Midterm_Zolboo
2023-04-16
R Markdown
This is an R Markdown document. Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see http://rmarkdown.rstudio.com.
When you click the Knit button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. You can embed an R code chunk like this:
# Q1
# Load required libraries
library(psych)
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.1 ✔ readr 2.1.4
## ✔ forcats 1.0.0 ✔ stringr 1.5.0
## ✔ ggplot2 3.4.2 ✔ tibble 3.2.1
## ✔ lubridate 1.9.2 ✔ tidyr 1.3.0
## ✔ purrr 1.0.1
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ ggplot2::%+%() masks psych::%+%()
## ✖ ggplot2::alpha() masks psych::alpha()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(tidymodels)## ── Attaching packages ────────────────────────────────────── tidymodels 1.0.0 ──
## ✔ broom 1.0.4 ✔ rsample 1.1.1
## ✔ dials 1.2.0 ✔ tune 1.1.1
## ✔ infer 1.0.4 ✔ workflows 1.1.3
## ✔ modeldata 1.1.0 ✔ workflowsets 1.0.1
## ✔ parsnip 1.1.0 ✔ yardstick 1.1.0
## ✔ recipes 1.0.5
## ── Conflicts ───────────────────────────────────────── tidymodels_conflicts() ──
## ✖ ggplot2::%+%() masks psych::%+%()
## ✖ scales::alpha() masks ggplot2::alpha(), psych::alpha()
## ✖ scales::discard() masks purrr::discard()
## ✖ dplyr::filter() masks stats::filter()
## ✖ recipes::fixed() masks stringr::fixed()
## ✖ dplyr::lag() masks stats::lag()
## ✖ yardstick::spec() masks readr::spec()
## ✖ recipes::step() masks stats::step()
## • Learn how to get started at https://www.tidymodels.org/start/library(vip)##
## Attaching package: 'vip'
##
## The following object is masked from 'package:utils':
##
## vilibrary(ISLR2)
# Load mtcars dataset
data(mtcars)
# Scatter plot of mpg against all predictors
ggplot(mtcars, aes(x = mpg)) +
geom_point(aes(y = hp)) + # Example plot for predictor hp
labs(x = "mpg", y = "hp") 
# Multiple regression on mpg across all predictors
model <- lm(mpg ~ ., data = mtcars)
summary(model)##
## Call:
## lm(formula = mpg ~ ., data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4506 -1.6044 -0.1196 1.2193 4.6271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337 18.71788 0.657 0.5181
## cyl -0.11144 1.04502 -0.107 0.9161
## disp 0.01334 0.01786 0.747 0.4635
## hp -0.02148 0.02177 -0.987 0.3350
## drat 0.78711 1.63537 0.481 0.6353
## wt -3.71530 1.89441 -1.961 0.0633 .
## qsec 0.82104 0.73084 1.123 0.2739
## vs 0.31776 2.10451 0.151 0.8814
## am 2.52023 2.05665 1.225 0.2340
## gear 0.65541 1.49326 0.439 0.6652
## carb -0.19942 0.82875 -0.241 0.8122
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared: 0.869, Adjusted R-squared: 0.8066
## F-statistic: 13.93 on 10 and 21 DF, p-value: 3.793e-07# Rerun multiple regression by excluding disp and/or cyl
# Exclude disp
model_excluding_disp <- lm(mpg ~ . - disp, data = mtcars)
summary(model_excluding_disp)##
## Call:
## lm(formula = mpg ~ . - disp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.7863 -1.4055 -0.2635 1.2029 4.4753
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.55052 18.52585 0.677 0.5052
## cyl 0.09627 0.99715 0.097 0.9240
## hp -0.01295 0.01834 -0.706 0.4876
## drat 0.92864 1.60794 0.578 0.5694
## wt -2.62694 1.19800 -2.193 0.0392 *
## qsec 0.66523 0.69335 0.959 0.3478
## vs 0.16035 2.07277 0.077 0.9390
## am 2.47882 2.03513 1.218 0.2361
## gear 0.74300 1.47360 0.504 0.6191
## carb -0.61686 0.60566 -1.018 0.3195
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.623 on 22 degrees of freedom
## Multiple R-squared: 0.8655, Adjusted R-squared: 0.8105
## F-statistic: 15.73 on 9 and 22 DF, p-value: 1.183e-07# Exclude disp and cyl
model_excluding_disp_cyl <- lm(mpg ~ . - disp - cyl, data = mtcars)
summary(model_excluding_disp_cyl)##
## Call:
## lm(formula = mpg ~ . - disp - cyl, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8187 -1.3903 -0.3045 1.2269 4.5183
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.80810 12.88582 1.072 0.2950
## hp -0.01225 0.01649 -0.743 0.4650
## drat 0.88894 1.52061 0.585 0.5645
## wt -2.60968 1.15878 -2.252 0.0342 *
## qsec 0.63983 0.62752 1.020 0.3185
## vs 0.08786 1.88992 0.046 0.9633
## am 2.42418 1.91227 1.268 0.2176
## gear 0.69390 1.35294 0.513 0.6129
## carb -0.61286 0.59109 -1.037 0.3106
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.566 on 23 degrees of freedom
## Multiple R-squared: 0.8655, Adjusted R-squared: 0.8187
## F-statistic: 18.5 on 8 and 23 DF, p-value: 2.627e-08# We can compare the results of the updated models with the original model to assess any improvements or changes in the regression results.
# ---
# Q2
# Load ISLR2 package
library(ISLR2)
# Fit multiple regression model
model <- lm(Sales ~ Price + Urban + US, data = Carseats)
summary(model)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16# Interpretation of coefficients in the model:
#The coefficients in the multiple regression model represent the estimated effect of each predictor on the outcome variable (Sales), while holding other predictors constant.
#Price: The coefficient for Price represents the estimated change in Sales for a one-unit increase in Price, holding Urban and US constant.
#Urban: The coefficient for Urban represents the estimated difference in Sales between Urban and Non-urban areas, holding Price and US constant. Since Urban is a qualitative variable (binary), the coefficient represents the estimated change in Sales when Urban changes from 0 to 1 (i.e., from Non-urban to Urban).
#US: The coefficient for US represents the estimated difference in Sales between US and Non-US stores, holding Price and Urban constant. Since US is a qualitative variable (binary), the coefficient represents the estimated change in Sales when US changes from 0 to 1 (i.e., from Non-US to US).
#
# The model in equation form, handling qualitative variables properly:
# The model equation can be written as:
# Sales = β0 + β1 * Price + β2 * Urban + β3 * US
# To test for the significance of each predictor, we can look at the p-values in the model summary. A low p-value (typically < 0.05) indicates that we can reject the null hypothesis (H0: βj = 0) and conclude that the corresponding predictor has a significant association with the outcome variable. In this case, we can assess the p-values for Price, Urban, and US in the model summary to determine which predictors are significant.
# Fit a smaller model that only uses predictors with evidence of association:
# Based on the p-values from the model summary, we can exclude any predictors with p-values greater than 0.05, indicating no significant association with the outcome variable. Let's fit a smaller model that only includes the significant predictors:
# Fit smaller model with significant predictors
model_smaller <- lm(Sales ~ Price + US, data = Carseats)
summary(model_smaller)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16# Assess the goodness of fit for the models in (a) and (e):
# We can assess the goodness of fit for the models by looking at the R-squared value in the model summary. R-squared represents the proportion of variance in the outcome variable that is explained by the predictors in the model. A higher R-squared value indicates a better fit to the data.
# Obtain 95% confidence intervals for coefficients in smaller model
conf_intervals <- confint(model_smaller, level = 0.95)
conf_intervals## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632# Check for outliers or high leverage observations in the smaller model:
# We can check for outliers or high leverage observations in the model by examining diagnostic plots, such as the residuals plot, Cook's distance plot, and leverage plot. These plots can help identify any data points that have a significant impact
# Zolboo Baatarkhuu 111035091Including Plots
You can also embed plots, for example:
Note that the echo = FALSE parameter was added to the
code chunk to prevent printing of the R code that generated the
plot.