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: The coefficient is about 0.1149, but it is
not statistically significant. (p = 0.510), which means working hours
doesn’t have a clear or consistent effect on job satisfaction in this
data.Job_Complexity: The coefficient is about 1.3376 and it
is significant. (p = 0.006). This means as job complexity increases by 1
unit, job satisfaction increases by about 1.34 points, assuming other
variables stay the same. So, more complex jobs are linked with slightly
higher satisfaction.Salary: The coefficient is 0.00008867, and it’s also
significant (p = 0.0000485). This means that higher salaries are
associated with higher job satisfaction. For every $1 in salary,
satisfaction goes up slightly.Dataset: You are provided with a dataset containing
Study_Hours, Attendance,
Parent_Education_Level, and GPA. Your task is
to predict GPA based on the other predictors.
Dataset Creation:
# Create the dataset with a larger sample size
set.seed(200)
data_ex2 <- data.frame(
Study_Hours = c(15, 12, 20, 18, 14, 17, 16, 13, 19, 14, 18, 16, 21, 13, 15, 20, 19, 18, 17, 16, 12, 14, 13, 20, 21, 22, 17, 19, 15, 16),
Attendance = c(90, 85, 95, 92, 88, 91, 89, 87, 93, 86, 91, 89, 95, 87, 90, 96, 94, 93, 89, 90, 85, 88, 87, 95, 96, 97, 92, 94, 88, 89),
Parent_Education_Level = factor(rep(c("High School", "College"), 15))
)
# Effect coding for Parent_Education_Level: -1 for High School, 1 for College
data_ex2$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_ex2 <- lm(GPA ~ Study_Hours + Attendance + Parent_Education_Level, data = data_ex2)
summary(model_ex2)##
## Call:
## lm(formula = GPA ~ Study_Hours + Attendance + Parent_Education_Level,
## data = data_ex2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.208221 -0.058093 -0.001553 0.040198 0.143841
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.79220 1.32615 1.351 0.1882
## Study_Hours 0.04868 0.02267 2.147 0.0413 *
## Attendance 0.04170 0.01862 2.239 0.0339 *
## Parent_Education_Level 0.40410 0.01574 25.669 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.08593 on 26 degrees of freedom
## Multiple R-squared: 0.9731, Adjusted R-squared: 0.97
## F-statistic: 313.4 on 3 and 26 DF, p-value: < 2.2e-16Study_Hours: The coefficient is about 0.0487, and it’s
statistically significant (p = 0.0413). This means for each extra hour
spent studying, GPA goes up by about 0.049 points. So, more study time =
better GPA.Attendance: The coefficient is 0.0417, and it’s also
significant (p= 0.0339). This shows that better attendance is connected
to higher GPA. For each unit increase in attendance, GPA goes up by
about 0.042 points.Parent_Education_Level: The coefficient is 0.4041, and
it’s very significant (p< 0.001). This means that students whose
parents went to college tend to have a GPA that’s about 0.40 points
higher than those whose parents only went to high school.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_ex3 <- lm(Health_Index ~ Exercise_Frequency + Diet_Quality + Sleep_Duration, data = data_ex3)
summary(model_ex3)##
## Call:
## lm(formula = Health_Index ~ Exercise_Frequency + Diet_Quality +
## Sleep_Duration, data = data_ex3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.11901 -1.17265 0.03783 1.31807 2.86568
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.3914 2.9595 15.675 9.15e-15 ***
## Exercise_Frequency 1.8387 0.2829 6.500 6.87e-07 ***
## Diet_Quality 1.8522 0.2685 6.898 2.53e-07 ***
## Sleep_Duration 1.3717 0.5448 2.518 0.0183 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.684 on 26 degrees of freedom
## Multiple R-squared: 0.9225, Adjusted R-squared: 0.9136
## F-statistic: 103.2 on 3 and 26 DF, p-value: 1.467e-14Exercise_Frequency: The coefficient is 1.8387, and it’s
highly significant (p < 0.001). This means that for every additional
day or unit of exercise, the health index increases by about 1.84
points. So, more exercise is strongly linked with better health.Diet_Quality: The coefficient is 1.8522, and it’s also
highly significant (p < 0.001). This says that better diet quality
leads to better health outcomes. For every 1-unit increase in diet
score, health index goes up about 1.85 points.Sleep_Duration: The coefficient is 1.3717, and it’s
significant (p = 0.0183). So, more sleep is also associated with a
higher health index. FOr each extra hour of sleep, the health index
rises by about 1.37 points.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: The coefficient is 3501.8, and it’s very significant (p. < 0.001). This means that for every additional year of work experience, salary increases by about $3,502.
Education_Level: The coefficient is 1239.9, and it’s also significant (p = 0.0006). This means that each additional unit of education increases salary by around $1,240.
Gender_Effect: The coefficient is 4823.7, and it’s very significant (p < 0.001). This shows that females earn about $4824 more than males, all else being equal.
Gender_Effect:Submission Instructions:
Ensure to knit your document to HTML format, checking that all content is correctly displayed before submission.