Replace “Your Name” with your actual name.

Instructions

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.

Exercise 1: Predicting Job Satisfaction

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               82

Task:

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?

# Fit the multiple regression model
model <- lm(Job_Satisfaction ~ Work_Hours + Job_Complexity + Salary, data = data_ex1)

# Summary of the model
summary(model)
## 
## 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-16
  • Interpretation of Main Effects:
    • Work_Hours: Coefficient: 0.30

Interpretation: For each additional hour worked per week, Job Satisfaction increases by 0.30 units, assuming other variables are held constant. - Job_Complexity: Coefficient: 2.00 Interpretation: For each one-unit increase in job complexity, Job Satisfaction increases by 2.00 units, holding other variables constant. - Salary: Coefficient: 0.0005

Interpretation: For every $1 increase in salary, Job Satisfaction increases by 0.0005 units. For every $1,000 increase, satisfaction goes up by 0.5 units.

Exercise 2: Predicting Student Performance with Effect Coding

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 6.808536

Task:

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?

# Fit the regression model
model2 <- lm(GPA ~ Study_Hours + Attendance + Parent_Education_Level, data = data_ex2)

# Show summary
summary(model2)
## 
## 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-16
  • Interpretation of Main Effects:
    • Study_Hours: Coefficient: 0.07 Interpretation: Each additional hour of study per week is associated with a 0.07 increase in GPA, controlling for other variables.
    • Attendance: Coefficient: 0.03 Interpretation: For each 1% increase in attendance, GPA increases by 0.03, holding other variables constant.
    • Parent_Education_Level: Coefficient: 0.40 (effect coded: -1 = High School, 1 = College) Interpretation: Students with college-educated parents score, on average, 0.40 GPA points higher than those whose parents only finished high school.

Exercise 3: Predicting Health Outcomes

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     83.13740

Task:

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?

# Fit the model
model3 <- lm(Health_Index ~ Exercise_Frequency + Diet_Quality + Sleep_Duration, data = data_ex3)

# Show the summary
summary(model3)
## 
## 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-14
  • Interpretation of Main Effects:
    • Exercise_Frequency: For each additional day of exercise per week, Health_Index increases by 2 points, holding other variables constant.

Diet_Quality: For each one-point improvement in diet quality, Health_Index increases by 1.5 points.

Sleep_Duration: Each additional hour of sleep adds 1 point to the Health_Index.

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 66794.78

Task:

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.

# Fit the model using effect coding for Gender
model4 <- lm(Salary ~ Work_Experience + Education_Level + Gender_Effect, data = data_ex4)

# Show the summary
summary(model4)
## 
## 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-16
  • Interpretation of Main Effects:

    • Work_Experience: Each additional year of experience adds $3,000 to salary.

    • Education_Level: Each additional year of education adds $1,500.

    • Gender_Effect: Females earn $5,000 more than the overall mean salary, while males earn $5,000 less than the overall mean, controlling for education and experience.

  • Interpretation of Categorical Variables with Effect Coding:
    • Gender_Effect: With effect coding, the intercept represents the grand mean (average across both genders).

The Gender_Effect coefficient represents deviation from the grand mean.

So, instead of comparing to a reference group (like dummy coding), you’re interpreting effects relative to the overall mean.

Submission Instructions:

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