COMMON MEASURES OF ASSOCIATION IN MEDICAL AND EPIDEMIOLOGIC RESEARCH: ODDS, RISK, & THE 2X2 TABLEPatrick BarlowUniversity of Tennessee Graduate School of Medicine

ON THE AGENDAPart I

Odds, risk, rate, & proportion, what’s the difference?

The 2x2 table explainedPart II

Calculating measures of association

SOME TERMS FOR PART I

Proportion Risk Odds Rate

The Basics

PART I: THE BASICSComparing probability, risk, rates, & odds

WHAT IS PROBABILITY?The probability of a favorable event is the fraction of times you expect to see that event in many trials.

Always range between 0 and 1For example…You record 25 heads on 50 flips of a coin, what is

the probability of a heads?

=0.5 or “50% chance of heads

A “risk” is simply the proportion of individuals in a certain group who had the outcome divided by the total number in that group.

WHAT ARE ODDS?An “odds” is a probability of a favorable event occurring vs. not occurring.

For example…What are the odds you will get a heads when

flipping a fair coin?

Odds of heads = Probability of heads / (1-Probability of heads) = .5 / (1-.5) = 1

“The odds of flipping heads to flipping tails is 1:1”

WHAT IS A RATE?The term “Rate” is often misused in medical literature as well as in everyday conversation.

Technically, a rate is a measure of occurrence per unit of time such as…

Miles Per Hour Words Per Minute

OR

WHAT IS A RATE?In the health sciences, rates are generally expressed as the number of deaths, cases, etc. per “person time”.

• For example: a study looking at the incidence of COPD exacerbations following a clinic-wide intervention had five participants…

Time in the study

(months)

COPD Exacerbation

Patient 1 3 YesPatient 2 11 YesPatient 3 12 NoPatient 4 12 NoPatient 5 4 NoTotal 42 2

WHAT IS A RATE?What is the rate of COPD Exacerbation in this sample?

Time in the study

(months)

COPD Exacerbation

Patient 1 3 YesPatient 2 11 YesPatient 3 12 NoPatient 4 12 NoPatient 5 4 NoTotal 42 2

𝑅𝑎𝑡𝑒= 𝑇𝑜𝑡𝑎𝑙 𝐸𝑣𝑒𝑛𝑡𝑠𝑇𝑜𝑡𝑎𝑙 𝑃𝑒𝑟𝑠𝑜𝑛𝑇𝑖𝑚𝑒=¿

𝑅𝑎𝑡𝑒=2 COPD  Exacerbations42 months =0.05𝐸𝑥𝑎𝑐𝑒𝑟𝑏𝑎𝑡𝑖𝑜𝑛𝑠𝑝𝑒𝑟 𝑝𝑒𝑟𝑠𝑜𝑛 h𝑚𝑜𝑛𝑡

THE BOTTOM LINE Proportions & risks are synonymous with one

another as the number of “occurrences” or the number at risk to develop the outcome (i.e. sample)

An “odds” is a probability of a favorable event occurring vs. not occurring. It is expressed as a ratio, for example, an odds of 1.00 means there is a 1:1 (1 to 1) odds of the event occurring vs. not occurring.

A rate differs from both proportions and odds because it is always expressed per a unit of time such as miles per hour. Health sciences usually express rates in terms of “person-time.”

PART II: CALCULATING COMMON MEASURES OF ASSOCIATION ON A 2X2

TABLEOdds Ratio (OR)

Relative Risk Ratio (RR)Attributable Risk (AR)

Absolute Risk Reduction (ARR)Number Needed to Treat (NNT)Number Needed to Harm (NNH)

SOME TERMS FOR PART IICommon Measures of

Association Odds Ratio (OR) Relative Risk Ratio

(RR) Attributable Risk

(AR) Absolute Risk

Reduction (ARR) Number Needed to

Treat (NNT) Number Needed to

Harm (NNH)

RELATIVE RISK VS. ODDS RATIOS Relative Risk (RR) is a more accurate measure

of incidence of an outcome of interest. Used in prospective studies or when the total

population are known What study designs would use RR? Mathematically, RR is calculated the same way as

an odds whereRelative Risk of an event = Odds of event occurring / Odds of event not occurring.

An odds ratio (OR) provides researchers with an estimate of RR in situations where the total population is unknown. What study designs would use ORs instead of RRs?

THE 2X2 TABLE The basis of nearly every common measure

of association in medical and epidemiologic research can be traced back to a 2x2 contingency table.

A BC D

THE 2X2 TABLE For every measure of association using the

2x2 table, your research question comes from the A cell.

A BC D

EXAMPLE What is the risk of myocardial infarction (MI)

if a patient is taking aspirin versus a placebo?

Had MI No MI

Aspirin A BPlacebo C D

What other research questions could be answered using this same

table?

RELATIVE RISK ON A 2X2 TABLE What is the risk of myocardial infarction (MI)

if a patient is taking aspirin versus a placebo?

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

RELATIVE RISK ON A 2X2 TABLE

What is the risk of MI if a patient is taking aspirin?

What is the risk of MI if a patient is taking placebo?

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

𝑅𝑖𝑠𝑘𝑜𝑓 𝑀𝐼 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=¿ h𝑤𝑖𝑡 𝑀𝐼

¿𝑜𝑛𝐴𝑠𝑝𝑖𝑟𝑖𝑛501080=0.048𝑜𝑟 4.8%

𝑅𝑖𝑠𝑘𝑜𝑓 𝑀𝐼 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=¿ h𝑤𝑖𝑡 𝑀𝐼

¿𝑜𝑛𝑃𝑙𝑎𝑐𝑒𝑏𝑜2001770=0.11𝑜𝑟 11%

RELATIVE RISK ON A 2X2 TABLE

So… What is the risk of myocardial infarction (MI) if

a patient is taking aspirin versus a placebo?

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

𝑅𝑅=(𝐴𝐴+𝐵 )

( 𝐶𝐶+𝐷 )

=𝑅𝑖𝑠𝑘𝑜𝑓 𝑀𝐼 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛𝑅𝑖𝑠𝑘𝑜𝑓 𝑀𝐼 𝑓𝑜𝑟 𝑃𝑙𝑎𝑐𝑒𝑏𝑜

RR

INTERPRETING ORS AND RRS: THE BASICS Odds/Risk ratio ABOVE 1.0 = Your exposure

INCREASES risk of the event occurring For OR/RRs between 1.00 and 1.99, the risk is

increased by (OR – 1)%. For OR/RRs 2.00 or higher, the risk is increased OR

times. Example:

Smoking is found to increase your odds of breast cancer by OR = 1.25. What is the increase in odds? You are 25% more likely to have breast cancer if you are a

smoker. Smoking is found to increase your risk of developing

lung cancer by RR = 4.8. What is the increase in risk? You are 4.8 times more likely to develop lung cancer if you

are a smoker vs. non-smoker.

INTERPRETING ORS AND RRS: THE BASICS Odds/Risk ratio BELOW 1.0 = Your exposure

DECREASES risk of the event occurring The risk is decreased by (1 – OR)% Often called a PROTECTIVE effect

Example: Addition of the new guidelines for pacemaker/ICD

interrogation produced an OR for device interrogation of OR = .30 versus the old guidelines. What is the reduction in odds? (1 – OR) = (1 – .30) = 70% reduction in odds.

YOUR TURN Work in pairs to calculate the RRs for each of

the 2x2 tables below.

RR = (79/79+157) / (100/100+375) = 1.59

1 PE No PE

DVT 79 157

No DVT 100 375

RR = (190/(190+450)) / (70/(70+700)) = 3.27

3 Lung Cancer

No Lung Cancer

Smoking Hx 190 450

No Smoking Hx 70 700

RR = (35/(35+170)) / (52/(52+160)) = .70

2 Glucose Tolerance Improved

Tolerance not

Improved

Lap Band 35 170

Gastric Bypass 52 160

RR = (25/(25+350)) / (65/(65+200)) = .27

4 DM Type II No DM Type II

BMI < 30 25 350

BMI > 30 65 200

ODDS RATIOS AND THE 2X2 TABLE Recall…

Odds ratios are used to estimate RR when the true population is unknown.

For discussion Why can’t we just use RR all the time? Will an OR and RR differ from one another? If so,

how? Odds ratios look at prevalence rather than

incidence of the event. Remember:

OR = “Odds of having the outcome” RR = “Risk of developing the outcome”

ODDS RATIOS AND THE 2X2 TABLE

What are the odds of myocardial infarction (MI) if a patient is taking aspirin versus a placebo? OR = A*D / B*C OR = 50*1570 / 1030 * 200 = .38 or 38%

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

OR = (25*200) / (350*65) = .21

4 DM Type II No DM Type II

BMI < 30 25 350

BMI > 30 65 200

OR = (35*160) / (170*52) = .63

2 Glucose Tolerance Improved

Tolerance not

Improved

Lap Band 35 170

Gastric Bypass 52 160

OR = (190*700) / (450*70) = 4.22

3 Lung Cancer

No Lung Cancer

Smoking Hx 190 450

No Smoking Hx 70 700

YOUR TURN Work in pairs to calculate the ORs for the same 2x2

tables as before. How do the ORs and RRs differ?

OR = (79*375) / (157*100) = 1.89

1 PE No PE

DVT 79 157

No DVT 100 375

INTERPRETING ORS AND RRS: THE BASICS So for our example…

OR = .39 What is the reduction in odds? So: “Taking aspirin provides a 61% reduction in the

odds of having an MI compared to a placebo.”

RR = .41 What is the reduction in risk? So: “Taking aspirin provides a 59% reduction in risk of

MI compared to a placebo.”

INTERPRET THE FOLLOWING OR/RRS OR = 3.00 OR = .39 RR = 1.50 OR = 1.00 RR = .22 RR = 18.99 OR = .78

What does the OR/RR say about the strength of relationship?

OR/RR AND CONFIDENCE INTERVALS The magnitude of the OR/RR only provides the

strength of the relationship, but not the accuracy 95% Confidence intervals are added to any

OR/RR calculation to provide an estimate on the accuracy of the estimation. 95% of the time the true value will fall within a given

rangeWide CI = weaker inferenceNarrow CI = stronger inferenceCI crosses over 1.0 = non-significant

An OR/RR is only as important as the confidence interval that comes with it

INTERPRET THESE 95% CIS OR 2.4 (95% CI 1.7 - 3.3) OR 6.7 (95% CI 1.4 - 107.2) OR 1.2 (95% CI .147 - 1.97) OR .37 (95% CI .22 - .56) OR .57 (95% CI .12 - .99) OR .78 (95% CI .36 – 1.65)

OTHER COMMON MEASURES OF ASSOCIATION EXAMPLE ONE: ABSOLUTE RISK REDUCTION Absolute Risk Reduction (ARR):

This is the difference between the risk (not RR) of the outcome in the control group minus the risk of the outcome in the study group. For Example…Recall the MI & Aspirin

study

What is the ARR of aspirin vs. the placebo?

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

𝑅𝑖𝑠𝑘𝑜𝑓 𝑀𝐼 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=¿ h𝑤𝑖𝑡 𝑀𝐼

¿𝑜𝑛𝐴𝑠𝑝𝑖𝑟𝑖𝑛=501080=0.048𝑜𝑟 4.8%

𝑅𝑖𝑠𝑘𝑜𝑓 𝑀𝐼 𝑓𝑜𝑟 𝑃𝑙𝑎𝑐𝑒𝑏𝑜=¿ h𝑤𝑖𝑡 𝑀𝐼

¿ 𝑜𝑛𝑃𝑙𝑎𝑐𝑒𝑏𝑜=2001770=0.11𝑜𝑟 11%

𝐴𝑅𝑅 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=𝑅𝑖𝑠𝑘 𝑓𝑜𝑟 𝑃𝑙𝑎𝑐𝑒𝑏𝑜−𝑅𝑖𝑠𝑘 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=11% −4.8%=6.2%  𝐴𝑅𝑅

OTHER COMMON MEASURES OF ASSOCIATION EXAMPLE TWO: ATTRIBUTABLE RISK Attributable Risk (AR) is the increase in risk

associated with a particular risk factor. It is the incidence in the exposed group minus the incidence in the unexposed group.

For Example…a classic example is a 1980s controversy between Aspirin and Rye’s syndrome.

What is the AR for Rye’s in children exposed to aspirin vs. not exposed?

Ryes (+) Ryes (–)

Aspirin (+) 600 9030

Aspirin (–) 150 10000

𝑅𝑖𝑠𝑘𝑜𝑓 𝑅𝑦𝑒𝑠 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=¿ h𝑤𝑖𝑡 𝑅𝑦𝑒𝑠

𝑇𝑜𝑡𝑎𝑙 𝐸𝑥𝑝𝑜𝑠𝑒𝑑=6009630=0.62𝑜𝑟 6.2%

𝑅𝑖𝑠𝑘𝑜𝑓 𝑅𝑦𝑒𝑠 𝑓𝑜𝑟 𝑃𝑙𝑎𝑐𝑒𝑏𝑜=¿ h𝑤𝑖𝑡 𝑅𝑦𝑒𝑠

𝑇𝑜𝑡𝑎𝑙 𝑁𝑜𝑡 𝐸𝑥𝑝𝑜𝑠𝑒𝑑=1501150=0.015𝑜𝑟 1.5%

for Aspirin exposure

OTHER COMMON MEASURES OF ASSOCIATION: NUMBER NEEDED TO TREAT / HARM Number needed to treat (NNT) is the number of patients

that would need to be treated in order to prevent a single event. It is the inverse of ARR.

Conversely, number need to harm (NNH) is the number of patients that would need to be exposed to the risk factor before someone had an event. It is the inverse of AR.

𝐴𝑅𝑅 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=𝑅𝑖𝑠𝑘 𝑓𝑜𝑟 𝑃𝑙𝑎𝑐𝑒𝑏𝑜−𝑅𝑖𝑠𝑘 𝑓𝑜𝑟 𝐴𝑠𝑝𝑖𝑟𝑖𝑛=11% −4.8%=6.2%  𝐴𝑅𝑅

𝑁𝑁𝑇=1𝐴𝑅𝑅

𝑁𝑁𝐻=1𝐴𝑅

In our aspirin and MI example (Example 1), the ARR for aspirin = 6.2%

and the AR = 4.7 𝑁𝑁𝑇=

1𝐴𝑅𝑅=

1.062=16𝑝𝑒𝑜𝑝𝑙𝑒𝑤𝑜𝑢𝑙𝑑𝑛𝑒𝑒𝑑𝑡𝑜𝑏𝑒 𝑡𝑟𝑒𝑎𝑡𝑒𝑑

people would need to be exposed

WHAT IS STATISTICAL INFERENCE?Causation, hypothesis testing & what it means to be “statistically significant”