# ADA2: Class 03, Ch 02 Introduction to Multiple Linear Regression

Advanced Data Analysis 2, Stat 428/528, Spring 2024, Prof. Erik Erhardt, UNM

Author

Published

January 22, 2024

# Auction selling price of antique grandfather clocks

The data include the selling price in pounds sterling at auction of 32 antique grandfather clocks, the age of the clock in years, and the number of people who made a bid. In the sections below, describe the relationship between variables and develop a model for predicting selling Price given Age and Bidders.

library(erikmisc)
Registered S3 methods overwritten by 'ggpp':
method                  from
heightDetails.titleGrob ggplot2
widthDetails.titleGrob  ggplot2
Registered S3 method overwritten by 'ggpmisc':
method                  from
as.character.polynomial polynom
Warning: replacing previous import 'ggplot2::annotate' by 'ggpp::annotate' when
loading 'erikmisc'
── Attaching packages ─────────────────────────────────────── erikmisc 0.2.12 ──
✔ tibble 3.2.1     ✔ dplyr  1.1.4
── Conflicts ─────────────────────────────────────────── erikmisc_conflicts() ──
✖ dplyr::lag()    masks stats::lag()
erikmisc, solving common complex data analysis workflows
by Dr. Erik Barry Erhardt <erik@StatAcumen.com>
library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ forcats   1.0.0     ✔ readr     2.1.5
✔ ggplot2   3.4.4     ✔ stringr   1.5.1
✔ lubridate 1.9.3     ✔ tidyr     1.3.0
✔ purrr     1.0.2     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
dat_auction <- read_csv("ADA2_CL_03_auction.csv")
Rows: 32 Columns: 3
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (3): Age, Bidders, Price

ℹ Use spec() to retrieve the full column specification for this data.
ℹ Specify the column types or set show_col_types = FALSE to quiet this message.
str(dat_auction)
spc_tbl_ [32 × 3] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
$Age : num [1:32] 127 115 127 150 156 182 156 132 137 113 ...$ Bidders: num [1:32] 13 12 7 9 6 11 12 10 9 9 ...
\$ Price  : num [1:32] 1235 1080 845 1522 1047 ...
- attr(*, "spec")=
.. cols(
..   Age = col_double(),
..   Bidders = col_double(),
..   Price = col_double()
.. )
- attr(*, "problems")=<externalptr> 
summary(dat_auction)
      Age           Bidders           Price
Min.   :108.0   Min.   : 5.000   Min.   : 729
1st Qu.:117.0   1st Qu.: 7.000   1st Qu.:1053
Median :140.0   Median : 9.000   Median :1258
Mean   :144.9   Mean   : 9.531   Mean   :1327
3rd Qu.:168.5   3rd Qu.:11.250   3rd Qu.:1561
Max.   :194.0   Max.   :15.000   Max.   :2131  

## (1 p) Scatterplot matrix

In a scatterplot matrix below interpret the relationship between each pair of variables. If a transformation is suggested by the plot (that is, because there is a curved relationship), also plot the data on the transformed scale and perform the following analysis on the transformed scale. Otherwise indicate that no transformation is necessary.

library(ggplot2)
library(GGally)
Registered S3 method overwritten by 'GGally':
method from
+.gg   ggplot2
p <- ggpairs(dat_auction)
print(p)

## (1 p) Correlation matrix

Below is the correlation matrix and tests for the hypothesis that each correlation is equal to zero. Interpret the hypothesis tests and relate this to the plot that you produced above.

# correlation matrix and associated p-values testing "H0: rho == 0"
#library(Hmisc)
Hmisc::rcorr(as.matrix(dat_auction))
          Age Bidders Price
Age      1.00   -0.25  0.73
Bidders -0.25    1.00  0.39
Price    0.73    0.39  1.00

n= 32

P
Age    Bidders Price
Age            0.1611  0.0000
Bidders 0.1611         0.0254
Price   0.0000 0.0254        

## (1 p) Plot interpretation

Below are two plots. The first has $$y =$$ Price, $$x =$$ Age, and colour = Bidders, and the second has $$y =$$ Price, $$x =$$ Bidders, and colour = Age. Interpret the relationships between all three variables, simultaneously. For example, say how Price relates to Age, then also how Price relates to Bidders conditional on Age being a specific value.


Attaching package: 'gridExtra'
The following object is masked from 'package:dplyr':

combine
geom_smooth() using formula = 'y ~ x'
Warning: The following aesthetics were dropped during statistical transformation: label
ℹ This can happen when ggplot fails to infer the correct grouping structure in
the data.
ℹ Did you forget to specify a group aesthetic or to convert a numerical
variable into a factor?
geom_smooth() using formula = 'y ~ x'
Warning: The following aesthetics were dropped during statistical transformation: label
ℹ This can happen when ggplot fails to infer the correct grouping structure in
the data.
ℹ Did you forget to specify a group aesthetic or to convert a numerical
variable into a factor?

## (2 p) Multiple regression assumptions (assessing model fit)

Below the multiple regression is fit. Start by assessing the model assumptions by interpretting what you learn from the first six plots (save the added variable plots for the next question). If assumptions are not met, attempt to address by transforming a variable and restart at the beginning using the new transformed variable.

# fit the simple linear regression model
lm_p_a_b <- lm(Price ~ Age + Bidders, data = dat_auction)

Plot diagnostics.

# plot diagnostics
e_plot_lm_diagnostics(lm_p_a_b, sw_plot_set = "simpleAV")
Warning in plot.xy(xy.coords(x, y), type = type, ...): "add" is not a graphical
parameter

### Solution

From the diagnostic plots above,

## (1 p) Added variable plots

Use partial regression residual plots (added variable plots) to check for the need for transformations. If linearity is not supported, address and restart at the beginning.

## (1 p) Multiple regression hypothesis tests

State the hypothesis test and conclusion for each regression coefficient.

# fit the simple linear regression model
lm_p_a_b <- lm(Price ~ Age + Bidders, data = dat_auction)
# use summary() to get t-tests of parameters (slope, intercept)
summary(lm_p_a_b)

Call:
lm(formula = Price ~ Age + Bidders, data = dat_auction)

Residuals:
Min     1Q Median     3Q    Max
-207.2 -117.8   16.5  102.7  213.5

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1336.7221   173.3561  -7.711 1.67e-08 ***
Age            12.7362     0.9024  14.114 1.60e-14 ***
Bidders        85.8151     8.7058   9.857 9.14e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 133.1 on 29 degrees of freedom
Multiple R-squared:  0.8927,    Adjusted R-squared:  0.8853
F-statistic: 120.7 on 2 and 29 DF,  p-value: 8.769e-15

## (1 p) Multiple regression interpret coefficients

Interpret the coefficients of the multiple regression model.

## (1 p) Multiple regression $$R^2$$

Interpret the Multiple R-squared value.

## (1 p) Summary

Summarize your findings in one sentence.

### Solution

## Aside: I generally recommend against 3D plots for a variety of reasons.
## However, here's a 3D version of the plot so you can visualize the surface fit in 3D.
## I will point out a feature in this plot that we wouldn't see in other plots
## and it would typically only be detected by careful consideration
## of a "more complicated" second-order model that includes curvature.

# library(rgl)
# library(car)
# scatter3d(Price ~ Age + Bidders, data = dat_auction)