assignment1
B09302118 政治三 黃柏叡
2023-03-18
1. Load required packages (ISLR) and data (Carseats)
require(ISLR)
Carseats <- read.csv(file = 'C:/Users/user0522/Downloads/dataset-11424.csv')I made a copy of Carseats.
Because library() does not work on R
markdown, instead, I use require().
2. Do the followings.
(2a) Data dimension: How many rows (number of observations) and columns (number of variables) are in this data?
dim(Carseats) ## [1] 400 11Total rows are 400 and total columns are 11.
(2b) How many of variables are in the data? What are they?
names(Carseats)## [1] "Sales" "CompPrice" "Income" "Advertising" "Population"
## [6] "Price" "ShelveLoc" "Age" "Education" "Urban"
## [11] "US"(2c) Using summary() function to check out data distribution
summary(Carseats)## Sales CompPrice Income Advertising
## Min. : 0.000 Min. : 77 Min. : 21.00 Min. : 0.000
## 1st Qu.: 5.390 1st Qu.:115 1st Qu.: 42.75 1st Qu.: 0.000
## Median : 7.490 Median :125 Median : 69.00 Median : 5.000
## Mean : 7.496 Mean :125 Mean : 68.66 Mean : 6.635
## 3rd Qu.: 9.320 3rd Qu.:135 3rd Qu.: 91.00 3rd Qu.:12.000
## Max. :16.270 Max. :175 Max. :120.00 Max. :29.000
## Population Price ShelveLoc Age
## Min. : 10.0 Min. : 24.0 Length:400 Min. :25.00
## 1st Qu.:139.0 1st Qu.:100.0 Class :character 1st Qu.:39.75
## Median :272.0 Median :117.0 Mode :character Median :54.50
## Mean :264.8 Mean :115.8 Mean :53.32
## 3rd Qu.:398.5 3rd Qu.:131.0 3rd Qu.:66.00
## Max. :509.0 Max. :191.0 Max. :80.00
## Education Urban US
## Min. :10.0 Length:400 Length:400
## 1st Qu.:12.0 Class :character Class :character
## Median :14.0 Mode :character Mode :character
## Mean :13.9
## 3rd Qu.:16.0
## Max. :18.03. Fit three linear regression models of the following specifications:
(3a) Model 1: Using Sales as the dependent variable, Advertising and Price as the two independent variables, print out the result using summary() or print()
lm.model_1 <- lm(Carseats$Sales ~ Carseats$Price + Carseats$Advertising)
summary(lm.model_1)##
## Call:
## lm(formula = Carseats$Sales ~ Carseats$Price + Carseats$Advertising)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.9011 -1.5470 -0.0223 1.5361 6.3748
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.003427 0.606850 21.428 < 2e-16 ***
## Carseats$Price -0.054613 0.005078 -10.755 < 2e-16 ***
## Carseats$Advertising 0.123107 0.018079 6.809 3.64e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.399 on 397 degrees of freedom
## Multiple R-squared: 0.2819, Adjusted R-squared: 0.2782
## F-statistic: 77.91 on 2 and 397 DF, p-value: < 2.2e-16(3b) Model 2: Sales as the dependent variable, Advertising, Price, and Education as the three independent variables, also, limit your estimation on the first 200 observations, again, print out the result
lm.model_2 <- lm(Sales ~ Price + Advertising + Education, data = Carseats[1:200,])
summary(lm.model_2)##
## Call:
## lm(formula = Sales ~ Price + Advertising + Education, data = Carseats[1:200,
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.1095 -1.4929 -0.1362 1.3478 6.3480
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.882594 1.285484 10.800 < 2e-16 ***
## Price -0.057705 0.007311 -7.893 2.05e-13 ***
## Advertising 0.152776 0.028626 5.337 2.60e-07 ***
## Education -0.054431 0.065058 -0.837 0.404
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.447 on 196 degrees of freedom
## Multiple R-squared: 0.3201, Adjusted R-squared: 0.3097
## F-statistic: 30.76 on 3 and 196 DF, p-value: 2.432e-16(3c) Visualize Model 1 and Model 2 using the coefplot() function
require(arm)
coefplot(lm.model_1)
coefplot(lm.model_2)
(3d) For both models, explain what you find. (i) which variable(s) is/are significant or insignificant? If the variable is significant (or insignificant), at what confidence level can you make this claim? For those variables that are significant, what are the sign (+ or -) of their effects on the outcome variable (Sales)?
- For model 1, both price and advertising are significant. The P value of price and advertising are almost close to 0. Both are less than .05 which means it does explain variation of sales to some extent. As for model 2, only education is not significant, because its P value is 0.404, price and advertising still matter in model 2. Meanwhile, coefplots above show that the interval of price and advertising do not cross 0.00 while education touches 0.00 in coefplot 2. Price negatively affects sales but advertising positively affects sales. As a matter of fact, the regression outcome is pretty intuitive.
(bonus) Describe how each of the significant variables influence the car Sales
(1) model 1
- One unit increase in price decreases car sales by 0.054613 unit.
- One unit increase in advertising increases car sales by 0.123107 unit.
(2) model 2
- One unit increase in price decreases car sales by 0.057705 unit.
- One unit increase in advertising increases car sales by 0.152776 unit.
4. From the same Carseats data, fit a linear regression model using Sales as the dependent variable, with Income, Age, Education, and Price and the independent variables. Save the result as model3:
(4a) Calculate the MSE, RMSE, RSS, RSE, and TSS from the model3
lm.model_3 <- lm(Carseats$Sales ~ Carseats$Income + Carseats$Age + Carseats$Education + Carseats$Price)
summary(lm.model_3)##
## Call:
## lm(formula = Carseats$Sales ~ Carseats$Income + Carseats$Age +
## Carseats$Education + Carseats$Price)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.907 -1.611 -0.228 1.705 7.189
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.242705 1.042763 15.577 < 2e-16 ***
## Carseats$Income 0.012319 0.004285 2.875 0.00426 **
## Carseats$Age -0.048570 0.007417 -6.548 1.81e-10 ***
## Carseats$Education -0.040663 0.045687 -0.890 0.37399
## Carseats$Price -0.055590 0.005083 -10.936 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.387 on 395 degrees of freedom
## Multiple R-squared: 0.2925, Adjusted R-squared: 0.2853
## F-statistic: 40.82 on 4 and 395 DF, p-value: < 2.2e-16#store output that will be printed later.
outputList <- list()
# Mean Squared Error
mse <- mean((lm.model_3$residuals)^2)
outputList <- append(outputList, c("MSE", mse))
# root Mean Squared Error
rmse <- sqrt(mse)
outputList <- append(outputList, c("RMSE", rmse))
# RSS (residual sum of squares)
rss <- sum((lm.model_3$residuals)^2)
outputList <- append(outputList, c("RSS", rss))
# RSE (Residual Standard Error)
numerator <- sum((lm.model_3$residuals)^2)
df <- length(lm.model_3$residuals) - (length(lm.model_3$coefficients) - 1) - 1
rse <- sqrt(numerator / df)
outputList <- append(outputList, c("RSE", rse))
# TSS(total sum of square)
rSquared <- 0.2925
tss <- rss / (1 - rSquared)
# IF I put: tss <- sum((Carseats$Sales - mean(Carseats$Sales))^2)
# tss == 3182.275
outputList <- append(outputList, c("TSS", tss))
for (elements in outputList){
print(elements)
}## [1] "MSE"
## [1] "5.62888456473612"
## [1] "RMSE"
## [1] "2.37252704193991"
## [1] "RSS"
## [1] "2251.55382589445"
## [1] "RSE"
## [1] "2.38749581530255"
## [1] "TSS"
## [1] "3182.40823447979"(4b) Why is RSS/TSS calculated as 1 - R2?
- TSS represents total variation. RSS represents the part which is not explained.
- RSS/TSS means how much of the variation of dependent variable is not explained.
- R^2 describes how much of the variation of dependent variable is explained by independent variables
- R^2 = 1 - (RSS / TSS)
(4c) Calculate the ANOVA of model3, print out the result, and answer this question: Which variable accounts for the largest variance in the model?
anova(lm.model_3)## Analysis of Variance Table
##
## Response: Carseats$Sales
## Df Sum Sq Mean Sq F value Pr(>F)
## Carseats$Income 1 73.48 73.48 12.8902 0.0003719 ***
## Carseats$Age 1 169.97 169.97 29.8184 8.405e-08 ***
## Carseats$Education 1 5.60 5.60 0.9823 0.3222376
## Carseats$Price 1 681.68 681.68 119.5896 < 2.2e-16 ***
## Residuals 395 2251.55 5.70
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1price accounts for the largest variance in the model.
5. Bias-Variance tradeoff
(5a) What is “bias”? What is “variance”? What is the consequence of high “bias” when you try to fit a regression line to a set of data points? What is the consequence of high “variance” when you try to fit a regression line to a set of data points?
Bias is the difference between the expected value of the estimator and the actual value. Variance here is the variance of estimated value. If high bias exists, it leads to high probability of wrong prediction. Instead, if high variance exists, it might make some good prediction but each prediction points count possibly varies a lot.
(5b) Fit two linear models of the following specifications:
Calculate and print out the two models’ bias-variance tradeoff using the code given to you in Rpubs, and compare the result: which model has higher (lower) “bias” and lower (higher) “variance”?
lm.model_4 <- lm(Carseats$Sales ~ Carseats$Income + Carseats$Age)
lm.model_5 <- lm(Carseats$Sales ~ Carseats$Income + Carseats$Age + Carseats$Education + Carseats$Price)
#define function to calculate bias
calc_bias <- function(predicted, actual) {
mean(sum((predicted - actual)^2))
}
#bias
bias_model4 <- calc_bias(lm.model_4$fitted.values, Carseats$Sales)
bias_model5 <- calc_bias(lm.model_5$fitted.values, Carseats$Sales)
# variance (the sum of variance-covariance matrix)
var_model4 <- sum(vcov(lm.model_4))
var_model5 <- sum(vcov(lm.model_5))
# dataframe
table <- data.frame("Model" = c("Model 4", "Model 5"), "Bias" = c(bias_model4, bias_model5), "Variance" = c(var_model4, var_model5))
#install.package("gridExtra")
require(gridExtra)
require(grid)
grid.table(table)
model4 (sales = income + age)
- higher bias, lower variance
model5 (sales = income + age + education + price)
- lower bias, higher variance
It literally shows that if more independent variables involve in a regression model the fitted values make better prediction while the variance of fitted values is higher. That is, there’s a trade-off.