library("car")
data(Mroz)

#Now that we have the data loaded into R environment let's have a look at the quick summary and dimentions of the data

dim(Mroz)

#We can see that there are 753 observations in the data and 8 columns, let's deep dive into the summary

summary(Mroz)

#Description about the dataset: The Mroz data is from the Panel Study of Income Dynamics (PSID), for married women.
#Column description below:
# lfp: labor-force participation; a factor with levels: no; yes, from the above summary we can see that 428 Women participaded and 325 women did not participate.
# 
# k5: number of children 5 years old or younger, from the above summary we can see that there's maximum of 3 childrens.
# 
# k618: number of children 6 to 18 years old, from the above summary we can see that maximum number of childrens is 8 pertaining to a family.
# 
# age: in years, we can see from the summary that the average age is 43.
# 
# wc: wife's college attendance; a factor with levels: no; yes.
# 
# hc: husband's college attendance; a factor with levels: no; yes.
# 
# lwg: log expected wage rate; for women in the labor force, the actual wage rate; for women not in the labor force, an imputed value based on the regression of lwg on the other variables.
# 
# inc: family income exclusive of wife's income.

### Question 1. (a): using Logit regression to predict the probability to work given age, number of children 5 years old or younger, number of children 6 to 18 years old and a factor indicating if the woman has college attendance. Thus, run the binary regression to obtain estimates of the coefficients on these variables.

### Answer 1. (a)
# Using Logit regression to predict the probability to work(lfp) given age(age), number of children 5 years old or younger(k5), number of children 6 to 18 years old(k618) and a factor indicating if the woman has college attendance(wc).

# Let's modify the Response variable and the wc variable as we might want to multiply them ahead in the logit function

Mroz$lfpModi<-ifelse(Mroz$lfp=="yes",1,0)
Mroz$wcModi<-ifelse(Mroz$wc=="yes",1,0)

LogFit1<-glm(lfpModi~age+k5+k618+wc,data=Mroz,family=binomial(link='logit'))
summary(LogFit1)

# From the above summary of the model we can see that age, k5 and wc variables are coming out to be significant. The variable k618 is not significant but it is quite close to be significant. Estiated coefficients for the model are given as:
#
# Variable Estimate
# (Intercept) 3.44759
# age -0.06778
# k5 -1.45700
# k618 -0.10885
# wcyes 0.81433
#
# Also, let's look at the confusion matrix i.e. sensitivity and specificity of the model.

predicted <-ifelse(predict(LogFit1, type = 'response')>0.5,"yes p","no p")
table(predicted,Mroz$lfp)

TNR<-161/(161+90) # True Negative rate
TPR<-338/(338+164) # True Positive rate

# From the above matrix we can see that True negative rate (TNR) is coming out to be 64% meaning that Negatives are being predicted right for 64% of the time and True positive rate is 67% meaning that Trues are being predicted correctly 67% of the time.

### Question 1. (b): Sue is 30 years old, has college attendance and one child, Tom, he is 5 years old. WHat is the predicted probability that Sue works given the logit regression?

### Answer 1. (b)
# Let's make a dataframe from the above inputs and call it sue.

sue<-data.frame(age=30, wc="yes", k5=1, k618=0)

# Using the predict function to predict the probability
predict(LogFit1,sue)

# We can see from the above prediction that the probablity that Sue works is ~0.77

### Question 1. (c): Betty is 25 years old, without college attendance but with a 3 year old girl. What is the predicted probability to work for Betty?

### Answer 1. (c)
# Let's make a dataframe from the above inputs and call it betty.

betty<-data.frame(age=25, wc="no", k5=1, k618=0)

# Using the predict function to predict the probability
predict(LogFit1,betty)

# The predicted probability for Betty to work is ~0.3

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