Logistic Regression in R

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# Logistic Regression in R
## Zahra Moslemi

Adapted from slides by Mine Dogucu and Sam Behseta
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---
* Remember from the lecture that we are fitting a regression model with a binary outcome.

* As such, the model is as follows:

`$$\begin{eqnarray*}
\log \Big(\frac{\hat{p}}{1- \hat{p}} \Big) & = & a + b_{1}x_1 + \ldots + b_{q}x_{q}
\end{eqnarray*}$$`

* The left hand side of this model is the logarithm of the odds of success.

* Thereby, the probability of success of `$\pi$` can be written as follows:

`$$\begin{eqnarray*}
\hat{p} & = & \frac{\exp(a + b_{1}x_1 + \ldots + b_{q}x_{q})}{1 + \exp(a + b_{1}x_1 + \ldots + b_{q}x_{q})}
\end{eqnarray*}$$`

* The above means once we estimate the coefficients of the model, we can estimate the probability of success of the outcome of interest.

---
* Let's revisit Alzheimer's data set, and consider the task of building a logistic regression model with diagnosis as its response variable and variables age, education, naccicv, and female as its predictors.

* Let's begin by transforming the response to a new feature with two categories: no symptoms (0) versus mild or strong
symptoms (1):

```r
alzheimer_data <- alzheimer_data %>% 
  mutate(diag = ifelse(diagnosis %in% c(1, 2), "1", "0"),
         diag = as.factor(diag))
```

* Running a logistic regression model in R is pretty straightforward. Before we do that, we should notice female is a binary variable as well. As such, we should make sure R recognizes that feature as a factor variable.

```r
alzheimer_data <- alzheimer_data %>%
  mutate(female=as.factor(female))
```
---

```r
logistic_model <-glm(diag ~ educ + age + naccicv + female, family=binomial, data=alzheimer_data)
summary(logistic_model)
```

```
## 
## Call:
## glm(formula = diag ~ educ + age + naccicv + female, family = binomial, 
##     data = alzheimer_data)
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.1637929  0.6481630  -1.796   0.0726 .  
## educ        -0.0707890  0.0128192  -5.522 3.35e-08 ***
## age          0.0437751  0.0039941  10.960  < 2e-16 ***
## naccicv     -0.0004503  0.0003726  -1.209   0.2268    
## female1     -0.8928635  0.0989969  -9.019  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 3692.7  on 2699  degrees of freedom
## Residual deviance: 3395.6  on 2695  degrees of freedom
## AIC: 3405.6
## 
## Number of Fisher Scoring iterations: 4
```

---

```r
glm(diag ~ educ + age + naccicv + female, family=binomial, data=alzheimer_data) %>%
  tbl_regression(estimate_fun = function(x) style_number(x, digits = 3), exponentiate = TRUE)
```

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      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="<strong>Characteristic</strong>"><strong>Characteristic</strong></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>OR</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>OR</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>95% CI</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span>"><strong>95% CI</strong><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="<strong>p-value</strong>"><strong>p-value</strong></th>
    </tr>
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    <tr><td headers="label" class="gt_row gt_left">educ</td>
<td headers="estimate" class="gt_row gt_center">0.932</td>
<td headers="ci" class="gt_row gt_center">0.908, 0.955</td>
<td headers="p.value" class="gt_row gt_center"><0.001</td></tr>
    <tr><td headers="label" class="gt_row gt_left">age</td>
<td headers="estimate" class="gt_row gt_center">1.045</td>
<td headers="ci" class="gt_row gt_center">1.037, 1.053</td>
<td headers="p.value" class="gt_row gt_center"><0.001</td></tr>
    <tr><td headers="label" class="gt_row gt_left">naccicv</td>
<td headers="estimate" class="gt_row gt_center">1.000</td>
<td headers="ci" class="gt_row gt_center">0.999, 1.000</td>
<td headers="p.value" class="gt_row gt_center">0.2</td></tr>
    <tr><td headers="label" class="gt_row gt_left">female</td>
<td headers="estimate" class="gt_row gt_center"></td>
<td headers="ci" class="gt_row gt_center"></td>
<td headers="p.value" class="gt_row gt_center"></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    0</td>
<td headers="estimate" class="gt_row gt_center">—</td>
<td headers="ci" class="gt_row gt_center">—</td>
<td headers="p.value" class="gt_row gt_center"></td></tr>
    <tr><td headers="label" class="gt_row gt_left">    1</td>
<td headers="estimate" class="gt_row gt_center">0.409</td>
<td headers="ci" class="gt_row gt_center">0.337, 0.497</td>
<td headers="p.value" class="gt_row gt_center"><0.001</td></tr>
  </tbody>
  
  <tfoot class="gt_footnotes">
    <tr>
      <td class="gt_footnote" colspan="4"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span> OR = Odds Ratio, CI = Confidence Interval</td>
    </tr>
  </tfoot>
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</div>

---

### Logistic Regression Model Evaluation:

To split the data into training and validation sets using the rsample package in R, you can use the initial_split() function. Here's an example of how you can split the data:

```r
library(rsample)

set.seed(0)
data_split <- initial_split(alzheimer_data, prop = 0.7)

train_data <- training(data_split)
test_data <- testing(data_split)
```

#### As we saw, next step after splitting the data into train and test would be training the model using training data:

```r
logistic_model2 <- glm(diag ~ educ + age + naccicv + female, family=binomial, data=train_data)
summary(logistic_model2)
```

```
## 
## Call:
## glm(formula = diag ~ educ + age + naccicv + female, family = binomial, 
##     data = train_data)
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -0.9952282  0.7730543  -1.287    0.198    
## educ        -0.0759332  0.0151489  -5.012 5.37e-07 ***
## age          0.0430518  0.0047803   9.006  < 2e-16 ***
## naccicv     -0.0004883  0.0004437  -1.101    0.271    
## female1     -0.9058400  0.1191447  -7.603 2.90e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2578.5  on 1888  degrees of freedom
## Residual deviance: 2371.0  on 1884  degrees of freedom
## AIC: 2381
## 
## Number of Fisher Scoring iterations: 4
```
---

```r
glm(diag ~ educ + age + naccicv + female, family=binomial, data=train_data) %>%
  tbl_regression(estimate_fun = function(x) style_number(x, digits = 3), exponentiate = TRUE)
```

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<td headers="estimate" class="gt_row gt_center">—</td>
<td headers="ci" class="gt_row gt_center">—</td>
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      <td class="gt_footnote" colspan="4"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;"><sup>1</sup></span> OR = Odds Ratio, CI = Confidence Interval</td>
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</div>
---

* Followed, by testing it via the validation set. This means to calculate the probability of success for each subject in the test set:

```r
pred_prob <- logistic_model2 %>% 
  predict(test_data,type="response")
```

* We are now ready to calculate the accuracy of our trained model. To accomplish this, we translate all probabilities of success above to 0.5 to a 1 and otherwise to a 0, followed by tracking the number of correct predictions (1's correctly predicted as 1's and 0's correctly predicted as 0's).

```r
predicted.classes <- ifelse(pred_prob > 0.5, "1", "0")
mean(predicted.classes == test_data$diag)
```

```
## [1] 0.6510481
```

* This model yields a 65% accuracy rate!