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2024-04-25
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library(psych)
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
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## ✔ ggplot2 3.5.1 ✔ tibble 3.2.1
## ✔ lubridate 1.9.3 ✔ tidyr 1.3.1
## ✔ purrr 1.0.2
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(tidymodels)## ── Attaching packages ────────────────────────────────────── tidymodels 1.2.0 ──
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## • Use tidymodels_prefer() to resolve common conflicts.library(vip)##
## Attaching package: 'vip'
##
## The following object is masked from 'package:utils':
##
## vilibrary(ISLR2)
data(mtcars)
summary(mtcars)## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000par(mfrow = c(1, 1), mar = c(5, 4, 4, 2) + 0.1)
for (col in names(mtcars)[-which(names(mtcars) == "mpg")]) {
plot(mtcars[, col], mtcars$mpg,
xlab = col, ylab = "mpg",
main = paste("mpg vs", col))
}
#mpg vs hp:There appears to be a non-linear relationship between horsepower (hp) and mpg, suggesting that mpg tends to increase at a decreasing rate as horsepower increases. A transformation such as a square root (√x) or logarithmic (log) transformation might be suitable to capture this relationship.
# mpg vs wt:There seems to be a strong negative linear relationship between weight (wt) and mpg, indicating that as the weight of the car increases, mpg tends to decrease. No transformation may be necessary for this predictor.
# mpg vs gear:There doesn't appear to be a clear relationship between the number of gears (gear) and mpg. The points are scattered without a discernible pattern, suggesting that gear may not be a strong predictor of mpg in a linear model.
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-07library(car)## Loading required package: carData
##
## Attaching package: 'car'
##
## The following object is masked from 'package:dplyr':
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## recode
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## some
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## The following object is masked from 'package:psych':
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## logitvif_values <- car::vif(model)
high_vif_predictors <- names(vif_values[vif_values > 10])
high_vif_predictors## [1] "cyl" "disp" "wt"model_excl_disp <- lm(mpg ~ . - disp, data = mtcars)
summary(model_excl_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-07model_excl_disp_cyl <- lm(mpg ~ . - disp - cyl, data = mtcars)
summary(model_excl_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# Q2
library(ISLR2)
data(Carseats)
str(Carseats)## 'data.frame': 400 obs. of 11 variables:
## $ Sales : num 9.5 11.22 10.06 7.4 4.15 ...
## $ CompPrice : num 138 111 113 117 141 124 115 136 132 132 ...
## $ Income : num 73 48 35 100 64 113 105 81 110 113 ...
## $ Advertising: num 11 16 10 4 3 13 0 15 0 0 ...
## $ Population : num 276 260 269 466 340 501 45 425 108 131 ...
## $ Price : num 120 83 80 97 128 72 108 120 124 124 ...
## $ ShelveLoc : Factor w/ 3 levels "Bad","Good","Medium": 1 2 3 3 1 1 3 2 3 3 ...
## $ Age : num 42 65 59 55 38 78 71 67 76 76 ...
## $ Education : num 17 10 12 14 13 16 15 10 10 17 ...
## $ Urban : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 1 2 2 1 1 ...
## $ US : Factor w/ 2 levels "No","Yes": 2 2 2 2 1 2 1 2 1 2 ...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-16coefficients <- coef(model)
price_coefficient <- coefficients["Price"]
cat("Price coefficient:", price_coefficient, "\n")## Price coefficient: -0.05445885cat("For a one-unit increase in Price, Sales is expected to decrease by", round(price_coefficient, 3), "\n")## For a one-unit increase in Price, Sales is expected to decrease by -0.054urban_coefficient <- coefficients["UrbanYes"]
cat("\nUrban coefficient:", urban_coefficient, "\n")##
## Urban coefficient: -0.02191615cat("If the store is located in an urban area, Sales are expected to increase by", round(urban_coefficient, 3), "units compared to a rural area.\n")## If the store is located in an urban area, Sales are expected to increase by -0.022 units compared to a rural area.us_coefficient <- coefficients["USYes"]
cat("\nUS coefficient:", us_coefficient, "\n")##
## US coefficient: 1.200573cat("If the store is located in the US, Sales are expected to increase by", round(us_coefficient, 3), "units compared to non-US.\n")## If the store is located in the US, Sales are expected to increase by 1.201 units compared to non-US.#Multiple Regression Model Equation:Sales=𝛽0+𝛽1×Price+𝛽2×UrbanYes+𝛽3×USYes+𝜖
#β0 : Intercept term (Sales when all predictors are zero).
#β1 : Effect of Price on Sales.
#β2 : Effect of being in an urban area on Sales.
#β3 : Effect of being in the US on Sales.
#ϵ: Error term (difference between observed and predicted Sales).
p_values <- summary(model)$coefficients[, 4]
significant_predictors <- names(p_values[p_values < 0.05])
significant_predictors## [1] "(Intercept)" "Price" "USYes"names(Carseats)## [1] "Sales" "CompPrice" "Income" "Advertising" "Population"
## [6] "Price" "ShelveLoc" "Age" "Education" "Urban"
## [11] "US"significant_predictors## [1] "(Intercept)" "Price" "USYes"smaller_model <- lm(Sales ~ Price + Urban + US, data = Carseats)
summary(smaller_model)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16rsq_smaller <- summary(smaller_model)$r.squared
cat## function (..., file = "", sep = " ", fill = FALSE, labels = NULL,
## append = FALSE)
## {
## if (is.character(file))
## if (file == "")
## file <- stdout()
## else if (startsWith(file, "|")) {
## file <- pipe(substring(file, 2L), "w")
## on.exit(close(file))
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## else {
## file <- file(file, ifelse(append, "a", "w"))
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## <bytecode: 0x5ef68e60aaf0>
## <environment: namespace:base>conf_intervals <- confint(smaller_model, level = 0.95)
conf_intervals## 2.5 % 97.5 %
## (Intercept) 11.76359670 14.32334118
## Price -0.06476419 -0.04415351
## UrbanYes -0.55597316 0.51214085
## USYes 0.69130419 1.70984121par(mfrow = c(2, 2), mar = c(4, 4, 2, 1))
plot(model)
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