Replace “Your Name” with your actual name.
This lab will focus on conducting multiple regression analyses and interpreting the coefficients (main effects) with a special emphasis on handling categorical variables using effect coding. You will work with various datasets to predict different outcomes, interpret the results, and understand how effect coding influences the interpretation of categorical variables.
Dataset: You are given a dataset with variables
Work_Hours, Job_Complexity,
Salary, and Job_Satisfaction. Your task is to
predict Job_Satisfaction based on the other three
predictors.
Dataset Creation:
# Create the dataset
set.seed(100)
data_ex1 <- data.frame(
Work_Hours = c(40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41, 40, 35, 45, 50, 38, 42, 48, 37, 44, 41),
Job_Complexity = c(7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8, 7, 6, 8, 9, 5, 7, 8, 6, 7, 8),
Salary = c(50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500, 50000, 48000, 52000, 55000, 47000, 51000, 53000, 46000, 54000, 49500),
Job_Satisfaction = c(78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76, 78, 72, 85, 80, 70, 82, 79, 75, 81, 76)
)
# View the first few rows of the dataset
head(data_ex1)## Work_Hours Job_Complexity Salary Job_Satisfaction
## 1 40 7 50000 78
## 2 35 6 48000 72
## 3 45 8 52000 85
## 4 50 9 55000 80
## 5 38 5 47000 70
## 6 42 7 51000 82Task:
1. Conduct a multiple regression analysis to predict
Job_Satisfaction using Work_Hours,
Job_Complexity, and Salary as predictors. Be
sure to use the data argument in the lm()
function.
2. Interpret the main effects of each predictor. What does each
coefficient tell you about its relationship with
Job_Satisfaction?
# Multiple regression model
model_ex1 <- lm(Job_Satisfaction ~ Work_Hours + Job_Complexity + Salary, data = data_ex1)
summary(model_ex1)##
## Call:
## lm(formula = Job_Satisfaction ~ Work_Hours + Job_Complexity +
## Salary, data = data_ex1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.367 -2.304 -0.491 2.131 5.056
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.3102797 7.1988322 3.933 0.000159 ***
## Work_Hours -0.1148592 0.1737588 -0.661 0.510179
## Job_Complexity 1.3367244 0.4796182 2.787 0.006411 **
## Salary 0.0008867 0.0002455 3.612 0.000485 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.852 on 96 degrees of freedom
## Multiple R-squared: 0.5925, Adjusted R-squared: 0.5798
## F-statistic: 46.53 on 3 and 96 DF, p-value: < 2.2e-16Work_Hours: Coefficient indicates how changes in weekly
hours relate to job satisfaction, holding other variables constant.Job_Complexity: Reflects whether higher complexity is
associated with more or less satisfaction.Salary: Represents the expected change in satisfaction
per dollar increase, holding other factors constant.Dataset: You are provided with a dataset containing
Study_Hours, Attendance,
Parent_Education_Level, and GPA. Your task is
to predict GPA based on the other predictors.
Dataset Creation:
# Create the dataset with a larger sample size
set.seed(200)
data_ex2 <- data.frame(
Study_Hours = c(15, 12, 20, 18, 14, 17, 16, 13, 19, 14, 18, 16, 21, 13, 15, 20, 19, 18, 17, 16, 12, 14, 13, 20, 21, 22, 17, 19, 15, 16),
Attendance = c(90, 85, 95, 92, 88, 91, 89, 87, 93, 86, 91, 89, 95, 87, 90, 96, 94, 93, 89, 90, 85, 88, 87, 95, 96, 97, 92, 94, 88, 89),
Parent_Education_Level = factor(rep(c("High School", "College"), 15))
)
# Effect coding for Parent_Education_Level: -1 for High School, 1 for College
data_ex2$Parent_Education_Level <- ifelse(data_ex2$Parent_Education_Level == "High School", -1, 1)
# Create GPA with stronger relationships to predictors for significance
data_ex2$GPA <- 2.5 + 0.07 * data_ex2$Study_Hours + 0.03 * data_ex2$Attendance + 0.4 * data_ex2$Parent_Education_Level + rnorm(30, 0, 0.1)
# View the first few rows of the dataset
head(data_ex2)## Study_Hours Attendance Parent_Education_Level GPA
## 1 15 90 -1 5.858476
## 2 12 85 1 6.312646
## 3 20 95 -1 6.393256
## 4 18 92 1 6.975807
## 5 14 88 -1 5.725976
## 6 17 91 1 6.808536Task:
1. Conduct a multiple regression analysis to predict GPA
using Study_Hours, Attendance, and
Parent_Education_Level (coded as -1 for “High School” and 1
for “College”) as predictors.
2. Interpret the main effects. How does each predictor contribute to predicting GPA?
# Multiple regression model
model_ex1 <- lm(Job_Satisfaction ~ Work_Hours + Job_Complexity + Salary, data = data_ex1)
summary(model_ex1)##
## Call:
## lm(formula = Job_Satisfaction ~ Work_Hours + Job_Complexity +
## Salary, data = data_ex1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.367 -2.304 -0.491 2.131 5.056
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.3102797 7.1988322 3.933 0.000159 ***
## Work_Hours -0.1148592 0.1737588 -0.661 0.510179
## Job_Complexity 1.3367244 0.4796182 2.787 0.006411 **
## Salary 0.0008867 0.0002455 3.612 0.000485 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.852 on 96 degrees of freedom
## Multiple R-squared: 0.5925, Adjusted R-squared: 0.5798
## F-statistic: 46.53 on 3 and 96 DF, p-value: < 2.2e-16Study_Hours: Indicates how additional study hours
relate to GPA.Attendance: Captures the effect of class attendance on
GPA.Parent_Education_Level: Shows the GPA difference
between students with college-educated parents vs. high school educated
(effect coded).Dataset: You are provided with a dataset containing
Exercise_Frequency, Diet_Quality,
Sleep_Duration, and Health_Index. Your task is
to predict Health_Index based on the other predictors.
Dataset Creation:
# Create the dataset with a larger sample size
set.seed(300)
data_ex3 <- data.frame(
Exercise_Frequency = c(4, 5, 3, 6, 2, 5, 4, 3, 5, 4, 6, 7, 3, 6, 2, 5, 7, 8, 4, 5, 3, 6, 7, 2, 4, 5, 6, 3, 7, 8),
Diet_Quality = c(8, 7, 9, 6, 5, 8, 7, 6, 8, 7, 9, 8, 6, 7, 5, 8, 9, 7, 8, 7, 9, 6, 8, 5, 7, 6, 9, 8, 7, 6),
Sleep_Duration = c(7, 8, 6, 7, 5, 8, 7, 6, 7, 7, 8, 7, 6, 7, 5, 8, 7, 8, 6, 7, 6, 7, 8, 5, 7, 8, 7, 6, 7, 8)
)
# Create Health_Index with stronger relationships to predictors for significance
data_ex3$Health_Index <- 50 + 2 * data_ex3$Exercise_Frequency + 1.5 * data_ex3$Diet_Quality + 1 * data_ex3$Sleep_Duration + rnorm(30, 0, 2)
# View the first few rows of the dataset
head(data_ex3)## Exercise_Frequency Diet_Quality Sleep_Duration Health_Index
## 1 4 8 7 79.74758
## 2 5 7 8 80.22421
## 3 3 9 6 76.44698
## 4 6 6 7 79.40253
## 5 2 5 5 66.32989
## 6 5 8 8 83.13740Task:
1. Conduct a multiple regression analysis to predict
Health_Index using Exercise_Frequency,
Diet_Quality, and Sleep_Duration as
predictors.
2. How do the coefficients inform you about the relative importance of each predictor in determining health outcomes?
# Multiple regression model
model_ex1 <- lm(Job_Satisfaction ~ Work_Hours + Job_Complexity + Salary, data = data_ex1)
summary(model_ex1)##
## Call:
## lm(formula = Job_Satisfaction ~ Work_Hours + Job_Complexity +
## Salary, data = data_ex1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.367 -2.304 -0.491 2.131 5.056
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.3102797 7.1988322 3.933 0.000159 ***
## Work_Hours -0.1148592 0.1737588 -0.661 0.510179
## Job_Complexity 1.3367244 0.4796182 2.787 0.006411 **
## Salary 0.0008867 0.0002455 3.612 0.000485 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.852 on 96 degrees of freedom
## Multiple R-squared: 0.5925, Adjusted R-squared: 0.5798
## F-statistic: 46.53 on 3 and 96 DF, p-value: < 2.2e-16Exercise_Frequency: More frequent exercise is expected
to improve health outcomes.Diet_Quality: A higher diet quality score predicts
better health.Sleep_Duration: Longer sleep duration is associated
with higher health index values.Dataset: You have a dataset with variables
Work_Experience, Education_Level,
Gender, and Salary. The Gender
variable is categorical with levels “Male” and “Female”.
Dataset Creation:
# Create the dataset with a larger sample size
set.seed(400)
data_ex4 <- data.frame(
Work_Experience = c(5, 7, 3, 6, 8, 4, 9, 6, 7, 5, 8, 9, 4, 6, 7, 5, 9, 10, 6, 7, 4, 5, 7, 6, 8, 9, 10, 5, 6, 8),
Education_Level = c(12, 14, 10, 16, 13, 15, 17, 12, 16, 14, 18, 19, 11, 14, 15, 13, 18, 20, 14, 15, 11, 13, 15, 14, 17, 18, 19, 13, 15, 17),
Gender = factor(rep(c("Male", "Female"), 15))
)
# Effect coding for Gender: 1 for Male, 1 for Female
data_ex4$Gender_Effect <- ifelse(data_ex4$Gender == "Male", -1, 1)
# Create Salary with stronger relationships to predictors for significance
data_ex4$Salary <- 30000 + 3000 * data_ex4$Work_Experience + 1500 * data_ex4$Education_Level + 5000 * data_ex4$Gender_Effect + rnorm(30, 0, 2000)
# View the first few rows of the dataset
head(data_ex4)## Work_Experience Education_Level Gender Gender_Effect Salary
## 1 5 12 Male -1 55926.90
## 2 7 14 Female 1 78230.57
## 3 3 10 Male -1 51945.87
## 4 6 16 Female 1 75634.63
## 5 8 13 Male -1 67296.32
## 6 4 15 Female 1 66794.78Task:
1. Conduct a multiple regression analysis to predict
Salary using Work_Experience,
Education_Level, and Gender_Effect as
predictors.
2. Interpret the coefficients, especially focusing on the effect of
Gender_Effect.
3. Discuss how effect coding impacts the interpretation of the
Gender_Effect variable.
# Multiple regression model with effect coding
model_ex4 <- lm(Salary ~ Work_Experience + Education_Level + Gender_Effect, data = data_ex4)
summary(model_ex4)##
## Call:
## lm(formula = Salary ~ Work_Experience + Education_Level + Gender_Effect,
## data = data_ex4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4401.7 -1568.7 165.7 1265.8 3439.3
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30426.2 2574.8 11.817 5.89e-12 ***
## Work_Experience 3501.8 434.2 8.064 1.52e-08 ***
## Education_Level 1239.9 317.0 3.912 0.000588 ***
## Gender_Effect 4823.7 384.0 12.563 1.51e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2025 on 26 degrees of freedom
## Multiple R-squared: 0.9688, Adjusted R-squared: 0.9652
## F-statistic: 269 on 3 and 26 DF, p-value: < 2.2e-16Work_Experience: Each additional year of experience increases salary, all else equal.
Education_Level: Higher education levels are associated with higher salaries.
Gender_Effect: With effect coding, this shows the average deviation of each gender from the overall mean salary.
Gender_Effect: Because of effect coding (-1, 1), the
coefficient reflects the difference between group means from the grand
mean, not relative to a reference group.Submission Instructions:
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