• Linear regression is when you fit a line to data
• We use it to predict one variable from another
• Example: predicting exam scores from study hours for ASU students
• This is a simple example to understand the basics

The equation for simple linear regression is:

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

Where: - \(y\) is the dependent variable (what we’re predicting) - \(x\) is the independent variable
- \(\beta_0\) is the intercept - \(\beta_1\) is the slope - \(\varepsilon\) is the error term

In our example: \(y\) = Exam Score, \(x\) = Study Hours

We use Ordinary Least Squares (OLS) to find the best line:

\[ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]

\[ \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \]

The R-squared tells us how good the fit is:

\[ R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \]

# load packages
library(ggplot2)
library(plotly)

# study hours and exam scores
study_hours <- c(2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 3, 5, 7, 9, 11, 13, 15, 4, 6, 8, 10, 12, 14, 16, 5, 7, 9)
exam_scores <- c(52, 61, 68, 71, 74, 77, 79, 81, 84, 87, 89, 91, 93, 58, 66, 73, 78, 83, 88, 92, 63, 70, 76, 80, 85, 90, 95, 67, 75, 82)

# putting all of the dats on the df 
data <- data.frame(study_hours = study_hours, exam_scores = exam_scores)

# display and simply check data
head(data)

##   study_hours exam_scores
## 1           2          52
## 2           4          61
## 3           5          68
## 4           6          71
## 5           7          74
## 6           8          77

# fitting linear regression model
model <- lm(exam_scores ~ study_hours, data = data)

# look at results
summary(model)

## 
## Call:
## lm(formula = exam_scores ~ study_hours, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.0910 -0.9077  0.3260  1.1500  4.4000 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  52.5170     0.9556   54.95   <2e-16 ***
## study_hours   2.7870     0.0976   28.55   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.061 on 28 degrees of freedom
## Multiple R-squared:  0.9668, Adjusted R-squared:  0.9656 
## F-statistic: 815.4 on 1 and 28 DF,  p-value: < 2.2e-16

# get coefficients
coef(model)

## (Intercept) study_hours 
##   52.517040    2.786996

# R squared
summary(model)$r.squared

## [1] 0.9667993

# making the scatterplot
ggplot(data, aes(x = study_hours, y = exam_scores)) +
  geom_point() +
  geom_smooth(method = "lm", se = TRUE) +
  labs(title = "Exam Score vs Study Hours",
       x = "Study Hours",
       y = "Exam Score") +
  theme_minimal()

# calculating residuals
fitted_values <- predict(model)
residuals_calc <- data$exam_scores - fitted_values


# making the dataframe for plotting
plot_data <- data.frame(
  fitted = fitted_values,
  residuals = residuals_calc
)

# plot residuals
ggplot(plot_data, aes(x = fitted, y = residuals)) +
   geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  labs(title = "Residual Plot",
       x = "Fitted Values",
       y = "Residuals") +
  theme_minimal()

# add residuals to data for 3d plot
data$residuals = residuals_calc

data$fitted_vals = fitted_values

# making the 3d plot vi
p <- plot_ly(data = data, 
             x = ~study_hours, 
              y = ~exam_scores, 
             z = ~residuals,
              type = "scatter3d", 
             mode = "markers",
             marker = list(size = 4, color = ~residuals))

# adding lables
p <- p %>% layout(title = "3D Plot: Study Hours, Exam Scores, and Residuals",
                 scene = list(
                   xaxis = list(title = "Study Hours"),
                   yaxis = list(title = "Exam Score"),
                   zaxis = list(title = "Residuals")
                 ))

p

library(ggplot2)
library(plotly)


study_hours <- c(2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 3, 5, 7, 9, 11, 13, 15, 4, 6, 8, 10, 12, 14, 16, 5, 7, 9)
exam_scores <- c(52, 61, 68, 71, 74, 77, 79, 81, 84, 87, 89, 91, 93, 58, 66, 73, 78, 83, 88, 92, 63, 70, 76, 80, 85, 90, 95, 67, 75, 82)
data <- data.frame(study_hours = study_hours, exam_scores = exam_scores)

# fit model
model <- lm(exam_scores ~ study_hours, data = data)

# first plot
ggplot(data, aes(study_hours, exam_scores)) + 
  geom_point() + 
  geom_smooth(method = "lm")

# residuals plot  
residuals_calc <- data$exam_scores - predict(model)
 plot_data <- data.frame(fitted = predict(model), 
                       residuals = residuals_calc)
ggplot(plot_data, aes(fitted, residuals)) + geom_point()

# 3d plot
data$residuals <- residuals_calc
plot_ly(data, x = ~study_hours, y = ~exam_scores, z = ~residuals, 
        type = "scatter3d", mode = "markers")

• The more sutdents study, the better the test results.
• The model shows study hours is an important factor and predictor.
• R-squared tells us how much variation is explained.
• Residual plots helped me check if assumptions are met.