Binary Logistic Regression is used to explain the relationship between the categorical dependent variable and one or more independent variables. When the dependent variable is dichotomous, we use binary logistic regression. However, by default, a binary logistic regression is almost always called logistics regression.
Overview – Binary Logistic Regression
The logistic regression model is used to model the relationship between a binary target variable and a set of independent variables. These independent variables can be either qualitative or quantitative. In logistic regression, the model predicts the logit transformation of the probability of the event. The following mathematical formula is used to generate the final output.
In the above equation, p represents the odds ratio, and the formula for the odds ratio is as given below:
Case Study – What is UCI Adult Income?
In this tutorial, we will be using Adult Income data from the UCI machine learning repository to predict the income class of an individual based upon the information provided in the data. You can download this Adult Income data from the UCI repository.
Beta coefficient in logistics regression are chosen based upon maximum likelihood estimates.
The idea here is to give you a fair idea about how a data scientist or a statistician builds a predictive model. So, we will try to demonstrate all the essential tasks which are part of model building exercise. However, for the demo purpose, we will be using only three variables from the whole dataset.
Getting the data
The adult dataset is fairly large, and to read it faster, I will be using
readr package to load the data from my local machine.
library(readr) adult <- read_csv("./static/data/adult.csv") # Checking the structure of adult data str(adult)
# Output Classes ‘tbl_df’, ‘tbl’ and 'data.frame': 48842 obs. of 15 variables: $ Age : int 25 38 28 44 18 34 29 63 24 55 ... $ Workclass : chr "Private" "Private" "Local-gov" "Private" ... $ Fnlwgt : int 226802 89814 336951 160323 103497 198693 227026 104626 369667 104996 ... $ Education : chr "11th" "HS-grad" "Assoc-acdm" "Some-college" ... $ Education-num : int 7 9 12 10 10 6 9 15 10 4 ... $ Marital-status: chr "Never-married" "Married-civ-spouse" "Married-civ-spouse" "Married-civ-spouse" ... $ Occupation : chr "Machine-op-inspct" "Farming-fishing" "Protective-serv" "Machine-op-inspct" ... $ Relationship : chr "Own-child" "Husband" "Husband" "Husband" ... $ Race : chr "Black" "White" "White" "Black" ... $ Sex : chr "Male" "Male" "Male" "Male" ... $ Capital-gain : int 0 0 0 7688 0 0 0 3103 0 0 ... $ Capital-loss : int 0 0 0 0 0 0 0 0 0 0 ... $ Hours-per-week: int 40 50 40 40 30 30 40 32 40 10 ... $ Native-country: chr "United-States" "United-States" "United-States" "United-States" ... $ Class : chr "<=50K" "<=50K" ">50K" ">50K" ...
As mentioned earlier, we will be using three variables; WorkClass, Marital-status and Age to build the model. Out of these three variables – WorkClass and Marital-status are categorical variables where as Age is a continuous variable.
# Subsetting the data and keeping the required variables adult <- adult[ ,c("Workclass", "Marital-status", "Age", "Class")] # Checking the dim dim(adult)
# Output  48842 4
The new dataset has 48842 observations and only 4 variables
We cannot use categorical variables directly in the model. So for these variables, we need to create dummy variables. A dummy variable takes the value of 0 or 1 to indicate the absence or presence of a particular level. In our example, the function will automatically create dummy variables.
Summarizing categorical variable
The best way to summarize the categorical variable is to create the frequency table, and that is what we will do using
# Generating the frequency table table(adult$Workclass)
# Output ? Federal-gov Local-gov Never-worked 2799 1432 3136 10 Private Self-emp-inc Self-emp-not-inc State-gov 33906 1695 3862 1981 Without-pay 21
The table suggests that there are some 2799 missing values in this variable, which are represented by the (?) symbol. Also, the data is not uniformly distributed. Some of the levels have very few observations and looks like we have an opportunity to combine similar looking levels.
# Combining levels adult$Workclass[adult$Workclass == "Without-pay" | adult$Workclass == "Never-worked"] <- "Unemployed" adult$Workclass[adult$Workclass == "State-gov" | adult$Workclass == "Local-gov"] <- "SL-gov" adult$Workclass[adult$Workclass == "Self-emp-inc" | adult$Workclass == "Self-emp-not-inc"] <- "Self-employed" # Checking the table again table(adult$Workclass)
# Output ? Federal-gov Private Self-employed 2799 1432 33906 5557 SL-gov Unemployed 5117 31
Let us do a similar treatment for our other categorical variable
# Generating the frequency table table(adult$Marital-status)
# Output Divorced Married-AF-spouse 6633 37 Married-civ-spouse Married-spouse-absent 22379 628 Never-married Separated 16117 1530 Widowed 1518
We can reduce the above levels to never married, married and never married.
# Combining levels adult$Marital-status[adult$Marital-status == "Married-AF-spouse" | adult$Marital-status == "Married-civ-spouse" | adult$Marital-status == "Married-spouse-absent"] <- "Married" adult$Marital-status[adult$Marital-status == "Divorced" | adult$Marital-status == "Separated" | adult$Marital-status == "Widowed"] <- "Not-Married" # Checking the table again table(adult$Marital-status)
# Output Married Never-married Not-Married 23044 16117 9681
This variable looks well-distributed then Workclass. Now, we must convert them to factor variables using as.factor() function.
# Converting to factor variables adult$Workclass <- as.factor(adult$Workclass) adult$Marital-status <- as.factor(adult$Marital-status) adult$Class <- as.factor(adult$Class)
Deleting the missing values
We will first convert all ? to NA and then use
na.omit() to keep the complete observation.
# Converting ? to NA adult[adult == "?"] <- NA # Keeping only the na.omit() function adult <- na.omit(adult)
Finally taking a look into the target variable
To save time, I will directly be going forward with the bivariate analysis. Let us see how the distribution of age looks for the two income groups.
library(ggplot2) ggplot(adult, aes(Age)) + geom_histogram(aes(fill = Class), color = "black", binwidth = 2)
Data looks much more skewed for the lower-income people as compared to the high-income group.
Building the Model
We will be splitting the data into the test and train using the
createDataPartition() function from the
caret package in R. We will train the model using the training dataset and predict the values on the test dataset. To train the logistic model, we will be using
# Loading caret library require(caret) # Splitting the data into train and test index <- createDataPartition(adult$Class, p = .70, list = FALSE) train <- adult[index, ] test <- adult[-index, ] # Training the model logistic_model <- glm(Class ~ ., family = binomial(), train) # Checking the model summary(logistic_model)
# Output Call: glm(formula = Class ~ ., family = binomial(), data = train) Deviance Residuals: Min 1Q Median 3Q Max -1.6509 -0.8889 -0.3380 -0.2629 2.5834 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.591532 0.094875 -6.235 0.00000000045227 *** WorkclassPrivate -0.717277 0.077598 -9.244 < 0.0000000000000002 *** WorkclassSelf-employed -0.575340 0.084055 -6.845 0.00000000000766 *** WorkclassSL-gov -0.445104 0.086089 -5.170 0.00000023374732 *** WorkclassUnemployed -2.494210 0.766488 -3.254 0.00114 ** `Marital-status`Never-married -2.435902 0.051187 -47.589 < 0.0000000000000002 *** `Marital-status`Not-Married -2.079032 0.045996 -45.200 < 0.0000000000000002 *** Age 0.023362 0.001263 18.503 < 0.0000000000000002 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 36113 on 32230 degrees of freedom Residual deviance: 28842 on 32223 degrees of freedom AIC: 28858 Number of Fisher Scoring iterations: 5
Interpreting Logistic Regression Output
All the variables in the above output have turned out to be significant(p values are less than 0.05 for all the variables). If you look at the categorical variables, you will notice that n – 1 dummy variables are created for these variables. Here, n represents the total number of levels. The one level which is left is considered as the reference variable, and all other variable levels are interpreted in reference to this level.
1. Null and Residual deviance – Null deviance suggests the response by the model if we only consider the intercept; lower the value better is the model. The Residual deviance indicates the response by the model when all the variables are included; again, lower the value, better is the model.
2. (Intercept) – Intercept(β0) indicates the log of odds of the whole population of interest to be on higher-income class with no predictor variables in the model. We can convert the log of odds back to simple probabilities by using sigmoid function.
Sigmoid function, p = exp(-0.591532)/(1+exp(-0.591532))
The other way is to convert this logit of odds to simple odds by taking exp(-0.591532) = 0.5534. The number indicates that the odds of an individual being in the high-income group decreases by 45% if we have no predictor variables.
3. WorkclassPrivate – The beta coefficient against this variable is -0.717277. Let us convert this value into odds by taking the exp(-0.717277) = 0.4880795. The value indicates that the odds of an individual with Private work-class being in the high-income group decreases by 52% than the one in a Federal-gov job.
Out of 5 levels, the Federal-gov level became the reference, and thus all other levels of workclass variables are inferred in comparison to the referenced variable. That is how we interpret the categorical variables.
4. Age – The beta coefficient of the age variable is 0.023362, which is in the logit of odds terms. When we convert this to odds by taking exp(0.023362) we get 1.023. The value indicates that as age increase by one more unit, then the odds of an individual being in the high-income group will increase by 2%.
Odds value is never negative, and the value of 1 indicates that this variable has no impact on the target variables. If the value is less than one then the value is read as (1 – value) as a decrease in odds and a value greater than one indicates an increase in the odds.
Predicting Dependent Variable(Y) in Test Dataset
To predict the target variable in the unseen data, we use
predict function. The output of the predict function is the probability.
# Predicting in the test dataset pred_prob <- predict(logistic_model, test, type = "response")
Evaluating Logistic Regression Model
There are number of ways in which we can validate our logistic regression model. We have picked all the popular once which you can use to evaluate the model. Let’s discuss and see how to run those in R.
1. Classification Table – I would say this one is the most popular validation technique among all the known validation methods of the logistic model. It’s basically a contingency table that we draw between the actual values and the predicted values. The table is then used to dig in many other estimates like Accuracy, Misclassification Rate, True Positive Rate, also known as recall, True Negative Rate, and Precision.
Here is the representation of the contingency table marking essential terms.
Before we create a contingency table, we need to convert the probability into the two levels IE class <=50K and >50K. To get these values, we will be using a simple
ifelse() function and will create a new variable in the train data by the name pred_class.
We have to repeat the below steps for both the test and train dataset.
Converting probability to class values in the training dataset
# Converting from probability to actual output train$pred_class <- ifelse(logistic_model$fitted.values >= 0.5, ">50K", "<=50K") # Generating the classification table ctab_train <- table(train$Class, train$pred_class) ctab_train
# Output <=50K >50K <=50K 1844 22391 >50K 1697 6299
Training dataset converting from probability to class values
# Converting from probability to actual output test$pred_class <- ifelse(pred_prob >= 0.5, ">50K", "<=50K") # Generating the classification table ctab_test <- table(test$Class, test$pred_class) ctab_test
# Output <=50K >50K <=50K 9602 784 >50K 2676 750
Accuracy is calculated by adding the diagonal elements and dividing it by the sum of all the elements of the contingency table. We will also compare the accuracy of the training dataset with the test dataset to see if our results are holding in the unseen data or not.
Accuracy = (TP + TN)/(TN + FP + FN + TP)
# Accuracy in Training dataset accuracy_train <- sum(diag(ctab_train))/sum(ctab_train)*100 accuracy_train
#Output  74.7355
Our logistics model is able to classify 74.7% of all the observations correctly in the training dataset.
# Accuracy in Test dataset accuracy_test <- sum(diag(ctab_test))/sum(ctab_test)*100 accuracy_test
#Output  74.94932
The over all correct classification accuracy in test dataset is 74.9% which is comparable to train dataset. This shows that our model is performing good.
A model is considered fairly good if the model accuracy is greater than 70%.
Misclassification Rate indicates how often is our predicted values are False.
Misclassification Rate = (FP+FN)/(TN + FP + FN + TP)
True Positive Rate – Recall or Sensitivity
Recall or TPR indicates how often does our model predicts actual TRUE from the overall TRUE events.
Recall Or TPR = TP/(FN + TP)
# Recall in Train dataset Recall <- (ctab_train[2, 2]/sum(ctab_train[2, ]))*100 Recall
# Output  21.22311
True Negative Rate
TNR indicates how often does our model predicts actual nonevents from the overall nonevents.
TNR = TN/(TN + FP)
# TNR in Train dataset TNR <- (ctab_train[1, 1]/sum(ctab_train[1, ]))*100 TNR
Precision indicates how often does your predicted TRUE values are actually TRUE.
Precision = TP/FP + TP
# Precision in Train dataset Precision <- (ctab_train[2, 2]/sum(ctab_train[, 2]))*100 Precision
#Output  47.92432
F-Score is a harmonic mean of recall and precision. The score value lies between 0 and 1. The value of 1 represents perfect precision & recall. The value 0 represents the worst case.
F_Score <- (2 * Precision * Recall / (Precision + Recall))/100 F_Score
#Output  0.2941839
The area under the curve(AUC) is the measure that represents ROC(Receiver Operating Characteristic) curve. This ROC curve is a line plot that is drawn between the Sensitivity and (1 – Specificity) Or between TPR and TNR. This graph is then used to generate the AUC value. An AUC value of greater than .70 indicates a good model.
library(pROC) roc <- roc(train$Class, logistic_model$fitted.values) auc(roc)
# Output Area under the curve: 0.7965
Concordance In how many pairs does the probability of ones is higher than the probability of zeros divided by the total number of possible pairs. The higher the values better is the model. The value of concordance lies between 0 and 1.
Similar to concordance, we have disconcordance which states in how many pairs the probability of ones was less than zeros. If the probability of ones is equal to 1 we say it is a tied pair.
# Output $`Concordance`  0.7943923 $Discordance  0.2056077 $Tied  0 $Pairs  193783060