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This is an updated version of my Common Measures of Association presentation. I've updated it to include (1) more detail on rates, risks, and proportions, (2) Absolute Risk Reduction (ARR), Attributable Risk (AR), Number Needed to Treat (NNT) and Number Needed to Harm (NNH). Feel free to email me for a full version of the slideshow.
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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, whatโs 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 exampleโฆYou record 25 heads on 50 flips of a coin, what is
the probability of a heads?
=0.5 or โ50% chance of heads
A โriskโ 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 โoddsโ is a probability of a favorable event occurring vs. not occurring.
For exampleโฆWhat 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
โThe odds of flipping heads to flipping tails is 1:1โ
WHAT IS A RATE?The term โRateโ 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 asโฆ
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 โperson timeโ.
โข For example: a study looking at the incidence of COPD exacerbations following a clinic-wide intervention had five participantsโฆ
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
๐ ๐๐ก๐= ๐๐๐ก๐๐ ๐ธ๐ฃ๐๐๐ก๐ ๐๐๐ก๐๐ ๐๐๐๐ ๐๐๐๐๐๐=ยฟ
๐ ๐๐ก๐=2 COPD Exacerbations42 months =0.05๐ธ๐ฅ๐๐๐๐๐๐๐ก๐๐๐๐ ๐๐๐ ๐๐๐๐ ๐๐ h๐๐๐๐ก
THE BOTTOM LINE Proportions & risks are synonymous with one
another as the number of โoccurrencesโ or the number at risk to develop the outcome (i.e. sample)
An โoddsโ 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 โperson-time.โ
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
๐ ๐๐ ๐๐๐ ๐๐ผ ๐๐๐ ๐ด๐ ๐๐๐๐๐=ยฟ h๐ค๐๐ก ๐๐ผ
ยฟ๐๐๐ด๐ ๐๐๐๐๐501080=0.048๐๐ 4.8%
๐ ๐๐ ๐๐๐ ๐๐ผ ๐๐๐ ๐ด๐ ๐๐๐๐๐=ยฟ h๐ค๐๐ก ๐๐ผ
ยฟ๐๐๐๐๐๐๐๐๐2001770=0.11๐๐ 11%
RELATIVE RISK ON A 2X2 TABLE
Soโฆ 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
๐ ๐ =(๐ด๐ด+๐ต )
( ๐ถ๐ถ+๐ท )
=๐ ๐๐ ๐๐๐ ๐๐ผ ๐๐๐ ๐ด๐ ๐๐๐๐๐๐ ๐๐ ๐๐๐ ๐๐ผ ๐๐๐ ๐๐๐๐๐๐๐
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 โ 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 โ 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 โ OR) = (1 โ .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 Recallโฆ
Odds ratios are used to estimate RR when the true population is unknown.
For discussion Why canโt 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 = โOdds of having the outcomeโ RR = โRisk of developing the outcomeโ
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 exampleโฆ
OR = .39 What is the reduction in odds? So: โTaking aspirin provides a 61% reduction in the
odds of having an MI compared to a placebo.โ
RR = .41 What is the reduction in risk? So: โTaking aspirin provides a 59% reduction in risk of
MI compared to a placebo.โ
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 โ 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 ExampleโฆRecall the MI & Aspirin
study
What is the ARR of aspirin vs. the placebo?
Had MI No MI
Aspirin 50 1030
Placebo 200 1570
๐ ๐๐ ๐๐๐ ๐๐ผ ๐๐๐ ๐ด๐ ๐๐๐๐๐=ยฟ h๐ค๐๐ก ๐๐ผ
ยฟ๐๐๐ด๐ ๐๐๐๐๐=501080=0.048๐๐ 4.8%
๐ ๐๐ ๐๐๐ ๐๐ผ ๐๐๐ ๐๐๐๐๐๐๐=ยฟ h๐ค๐๐ก ๐๐ผ
ยฟ ๐๐๐๐๐๐๐๐๐=2001770=0.11๐๐ 11%
๐ด๐ ๐ ๐๐๐ ๐ด๐ ๐๐๐๐๐=๐ ๐๐ ๐ ๐๐๐ ๐๐๐๐๐๐๐โ๐ ๐๐ ๐ ๐๐๐ ๐ด๐ ๐๐๐๐๐=11% โ4.8%=6.2% ๐ด๐ ๐
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 Exampleโฆa classic example is a 1980s controversy between Aspirin and Ryeโs syndrome.
What is the AR for Ryeโs in children exposed to aspirin vs. not exposed?
Ryes (+) Ryes (โ)
Aspirin (+) 600 9030
Aspirin (โ) 150 10000
๐ ๐๐ ๐๐๐ ๐ ๐ฆ๐๐ ๐๐๐ ๐ด๐ ๐๐๐๐๐=ยฟ h๐ค๐๐ก ๐ ๐ฆ๐๐
๐๐๐ก๐๐ ๐ธ๐ฅ๐๐๐ ๐๐=6009630=0.62๐๐ 6.2%
๐ ๐๐ ๐๐๐ ๐ ๐ฆ๐๐ ๐๐๐ ๐๐๐๐๐๐๐=ยฟ h๐ค๐๐ก ๐ ๐ฆ๐๐
๐๐๐ก๐๐ ๐๐๐ก ๐ธ๐ฅ๐๐๐ ๐๐=1501150=0.015๐๐ 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.
๐ด๐ ๐ ๐๐๐ ๐ด๐ ๐๐๐๐๐=๐ ๐๐ ๐ ๐๐๐ ๐๐๐๐๐๐๐โ๐ ๐๐ ๐ ๐๐๐ ๐ด๐ ๐๐๐๐๐=11% โ4.8%=6.2% ๐ด๐ ๐
๐๐๐=1๐ด๐ ๐
๐๐๐ป=1๐ด๐
In our aspirin and MI example (Example 1), the ARR for aspirin = 6.2%
and the AR = 4.7 ๐๐๐=
1๐ด๐ ๐ =
1.062=16๐๐๐๐๐๐๐ค๐๐ข๐๐๐๐๐๐๐ก๐๐๐ ๐ก๐๐๐๐ก๐๐
people would need to be exposed
WHAT IS STATISTICAL INFERENCE?Causation, hypothesis testing & what it means to be โstatistically significantโ