Homework 3
Discussion Group 1
04/09/2021
Overview
In this homework assignment, you will explore, analyze and model a data set containing information on crime for various neighborhoods of a major city. Each record has a response variable indicating whether or not the crime rate is above the median crime rate (1) or not(0).
Your objective is to build a binary logistic regression model on the training data set to predict whether the neighborhood will be at risk for high crime levels. You will provide classifications and probabilities for the evaluation data set using your binary logistic regression model. You can only use the variables given to you (or variables that you derive from the variables provided). Below is a short description of the variables of interest in the data set:
- zn: proportion of residential land zoned for large lots (over 25000 square feet) (predictor variable)
- indus: proportion of non-retail business acres per suburb (predictor variable)
- chas: a dummy var. for whether the suburb borders the Charles River (1) or not (0) (predictor variable)
- nox: nitrogen oxides concentration (parts per 10 million) (predictor variable)
- rm: average number of rooms per dwelling (predictor variable)
- age: proportion of owner-occupied units built prior to 1940 (predictor variable)
- dis: weighted mean of distances to five Boston employment centers (predictor variable)
- rad: index of accessibility to radial highways (predictor variable)
- tax: full-value property-tax rate per $10,000 (predictor variable)
- ptratio: pupil-teacher ratio by town (predictor variable)
- black: 1000(Bk - 0.63)2 where Bk is the proportion of blacks by town (predictor variable)
- lstat: lower status of the population (percent) (predictor variable)
- medv: median value of owner-occupied homes in $1000s (predictor variable)
- target: whether the crime rate is above the median crime rate (1) or not (0) (response variable)
Data Exploration
Load Input Datasets
training <- read.csv('./crime-training-data_modified.csv')
training2 <- training # for melting and box plot
evaluation <- read.csv('./crime-evaluation-data_modified.csv')
training %>% head() %>% kable() | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | lstat | medv | target |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 19.58 | 0 | 0.605 | 7.929 | 96.2 | 2.0459 | 5 | 403 | 14.7 | 3.70 | 50.0 | 1 |
| 0 | 19.58 | 1 | 0.871 | 5.403 | 100.0 | 1.3216 | 5 | 403 | 14.7 | 26.82 | 13.4 | 1 |
| 0 | 18.10 | 0 | 0.740 | 6.485 | 100.0 | 1.9784 | 24 | 666 | 20.2 | 18.85 | 15.4 | 1 |
| 30 | 4.93 | 0 | 0.428 | 6.393 | 7.8 | 7.0355 | 6 | 300 | 16.6 | 5.19 | 23.7 | 0 |
| 0 | 2.46 | 0 | 0.488 | 7.155 | 92.2 | 2.7006 | 3 | 193 | 17.8 | 4.82 | 37.9 | 0 |
| 0 | 8.56 | 0 | 0.520 | 6.781 | 71.3 | 2.8561 | 5 | 384 | 20.9 | 7.67 | 26.5 | 0 |
Numerical Summaries
summary(training)## zn indus chas nox
## Min. : 0.00 Min. : 0.460 Min. :0.00000 Min. :0.3890
## 1st Qu.: 0.00 1st Qu.: 5.145 1st Qu.:0.00000 1st Qu.:0.4480
## Median : 0.00 Median : 9.690 Median :0.00000 Median :0.5380
## Mean : 11.58 Mean :11.105 Mean :0.07082 Mean :0.5543
## 3rd Qu.: 16.25 3rd Qu.:18.100 3rd Qu.:0.00000 3rd Qu.:0.6240
## Max. :100.00 Max. :27.740 Max. :1.00000 Max. :0.8710
## rm age dis rad
## Min. :3.863 Min. : 2.90 Min. : 1.130 Min. : 1.00
## 1st Qu.:5.887 1st Qu.: 43.88 1st Qu.: 2.101 1st Qu.: 4.00
## Median :6.210 Median : 77.15 Median : 3.191 Median : 5.00
## Mean :6.291 Mean : 68.37 Mean : 3.796 Mean : 9.53
## 3rd Qu.:6.630 3rd Qu.: 94.10 3rd Qu.: 5.215 3rd Qu.:24.00
## Max. :8.780 Max. :100.00 Max. :12.127 Max. :24.00
## tax ptratio lstat medv
## Min. :187.0 Min. :12.6 Min. : 1.730 Min. : 5.00
## 1st Qu.:281.0 1st Qu.:16.9 1st Qu.: 7.043 1st Qu.:17.02
## Median :334.5 Median :18.9 Median :11.350 Median :21.20
## Mean :409.5 Mean :18.4 Mean :12.631 Mean :22.59
## 3rd Qu.:666.0 3rd Qu.:20.2 3rd Qu.:16.930 3rd Qu.:25.00
## Max. :711.0 Max. :22.0 Max. :37.970 Max. :50.00
## target
## Min. :0.0000
## 1st Qu.:0.0000
## Median :0.0000
## Mean :0.4914
## 3rd Qu.:1.0000
## Max. :1.0000Missing Data Check
Missmap Plot illustrates there are no missing values in the Input Dataset. Each column has complete values and lets check the skewness of the values in the input columns.
missmap(training, main="Missing Values") 
colSums(is.na(training))## zn indus chas nox rm age dis rad tax ptratio
## 0 0 0 0 0 0 0 0 0 0
## lstat medv target
## 0 0 0Skewness Check
The histograms and density plots provides us a better understanding on the distribution of values in the columns.
The variables zn, age, dis, ptratio, black and lstat are heavily skewed.
nonbinary <- c(1:2, 4:13)
X <- training[ , -14]
par(mfrow = c(3,4))
for (i in nonbinary) {
hist(X[ ,i], xlab = names(X[i]), main = names(X[i]))
d <- density(X[,i])
plot(d, main = names(X[i]))
polygon(d, col="red")
}
# Converting to factor
var <- c("chas","target")
training[,var] <- lapply(training[,var], as.factor)
evaluation$chas <- as.factor(evaluation$chas)
# Density plot to check normality
melt(training2, id.vars='target') %>% mutate(target = as.factor(target)) %>%
ggplot(., aes(x=value))+geom_density(fill='gray')+facet_wrap(~variable, scales='free')+
labs(title="Density Plot for Normality and Skewness") +
theme_classic()
# Skewness and outliers
sapply(training2, skewness, function(x) skewness(x))## zn indus chas nox rm age
## 2.17681518 0.28854503 3.33548988 0.74632807 0.47932023 -0.57770755
## dis rad tax ptratio lstat medv
## 0.99889262 1.01027875 0.65931363 -0.75426808 0.90558642 1.07669198
## target
## 0.03422935Box Plot Distributions
Box Plot to see how target variable is distributed
# Boxplot to see distributions with target variable
melt(training2, id.vars='target') %>% mutate(target = as.factor(target)) %>%
ggplot(., aes(x=variable, y=value))+geom_boxplot(aes(fill=target))+facet_wrap(~variable, dir='h',scales='free')+ labs(title="BoxPlot - Predictors Data Distribution with Target Variable")
Correlation Plot
Correlation Plot illustrates the relationship between the variables in the input dataset.
# Correlation matrix among variables
training2 %>%
cor(., use = "complete.obs") %>%
corrplot(., method = "color", type = "upper", tl.col = "black", tl.cex=.8, diag = FALSE)
# Correlation table
correlation <- training2 %>%
cor(., use = "complete.obs") %>%
as.data.frame() %>%
rownames_to_column()%>%
gather(Variable, Correlation, -rowname)
correlation %>%
filter(Variable == "target") %>%
arrange(desc(Correlation)) %>%
kable() | rowname | Variable | Correlation |
|---|---|---|
| target | target | 1.0000000 |
| nox | target | 0.7261062 |
| age | target | 0.6301062 |
| rad | target | 0.6281049 |
| tax | target | 0.6111133 |
| indus | target | 0.6048507 |
| lstat | target | 0.4691270 |
| ptratio | target | 0.2508489 |
| chas | target | 0.0800419 |
| rm | target | -0.1525533 |
| medv | target | -0.2705507 |
| zn | target | -0.4316818 |
| dis | target | -0.6186731 |
Data Preparation
Data Splitting
The Input dataset is split into training and test data using createDataPartition function. The ratio of splitup is 70% for training and 30% for test data.
# Data splitting into train and test datasets out of training2
set.seed(1003)
training_partition <- createDataPartition(training2$target, p=0.7, list = FALSE, times=1)
train2 <- training2[training_partition, ]
test2 <- training2[-training_partition, ]
sapply(training2, skewness, function(x) skewness(x))## zn indus chas nox rm age
## 2.17681518 0.28854503 3.33548988 0.74632807 0.47932023 -0.57770755
## dis rad tax ptratio lstat medv
## 0.99889262 1.01027875 0.65931363 -0.75426808 0.90558642 1.07669198
## target
## 0.03422935Log transformation
The Predictor variable zn ( proportion of residential land zoned ) is transformed using log10 function and the transformation is applied to both training and test data.
train_log <- train2 # copy of basic model for log transformation
test_log <- test2
train_log$zn <- log10(train_log$zn + 1)
test_log$zn <- log10(test_log$zn + 1)
# Plot and check skewness
sapply(train_log, skewness, function(x) skewness(x))## zn indus chas nox rm age dis
## 1.1954317 0.3708873 3.2567321 0.8108244 0.4843153 -0.5322325 0.9721716
## rad tax ptratio lstat medv target
## 1.1317640 0.7385414 -0.8218609 0.9700911 0.9700927 0.1036391ggplot(melt(train_log), aes(x=value))+geom_density()+facet_wrap(~variable, scales='free') + labs(title="Log Transformation")
BoxCox Transformation
The three methods BoxCox, center and scale is used for preprocessing the dataset.
The BoxCox transformation is used for transforming a non-normally distributed data set into a normal distributed.
# Copy of train and test
train_boxcox <- train2
test_boxcox <- test2
# Preprocessing
preproc_value <- preProcess(train2[,-1] , c("BoxCox", "center", "scale"))
# Transformation on both train and test datasets
train_boxcox_transformed <- predict(preproc_value, train_boxcox)
test_boxcox_transformed <- predict(preproc_value, test_boxcox)
ggplot(melt(train_boxcox_transformed), aes(x=value))+geom_density()+facet_wrap(~variable, scales='free') + labs(title="BoxCox Transformation")
sapply(train_boxcox_transformed, function(x) skewness(x))## zn indus chas nox rm age
## 2.353729370 -0.120195838 3.256732134 0.068418489 0.026658478 -0.366340589
## dis rad tax ptratio lstat medv
## 0.106811243 0.402503618 0.069764777 -0.613098264 -0.002592159 -0.039697233
## target
## 0.103639097Derive New Variables
New variables are created by applying some transformation logic on the existing values. The transformation list can be found in the R code below. The transformation is applied on both training and test data set.
# Copying test and train subset to unique variable name
train_M2 <- train2
test_M2 <- test2
# Calcuated vars on test set
test_M2$target <- as.factor(test_M2$target)
test_M2$cfas <- as.factor(test_M2$chas)
test_M2$business<-test_M2$tax*(1-test_M2$indus)
test_M2$apartment<-test_M2$rm/test_M2$tax
test_M2$pollution<-test_M2$nox*test_M2$indus
test_M2$zndum <- ifelse(test_M2$zn>0,1,0)
test_M2$rmlog<-log(test_M2$rm)
test_M2$raddum <- ifelse(test_M2$rad>23,1,0)
test_M2$lstatptratio<-((1-test_M2$lstat)*test_M2$ptratio)#/test_M2$rm
# Calcuated vars on test set
train_M2$target <- as.factor(train_M2$target)
train_M2$cfas <- as.factor(train_M2$chas)
train_M2$business<-train_M2$tax*(1-train_M2$indus)
train_M2$apartment<-train_M2$rm/train_M2$tax
train_M2$pollution<-train_M2$nox*train_M2$indus
train_M2$zndum <- ifelse(train_M2$zn>0,1,0)
train_M2$rmlog<-log(train_M2$rm)
train_M2$raddum <- ifelse(train_M2$rad>23,1,0)
train_M2$lstatptratio<-((1-train_M2$lstat)*train_M2$ptratio)#/train_M2$rm
ggplot(melt(train_M2), aes(x=value))+geom_density()+facet_wrap(~variable, scales='free') + labs(title="New Variables derivation")## Using target, cfas as id variables
Lasso Transformation
# Copying test and train subset to unique variable name
test_M3 <- test2
train_M3 <- train2
test_M3$target <- as.factor(test_M3$target)
train_M3$target <- as.factor(train_M3$target)
trainx = model.matrix(~.-target,data=train_M3)
newx = model.matrix(~.-target,data=test_M3)
ggplot(melt(train_M3), aes(x=value))+geom_density()+facet_wrap(~variable, scales='free') + labs(title="Lasso Transformation")## Using target as id variables
Build Models
Model 1 - Glmulti
The model glmulti is similar to Generalized Linear Model but it has ability to find confidence set of models (best models) from the list of all possible models (candidate models). Models are fitted with the specified fitting function (glm) and are ranked with the criterion ‘aic’
The model takes training dataset and linear regression is calculated for response variable (target) and other explanatory variables. Summary of the model is displayed on the output and AUC (Area under the curve) is calcluated
Model by Forhad Akbar
summary(model1@objects[[1]]) ##
## Call:
## fitfunc(formula = as.formula(x), family = ..1, data = data)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7621 -0.2870 -0.0080 0.0057 3.2397
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -36.411735 6.701072 -5.434 5.52e-08 ***
## zn -0.052983 0.034824 -1.521 0.12815
## indus -0.069985 0.048870 -1.432 0.15213
## nox 44.712077 8.392153 5.328 9.94e-08 ***
## age 0.032535 0.012388 2.626 0.00863 **
## dis 0.749006 0.262842 2.850 0.00438 **
## rad 0.569547 0.159955 3.561 0.00037 ***
## tax -0.005851 0.003072 -1.905 0.05683 .
## ptratio 0.268150 0.129009 2.079 0.03766 *
## medv 0.089608 0.039255 2.283 0.02245 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.43 on 326 degrees of freedom
## Residual deviance: 152.03 on 317 degrees of freedom
## AIC: 172.03
##
## Number of Fisher Scoring iterations: 8test_M1$predictions<- predict(model1@objects[[1]], test_M1, type="response")
test_M1$predicted = as.factor(ifelse(test_M1$predictions >= 0.5, 1, 0))
test_M1$target <- as.factor(test_M1$target)
confusionMatrix(test_M1$predicted, test_M1$target, positive = '1')## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 60 5
## 1 5 69
##
## Accuracy : 0.9281
## 95% CI : (0.8717, 0.965)
## No Information Rate : 0.5324
## P-Value [Acc > NIR] : <2e-16
##
## Kappa : 0.8555
##
## Mcnemar's Test P-Value : 1
##
## Sensitivity : 0.9324
## Specificity : 0.9231
## Pos Pred Value : 0.9324
## Neg Pred Value : 0.9231
## Prevalence : 0.5324
## Detection Rate : 0.4964
## Detection Prevalence : 0.5324
## Balanced Accuracy : 0.9278
##
## 'Positive' Class : 1
## proc = roc(test_M1$target, test_M1$predictions)
plot(proc)
print(proc$auc)## Area under the curve: 0.9838Model 2 - Stepwise Regression and Calculated Variables
The stepwise regression takes the predictors and adds/removes based on the significance of the predictors. At first the model is run with 0 predictors and the predictors are added in sequence based on its significance. Since the model chooses the predictors by itself all predictors (explanator variables) are considered for model against target variable.
Adding to the stepwise regression we are also considering the transformed dataset with new variables derived from the existing variables.
Model by Adam Gersowitz
Model_2 <- glm(target~., data = train_M2, family = "binomial") %>%
stepAIC(trace = FALSE)
summary(Model_2)##
## Call:
## glm(formula = target ~ nox + rm + dis + rad + tax + lstat + business +
## apartment + pollution + rmlog + raddum + lstatptratio, family = "binomial",
## data = train_M2)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.3124 -0.0448 0.0000 0.0001 4.6220
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.984e+02 9.493e+01 3.144 0.001668 **
## nox 9.004e+01 2.087e+01 4.314 1.60e-05 ***
## rm 6.034e+01 1.777e+01 3.395 0.000686 ***
## dis 4.546e-01 3.080e-01 1.476 0.139912
## rad 8.286e-01 2.684e-01 3.088 0.002018 **
## tax -3.586e-01 7.635e-02 -4.697 2.64e-06 ***
## lstat -7.261e-01 2.727e-01 -2.663 0.007748 **
## business -7.089e-03 1.862e-03 -3.808 0.000140 ***
## apartment -4.511e+03 1.124e+03 -4.014 5.96e-05 ***
## pollution -3.748e+00 1.135e+00 -3.303 0.000957 ***
## rmlog -2.867e+02 1.010e+02 -2.839 0.004529 **
## raddum 3.732e+01 1.578e+03 0.024 0.981133
## lstatptratio -3.801e-02 1.390e-02 -2.734 0.006256 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.434 on 326 degrees of freedom
## Residual deviance: 80.282 on 314 degrees of freedom
## AIC: 106.28
##
## Number of Fisher Scoring iterations: 19test_M2$predictions<-predict(Model_2, test_M2, type="response")
test_M2$predicted = as.factor(ifelse(test_M2$predictions >= 0.5, 1, 0))
confusionMatrix(test_M2$predicted, test_M2$target, positive = '1')## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 65 4
## 1 0 70
##
## Accuracy : 0.9712
## 95% CI : (0.928, 0.9921)
## No Information Rate : 0.5324
## P-Value [Acc > NIR] : <2e-16
##
## Kappa : 0.9424
##
## Mcnemar's Test P-Value : 0.1336
##
## Sensitivity : 0.9459
## Specificity : 1.0000
## Pos Pred Value : 1.0000
## Neg Pred Value : 0.9420
## Prevalence : 0.5324
## Detection Rate : 0.5036
## Detection Prevalence : 0.5036
## Balanced Accuracy : 0.9730
##
## 'Positive' Class : 1
## proc = roc(test_M2$target, test_M2$predictions)
plot(proc)
print(proc$auc)## Area under the curve: 0.9956Model 3 - Lasso
Model by David Blumenstiel
library(glmnet)
glmnetmodel <- cv.glmnet(x = trainx, #Predictor variables
y = train_M3[,names(train_M3) == "target"], #Responce variable
family = "binomial", #Has it do logistic regression
nfolds = 10, #10 fold cv
type.measure = "class", #uses missclassification error as loss
gamma = seq(0,1,0.1), #Values to use for relaxed fit
relax = TRUE,#Mixes relaxed fit with regluarized fit
alpha = 1) #Basically a choice betwen lasso, ridge, or elasticnet regression. Alpha = 1 is lasso.
#Predicts the probability that the target variable is 1
predictions <- predict(glmnetmodel, newx = newx, type = "response", s=glmnetmodel$lambda.min)
print(coef.glmnet(glmnetmodel, s = glmnetmodel$lambda.min))## 14 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) -33.18931637
## (Intercept) .
## zn -0.03787292
## indus -0.08199683
## chas 0.85574122
## nox 40.35258411
## rm .
## age 0.02436244
## dis 0.59880268
## rad 0.43866864
## tax -0.00406803
## ptratio 0.25457861
## lstat 0.04079768
## medv 0.09205576confusionMatrix(as.factor(ifelse(predictions >= 0.5, 1, 0)), test_M3$target, positive = '1')## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 61 7
## 1 4 67
##
## Accuracy : 0.9209
## 95% CI : (0.8628, 0.9598)
## No Information Rate : 0.5324
## P-Value [Acc > NIR] : <2e-16
##
## Kappa : 0.8415
##
## Mcnemar's Test P-Value : 0.5465
##
## Sensitivity : 0.9054
## Specificity : 0.9385
## Pos Pred Value : 0.9437
## Neg Pred Value : 0.8971
## Prevalence : 0.5324
## Detection Rate : 0.4820
## Detection Prevalence : 0.5108
## Balanced Accuracy : 0.9219
##
## 'Positive' Class : 1
## proc = roc(test_M3$target, predictions)
plot(proc)
print(proc$auc)## Area under the curve: 0.9844Model Selection
We selected Model 2 (Stepwise Regression with New Derived Variables) because of two key factors mentioned below.
- Confusion Matrix
The Confusion matrix values (accuracy, Sensitivity etc.) suggests the model 2 has higher values comparing to other models. For e.g. Accuracy of Model 2 is 97% while the glmulti and lasso model is around 92%. The other values illustrates the same result.
- AUC
AUC provides aggregate measure of performance across all threshold values. The AUC has to be higher for a model to perform better. The higher AUC value of 99% suggests the Stepwise model performs way better than other models.
Rerun model on entire training set
evaluation_F <- evaluation
training_F <- training
evaluation_F$cfas <- as.factor(evaluation_F$chas)
evaluation_F$business<-evaluation_F$tax*(1-evaluation_F$indus)
evaluation_F$apartment<-evaluation_F$rm/evaluation_F$tax
evaluation_F$pollution<-evaluation_F$nox*evaluation_F$indus
evaluation_F$zndum <- ifelse(evaluation_F$zn>0,1,0)
evaluation_F$rmlog<-log(evaluation_F$rm)
evaluation_F$raddum <- ifelse(evaluation_F$rad>23,1,0)
evaluation_F$lstatptratio<-((1-evaluation_F$lstat)*evaluation_F$ptratio)#/evaluation_F$rm
training_F$target <- as.factor(training_F$target)
training_F$cfas <- as.factor(training_F$chas)
training_F$business<-training_F$tax*(1-training_F$indus)
training_F$apartment<-training_F$rm/training_F$tax
training_F$pollution<-training_F$nox*training_F$indus
training_F$zndum <- ifelse(training_F$zn>0,1,0)
training_F$rmlog<-log(training_F$rm)
training_F$raddum <- ifelse(training_F$rad>23,1,0)
training_F$lstatptratio<-((1-training_F$lstat)*training_F$ptratio)#/training_F$rm
Model_F <- glm(target~., data = training_F, family = "binomial") %>%
stepAIC(trace = FALSE)
summary(Model_F)##
## Call:
## glm(formula = target ~ nox + rm + rad + tax + lstat + business +
## apartment + pollution + rmlog + raddum + lstatptratio, family = "binomial",
## data = training_F)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.9036 -0.0290 0.0000 0.0001 4.7310
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.641e+02 8.741e+01 3.022 0.002514 **
## nox 7.917e+01 1.462e+01 5.415 6.14e-08 ***
## rm 5.390e+01 1.664e+01 3.240 0.001196 **
## rad 9.930e-01 2.473e-01 4.016 5.92e-05 ***
## tax -3.459e-01 6.765e-02 -5.114 3.16e-07 ***
## lstat -8.961e-01 2.498e-01 -3.588 0.000333 ***
## business -6.240e-03 1.423e-03 -4.384 1.16e-05 ***
## apartment -4.506e+03 1.048e+03 -4.302 1.69e-05 ***
## pollution -3.239e+00 8.585e-01 -3.773 0.000161 ***
## rmlog -2.447e+02 9.322e+01 -2.625 0.008663 **
## raddum 3.421e+01 1.257e+03 0.027 0.978292
## lstatptratio -5.001e-02 1.322e-02 -3.784 0.000154 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 645.88 on 465 degrees of freedom
## Residual deviance: 103.06 on 454 degrees of freedom
## AIC: 127.06
##
## Number of Fisher Scoring iterations: 19Predict Test Set / Export results
predicted_probability <- predict(Model_F, evaluation_F, type = "response")
predicted_class <- as.factor(ifelse(predicted_probability >= 0.5, 1, 0))
predictions <- data.frame(predicted_class,predicted_probability)
colnames(predictions) <- c("predicted_class","predicted_probability")
#write.csv(predictions, file = "predictions.csv")