In this lecture we study relationship between a binary variable, such as having a disease vs. being healthy, and one or more predictors, such as risk factors of a disease. We extend the linear regression model to binary outcomes by a modeling approach called **logistic regression**.

When the outcome \(Y\) of an experiment/observational study is binary (0 or 1), we typically want to model how the probability \(p_i\), that the outcome is 1 for individual \(i\), depends on the available knowledge of the values of predictor variables \(X_1,\ldots,X_k\) for individual \(i\). We call this probability “risk” although in general it can be any probability, not necessarily related to a disease risk.

Intuitively, extending our experience from the linear regression model, we would like to have a model like \[p_i = a + b\, x_i + \varepsilon_i,\] from which we could estimate how a change in the predictor value \(x_i\) changes the risk \(p_i\). This has two problems. First, we can’t measure \(p_i\) but we only observe binary outcome \(y_i\) which is either 0 or 1 (e.g. for patients we set \(y_i=1\) and for healthy \(y_i=0\)). Second, the risk value \(p_i\) is a probability and hence always between 0 and 1, whereas the linear model above doesn’t impose such a restriction to the right hand side of the equation, which could result in inacceptable risk values from the model.

In this lecture, we leave the first problem for R to handle. The second problem is solved by transforming the outcome value \(p_i\) from the interval (0,1) to the whole real line by the **log-odds transformation**, where \(p_i\) is replaced by \[ \log\left(\frac{p_i}{1-p_i}\right).\]

```
par(mfrow = c(1,2))
p = seq(0.001, 0.999, 0.001)
y = log( p/(1-p) )
plot(p, y, xlab = "risk p", ylab = "log-odds: log( p/(1-p) )", t = "l", lwd = 1.5)
plot(y, p, xlab = "log-odds: log( p/(1-p) )", ylab = "risk p", t = "l", lwd = 1.5)
```

When \(p_i\) is the probability that an event of interest happens, then \(1-p_i\) is the probability that the event does not happen. The **odds** of the event are \(p_i / (1-p_i)\) and they tell how many times more likely the event is to happen than not to happen. For example, if risk is 50%, then odds are 1:1 = 1 and if risk is 2/3 then odds are 2:1 = 2. Large odds (and also large log-odds) correspond to very probable events and small odds (and also small log-odds) correspond to very unlikely events. Mathematically, the inverse of log-odds transformation \[ \ell = \log\left( \frac{p}{1-p}\right) \qquad \textrm{ is } \qquad p = \frac{\exp(\ell)}{1+\exp(\ell)}.\]

By taking the logarithm of odds, we have a quantity that can take any real number as its value. Hence, it is conceptually valid to use a regression model where \[ \log\left(\frac{p_i}{1-p_i}\right) = a + b\,x_i. \] If \(b\) in this model is positive, then, as \(x\) increases, also the odds of the event increases and therefore also the probability of the event increases. If \(b\) is negative, then as \(x\) increases, also the odds of the event and hence also the probability of the event decreases. With the logistic regression model, for each pair of estimates of \(a\) and \(b\), we can do a prediction of risk \(p_i\) for individual \(i\) given his/her value for predictor \(x_i\).

- Risk of a disease is 10%. What are the odds of the disease? And log-odds of the disease? How does odds and log-odds change when risk changes from 10% to (100% - 10%) = 90%?

Odds are \(p/(1-p) = 0.1/(1-0.1) = 0.1/0.9 = 1/9 \approx 0.111\). Log-odds are \(\log(1/9) = -2.197.\) If risk changes to 90%, then odds become \(9/1 = 9\) and log-odds become \(\log(9) = -\log(1/9) = 2.197\). Thus odds change to their inverse value, and log-odds changes the sign.

- Suppose that log-odds of a disease follows a linear model \(a + b\, x\), where \(x\) is measured BMI and parameter values are \(a=-1\) and \(b=0.01\). Is increasing BMI a risk or protective factor for the disease? What is the risk of disease for an individual with bmi = 30? What about bmi = 40?

Since larger log-odds mean larger odds mean larger risk, a positive value of \(b\) means that increasing BMI increases the risk.

```
a = -1
b = 0.01
x = c(30, 40)
logodds = a + b*x
p = exp(logodds)/(1 + exp(logodds))
data.frame(bmi = x, risk = p)
```

```
## bmi risk
## 1 30 0.3318122
## 2 40 0.3543437
```

Let’s use a dataset `Pima.tr`

on 200 Pima indian women from `MASS`

package.

```
library(MASS)
y = Pima.tr
str(y)
```

```
## 'data.frame': 200 obs. of 8 variables:
## $ npreg: int 5 7 5 0 0 5 3 1 3 2 ...
## $ glu : int 86 195 77 165 107 97 83 193 142 128 ...
## $ bp : int 68 70 82 76 60 76 58 50 80 78 ...
## $ skin : int 28 33 41 43 25 27 31 16 15 37 ...
## $ bmi : num 30.2 25.1 35.8 47.9 26.4 35.6 34.3 25.9 32.4 43.3 ...
## $ ped : num 0.364 0.163 0.156 0.259 0.133 ...
## $ age : int 24 55 35 26 23 52 25 24 63 31 ...
## $ type : Factor w/ 2 levels "No","Yes": 1 2 1 1 1 2 1 1 1 2 ...
```

This data frame contains the following columns:

Variable | Explanation |
---|---|

npreg | number of pregnancies |

glu | plasma glucose concentration in an oral glucose tolerance test |

bp | diastolic blood pressure (mm Hg) |

skin | triceps skin fold thickness (mm) |

bmi | body mass index |

ped | diabetes pedigree function |

age | age in years |

type | Yes or No, for diabetic according to WHO criteria |

Our outcome variable is the binary diabetes status `type`

.

`table(y$type)`

```
##
## No Yes
## 132 68
```

Let’s start by seeing how `bmi`

associates to `type`

using a boxplot and a t-test. Since `type`

is a factor (see above output from `str()`

), we can make boxplots and t-test of `bmi`

with respect to the two levels of the factor by using notation `bmi ~ type`

.

```
boxplot(bmi ~ type, data = y, xlab = "bmi", ylab = "Diabetes",
horizontal = TRUE, col=c("blue", "red")) #turns boxplot horizontal
```

`t.test(bmi ~ type, data = y)`

```
##
## Welch Two Sample t-test
##
## data: bmi by type
## t = -4.512, df = 171.46, p-value = 1.188e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -5.224615 -2.044547
## sample estimates:
## mean in group No mean in group Yes
## 31.07424 34.70882
```

There seems to be a difference of about 3.5 kg/m^2, 95%CI (2.0..5.2) between the groups.

If we are given a value of BMI, what would be a risk of diabetes (in these data)? Let’s use logistic regression. We use `glm( )`

function that fits “generalized linear models”, and we specify logistic regression by parameter `family = "binomial"`

. This means that the outcome variable in regression is modeled as having a binomial distribution, where the success probability may depend on the values of predictors. For model formula the syntax is similar to the linear model `lm()`

.

**Important**: If you forget to specify `family = "binomial"`

in `glm( )`

, then the function fits a linear model, not a logistic model!

```
glm.1 = glm(type ~ bmi, data = y, family = "binomial")
summary(glm.1) #Always check from summary that you have fitted correct model
```

```
##
## Call:
## glm(formula = type ~ bmi, family = "binomial", data = y)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5797 -0.9235 -0.6541 1.2506 1.9377
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.11156 0.92806 -4.430 9.41e-06 ***
## bmi 0.10482 0.02738 3.829 0.000129 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 256.41 on 199 degrees of freedom
## Residual deviance: 239.97 on 198 degrees of freedom
## AIC: 243.97
##
## Number of Fisher Scoring iterations: 4
```

Coefficient for `bmi`

is the increase in log-odds of diabetes per one unit increase in `bmi`

and shows a clear statistical association with P ~ 1e-4. Let’s demonstrate what this means in terms of the risk of diabetes by applying `predict( )`

function for a sequence of `bmi`

values. The default output from `predict( )`

on logistic regression object is in log-odds but we can ask the prediction directly as probabilities by specifying `type = "response"`

.

```
bmi.grid = 18:55
data.in = data.frame(bmi = bmi.grid)
y.pred = predict(glm.1, newdata = data.in, type="response") #type = "response" predicts probabilities
```

Let’s plot these predictions as a function of bmi, and let’s add to the figure also the bmi distributions of the observed cases and controls of diabetes using `stripchart()`

.

```
plot(bmi.grid, y.pred, ylim = c(-0.05,1.05), t = "l",
main = "prediction", ylab = "risk of diabetes", xlab = "bmi")
stripchart(y$bmi[y$type == "No"], at = -0.05, add = T, method = "jitter", jitter = 0.02, col = "blue")
stripchart(y$bmi[y$type == "Yes"], at = 1.05, add = T, method = "jitter", jitter = 0.02, col = "red")
```