COMMON MEASURES OF ASSOCIATION IN MEDICAL AND EPIDEMIOLOGIC RESEARCH: ODDS, RISK, & THE 2X2 TABLEPatrick Barlow

PhD. Student in Evaluation, Statistics, & Measurement

The University of Tennessee

ON THE AGENDA

What are odds/risks?The 2x2 table explainedCalculating measures of associationOdds RatioRisk Ratio

Interpreting measures of associationMagnitude of the relationshipAccuracy of the inferenceThe P-value fallacy

SOME TERMS

2x2 table Proportion Odds Risk Odds Ratio (OR) Relative Risk Ratio (RR)

WHAT IS PROBABILITY?The probability of a favorable event is the fraction of times you expect to see that event in many trials. In epidemiology, a “risk” is considered a probability.

For example…

You record 25 heads on 50 flips of a coin, what is the probability of a heads?

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐻𝑒𝑎𝑑𝑠=¿𝐻𝑒𝑎𝑑𝑠¿𝑇𝑟𝑖𝑎𝑙𝑠

=2550

=.50 𝑜𝑟 50%

Remember: a probability should never exceed 1.0 or 100%.

WHAT ARE ODDS?

An “odds” is a probability of a favorable event occurring vs. not occurring.

In clinical and epidemiologic research, we use a ratio of two odds, or Odds Ratio (OR) and Relative Risk Ratio (RR), to express the strength of relationship between two variables.

For example…

What are the odds you will get a heads when flipping a fair coin?

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

1

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?

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

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? Risk of MI for aspirin = Number with MI / Number on

Aspirin = 50 / 1080 = .048 or 4.8% What is the risk of MI if a patient is taking placebo?

Risk of MI for placebo = Number with MI / Number on placebo = 200 / 1770 = .11 or 11%

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

RELATIVE RISK ON A 2X2 TABLE

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

patient is taking aspirin versus a placebo? RR = (A / A+B) / (C / C+D) RR = Risk of MI for Aspirin / Risk of MI for Placebo RR = .048 / .11 = .41 or 41%

Had MI No MI

Aspirin 50 1030

Placebo 200 1570

YOUR TURN

Work in pairs to calculate the RRs for each of the 2x2 tables below.

1 PE No PE

DVT 79 157

No DVT 100 375

3 Lung Cancer

No Lung Cancer

Smoking Hx 190 450

No Smoking Hx 70 700

2Glucose

Tolerance Improved

Tolerance not

Improved

Lap Band 35 170

Gastric Bypass 52 160

4 DM Type II No DM Type II

BMI < 30 25 350

BMI > 30 65 200

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

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

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

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

YOUR TURN

Work in pairs to calculate the RRs for each of the 2x2 tables below.

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

2Glucose

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

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

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

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

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

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, but you could also still use (OR – 1)%. 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.

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)

THE P-VALUE FALLACY

What is a p-value? The probability that the observed statistics would

occur due to chance. Alpha, usually set to .05 Values below .05 indicate a statistically

significant relationship exists. What influences p-values?

Sample size Chance Effect size Statistical power

Is a p-value of .001 a more significant relationship than a value of .03?

GOING BEYOND THE P-VALUE

The OR/RR provides a far more vivid description of the magnitude of the relationship. Can you say an OR of 4.30 is stronger than an

OR of 1.50? What about RR = .25 vs. RR = .56?

The 95% CI provides far more information on the accuracy of the inference. Which is more accurate?

OR = 2.5 (95% CI = 1.2 – 10.0) vs. OR = 2.5 (95% CI = 1.2 – 3.1)