Measures of AssociationEric T. Roberts MPH

Recall the clowns

Χ2(1) = 6.081

How do we interpret this?

How do we interpret this?

• Χ2 test statistic = 6.081 > 3.84 = Χ2

critical value

• Reject the null hypothesis of independence

• Conclude: There is a statistically significant association between clown therapy and becoming pregnant [Χ2(1) = 6.081, p-value < 0.05] such that 35% of women exposed to clown therapy became pregnant compared to 19% of women not exposed to clown therapy.

Measurement of effect or association• Epidemiologic studies strive to determine difference in

measures of disease occurrence between populations

• Populations typically considered as “exposed” vs “unexposed”

• Measures of association/effect measures• Determine association between “exposure” and disease

“outcome”

• Quantity that measures effect of a factor on frequency/risk of health outcome

Measurement of effect or association• Risk ratio (cumulative incidence ratio)

• Odds ratio

• Incidence rate ratio

What is risk

• Probability of developing a given disease

Risk =Number of new cases of disease

Number of persons followed/at risk

over a time period

Relative risk (risk ratio)• The ratio of risks for two populations

exp

exp

osed

un osed

RRR

R

Ranges from 0 to + , has no units

Null hypothesis: RR = 1

Same formula for incidence rate ratio (IRR) = rate among exposed / rate among unexposed

The “2x2” table

Disease No disease Total

Exposed a b a+b

Not exposed c d c+d

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

Relative risk, i.e., risk ratio

exp

exp

osed

un osed

aR

a bc

Rc d

aa bRRcc d

DiseaseNo

disease Total

Exposed a b a+b

Not exposed c d c+d

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

Example

• In a particular study out of 100 exposed persons, 20 develop disease; out of 200 unexposed, 25 develop disease

Disease No disease Total

Exposed 20 80 100

Not exposed 25 175 200

Total 45 255 300

Example (cont.)

20100 1.6025200

RR

Disease No disease Total

Exposed 20 80 100

Not exposed 25 175 200

Total 45 255 300

Example (cont.)• We calculated the RR as the risk of disease amongst the

exposed compared to the risk of disease amongst the unexposed.

• Therefore, we say that the exposed group has X times the risk of disease compared to the unexposed.

• Exposed individuals were 1.6 times more likely to develop disease compared to unexposed individuals.

Odds Ratios Overview

• An odds ratio is a measure of association between two variables. • An alternative to stating the percent amongst the exposed

and the unexposed that have the outcome.• The odds of an event is the probability of “success”

divided by the probability of “failure”• OE = OE =

• Odds ratio is a ratio of the odds of an event under contrasting conditions (exposed and unexposed)

• OR =

OR =

OE = OE =

OD|E = OD|E =

OD|UE = OD|UE =

Odds Ratio FormulaDisease

(D)No disease

(ND) Total

Exposed (E) a b a+b

Not exposed (UE) c d c+d

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

Disease (D)

No disease (ND) Total

Exposed (E) a b a+b

Not exposed (UE) c d c+d

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

OD|E = ( / (= ( / (=

[a/(a+b)] / (1- [a/(a+b)]) = [a/(a+b)] / [b/(a+b)] = a/b

OD|UE = ( / (= ( / (=

OR = =

Odds Ratio – putting algebra to work

Odds ratio example• How strong is the association between lifetime marijuana use

is different and race (white vs non-white)?white * MARI Crosstabulation

MARI Total

0 no 1 yes

white

0

Count 259 316 575

% within white 45.0% 55.0% 100.0%

% of Total 25.4% 31.0% 56.3%

1

Count 94 352 446

% within white 21.1% 78.9% 100.0%

% of Total 9.2% 34.5% 43.7%

Total

Count 353 668 1021

% within white 34.6% 65.4% 100.0%

% of Total 34.6% 65.4% 100.0%

Odds ratio examplewhite * MARI Crosstabulation

MARI Total

0 no 1 yes

white

0

Count 259 316 575

% within white 45.0% 55.0% 100.0%

% of Total 25.4% 31.0% 56.3%

1

Count 94 352 446

% within white 21.1% 78.9% 100.0%

% of Total 9.2% 34.5% 43.7%

Total

Count 353 668 1021

% within white 34.6% 65.4% 100.0%

% of Total 34.6% 65.4% 100.0%

352 (a) 94 (b) a+b

316 (c) 259 (d) c+d

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

Diseased (D) Not Diseased (ND)

Exposed (E)

Unexposed (UE)

DND

UE

E

OR = (a*d) / (c*d)

OR = (352*259) / (94*316)

OR = 3.07

Ratio Measures• Risk ratios and odds ratios range from 0 - ∞• Null value is 1

0 1∞

The exposed has more disease.A deleterious effect.

The exposed and unexposed have the same risk/odds of disease. No effect.

The unexposed has more disease. A protective effect.

Odds Ratio - example• How strong is the association between gender and lifetime

marijuana use?

sex * MARI CrosstabulationMARI Total

0 no 1 yes

sex

0 male

Count 159 316 475

% within sex 33.5% 66.5% 100.0%

% of Total 15.6% 31.0% 46.5%

1 female

Count 194 352 546

% within sex 35.5% 64.5% 100.0%

% of Total 19.0% 34.5% 53.5%

Total

Count 353 668 1021

% within sex 34.6% 65.4% 100.0%

% of Total 34.6% 65.4% 100.0%

Study design determines which measure of association• Our ultimate goal is to determine whether an exposure causes

disease or in the case of an RCT if a treatment produces the desired outcome.

• The study design we employ influences the type of data we collect and therefore determines which measure of association we can calculate.

Study Designs

• Allow us to examine the associations of interest (E D; X Y; or E1, E2… Y; X1,X2..Y)

• Two main types of designs• Observational

• Cross sectional• Cohort• Case Control

• Experimental• RCT (Randomized Control Trial)

Cross Sectional Study

• Observational study that involves data collection from a population, or a representative subset, at one specific point in time

• e.g. A study to examine whether weekly drinking to intoxication is related to frequent absenteeism from class

Concurrent assessment

Defined Population

Exposed

Unexposed

Disease

Disease

No Disease

No Disease

2015

Cohort StudyPROSPECTIVE

Defined Population

Exposed

Unexposed

Disease

Disease

No Disease

No Disease

2015 2020D

isea

se F

ree

Sub-

sam

ple

Cohort study:observational study that is longitudinal and involves data collection from a group of people over many points to examine factors related to an outcome (e.g. disease) Prospective-Retrospectivee.g. A study of HIV-negative gay men who are ages 18 at baseline over time to examine the factors (exposures) which are related to HIV seronconversion

Case Control Study

First, start with:

a

d

c

b

Diseased

Non - Diseased

Exposed

Unexposed

Exposed

Unexposed

“cases”

“controls”

Then, assess whether:

Case Control studyobservational study in which two existing groups differing in outcome (e,g., disease) are identified and compared on the basis of some supposed causal attribute (i.e., exposure)

Randomized Controlled TrialPROSPECTIVE

Defined Population

Treatment A

Treatment B

Disease

Disease

No Disease

No Disease

2015 2020Ra

ndom

trea

tmen

t as

sign

men

t

RCT:Experimental study that is longitudinal. Participants are randomly assigned to their treatment to control for confounding – the randomization balances the distribution of all potential confounders between groups. Participants are followed over time for the development of the outcome.e.g. A study to examine the efficacy of an antiretroviral gel to prevent the

Measures of association that can be calculated from each study• Cross sectional

• Prevalence odds ratio• Cohort

• Risk ratio• Case-Control

• Odds ratio• Randomized Controlled Trial

• Risk ratio

Case Control Study and the OR

• In a case control study we enroll based on disease – therefore we cannot estimate the probability of disease given exposure.

• We can estimate the exposure odds ratio• Bayes’ Theorem: P(A|B) = • Using Bayes’ Theorem we can prove the exposure odds ratio

is the same as the disease odds ratio

Disease (D)

No disease (ND) Total

Exposed (E) a b a+b

Not exposed (UE) c d c+d

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

Practice Question 1• You conduct an RCT to evaluate the efficacy of a new cancer

drug. You randomize 200 people to receive the therapy and 200 people to the usual care group. 73 people in the usual care group go into remission and 112 people in the intervention group go into remission . What is the appropriate measure of association? Quantify the association between the new drug and the likelihood of going into remission.

Practice Question 2• You want to assess whether there is an association between

traumatic head injuries in youth and pediatric stroke. The reported incidence of combined ischemic and hemorrhagic pediatric stroke ranges from 1.2 to 13 cases per 100,000 children under 18 years of age. You enroll all 72 cases of pediatric strokes at NYU Langone Medical Center from 2014 and 2015 and assess whether or not they had a traumatic head injury. You enroll 2 controls for every case from a sample of children recruited from the vaccine clinic. You find that 43 of the cases report a traumatic brain injury and that 69 of the controls had a traumatic brain injury. What is the appropriate measure of association? Quantify the association between the traumatic head injuries in youth and pediatric stroke.

Practice Question 3• A refinery in New Jersey reports to the EPA that there was a

massive leak of a toxic chemical into the ground water supply over a 3 month period. Based on water samples the EPA determines that there are four zones of contamination: severe, high, medium and low with 5%, 10%, 33% and 52% of the town’s population respectively. You bring in all 7,000 people in the town for a health assessment and then monitor hospital records for cases of myocardial infarction. In the two years after the spill you observe 100 MI’s in the severe group, 133 in the high group, 254 in the medium group and 156 in the low group. What is the appropriate measure of association? Quantify the association between the exposure to the toxin and having an MI.

Practice Question 4• The Mayor’s office is worried about a highly publicized

incident of violence perpetrated by someone using bath salts. They don’t know much about who use bath salts or why or how. They charge the NYCDOHMH with investigating. As part of the NYC HANES survey you include a question about bath salt use. Of the 10,000 people surveyed 157 report ever using bath salts of which 114 are white, 22 are Black and 21 are Hispanic. NYC HANES was representative of NYC, which is 44% white, 28% Hispanic, and 25% Black. What is the appropriate measure of association? Is race associated with bath salt use?

Participant ID

Gender Race PM2.5 Nitrogen oxides

DM

001 1 1 1.4 25 1

002 . 2 2.3 32 0

003 2 2 1.7 9 0

004 2 . 3.3 10 1

005 1 1 2.8 31 1

006 1 3 0.8 8 0

007 2 3 1.3 25 0

008 1 4 2.6 34 0

009 2 . 0.6 27 1

010 2 1 4.1 13 0