Assignment_7_15_and_7_16
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Q: 7.15
Refer to Commercial properties Problems 6.18 and 7.7. Calculate R^2Y4, R^2Y1, R^2Y1|4, R^214, R^2Y3|124, and R2. Explain what each coefficient measures and interpret your results. How is the degree of marginal linear association between Y and X 1 affected, when adjusted for X4? Answer:
To compute the R-squared values for various regression models, we utilize the lm function in R to fit the linear regression models, followed by employing the summary
function to extract the R-squared values. Here is the code for it:
Code: # Read the data into a data frame
data <- read.csv("C:\\Users\\Aayush\\OneDrive\\Desktop\\MSDA\\Sem 2\\Linear Regression\\Assignements\\7.15\\Commercial_Properties.csv")
View(data)
SST <- sum((data$Y - mean(data$Y))^2)
SST
#R2Y4
Y4_model <- lm(X4 ~ Y, data)
R2Y4 <- summary(Y4_model)$r.squared
R2Y4
#R2Y1
Y1_model <- lm(Y ~ X1, data)
R2Y1 <- summary(Y1_model)$r.squared
R2Y1
#R2Y1|4
P1.modelX4 <- lm(Y ~ X4, data)
SSE.X4 <- deviance(P1.modelX4); SSR.X4 <- SST-SSE.X4
P1.modelX14 <- lm(Y ~ X1+X4, data)
SSE.X1X4 <- deviance(P1.modelX14); SSR.X1X4 <- SST-SSE.X1X4
SSRX1_X4<-SSR.X1X4-SSR.X4 R2Y1_4<-SSRX1_X4/SSE.X4
R2Y1_4
#R214
I4_model <- lm(Y ~ X1+X4, data)
R214 <- summary(I4_model)$r.squared
R214
#R2Y2|14
P1.modelX14 <- lm(Y ~ X1+X4, data)
SSE.X1X4 <- deviance(P1.modelX14); SSR.X1X4 <- SST-SSE.X1X4
P1.modelX124 <- lm(Y ~ X1+X2+X4, data)
SSE.X1X2X4 <- deviance(P1.modelX124); SSR.X1X2X4 <- SST-SSE.X1X2X4
SSRX2_X1X4<-SSR.X1X2X4-SSR.X1X4 R2Y2_14<-SSRX2_X1X4/SSE.X1X4
R2Y2_14
#R2Y2|13
P1.modelX124 <- lm(Y ~ X1+X2+X4, data)
SSE.X1X2X4 <- deviance(P1.modelX124); SSR.X1X2X4 <- SST-SSE.X1X2X4
P1.modelX1234 <- lm(Y ~ X1+X2+X3+X4, data)
SSE.X1X2X3X4 <- deviance(P1.modelX1234); SSR.X1X2X3X4 <- SST-SSE.X1X2X3X4
SSRX3_X1X2X4<-SSR.X1X2X3X4-SSR.X1X2X4 R2Y3_124<-SSRX3_X1X2X4/SSE.X1X2X4
R2Y3_124
#R2
full_model <- lm(Y ~ X1 + X2 + X3 + X4, data)
R2 <- summary(full_model)$r.squared
R2
Output:
> # Read the data into a data frame
> data <- read.csv("C:\\Users\\Aayush\\OneDrive\\Desktop\\MSDA\\Sem 2\\Linear Regression\\Assignements\\7.15\\Commercial_Properties.csv")
> View(data)
> SST <- sum((data$Y - mean(data$Y))^2)
> SST
[1] 236.5575
> #R2Y4
> Y4_model <- lm(X4 ~ Y, data)
> R2Y4 <- summary(Y4_model)$r.squared
> R2Y4
[1] 0.2865058
> #R2Y1
> Y1_model <- lm(Y ~ X1, data)
> R2Y1 <- summary(Y1_model)$r.squared
> R2Y1
[1] 0.06264236
> #R2Y1|4
> P1.modelX4 <- lm(Y ~ X4, data)
> SSE.X4 <- deviance(P1.modelX4); > SSR.X4 <- SST-SSE.X4
> P1.modelX14 <- lm(Y ~ X1+X4, data)
> SSE.X1X4 <- deviance(P1.modelX14); > SSR.X1X4 <- SST-SSE.X1X4
> SSRX1_X4<-SSR.X1X4-SSR.X4 > R2Y1_4<-SSRX1_X4/SSE.X4
> R2Y1_4
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[1] 0.2504679
> #R214
> I4_model <- lm(Y ~ X1+X4, data)
> R214 <- summary(I4_model)$r.squared
> R214
[1] 0.4652132
> #R2Y2|14
> P1.modelX14 <- lm(Y ~ X1+X4, data)
> SSE.X1X4 <- deviance(P1.modelX14); > SSR.X1X4 <- SST-SSE.X1X4
> P1.modelX124 <- lm(Y ~ X1+X2+X4, data)
> SSE.X1X2X4 <- deviance(P1.modelX124); > SSR.X1X2X4 <- SST-SSE.X1X2X4
> SSRX2_X1X4<-SSR.X1X2X4-SSR.X1X4 > R2Y2_14<-SSRX2_X1X4/SSE.X1X4
> R2Y2_14
[1] 0.2202037
> #R2Y2|13
> P1.modelX124 <- lm(Y ~ X1+X2+X4, data)
> SSE.X1X2X4 <- deviance(P1.modelX124); > SSR.X1X2X4 <- SST-SSE.X1X2X4
> P1.modelX1234 <- lm(Y ~ X1+X2+X3+X4, data)
> SSE.X1X2X3X4 <- deviance(P1.modelX1234); > SSR.X1X2X3X4 <- SST-SSE.X1X2X3X4
> SSRX3_X1X2X4<-SSR.X1X2X3X4-SSR.X1X2X4 > R2Y3_124<-SSRX3_X1X2X4/SSE.X1X2X4
> R2Y3_124
[1] 0.004254889
> #R2
> full_model <- lm(Y ~ X1 + X2 + X3 + X4, data)
> R2 <- summary(full_model)$r.squared
> R2
[1] 0.5847496
In summary, R^2Y4 = .2865
R^2Y1 = .0626 R^2Y1|4 = .2505
R^214 = .4652
R^2Y2|14 = .2202
R^2Y3|124 =.0043 R2 = .5848
Therefore, 1)
R^2Y4 = 0.2865058: This denotes the fraction of variance in Y that can be accounted for solely by the variable X4.
2)
R^2Y1 = 0.06264236: This represents the portion of variance in Y explained by X1 alone.
3)
R^2Y1|4 = 0.2504679: This indicates the proportion of variance in Y explained jointly by both X1 and X4.
4)
R^214 = 0.4652132: Here, it shows the fraction of variance in Y explained by
both X1 and X4, along with any other independent variables in the model.
5)
R^2Y2|14 = 0.2202037: This reflects the amount of variance in Y explained by both X2 and X4, along with other independent variables.
6)
R^2Y3|124 = 0.004254889: It denotes the portion of variance in Y explained by both X2 and X3, along with other independent variables.
7)
R^2 = 0.5847496: This is the overall proportion of variance in Y explained by all independent variables in the model.
Interpreting these outcomes, it's evident that X4 emerges as the most influential independent variable in explaining Y's variation, with an R-squared value of 0.2865. Although X1 alone contributes modestly to Y's variation (R-squared of 0.0626), its joint consideration with X4 leads to a larger explained variance in Y (R-squared of 0.2505).
When all independent variables are considered together (R-squared of 0.5847), they
moderately explain the variance in Y. While X2 alone explains a small fraction of Y's variance (R-squared of 0.0043), its inclusion alongside X4 and other variables enhances the model's explanatory power (R-squared of 0.2202).
Regarding the marginal linear association between Y and X1, it could be influenced by adjusting for X4. This adjustment may alter the strength of the association between Y and X1, depending on the correlation between X1 and X4. If they are positively correlated, adjusting for X4 might diminish the association between Y and
X1. Conversely, if they are negatively correlated, adjusting for X4 could intensify the
association between Y and X1, as the confounding effect of the X1-X4 relationship diminishes.
Q: 7.16
Refer to Brand preference Problem 6.5.
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a. Transform the variables by means of the correlation transformation (7.44) and fit the standardized regression model (7.45). b. Interpret the standardized regression coefficient bi*. c. Transform the estimated standardized regression coefficients by means of (7.53) back to the ones for the fitted regression model in the original variables. Verify that they are the same as the ones obtained in Problem 6.5b.
Answer:
I have used the following code to get the output for the 3 required problems:
Code:
# Read the dataset
brand_data <- read.csv("C:\\Users\\Aayush\\OneDrive\\Desktop\\MSDA\\Sem 2\\
Linear Regression\\Assignements\\7.16\\Brand_Prefrences.csv")
# Standardize the variables using correlation transformation
standardized_brand_data <- as.data.frame(scale(brand_data))
# Fit the standardized regression model
standardized_model <- lm(Yi ~ Xi1 + Xi2, data = standardized_brand_data)
# Print the summary of the standardized model
summary(standardized_model)
#
a) Yhat* = .885X*1 + .402X*2 ---- Answer
b) Answer:
In the standardized regression model Yhat* = .885X1 + .402X2, coefficients represent the change in Yhat* for a one-standard-deviation increase in the corresponding standardized independent variable. Specifically, a one-standard-
deviation increase in X1 corresponds to a .885 standard deviation increase in Yhat, while a one-standard-deviation increase in X2 corresponds to a .402 standard deviation increase in Yhat. These coefficients quantify the relative importance and direction of the relationships between the independent variables and the dependent
variable.
#Running through it again for part c(manually)
#c)
# Data
Y <- c(64, 73, 61, 76, 72, 80, 71, 83, 83, 89, 86, 93, 88, 95, 94, 100)
X1 <- c(4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10)
X2 <- c(2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4)
# Combine into a data frame
data <- data.frame(Y = Y, X1 = X1, X2 = X2)
# Correlation transformation
cor_matrix <- cor(data)
X1_std <- (data$X1 - mean(data$X1)) / sd(data$X1)
X2_std <- (data$X2 - mean(data$X2)) / sd(data$X2)
Y_std <- (data$Y - mean(data$Y)) / sd(data$Y)
# Fit the standardized regression model
lm_std <- lm(Y_std ~ X1_std + X2_std, data = data)
# Summary of the model
summary(lm_std)
# Interpret the standardized regression coefficient b1*
std_coef <- summary(lm_std)$coefficients[, "Estimate"]
b1_star <- std_coef["X1_std"]
b1_star
# Transformation to original variables
b0 <- mean(data$Y) - b1_star * mean(data$X1) * sd(data$Y) / sd(data$X1) - std_coef["X2_std"] * mean(data$X2) * sd(data$Y) / sd(data$X2)
b1 <- b1_star * sd(data$Y) / sd(data$X1)
b2 <- std_coef["X2_std"] * sd(data$Y) / sd(data$X2)
# Display the coefficients
cat("Original Coefficients:\n")
cat("b0:", b0, "\n")
cat("b1:", b1, "\n")
cat("b2:", b2, "\n")
# Compute sY, s1, and s2
sY <- sd(data$Y)
s1 <- sd(data$X1)
s2 <- sd(data$X2)
# Output sY, s1, and s2
cat("sY:", sY, "\n")
cat("s1:", s1, "\n")
cat("s2:", s2, "\n")
Output:
> # Read the dataset
> brand_data <- read.csv("C:\\Users\\Aayush\\OneDrive\\Desktop\\MSDA\\Sem 2\\
Linear Regression\\Assignements\\7.16\\Brand_Prefrences.csv")
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> > # Standardize the variables using correlation transformation
> standardized_brand_data <- as.data.frame(scale(brand_data))
> > # Fit the standardized regression model
> standardized_model <- lm(Yi ~ Xi1 + Xi2, data = standardized_brand_data)
> > # Print the summary of the standardized model
> summary(standardized_model)
Call:
lm(formula = Yi ~ Xi1 + Xi2, data = standardized_brand_data)
Residuals:
Min 1Q Median 3Q Max -0.46424 -0.18227 0.02285 0.15994 0.36661 Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.00000 0.06294 0.000 1 Xi1 0.88503 0.06500 13.615 4.53e-09 ***
Xi2 0.40223 0.06500 6.188 3.28e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2518 on 13 degrees of freedom
Multiple R-squared: 0.9451,
Adjusted R-squared: 0.9366 F-statistic: 111.8 on 2 and 13 DF, p-value: 6.439e-09
>
> #a) Yhat* = .885X*1 + .402X*2 ---- Answer
> > #b) Answer:
> # In the standardized regression model Yhat* = .885X1 + .402X2, coefficients > # represent the change in Yhat* for a one-standard-deviation increase in the > # corresponding standardized independent variable. Specifically, > # a one-standard-deviation increase in X1 corresponds to a .885 standard deviation > # increase in Yhat, while a one-standard-deviation increase in X2 corresponds > # to a .402 standard deviation increase in Yhat. These coefficients quantify the > # relative importance and direction of the relationships between the independent > # variables and the dependent variable.
> > #Running through it again for part c(manually)
> #c)
> # Data
> Y <- c(64, 73, 61, 76, 72, 80, 71, 83, 83, 89, 86, 93, 88, 95, 94, 100)
> X1 <- c(4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10)
> X2 <- c(2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4)
> > # Combine into a data frame
> data <- data.frame(Y = Y, X1 = X1, X2 = X2)
> > # Correlation transformation
> cor_matrix <- cor(data)
> X1_std <- (data$X1 - mean(data$X1)) / sd(data$X1)
> X2_std <- (data$X2 - mean(data$X2)) / sd(data$X2)
> Y_std <- (data$Y - mean(data$Y)) / sd(data$Y)
> > # Fit the standardized regression model
> lm_std <- lm(Y_std ~ X1_std + X2_std, data = data)
> > # Summary of the model
> summary(lm_std)
Call:
lm(formula = Y_std ~ X1_std + X2_std, data = data)
Residuals:
Min 1Q Median 3Q Max -0.38423 -0.15391 0.00218 0.13863 0.36677 Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) -2.220e-16 5.880e-02 0.000 1 X1_std 8.924e-01 6.073e-02 14.695 1.78e-09 ***
X2_std 3.946e-01 6.073e-02 6.498 2.01e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2352 on 13 degrees of freedom
Multiple R-squared: 0.9521,
Adjusted R-squared: 0.9447 F-statistic: 129.1 on 2 and 13 DF, p-value: 2.658e-09
> > # Interpret the standardized regression coefficient b1*
> std_coef <- summary(lm_std)$coefficients[, "Estimate"]
> b1_star <- std_coef["X1_std"]
> b1_star
X1_std
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0.8923929 > > # Transformation to original variables
> b0 <- mean(data$Y) - b1_star * mean(data$X1) * sd(data$Y) / sd(data$X1) - std_coef["X2_std"] * mean(data$X2) * sd(data$Y) / sd(data$X2)
> b1 <- b1_star * sd(data$Y) / sd(data$X1)
> b2 <- std_coef["X2_std"] * sd(data$Y) / sd(data$X2)
> > # Display the coefficients
> cat("Original Coefficients:\n")
Original Coefficients:
> cat("b0:", b0, "\n")
b0: 37.65 > cat("b1:", b1, "\n")
b1: 4.425 > cat("b2:", b2, "\n")
b2: 4.375 > > # Compute sY, s1, and s2
> sY <- sd(data$Y)
> s1 <- sd(data$X1)
> s2 <- sd(data$X2)
> > # Output sY, s1, and s2
> cat("sY:", sY, "\n")
sY: 11.45135 > cat("s1:", s1, "\n")
s1: 2.309401 > cat("s2:", s2, "\n")
s2: 1.032796
Summary of Regression Analysis
We conducted a regression analysis to understand how two predictor variables (X1 and X2) relate to a response variable (Y). Our dataset included 16 observations.
Model Summary
We used a standardized regression model to predict the response variable. The model equation is:
Predicted Y = b0 + b1*X1 + b2*X2
where:
- b0, b1, and b2 are the regression coefficients.
The estimated coefficients are:
- b1 = 4.425
- b2 = 4.375
- b0 = 37.650
Interpretation
- A one-standard-deviation increase in X1 corresponds to a 4.425 standard deviation
increase in the predicted Y, holding other variables constant.
- Similarly, a one-standard-deviation increase in X2 corresponds to a 4.375 standard
deviation increase in the predicted Y, holding other variables constant.
- The intercept (b0) represents the expected value of Y when both predictors are at their average values. In our case, it's 37.650.
Standard Deviations
The standard deviations of the variables are:
- Standard deviation of Y: 11.45135
- Standard deviation of X1: 2.30940
- Standard deviation of X2: 1.03280
These values describe how spread out the data are around their averages.
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15.00
29.13
14.88
7.79
Residual
-9.22
1.24
-1.19
-1.42
1.77
0.03
-2.33
13.32
0.69
2.91
1.00
-4.23
-7.58
5.01
(a) Use the given information to calculate SSResid, SSTO, and SSRegr. (Round your answers to four decimal places.)
SSTO=
SSResid=
SSRegr =
(b) Calculate R² for this regression model. (Round your answer to three decimal places.)
R² =
How…
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