In npaterno/stat231: Course Materials for Stat 231

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(Stat231)
library(tidyverse)
library(moderndive)
library(openintro)

Building a Linear Model

For this lab we will be trying to predict the difference between the starting price and final price (not including shipping) for a Mario Kart game auctioned on ebay. Let's take a look at our data set.

glimpse(mariokart)

We want to predict the difference of the start and final price of the game itself. Looking at the pricing variables given we have start_Pr (starting price), ship_Pr (shipping price) and total_Pr (total price). This means we need to do a bit of work to get the variable we want. Below I'll clean the data - to get rid of unnecessary variables - and add the new variable.

model_data <- mariokart %>% 
  select(n_bids, duration ,start_pr, ship_pr, total_pr) %>% #This line selects the variable we might need for our model.
  mutate(price_diff = total_pr-ship_pr-start_pr) #This line calculates the difference between start and final price for each auction.

Finding the Correlation Coefficient

Next, we'll use a function from the moderndive package to calculate a few linear correlation coefficients and see if there are any relationships worth modeling.

get_correlation(data = model_data, price_diff ~ duration)

get_correlation(data = model_data, price_diff ~ n_bids)

Looking for a Better Fit

Based on the correlation values, it would be a better choice to make our prediction based on the number of bids than the duration of the auction. While the correlation value is not spectacular, it is close enough to 1 based on the sample of 143 auctions. Before we build our model let's sketch a scatter plot.

ggplot(model_data, aes(n_bids, price_diff))+
  geom_point()

There is definitely a positive linear relationship in this data. There is also a clear outlier. Let's find that data point in the original data set. The auction title reveals that this was for a Wii with Guitar Hero 5 and Mario Kart bundled together. Let's remove this as the price of the console itself is heavily influencing the model. Once removed, we'll build a new model and compare their accuracy via correlation.

model_data_2 <- model_data %>% 
  filter(price_diff<300)

get_correlation(data = model_data_2, price_diff ~ n_bids)

This is a much better correlation. We'll use this dataset to build our model.

Building the Linear Model

price_model <- lm(data = model_data_2, price_diff ~ n_bids)

Next we'll overlay the model onto our scatter plot.

ggplot(model_data_2, aes(n_bids, price_diff))+
  geom_point()+
  geom_smooth(method = "lm",
              se = FALSE)

Analyzing the Model

Now we'll look at a summary of the model.

get_regression_table(price_model)

The intercept row is for the y-intercept of the line. Based on our data, the model shows an intercept of 12.702 and a standard error of 2.725; This leads to a confidence interval of (7.315, 18.090). For this row we aren't concerned about the (test) statistic or p_value.

The nBids row is for the slope of the line. Based on our data, the model shows a slope of 1.737 (meaning the price difference goes up $1.74 per bid) and a standard error of .186; This leads to a confidence interval of (1.370, 2.104). For this row, we are concerned with the (test) statistic and p_value. The null hypothesis would be that the variables are not correlated and the slope is 0. With a test statistic of 4.662 and p_value (close to) 0, we can definitely reject the null hypothesis and support the claim that there is a relationship between these variables with a nonzero slope.

Making Predictions

Now that our model is built, let's make some predictions. To do this, we'll use the linear_model_prediction function from our Stat231 package.

We'll make two predictions: an interpolation (within the original data cluster) and an extrapolation (a point away from the original data cluster).

Interpolation:

linear_model_prediction(price_model, 15)

Based on our model, an ebay auction for Mario Kart that receive 15 bids will have a $38.76 increase from the starting price to the final price.

Extrapolation:

linear_model_prediction(price_model, 100)

Based on our model, an ebay auction for Mario Kart that receive 100 bids will have a $186.40 increase from the starting price to the final price.

npaterno/stat231 documentation built on May 1, 2023, 6:07 p.m.