MATH 4140/5140

class: center, middle, inverse, title-slide

# MATH 4140/5140
## Simple Logistic Regression
### Ziwei Ma
### UTC
### updated: 2022-04-27

---

### Logistic regression example

> **Election**  Suppose that we are interested in the group membership of a hurricane, either tropical or non-tropical, based on the latitude of formation.

```
## [1] TRUE
```
- Load data

```r
library(readxl)
hurricanes <- read_excel("hurricanes.xlsx")
```

---

### Inspect the data

- First, we inspect the structure of the data set.

```r
str(hurricanes)
```

```
## tibble [337 x 12] (S3: tbl_df/tbl/data.frame)
##  $ RowNames: chr [1:337] "1" "2" "3" "4" ...
##  $ Number  : num [1:337] 430 432 433 436 437 438 440 441 445 449 ...
##  $ Name    : chr [1:337] "NOTNAMED" "NOTNAMED" "NOTNAMED" "NOTNAMED" ...
##  $ Year    : num [1:337] 1944 1944 1944 1944 1944 ...
##  $ Type    : num [1:337] 1 0 0 0 0 1 0 1 0 0 ...
##  $ FirstLat: num [1:337] 30.2 25.6 14.2 20.8 20 29.2 16.1 27.6 21.6 19 ...
##  $ FirstLon: num [1:337] -76.1 -74.9 -65.2 -58 -84.2 -55.8 -80.8 -85.6 -95.2 -56.6 ...
##  $ MaxLat  : num [1:337] 32.1 31 16.6 26.3 20.6 38 21.9 27.6 28.6 24.9 ...
##  $ MaxLon  : num [1:337] -74.8 -78.1 -72.2 -72.3 -84.9 -53.2 -82.9 -85.6 -96.1 -79.6 ...
##  $ LastLat : num [1:337] 35.1 32.6 20.6 42.1 19.1 50 28.4 31.7 29.5 28.9 ...
##  $ LastLon : num [1:337] -69.2 -78.2 -88.5 -71.5 -93.9 -46.5 -82.1 -79.1 -96 -81.8 ...
##  $ MaxInt  : num [1:337] 80 80 105 120 70 85 105 100 120 120 ...
```

There are 337 observations and 12 variables in the data set. We are primarily interested in the variable **Type**, which is our response variable and the variable **FirstLat**, which corresponds to the latitude of formation, and thus is our predictor variable.

---

- In order to get a sense for the data set we plot the number of hurricanes for each year as a bar plot using the ggplot2 package.

```r
library(ggplot2)
ggplot(hurricanes, aes(x = Year)) + geom_bar(stat = "count")
```

---

- Types of hurricanes

```r
ggplot(hurricanes, aes(x = Year, fill = factor(Type))) +
        geom_bar(stat = "count") +
        scale_fill_discrete(name = "Type of Hurricane",
        labels = c("tropical-only", "baroclinic influences",
                   "baroclinic initiation"))
```

---

For a numerical representation of the hurricane classes we summarize the data based on types.

```r
table(hurricanes$Type)
```

```
## 
##   0   1   3 
## 187  77  73
```

In **class 0**, tropical hurricanes, there are 187 observations, in **class 1**, *baroclinic influences*, there are 77 observations and in **class 3**, *baroclinic initiation*, there are 73 observations. In logistic regression we deal with dichotomous data; thus, for simplicity we re-code the classes and assign the class 1 and class 2, both being influenced by the outer tropics, the label 1.

```r
hurricanes$Type.new <- ifelse(test = hurricanes$Type==0, 
                              yes = 0, no = 1)
```
Now we have a binary response variable of two classes: Class 0 corresponds tropical hurricanes and class 1 to non-tropical hurricanes.

```r
table(hurricanes$Type.new)
```

```
## 
##   0   1 
## 187 150
```

---

### Geographical representation

- We plot the data on an interactive map.

```r
options(viewer=NULL)
library(leaflet)
m <- leaflet()
m <- addTiles(m)
m <- addProviderTiles(m, "Esri.OceanBasemap")

cols <- c("red", "navy")
m <- addCircleMarkers(m, 
                      lng = hurricanes$FirstLon, 
                      lat = hurricanes$FirstLat,
                      radius = 2.5,
                      color = cols[hurricanes$Type.new+1],
                      popup = paste('Year:', as.character(hurricanes$Year)))
m <- addLegend(m,
               "topright", 
               colors = cols, 
               labels = c('tropical', 'non-tropical'),
               title = "Type of Hurricane",
               opacity = 1)
```

---

### Geographical representation

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---

- **Fit Logistic regression model** We want to build a model that predicts the group membership of a hurricane, either tropical (0) or non-tropical (1), based on the latitude of formation of the hurricane. The response variable is the binary variable `Type.new` and the predictor variable is `FirstLat`

```r
log.model <- glm(Type.new ~ FirstLat, data = hurricanes, 
                 family = 'binomial')

# show the theoretical model
library(equatiomatic)
equatiomatic::extract_eq(log.model)
```

$$
\log\left[ \frac { P( \operatorname{Type.new} = \operatorname{1} ) }{ 1 - P( \operatorname{Type.new} = \operatorname{1} ) } \right] = \alpha + \beta_{1}(\operatorname{FirstLat})
$$
The fitted model is here

```r
# display the actual coefficients
equatiomatic::extract_eq(log.model, use_coefs = TRUE)
```

$$
\log\left[ \frac { \widehat{P( \operatorname{Type.new} = \operatorname{1} )} }{ 1 - \widehat{P( \operatorname{Type.new} = \operatorname{1} )} } \right] = -9.08 + 0.37(\operatorname{FirstLat})
$$

---

### Interpret the model

- The baseline odds ratio `$$e^{\alpha}=e^{-9.0826335}=1.1362198×10^{-4}$$` 
  - This is a very low number! And it makes perfectly sense, since it is very unlikely to observe a non-tropical hurricane at the equator.

- The odds ratio per unit increase of predictor
`$$e^{\beta}_1=e^{0.3728295}=1.451837$$`
Thus, for every degree increase in formation latitude the odds ratio increases on average by a constant factor of 1.4518368 (or 45%).

---

### Inference on the coefficients

- we construct a 95% confidence interval for the estimated model coefficient. We exponentiate the result to return the odds-ratio, which is a quantity that is easier to interpret than the log-odds.

- log odds ratio

```r
confint.default(log.model)[2,]
```

```
##     2.5 %    97.5 % 
## 0.2954770 0.4501821
```
  - odds ratio

```r
exp(confint.default(log.model)[2,])
```

```
##    2.5 %   97.5 % 
## 1.343767 1.568598
```

Thus, we can state that we are 95% confident that for every degree increase in formation latitude the observation of a non-tropical hurricane becomes on average more likely between 34 and 57%.

---

### Prediction

- Now, we apply that model to make predictions about the chance of observing a non-tropical hurricane given the latitude of formation.
 
In order to predict the probability that a hurricane picked at random, which forms at a given latitude is of type non-tropical we apply the `predict()` function.

```r
predict(log.model, newdata = list(FirstLat = c(10, 23.5,30)), 
        type = "response")
```

```
##           1           2           3 
## 0.004705352 0.420398055 0.891121908
```
**NOTE** We should indicate `type="response"` in `predict()` to get the probability. Otherwise, we will get the following results.

```r
predict(log.model, newdata = list(FirstLat = c(10, 23.5,30)))
```

```
##          1          2          3 
## -5.3543382 -0.3211396  2.1022524
```
**Question** Can we use the later results to get the desired probabilities.

---

### Visualization

- we create a plot of predictions across a range of latitudes to visualize our results .

The model predictions show that the probability of a non-tropical hurricane is less than 10% at latitudes south of 20∘N. However, by latitude 26∘N, the probability exceeds 50% and by latitude 33∘N the probability exceeds 90%.