---
title: "Problem Set 4"
author: '[YOUR NAME]'
date: "Due Date: 2025-11-14"
output:
  html_document: default
  pdf_document: default
subtitle: Regression Analysis
institute: Villanova University
type: homeworks
---

```{r,include=F}
knitr::opts_chunk$set(error=TRUE)
```


## Getting Set Up

Open `RStudio` and create a new RMarkDown file (`.Rmd`) by going to `File -> New File -> R Markdown...`.
Accept defaults and save this file as `[LAST NAME]_ps4.Rmd` to your `code` folder.

Copy and paste the contents of this `.Rmd` file into your `[LAST NAME]_ps4.Rmd` file. Then change the `author: [Your Name]` to your name.

All of the following questions should be answered in this `.Rmd` file. There are code chunks with incomplete code that need to be filled in. To submit, compile (i.e., `knit as pdf`) the completed problem set and upload the PDF file to Blackboard on Friday by midnight. Be sure to check your knitted PDF for mistakes before submitting!

This problem set is worth 7 total points, plus 2 extra credit points. The point values for each question are indicated in brackets below. To receive full credit, you must have the correct code. In addition, some questions ask you to provide a written response in addition to the code.

You will be deducted 1 point for each day late the problem set is submitted, and 1 point for failing to submit in the correct format (i.e., not knitting as a PDF).

You are free to rely on whatever resources you need to complete this problem set, including lecture notes, lecture presentations, Google, your classmates...you name it. However, the final submission must be complete by you. There are no group assignments. To submit, compile the completed problem set and upload the PDF file to Blackboard on Friday by midnight. If you use AI for help, choose to save your output as a PDF and submit this with the problem set as well. Also note that I will not respond to Campuswire messages after 2PM ET on Friday, so don't wait until the last minute to get started!

**Good luck!**

**If you collaborated with a colleague and/or used AI for any help on this problem set, document here.** Write the names of your classmates and/or upload a PDF of your AI prompt and output with your problem set:

## Question 0
Require `tidyverse` and load the [`mv.Rds`](https://github.com/rweldzius/PSC4175/raw/main/static/data/mv.Rds') data to an object called `movies`.
```{r}
# INSERT CODE HERE
```


## Question 1 [1 point]
In this problem set, we will answer the following research question: "Do movies that have a higher Bechdel Score (`bechdel_score`) make more money (`gross`)?" First, write out a **theory** that answers this question and transform it into a **hypothesis**. To learn more about the Bechdel Score, please read [this Wikipedia entry](https://en.wikipedia.org/wiki/Bechdel_test).

> Write answer here

## Question 2 [1.5 points]
Based on your theory, which variable is the $X$ variable (i.e., the independent variable or the predictor)? Which variable is the $Y$ variable (i.e., the dependent variable or the outcome)? Use **univariate** visualization to create two plots, one for each variable. Do you need to apply a log-transformation to either of these variables? Why? Then create a multivariate visualization of these two variables, making sure to put the independent variable on the x-axis and the dependent variable on the y-axis. Make sure to log the data if you determined this was necessary in the previous question! Does the visualization support your hypothesis?

```{r}
# Univariate visualization of predictor / X variable
movies %>%
  ggplot() + # Put the predictor on the x-axis
  geom_...() + # Choose the appropriate geom...()
  labs(title = '', # Make sure to give the plot intuitive labels!
       x = '',
       y = '')

# Univariate visualization of outcome / Y variable
movies %>%
  ggplot() + # Put the outcome on the x-axis
  geom_...() + # Choose the appropriate geom...()
  labs(title = '', # Make sure to give the plot intuitive labels!
       x = '',
       y = '')

# Multivariate visualization
movies %>%
  drop_na() %>% # Dropping missing observations from the variables
  mutate() %>% # Do you need to mutate anything?
  ggplot() + # Put the predictor on the x-axis and the outcome on the y-axis
  geom_...() + # Choose the appropriate geom...()
  labs(title = '', # Make sure to give the plot intuitive labels!
       x = '',
       y = '')
```

> Write answer here

## Question 3 [2 points]
Now estimate the regression using the `lm()` function. Describe the output of the model in English, talking about the intercept, the slope, and the statistical significance.

```{r}
# Data wrangling
movies_analysis <- movies %>%
  mutate() %>% # Do you need to mutate anything?
  drop_na() # Dropping missing observations from the variables

model_gross_bechdel_score <- lm(formula = , # Write the regression formula here
                                data = ) # Put the data here

summary(model_gross_bechdel_score) # Interpret the regression output. Check here for help: https://learneconomicsonline.com/blog/archives/1095
```


> Write answer here

## Question 4 [2.5 points]
Now calculate the model's prediction errors and create both a univariate and multivariate visualization of them. Based on these analyses, would you say that your model does a good job predicting how much money a movie makes? **Make sure to reference both the univariate and multivariate visualization of the errors!** Use the prediction errors to calculate the RMSE in the full data. Then calculate the RMSE using 100-fold cross validation with an 80-20 split and take the average of the 100 estimates.


```{r}
movies_analysis <- movies_analysis %>%
  mutate() %>% # Create new variable of predicted values from the model
  mutate() # Calculate the errors

# Univariate visualization of the errors
movies_analysis %>%
  ggplot() + # Put errors on the x-axis
  geom_...() + # Choose the apppropriate geom...()
  labs(title = '', # Make sure to give the plot intuitive labels!
       x = '',
       y = '')

# Multivariate
movies_analysis %>%
  ggplot() + # Put errors on the y-axis and the predictor on the x-axis
  geom_...() + # Choose the apppropriate geom...()
  geom_hline() + # Add a horizontal dashed line at zero
  labs(title = '', # Make sure to give the plot intuitive labels!
       x = '',
       y = '')

# RMSE Calculations
# RMSE Full Data
movies_analysis %>%
  mutate() %>% # Calculate the squared errors (SE)
  summarise() %>% # Calculate the mean of the squared errors (MSE)
  mutate() # Calculate the square root of the mean of the squared errors (RMSE)

# RMSE 100-fold CV
set.seed(123) # Set seed for consistency!
cvRes <- NULL # Instantiate an empty object to store data from the loop
for(i in 1:100) { # Loop 100 times
  inds <- sample() # Create a list of random row numbers from 80% of the data, without replacement
  
  train <- movies_analysis %>% slice() # Create the training dataset based on these rows
  test <- movies_analysis %>% slice() # Create the test dataset based on all other rows
  
  m <- lm(formula = ,
          data = ) # Calculate your regression on the training data
  
  test$preds <- predict(,
                        newdata = ) # Apply your model to the test data
  
  e <-  # Calculate the errors
  se <-  # Calculate the squared errors
  mse <- mean() # Calcualte the mean of the squared errors 
  rmse <- sqrt()# Calculate the RMSE
  cvRes <- c() # Save the RMSE to a list of numbers
} 

mean(cvRes)
mean(cvRes < 1.98)
```

> Write answer here

## Extra Credit [1 point]
Taking a step back, do you trust these results? What concerns might you have about the model? Can you propose a "control" to add that would speak to your concerns? Run the regression with this control and re-interpret the results.

> Write answer here

```{r}
# INSERT CODE HERE
```