Question Set 1

1. How many total posts are in the data set?

nrow(facebook)

2. How many ad posts of each Category (Action, Product, Inspiration) are in the data set?

table(facebook$Category)

3. Examine side-by-side boxplots of the number of people who liked the company’s page by whether a comment was posted. Describe the difference in the distribution of the number of likes.

boxplot(Page.total.likes ~ comment, data = facebook, col="red", xlab="Comment", ylab="Tal Likes")

4. Run the command prop.table(table(facebook$comment, facebook$Category), margin=2) to see the proportion of comments given by ad type. What can you say about the proportion of comments by ad type?

prop.table(table(facebook$comment, facebook$Category), margin=2)

Question Set 2

1. What is the distribution of the binary comment variable? What fraction of posts received at least one comment? Use the prop.table() function to answer this latter question.

table(facebook$comment)
prop.table(table(facebook$comment))

2. What fraction of posts were paid? Again, use the table command.

table(facebook$Paid)
prop.table(table(facebook$Paid))

3. As in Part 1, run the command prop.table(table(facebook$comment,facebook$Paid),margin=2) to see the fraction of comments posted to ads according to whether they were paid. What fraction of paid ads received comments? What fraction of unpaid ads received comments?

table(facebook$comment,facebook$Paid)
prop.table(table(facebook$comment,facebook$Paid),margin=2)
prop.table(table(facebook$comment, facebook$Paid), margin = 2)

4. How might you perform a hypothesis test to check whether the fractions are significantly different?

prop.test(table(facebook$Paid, facebook$comment), correct = F)
prop.test(c(274,120),c(360,140), correct=F)

Question Set 3

1. Run a simple logistic regression predicting comment from the Paid predictor. Interpret the sign of the coefficient. Is whether the ad was paid a significant predictor? How does this compare with the last question in the previous question set?

facebook1.glm <- 

facebook1.glm <- glm(comment ~ Paid , family = binomial, data = facebook)
summary(facebook1.glm)

2. Using the predict() function, predict the probability of a post receiving a comment for a paid ad. Do the same for an unpaid ad. How do these probability predictions compare to the fractions you computed inthe previous question set? What do you make of these comparisons?

p = 
out = 
  
  
p2 = 
out2 =

p = predict(facebook1.glm, newdata = data.frame(Paid=1), type = "response", se.fit=T)
out = c(p$fit, p$fit-2*p$se.fit, p$fit+2*p$se.fit)
names(out)=c("Fit", "Lower", "Upper")
out

p2 = predict(facebook1.glm, newdata = data.frame(Paid=0), type = "response", se.fit=T)
out2=c(p2$fit, p2$fit-2*p$se.fit, p2$fit+2*p$se.fit)
names(out2)=c("Fit", "Lower", "Upper")
out2

3. Run a simple logistic regression ofcommentonPost.Hour. Using the confint() function, obtain a 95% confidence interval for the coefficient of Post.Hour. Interpret this interval.

facebook2.glm <- 

facebook2.glm <- glm(comment ~ Post.Hour, family = binomial, data = facebook)
summary(facebook2.glm)
confint(facebook2.glm, "Post.Hour")
c.ph[1]
summary(facebook$Post.Hour)
plot(facebook$Post.Hour, facebook$comment)

4. Why might the logistic regression analysis including Post.Hour as a quantitative predictor variable be problematic?

Question Set 4

1. Convert the variables Post.Month, Post.Weekday and Post.Hour into categorical variables (factors) in the data frame. Call these new variables Post.Month.factor, Post.Weekday.factor, and Post.Hour.factor. You can use the factor() function to do so.

For example, if facebook was the name of the data frame, you can run the following code to create Post.Month.factor as the factor version of Post.Month:

facebook$Post.Month.factor = factor(facebook$Post.Month,levels = 1:12, labels = c("Jan", "Feb", "Mar", "Apr", "May", "Jun","Jul", "Aug", "Sep", "Oct", "Nov", "Dec"))

table(fb$Post.Hour)
facebook$Post.Month.factor = factor(facebook$Post.Month, levels= 1:12, labels = c("Jan", "Feb", "Mar", "Apr", "May", "Jun","Jul", "Aug", "Sep", "Oct", "Nov", "Dec"))
facebook$Post.Weekday.factor = factor(facebook$Post.Weekday, levels= 1:7, labels = c("Sun", "Mon", "Tue", "Wed", "Thu", "Fri", "Sat"))
facebook$Post.Hour.factor = factor(facebook$Post.Hour, levels= 1:24)
head(facebook)

Given that these variables will be used as predictors in a logistic regression, why does it make sense to include them as categorical variables?

2. Run a multiple logistic regression with the predictors Page.total.likes, Type, Category, Post.Month.factor, Post.Weekday.factor, Post.Hour.factor, and Paid. How does the coefficient for Paid correspond to the value in the simple logistic regression? What about the p-value for its significance?

facebook3.glm <- 

facebook3.glm <- glm(comment ~ Page.total.likes + Type + Category + Post.Month.factor + Post.Weekday.factor + Post.Hour.factor, family = binomial, data = facebook)
summary(facebook3.glm)

3. Perform a chi-squared test for assessing whether Post.Weekday.factor, Post.Month.factor and Post.Hour.factor are simultaneously significant predictors of the probability of a post receiving a comment.

facebook4.glm <-

facebook5.glm <- 

facebook4.glm <- glm(comment~Paid+Page.total.likes+Type+Category, data=facebook, family="binomial")
anova(facebook4.glm, facebook3.glm, test = "Chisq")

facebook5.glm <- glm(comment ~ Post.Month.factor + Post.Weekday.factor + Post.Hour.factor + Paid, family =binomial, data=facebook)

summary(facebook4.glm)

anova(facebook1.glm, facebook4.glm, test="Chisq")

Hint: The model in question 2 is the alternative hypothesis, but what is the model in the null hypothesis? Based on your analysis, should we include these three variables?

References

Kleinbaum, D. G., & Klein, M. (2010). Logistic regression: a self-learning text. Springer Science & Business Media. Can be downloaded here.

Faraway, J. J. (2016). Extending the linear model with R: generalized linear, mixed effects and non parametric regression models (Vol. 124). CRC press

Question Set 5

1. Perform backward stepwise model selection on the full model you fit in the previous question. Which predictor variables remain in the final model?

facebook.glm.reduced <-

facebook.glm.reduced <- step(facebook3.glm, direction="backward")
summary(facebook.glm.reduced)

2. How does the significance of Paid in the reduced model compare to that in the model with all of the predictors? What do you make of the difference?

3. It appears that none of the individual Post.Month.factor month variables are statistically significant, yet Post.Month.factor was retained as a predictive variable. Why might this be?

4. Construct a 95% confidence interval for the probability of a comment on a post where the ad was a photograph, the category was a product, the month was May, and the ad was unpaid. How do you interpret this interval? Does the width of the interval seem large? Further exposure to logistic regression.

newdata <- 
p <- 
out <- 

newdata = data.frame(Type="Photo", Category="Product", Post.Month.factor="May", Paid=0)
p = predict(facebook.glm.reduced, newdata, type="response", se.fit=T)
out=c(p$fit, p$fit-2*p$se.fit, p$fit+2*p$se.fit)
names(out)=c("Fit", "Lower", "Upper")
out

How many new connections have I made every year?

library(ggplot2)

# Show a bar chart with frequencies by year
li_cons %>% 
  ggplot() +
  geom_bar(aes(as.factor(connectedOnYear)), fill = c_col[1], color = "Black") +
  labs(title = "Number of connections per year",
       subtitle = paste("I startet using LinkedIn in", min(li_cons$connectedOnYear), "and it was the most active year concerning new connections"),
       x = "Years active on LinkedIn",
       y = "Number of new connections"
  ) 

How many new connections have I made every month?

# Show a bar chart with frequencies by year and quarter
li_cons %>% 
  ggplot() +
  geom_bar(
    aes(connectedOnYearQuarter),
    fill = c_col[1], color = "Black"
  ) +
  labs(
    title = "Number of connections per year and quarter",
    subtitle = paste("In the first month I connected to", count(li_cons, connectedOnYearQuarter)[1,2], "contacts"),
    x = "Years and quarters active on LinkedIn",
    y = "Number of new connections"
  ) +
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1, size = 12))

How many new connections have I made every month disregarding the first month after joining?

peak <- count(li_cons, connectedOnYearQuarter)%>%top_n(1)

# Show a bar chart with frequencies by year and quarter
li_cons %>% 
  filter(connectedOnYearMonth != min(li_cons$connectedOnYearMonth)) %>% 
  ggplot() +
  geom_bar(
    aes(connectedOnYearQuarter),
    fill = c_col[1], color = "Black"
  ) +
  labs(
    title = "Number of connections per year and quarter \n(without March 2012)",
    subtitle = paste("Disregarding the initial month, there is a connection peak in", zoo::as.yearqtr(peak$connectedOnYearQuarter)),
    x = "Years and quarters active on LinkedIn",
    y = "Number of new connections"
  ) +
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1, size = 12))

For which companies are my connections working for?

library(tidyr)
# Show frequencies of top-most companies
li_cons %>% 
  group_by(comp_clean) %>% 
  summarise(n = n()) %>% 
  drop_na() %>% 
  filter(n > 3) %>% 
  arrange(-n)  %>% 
  ggplot() +
  geom_col(
    aes(n, comp_clean),
    fill = c_col[1], color = "Black"
  ) +
  labs(
    title = "Number of connections per Company \n(with at least 4 connections)",
    subtitle = "Naturally, I have the most connections working for my employers, customers, and business partners",
    y = "Company name (with at least 4 connections)",
    x = "Number of connections"
  ) 

Are there any peak moments with top-companies?

# list of top companies, with at least 4 connections
li_top_companies <- li_cons %>% 
  group_by(comp_clean) %>% 
  summarise(n = n()) %>% 
  arrange(n) %>% 
  drop_na() %>% 
  filter(n > 5) %>% 
  pull(comp_clean)

li_top_companies

## [1] "University"

li_cons <- li_cons %>% 
  mutate(
    is_top_company = ifelse(comp_clean %in% li_top_companies, TRUE, FALSE),
    top_company = ifelse(comp_clean %in% li_top_companies, comp_clean, "Other")
  )

li_cons %>% 
  #filter(is_top_company) %>% 
  ggplot() + 
  geom_jitter(
    aes(connectedOnYear, top_company, color = top_company),
    size = 3,
    height = 0.2
  ) +
  labs(
    title = "Number of connections per company \n(with at least 6 connections per company)",
    subtitle = "Company information reflect my connections' actual employer and not the employer at the time \nwhen we connected",
    y = "Company name (with at least 6 connections)",
    x = "Years active on LinkedIn"
  ) +
  theme(legend.position = "none") + 
  scale_color_manual(values=c(c_col[1], c_col[2]))

What are my connections' positions?

# Show frequencies of top-most positions

# some reording based on frequency
levels4sorting <- li_cons %>% 
  group_by(pos_clean) %>% 
  summarise(n = n()) %>% 
  arrange(desc(n)) %>% 
  drop_na() %>% 
  top_n(2) %>% 
  pull(pos_clean)

li_cons_topPositions <- li_cons %>% 
  filter(pos_clean %in% levels4sorting) 

li_cons_topPositions$pos_clean <- factor(li_cons_topPositions$pos_clean, levels = levels4sorting)  

li_cons_topPositions %>% 
  group_by(pos_clean) %>% 
  summarise(
    n = n()
  ) %>% 
  ggplot() +
  geom_col(
    aes(n, pos_clean),
    fill = c_col[1], color = "Black"
  ) +
  labs(
    title = "Number of connections per position \n(with at least 4 positions)",
    subtitle = paste("I just realized I am the most connected with", levels4sorting,"!"),
    y = "Position name (with at least 4 connections)",
    x = "Number of connections"
  ) 

Are there any peak moments with top-positions?

# list of top positions, with at least 4 connections
li_top_pos <- li_cons %>% 
  group_by(pos_clean) %>% 
  summarise(n = n()) %>% 
  arrange(n) %>% 
  drop_na() %>% 
  filter(n >= 10) %>% 
  pull(pos_clean)

li_top_pos

## [1] "Consultant" "Analyst"

li_cons <- li_cons %>% 
  mutate(
    is_top_pos = ifelse(pos_clean %in% li_top_pos, TRUE, FALSE),
    top_pos = ifelse(pos_clean %in% li_top_pos, pos_clean, "Other")
  )

li_cons %>% 
  ggplot() + 
  geom_jitter(
    aes(connectedOnYear, top_pos, color = top_pos),
    height = 0.2
  ) +
  labs(
    title = "Number of connections per position \n(with at least 10 connections per position)",
    subtitle = "Position information reflect my connections' actual positions and not the positions at the time \nwhen we connected",
    y = "Position (with at least 10 connections)",
    x = "Years active on LinkedIn"
  ) +
  theme(legend.position = "none") + 
  scale_color_manual(values=c(c_col[1], c_col[2], c_col[3]))

Where are the top-positions working at the moment?

# Show relationship of top positions and top companies

li_cons %>% 
  ggplot() +
  geom_jitter(
    aes(top_company, top_pos, color = connectedOnYear),
    height = 0.2, width = 0.2
  ) +
  labs(
    title = "Relationship between top positions and \ntop companies",
    subtitle = paste("Many of my connections made it to", levels4sorting,". Congrats!"),
    y = "Position (with at least 10 connections)",
    x = "Company name (limited to companies with at least 6 connections)"
  ) +
  theme(legend.position = "none") +
  theme(axis.text.x = element_text(angle = 0, vjust = 0.5, hjust = 0.6, size = 12))