class: center, middle, inverse, title-slide # IDS 702: Module 2.1 ## Odds, odds ratios, and relative risks ### Dr. Olanrewaju Michael Akande --- ## Introduction - So far, our response variables have been continuous. -- - Sometimes, we would also like to build models for binary outcome variables. For example, + `\(Y = 1\)`: healthy, `\(Y = 0\)`: not healthy + `\(Y = 1\)`: employed, `\(Y = 0\)`: not employed + `\(Y = 1\)`: win, `\(Y = 0\)`: lose -- - Often, we want to predict or explain the binary outcome variable from several predictors. -- - Linear regression is NOT appropriate, because normality for the response variable (and errors) makes no sense in this case. -- - This brings us to .hlight[logistic regression], the most popular model for binary outcomes. -- - First let's review relative risk, odds and odds ratios. --- ## Absolute risk and relative risk - `\(Y\)`: binary response variable, `\(X\)`: binary predictor <br /> | `\(Y=1\)` | `\(Y=0\)` | :----------- | :---------: | :---------: | `\(X=1\)` | a | b | `\(X=0\)` | c | d | - .hlight[Absolute risk] of `\(Y=1\)` for level `\(X=1\)`: `\(\dfrac{a}{(a+b)}\)` - .hlight[Absolute risk] of `\(Y=1\)` for level `\(X=0\)`: `\(\dfrac{c}{(c+d)}\)` - .hlight[Relative risk (RR)]: `\(\dfrac{a/(a+b)}{c/(c+d)}\)` -- - Relative risk is a ratio of two probabilities. -- <div class="question"> Give an example of an application where you think relative risk might be useful. </div> --- ## Odds and odds ratio - `\(Y\)`: binary response variable, `\(X\)`: binary predictor <br /> | `\(Y=1\)` | `\(Y=0\)` | :----------- | :---------: | :---------: | `\(X=1\)` | a | b | `\(X=0\)` | c | d | - .hlight[Odds] of `\(Y=1\)` for level `\(X=1\)`: `\(\dfrac{a}{b}\)` - .hlight[Odds] of `\(Y=1\)` for level `\(X=0\)`: `\(\dfrac{c}{d}\)` - .hlight[Odds ratio (OR)]: `\(\dfrac{a/b}{c/d}\)` -- - Odds ratio is a ratio of two odds. -- <div class="question"> Give an example of an application where you think odds or odds ratio might be useful. </div> --- ## Probabilities and odds: motivating example - Physicians' Health Study (1989): randomized experiment with 22071 male physicians at least 40 years old. -- - Half the subjects were assigned to take aspirin every other day. -- - The other half were assigned to take a placebo pill. -- - Broad goal: determine whether aspirin decreases cardiovascular mortality. -- - Here are the number of people in each cell of the contingency table: <br /> | Heart attack | No heart attack | :------------- | :-----------: | :-------------: | Aspirin | 104 | 10933 | Placebo | 189 | 10845 | --- ## Absolute risk and relative risk for physicians health study - Physicians Health Study <br /> | Heart attack | No heart attack | :------------- | :-----------: | :-------------: | Aspirin | 104 | 10933 | Placebo | 189 | 10845 | - .block[Relative risk of a heart attack when taking aspirin versus when taking a placebo equals] `$$\textrm{RR} = \dfrac{104/(104+10933)}{189/(189+10845)} = 0.55$$` -- - .block[Odds of having a heart attack when taking aspirin over odds of a heart attach when talking a placebo (odds ratio)] `$$\textrm{OR} = \dfrac{104/10933}{189/10845} = 0.546$$` --- ## Interpreting odds ratios and relative risks <br /> | `\(Y=1\)` | `\(Y=0\)` | :----------- | :---------: | :---------: | `\(X=1\)` | a | b | `\(X=0\)` | c | d | - When the variables `\(X\)` and `\(Y\)` are independent `$$OR = 1; \ \ \ \ \ \ \ \ \ RR = 1$$` -- - When subjects with level `\(X=1\)` are more likely to have `\(Y=1\)` than subjects with level `\(X=0\)`, then `$$OR > 1; \ \ \ \ \ \ \ \ \ RR > 1$$` -- - When subjects with level `\(X=1\)` are less likely to have `\(Y=1\)` than subjects with level `\(X=0\)`, then `$$OR < 1; \ \ \ \ \ \ \ \ \ RR < 1$$` --- ## Relative risk vs. absolute risk: smoking and lung cancer - Small or large values of relative risk may or may not be significant depending on the base rate. -- - Thus, it can be more helpful or meaningful to present both the absolute risk and RR. -- - For example, + Percentage of smokers who get lung cancer: 8% (conservative estimate) + Relative risk of lung cancer for smokers: 800% + That is, getting lung cancer is not commonplace, even for smokers but, smokers’chances of getting lung cancer are much, much higher than non-smokers’ chances. + The absolute risk helps place the RR in context. --- class: center, middle # What's next? ### Move on to the readings for the next module!