hw 7
pbiner
Calculatingriskoddsrelativeriskandoddsratio.pptxRisk, odds, relative risk, and odds ratio in clinical studies
When analyzing whether one variable is a risk factor for the outcome (or risk) of the other variable, the variable for the risk factor is made the row variable and the variable for the risk is made the column variable in a 2-way table, as shown here:
The terms “risk” and “risk factor” are just another way to classify the row and column variables.
These terms are often used in clinical research to determine whether a behavior, a certain drug, or a genetic predisposition, etc. is a risk factor to develop a particular disease.
You already know other ways to classify row and column variables, such as predictor and outcome, or cause and effect.
Conventions for 2-way tables (or cross-tabs, contingency tables, 4-fold tables)
As a result of this convention, one only needs to analyze possible differences in the row proportions to determine whether the two variables are associated and thus, whether the row variable is indeed a risk factor for the column variable.
| Yes Risk | No Risk | Marginal row totals | |
| Yes Risk Factor | a | b | Σ R1 |
| No Risk Factor | c | d | Σ R2 |
| Marginal column totals | Σ C1 | Σ C2 | Grand total (T) |
| Row proportions For Yes Risk Factor | a/Σ R1 | b/Σ R1 | = 1 |
| Row proportions For No Risk Factor | c/Σ R2 | d/Σ R2 | = 1 |
Risk, (outcome, or effect)
Frequency
table
Risk factor, (predictor, or cause)
| Row variable (Risk factor, predictor, or cause) | Column variable (Risk, outcome, or effect) | ||
| Row 1 (“Yes” counts) | Row 2 (“No” counts) | Column 1 (“Yes” counts) | Column 2 (“No” counts) |
| Obesity (risk factor) | No obesity (No risk factor) | Hypertension (risk) | No hypertension (no risk) |
| Smoking (risk factor) | Not smoking (No risk factor) | Lung disease (risk) | No lung disease (no risk) |
| Taking a preventative medication, or supplement (no risk factor) | Not taking the medication (risk factor) | Developing a particular disease (risk) | Not developing a particular disease (no risk) |
Comments:
If we want to test whether obesity increases the risk to develop hypertension, the row with the “yes” answers for obesity is the risk factor, the column with the “yes” answers for hypertension is the risk.
If we want to test whether taking a preventive medicine reduces the risk to develop a certain disease, the row with the no answers for taking the medication is the risk factor. The column with the “yes” answers for the disease is still the risk.
One needs to use one’s common sense or expertise to decide which of the two row categories is the risk factor and which of the two column categories is the risk. Examples:
How to tell which row contains the risk factor and which column contains the risk?
Bottom line: one always has to pay attention to what question one is asking and which columns and rows contain the answers!
| Yes Risk | No Risk | Marginal row totals | |
| Yes Risk Factor | a | b | Σ R1 |
| No Risk factor | c | d | Σ R2 |
| Marginal column totals | Σ C1 | Σ C2 | Grand total (T) |
| Row proportions For Yes Risk Factor | a/Σ R1 | b/Σ R1 | = 1 |
| Row proportions For No Risk Factor | c/Σ R2 | d/Σ R2 | = 1 |
Calculating the risk (a proportion)
For risk calculations, one can ignore the column with the ”no risk” counts, shaded in grey here, as the information is redundant (as long as one knows the marginal totals)
Frequency
table
Risk factor, predictor, or cause
Risk, (outcome, or effect)
The risk of developing the disease, outcome, or effect in question is a proportion. It can also be expressed in %.
The risk (proportion) is nothing else than the respective row proportion of the “yes risk” column (see arrows above).
If you wonder why, read on. If it’s clear you can skip to the next slide.
We know that the number of subjects in the “yes risk factor” row (those who carry the risk factor) who also develop the disease is a, out of a+b (or ΣR1) total subjects that have the risk factor (that’s the study result from table).
If we assume these results apply to the population of subjects who carry the risk factor, that means that the population has a risk of a/(a+b) of developing the disease. Same reasoning applies the “No Risk Factor” population.
Risk of developing disease for people carrying the risk factor
Risk of developing disease for people NOT carrying the risk factor
| Yes Risk | No Risk | Marginal row totals | |
| Yes Risk Factor | a | b | Σ R1 |
| No Risk factor | c | d | Σ R2 |
| Marginal column totals | Σ C1 | Σ C2 | Grand total (T) |
| Row proportions For Yes Risk Factor | a/Σ R1 | b/Σ R1 | = 1 |
| Row proportions For No Risk Factor | c/Σ R2 | d/Σ R2 | = 1 |
Frequency
table
Risk factor, predictor, or cause
Risk, (outcome, or effect)
Risk of developing disease for people carrying the risk factor
Risk of developing disease for people NOT carrying the risk factor
Calculating the risk ratio (by dividing the 2 risk proportions)
Risk of developing disease for people carrying the risk factor
Risk of developing disease for people NOT carrying the risk factor
RISK RATIO =
=
a/Σ R1
c/Σ R2
Simply divide the row proportion for the “yes” risk factor by the row proportion for the “no” risk factor
| Yes Risk | No Risk | Marginal row totals | |
| Yes Risk Factor | a | b | Σ R1 |
| No Risk Factor | c | d | Σ R2 |
| Marginal column totals | Σ C1 | Σ C2 | Grand total (T) |
Risk, (outcome, or effect)
Frequency
table
Risk factor, (predictor, or cause)
| Odds to develop the disease (“risk”) for people with the risk factor (“Yes”): | a/b |
| Odds to develop the disease (“risk”) for people without the risk factor (“No”): | c/d |
Calculating odds and odds ratio
Odds and odds ratio are just another way to calculate and present the same kinds of results.
Sometimes you read about results presented in this way. So it is good to know what that means!
Definition: Odds are the probability of success divided by the probability of failure.
Or, in the terms we use here: The odds for developing a disease is equal to the number of people in a row with the “Yes risk” outcome divided by the number of people in that row with the “no risk” outcome.
So instead of dividing the number of people with the “success” or “yes risk” outcome (a) by the total number of people in the row (Σ R1), as we did for the row proportions (= the risk), we divide that number by the number of people with the “no risk” outcome, to obtain the odds.
If the risk increases to develop a disease if one carries the risk factor, then the odds also increase. They are just different numbers and you need to keep them straight.
ODDS RATIO =
a/b
c/d
Odds for people with risk factor
Odds for people without risk factor
=
Important: Pay attention to the use of language here!
Even though the odds are a ratio, we don’t call them odds ratio, just odds!
As you see here, the odds ratio is the ratio of two odds!
