyCohort Study

Diseased

Non-diseased

Totals: Risk odds

Exposure

Exposed a b a+b a / a+b a / b

Unexposed c d c+d c /c+d c / d

Totals:

Disease

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

Odds in Exposed and UnexposedOdds in Exposed and Unexposed

Odds in exposed=( a / a+b) / 1- (a / a+b )

=(a / a+b) / (b / a+b) = a/bOdds in unexposed=( c / c+d) / 1- (c / c+d )

=(c / c+d) / (d / c+d) = c/d

Relative RiskRelative Risk

RR= a / a+b / c / c+d

OR= a / b / c / d = a*d / b*cOdds ratio is a cross-product ratio

Rare Disease - MIRare Disease - MI

MI Free of MI Totals:

Exposure

High Blood

Pressure

180 9820 10000

Normal

Pressure

30 9970 10000

Probability + =q + = 180/10000 = 0.0180

Probability - = q - = 30/10000 = 0.0030

Odds dis +

= 180/9820 = 0.01833

Odds dis -

= 30/9970 = 0.00301

RR=6 OR=6.09

Common Disease – Vaccine ReactionsCommon Disease – Vaccine Reactions

Local

Reactions

Free of

Reactions

Totals:

Exposure

Vaccinated 650 1920 2570

Placebo 170 2240 2240

RR = 650 / 2570 / 170 / 2410 = 0.2529 / 0.0705 = 3.59

OR = 650 / 1920 / 170 / 2240 = 0.3385 / 0.0759 = 4.46

Built – in biasBuilt – in bias

OR =( q + / 1 - q +) / (q - / 1 - q –)

= q + / q - * (1 - q - / 1- q + ) = RR * (1 - q - / 1- q + )

Built – in biasBuilt – in bias

Use of the odds ratio as an estimate of the relative risk biases it in a direction opposite to the null hypothesis.

(1 - q - / 1- q + ) defines the bias responsible for the discrepancy between the RR & OR.

When the disease is relatively rare , this bias is negligible.

When the incidence is high, the bias can be substantial.

OR is a valuable measure of OR is a valuable measure of association :association :

1. It can be measured in case – control studies. 2. It is directly derived from logistic regression

models 3. The OR of an event is the exact reciprocal of

the OR of the nonevent. (survival or death OR both are informative)

4. when the baseline risk is not very small, RR can be meaningless.

Cross-sectional StudiesCross-sectional Studies

In the stationary population:Prevalent RR= Prev+ / prev-

= ( q+ * Dur+ * (1- prev+)) / ( q- * Dur- * (1- prev-))

PPR = RR x dur+ x {1-prev+}

dur- {1-prev-}

Cross-sectional StudiesCross-sectional Studies

A point prevalence ratio may be able to estimate the relative risk depending on – the ratio of the durations of disease among

• the exposed with disease+

• the unexposed with disease-

– the ratio of the values• 1-prevalence among the exposed+

• 1-prevalence among the unexposed-

The two bias factors that differentiate the PRR from the relative risk:

1. Dur+/Dur- survival or duration bias

2. (1- prev+/ 1- prev -) complement bias

1 & 2 (S.B) Incidence – prevalence bias

We can estimate RR in cross sectional study when the exposure don’t modify the duration of the disease and the disease is rare.

Since (1- prev+/ 1- prev -)< 1:

PRR underestimates RR

We should consider temporality

Case-Control StudyCase-Control Study

The OR of disease and the OR of exposure are mathematically equivalent.

In case control study we calculate the OR of exposure as it’s algebraically identical to the OR of disease.

OR exp = a /c / b/ d = a*d/ b*c = a / b / c / d = OR dis

Case-Control StudyCase-Control Study

The fact that the OR exp is identical to the OR dis

explains why the interpretation of the odds ratio in case control studies is prospective.

Odds Ratio as an Estimate of the Odds Ratio as an Estimate of the Relative Risk:Relative Risk:

The disease under study has low Incidence thus resulting in a small built-in bias : OR is an estimate of RR

The case – cohort approach allows direct estimation of RR by OR and does not have to rely on rarity assumption.

When the OR is used as a measure of association in itself, this assumption is obviously is not needed

Calculation of the OR when there are Calculation of the OR when there are more than two exposure categoriesmore than two exposure categories

To calculate the OR for different exposure categories , one is chosen as the reference category (biologically or largest sample size)

Cases of Craniosynostosis and normal Cases of Craniosynostosis and normal Control according to maternal ageControl according to maternal age

Maternal age

Cases Controls Odds exp in case

Odds exp in control

OR

<20 12 89 12/12 89/89 1

20-24 47 242 47/12 242/89 1.44

25-29 56 255 56/12 255/89 1.63

>29 58 173 58/12 173/89 2.49

When the multilevel exposure variable is ordinal, it may be of interest to perform a trend test