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<!DOCTYPE html>
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<head>
<title>Regression Modelling in R</title>
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<meta name="author" content="Chris Mainey - [email protected]" />
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</div>
# Regression Modelling in `R`
----
## **A bit of theory and application**
### Chris Mainey - [email protected]
### <svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:#005EB8;" xmlns="http://www.w3.org/2000/svg"> <path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"></path></svg>@chrismainey
#### 24/11/2022
---
# Workshop Overview
- Correlation
- Linear Regression
- Specifying a model with `lm`
- Interpreting the model output
- Assessing model fit
- Multiple Regression
- Prediction
- Generalized Linear Models using `glm`
- Logistic Regression
<br>
___Mixture of theory, examples and practical exercises___
---
# Relationships between variables
If two variables are related, we usually describe them as 'correlated'.
<br>
Usually interested in "strength" and "direction" of association
--
<br><br>
Two analysis techniques commonly used to investigate:
+ ___Correlation:___ shows direction, and strength of association
+ ___Regression:___ estimate how one (or more) variable(s) change in relation to each other. Usually:
+ `\(y\)` (the variable we're interested in) is the "dependent variable" or "outcome"
+ `\(x\)` (or more than one, `\(x_{i}\)`) as the "independent variables" or "predictors"
--
<br>
Sometimes the effects of other variables interact/mask this (___"confounding"___)
---
## Example:
<img src="Regression_modelling_files/figure-html/lm1-1.png" style="display: block; margin: auto;" />
---
# Correlation
+ Measured with a correlation coefficient ('Pearson' is the most common)
+ Range:
+ __-1 to 1:__ Perfect negative to Perfect positive Correlation
+ __0:__ No Correlation
--
<center>
<img src="https://upload.wikimedia.org/wikipedia/commons/d/d4/Correlation_examples2.svg" height="300" class="center">
</center>
.footnote[
Graphic from:
Wikipedia: [Correlation and dependence:](https://en.wikipedia.org/wiki/Correlation_and_dependence)
By DenisBoigelot, https://commons.wikimedia.org/w/index.php?curid=15165296 [Accessed 24 Sept 2019]
]
---
# Correlation in R
Lets check the correlation in our generated data:
```r
cor(x, y)
## [1] 0.8650106
cor.test(x,y)
##
## Pearson's product-moment correlation
##
## data: x and y
## t = 11.944, df = 48, p-value = 5.53e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.7727115 0.9214882
## sample estimates:
## cor
## 0.8650106
```
+ `cor.test` is a correlation and a t-test.
+ Different types of correlation coefficient, default is 'Pearson'
+ Doesn't work for different distributions, data types or more variables
---
### Regression models (1)
Regression gives us more options than correlation:
<img src="Regression_modelling_files/figure-html/lm3-1.png" width="1770" style="display: block; margin: auto;" />
`$$y= \alpha + \beta x + \epsilon$$`
---
### Regression models (2)
Zooming in...
<img src="Regression_modelling_files/figure-html/lm35-1.png" width="1770" style="display: block; margin: auto;" />
---
## Regression equation
`$$\large{y= \alpha + \beta_{i} x_{i} + \epsilon}$$`
<br>
.pull-left[
+ `\(y\)` - is our 'outcome', or 'dependent' variable
+ `\(\alpha\)` - is the 'intercept', the point where our line crosses y-axis
+ `\(\beta\)` - is a coefficient (weight) applied to `\(x\)`
+ `\(x\)` - is our 'predictor', or 'independent' variable
+ `\(i\)` - is our index, we can have `\(i\)` predictor variables, each with a coefficient
+ `\(\epsilon\)` - is the remaining ('residual') error
]
--
.pull-right[
We are making some assumptions:
+ Linear relations
+ Data points are independent (not correlated)
+ Normally distributed error
+ Homoskedastic (error doesn't vary across the range)
]
---
### Ordinary Least Squares 'OLS'
+ 'Residual' distance between prediction and data point ( `\(\epsilon\)` ).
--
+ Sum would be zero, so we 'square' (^2) it, and minimise the _'sum of the squares'_
--
<img src="Regression_modelling_files/figure-html/lm2-1.png" style="display: block; margin: auto;" />
---
## Regression models (3)
So now let's create a linear regression model. I prefer to create them as objects so I can use them again later. Let's call this one _model1_. That's fairly bad naming, but oh well....
<br><br><br>
Let's say we have a data.frame called _mydata_ and columns called _Y_ (that we are predicting) using a column called _X_
```r
model1 <- lm(Y ~ X, data = mydata)
```
<br><br>
We can then use other methods on this object, like `print()`, `summary()`, `plot()` and `predict()`.
<br><br>
The next two slides show the output of the summary function and plot.
---
## `lm` summary
```
##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9575 -2.2614 0.4444 2.4475 4.1663
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.4776 1.1386 4.811 1.53e-05 ***
## x 1.2507 0.1047 11.944 5.53e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.77 on 48 degrees of freedom
## Multiple R-squared: 0.7482, Adjusted R-squared: 0.743
## F-statistic: 142.7 on 1 and 48 DF, p-value: 5.53e-16
```
--
+ We can test fit using f-tests, prediction error, or the R<sup>2</sup> (the proportion of variation in `\(y\)`, explained by `\(x\)`).
---
## Interpretation (1)
So how do we interpret the output?
+ The intercept `\((\alpha)\)` = 5.48
+ The coefficient `\((\beta)\)` for `\(x\)` = 1.25
___"For each increase of 1 in `\(x\)`, `\(y\)` increases by 1.25, starting at 5.48."___
--
<br><br>
A common addition is to __"mean-centre and scale"__ our variables. So `\(x\)` becomes:
$$ \frac{(x - \bar{x})}{\sigma_x} $$
```r
model1_scaled <- lm(Y ~ scale(X), data = mydata)
```
---
## Interpretation (2)
.panelset[
.panel[.panel-name[Original]
```
##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9575 -2.2614 0.4444 2.4475 4.1663
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.4776 1.1386 4.811 1.53e-05 ***
## x 1.2507 0.1047 11.944 5.53e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.77 on 48 degrees of freedom
## Multiple R-squared: 0.7482, Adjusted R-squared: 0.743
## F-statistic: 142.7 on 1 and 48 DF, p-value: 5.53e-16
```
]
.panel[.panel-name[Scaled]
```
##
## Call:
## lm(formula = y ~ scale(x))
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9575 -2.2614 0.4444 2.4475 4.1663
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.2469 0.3917 46.59 < 2e-16 ***
## scale(x) 4.7256 0.3956 11.94 5.53e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.77 on 48 degrees of freedom
## Multiple R-squared: 0.7482, Adjusted R-squared: 0.743
## F-statistic: 142.7 on 1 and 48 DF, p-value: 5.53e-16
```
]
.panel[.panel-name[Interpretation]
<br><br>
This converts our interpretation:
+ The intercept becomes the average `\(y\)` value
+ `\(\beta\)` becomes the change in `\(y\)` for one standard deviation increase in `\(x\)`
<br>
So...
+ Average `\(x\)` is 18.24.69
+ Each increase of 1 standard deviation in `\(x\)` increase `\(y\)` by 4.7256
]
]
---
## Regression diagnostics (1)
A common check is to plot residuals:
<!-- -->
---
## Regression diagnostics (2)
A common check is to plot residuals:
<!-- -->
---
class: center, middle
## Exercise 1: Linear regression with a single predictor
---
# More than one predictor?
Our plots in earlier slides make sense in 2 dimensions, but regression is not limited to this.
--
<br><br>
If we add more predictors, our interpretation of each coefficient becomes: <br>
+ ___"The change in `\(y\)` whilst holding all others parameters constant"___
<br><br>
We can add more predictors with the `+`:
```r
lm(y ~ x1 + x2 + x3 + xi)
```
---
## Categorical variables
How do we enter categorical variables into a model?
+ Models won't understand text, and numbers are numeric, so we use `factor` variables?
--
<br>
`Factors` are 'dummy coded:'
+ _'pivotted' to binary columns_
+ Contain a reference level: with categories: "A", "B" & "C", we get:
```
## (Intercept) CategoryB CategoryC
## 1 1 0 0
## 2 1 1 0
## 3 1 0 1
## 4 1 0 1
## 5 1 0 0
## 6 1 1 0
## attr(,"assign")
## [1] 0 1 1
## attr(,"contrasts")
## attr(,"contrasts")$Category
## [1] "contr.treatment"
```
---
class: middle
## Exercise 2: Linear regression with multiple predictors
---
class: middle center
<div class="figure">
<img src="https://imgs.xkcd.com/comics/linear_regression.png" alt="https://xkcd.com/1725/" width="70%" />
<p class="caption">https://xkcd.com/1725/</p>
</div>
---
## What about non-linear data?
- Data are not necessarily linear. Death is binary, LOS is a count etc.
- We can use the __Generalized Linear Model (GLM)__:
`$$\large{g(\mu)= \alpha + \beta x}$$`
<br>
Where `\(\mu\)` is the _expectation_ of `\(Y\)`, and `\(g\)` is the link function
--
+ We assume a distribution from the [Exponential family](https://en.wikipedia.org/wiki/Exponential_family):
+ Binomial for binary, TRUE/FALSE, PASS/FAIL
+ Poisson for counts
--
+ The link function transforms the data before fitting a model
--
+ Can't use OLS for this, so we use 'maximum-likelihood' estimation, which is not exact.
--
+ Many of the methods, and `R` function, for `lm` are common to `glm`, but we can't use R<sup>2</sup>. Other measures include AUC ('C-statistic'), and AIC or likelihood ratio tests.
---
## Generalized Linear Models
Let's model the probably of death in a data set from US Medicaid.
+ The data are in the `COUNT` package, and are called `medpar`
+ Load the library and use the `data()` function to load it.
--
+ We'll use a `glm`, with a `binomial` distribution.
+ The `binomial` family automatically uses the `logit` link function: the log-odds of the event.
```r
library(COUNT)
data(medpar)
glm_binomial <- glm(died ~ factor(age80) + los + factor(type), data=medpar, family="binomial")
ModelMetrics::auc(glm_binomial)
## [1] 0.6372224
```
---
```r
summary(glm_binomial)
##
## Call:
## glm(formula = died ~ factor(age80) + los + factor(type), family = "binomial",
## data = medpar)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.590949 0.097351 -6.070 1.28e-09 ***
## factor(age80)1 0.656493 0.129180 5.082 3.73e-07 ***
## los -0.037483 0.007871 -4.762 1.92e-06 ***
## factor(type)2 0.418704 0.144611 2.895 0.00379 **
## factor(type)3 0.961028 0.230489 4.170 3.05e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1922.9 on 1494 degrees of freedom
## Residual deviance: 1857.8 on 1490 degrees of freedom
## AIC: 1867.8
##
## Number of Fisher Scoring iterations: 4
```
---
## Interactions
+ 'Interactions' are where predictor variables affect each other.
+ Allows us to separate effects into
+ Can add using `*` or `:` (check help for which to use)
--
```r
glm_binomial2 <- glm(died ~ factor(age80) * los + factor(type), data=medpar, family="binomial")
ModelMetrics::auc(glm_binomial2)
## [1] 0.6376572
```
---
```r
summary(glm_binomial2)
##
## Call:
## glm(formula = died ~ factor(age80) * los + factor(type), family = "binomial",
## data = medpar)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.561604 0.104479 -5.375 7.65e-08 ***
## factor(age80)1 0.525379 0.207818 2.528 0.01147 *
## los -0.040738 0.008995 -4.529 5.93e-06 ***
## factor(type)2 0.417439 0.144681 2.885 0.00391 **
## factor(type)3 0.964771 0.231118 4.174 2.99e-05 ***
## factor(age80)1:los 0.014507 0.017954 0.808 0.41909
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1922.9 on 1494 degrees of freedom
## Residual deviance: 1857.2 on 1489 degrees of freedom
## AIC: 1869.2
##
## Number of Fisher Scoring iterations: 4
```
---
# Interpretation
+ Our model coefficients in `lm` were straight-forward multipliers
+ `glm` is similar, but it is on the scale of the ___link-function___.
+ log scale for `poisson` models, or logit (log odds) scale for `binomial`
+ It is common to transform outputs back to the original ('response') scale.
+ This gives Incident Rate Ratios for `poisson`, or Odds Ratios for `binomial`.
```r
cbind(Link=coef(glm_binomial2), Response=exp(coef(glm_binomial2)))
```
```
## Link Response
## (Intercept) -0.56160376 0.5702937
## factor(age80)1 0.52537865 1.6910991
## los -0.04073769 0.9600809
## factor(type)2 0.41743940 1.5180694
## factor(type)3 0.96477095 2.6241865
## factor(age80)1:los 0.01450661 1.0146123
```
---
# Odds-what-now?
Odds is a concept commonly used in statistics, but often misunderstood. Lets' consider a '2 x 2 table':
<img src="./man/figures/2_by_2.png" width="60%" style="display: block; margin: auto;" />
.pull-left[
__Relative risk:__
+ _a / (a + b)_
+ _d / (c + d)_
]
.pull-right[
__Odds:__
+ _a / b_
+ _c / d_
]
---
# Odds Ratio
.pull-left[
<img src="./man/figures/2_by_2_pt2.png" width="100%" style="display: block; margin: auto;" />
]
.pull-right[
__Odds Ratio:__
+ _<span style="color: red;">(a / b)</span> / <span style="color: blue;">( c / d)</span>_
+ _= <span style="color: red;">a</span><span style="color: blue;">d</span> / <span style="color: blue;">c</span><span style="color: red;">b</span>_
<br><br>
+ If odds ratio = 1, chance of outcome the same in each group
+ If odds ratio >1 - greater chance of outcome in exposure group
+ If odds ratio <1 - lesser chance of outcome in exposure group
]
<br><br>
Great explainer: https://www.youtube.com/watch?v=ixKhS0Silb4
---
class: middle
## Exercise 3: Generalized Linear Model (GLM)
---
## Prediction (1)
- We can then use our model to predict our expected `\(Y\)`:
- Need to decide what scale to predict on: `link` or `response`
```r
library(dplyr)
medpar$preds <- predict(glm_binomial2, type="response")
top_n(medpar,5) %>% knitr::kable(format = "html")
```
<table>
<thead>
<tr>
<th style="text-align:left;"> </th>
<th style="text-align:right;"> los </th>
<th style="text-align:right;"> hmo </th>
<th style="text-align:right;"> white </th>
<th style="text-align:right;"> died </th>
<th style="text-align:right;"> age80 </th>
<th style="text-align:right;"> type </th>
<th style="text-align:right;"> type1 </th>
<th style="text-align:right;"> type2 </th>
<th style="text-align:right;"> type3 </th>
<th style="text-align:left;"> provnum </th>
<th style="text-align:right;"> preds </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"> 558 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030017 </td>
<td style="text-align:right;"> 0.7114250 </td>
</tr>
<tr>
<td style="text-align:left;"> 919 </td>
<td style="text-align:right;"> 5 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030061 </td>
<td style="text-align:right;"> 0.6894160 </td>
</tr>
<tr>
<td style="text-align:left;"> 1464 </td>
<td style="text-align:right;"> 2 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032000 </td>
<td style="text-align:right;"> 0.7060100 </td>
</tr>
<tr>
<td style="text-align:left;"> 1486 </td>
<td style="text-align:right;"> 5 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032002 </td>
<td style="text-align:right;"> 0.6894160 </td>
</tr>
<tr>
<td style="text-align:left;"> 1488 </td>
<td style="text-align:right;"> 4 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032002 </td>
<td style="text-align:right;"> 0.6950045 </td>
</tr>
</tbody>
</table>
---
## Prediction (2)
- Lets see the 10 cases with the highest predicted risk of death:
```r
medpar %>% arrange(desc(preds)) %>% top_n(10) %>% knitr::kable(format = "html")
```
<table>
<thead>
<tr>
<th style="text-align:left;"> </th>
<th style="text-align:right;"> los </th>
<th style="text-align:right;"> hmo </th>
<th style="text-align:right;"> white </th>
<th style="text-align:right;"> died </th>
<th style="text-align:right;"> age80 </th>
<th style="text-align:right;"> type </th>
<th style="text-align:right;"> type1 </th>
<th style="text-align:right;"> type2 </th>
<th style="text-align:right;"> type3 </th>
<th style="text-align:left;"> provnum </th>
<th style="text-align:right;"> preds </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"> 558 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030017 </td>
<td style="text-align:right;"> 0.7114250 </td>
</tr>
<tr>
<td style="text-align:left;"> 1464 </td>
<td style="text-align:right;"> 2 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032000 </td>
<td style="text-align:right;"> 0.7060100 </td>
</tr>
<tr>
<td style="text-align:left;"> 1488 </td>
<td style="text-align:right;"> 4 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032002 </td>
<td style="text-align:right;"> 0.6950045 </td>
</tr>
<tr>
<td style="text-align:left;"> 919 </td>
<td style="text-align:right;"> 5 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030061 </td>
<td style="text-align:right;"> 0.6894160 </td>
</tr>
<tr>
<td style="text-align:left;"> 1486 </td>
<td style="text-align:right;"> 5 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032002 </td>
<td style="text-align:right;"> 0.6894160 </td>
</tr>
<tr>
<td style="text-align:left;"> 955 </td>
<td style="text-align:right;"> 6 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030061 </td>
<td style="text-align:right;"> 0.6837716 </td>
</tr>
<tr>
<td style="text-align:left;"> 896 </td>
<td style="text-align:right;"> 9 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030061 </td>
<td style="text-align:right;"> 0.6665153 </td>
</tr>
<tr>
<td style="text-align:left;"> 941 </td>
<td style="text-align:right;"> 9 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030061 </td>
<td style="text-align:right;"> 0.6665153 </td>
</tr>
<tr>
<td style="text-align:left;"> 1084 </td>
<td style="text-align:right;"> 10 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 030069 </td>
<td style="text-align:right;"> 0.6606596 </td>
</tr>
<tr>
<td style="text-align:left;"> 1482 </td>
<td style="text-align:right;"> 11 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:right;"> 3 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 0 </td>
<td style="text-align:right;"> 1 </td>
<td style="text-align:left;"> 032000 </td>
<td style="text-align:right;"> 0.6547543 </td>
</tr>
</tbody>
</table>
---
class: middle
# Exercise 4: Predicting from models
---
# Summary
- Correlation shows the direction and strength of association
- Regression allows us to quantify the relationships
--
- We can use a single, or multiple, predictors
- Regression coefficients explain how much a change in `\(x\)` affects `\(y\)`
--
- R<sup>2</sup> is a common measure of in linear models, C-statistic/AUC/ROC in logistic models
--
- Generalized Linear Model (`glm`) allow linear models on a transformed scale, e.g. logistic regression for binary variables
- Interactions terms allow us to examine confounded predictors
--
- Consider back-transforming GLM coefficients for interpretability (e.g. odds ratios)
--
- We can predict from our model objects, but must remember the link-function scale in `glm`
---
class: middle
# Exercise 5: Predicting 10-year CHD risk in Framingham data
---
class: middle center
<div class="figure">
<img src="https://imgs.xkcd.com/comics/correlation.png" alt="https://xkcd.com/552/" width="70%" />
<p class="caption">https://xkcd.com/552/</p>
</div>
</textarea>
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