Introduction

In this section, we’ll delve into the details of a linear regression analysis conducted on our dataset. The aim is to understand the impact of both a categorical variable (Category) and a quantitative variable (QuantVar) on our response variable (ResponseVar).

Data Preparation

We’ve previously prepared the dataset, recoding the categorical variable and centering the quantitative variable. Let’s quickly review those steps.

# Data Preparation
set.seed(123)
n <- 100
df <- data.frame(
  Category = sample(c("Category_of_Interest", "Other_Category"), n, replace = TRUE),
  QuantVar = rnorm(n),
  ResponseVar = rnorm(n)
)

df$Category <- ifelse(df$Category == 'Category_of_Interest', 0, 1)
df$QuantVar <- df$QuantVar - mean(df$QuantVar)

Linear Regression Analysis

Now, let’s fit a linear regression model to examine the relationships between our variables.

# Fit linear regression model
model <- lm(ResponseVar ~ Category + QuantVar, data = df)

# Display regression results
summary(model)
## 
## Call:
## lm(formula = ResponseVar ~ Category + QuantVar, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.4635 -0.6559 -0.1982  0.5091  3.0627 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.09711    0.12396   0.783    0.435
## Category    -0.19062    0.18904  -1.008    0.316
## QuantVar    -0.08419    0.09750  -0.863    0.390
## 
## Residual standard error: 0.9359 on 97 degrees of freedom
## Multiple R-squared:  0.01784,    Adjusted R-squared:  -0.002411 
## F-statistic: 0.881 on 2 and 97 DF,  p-value: 0.4177

The linear regression model is represented by the equation:

\[ \text{ResponseVar} = \beta_0 + \beta_1 \times \text{Category} + \beta_2 \times \text{QuantVar} + \epsilon \]

where: - \(\beta_0\) is the intercept, - \(\beta_1\) and \(\beta_2\) are the coefficients for Category and QuantVar respectively, - \(\epsilon\) is the error term.

Regression Coefficients

The estimated coefficients are as follows:

  • Intercept (\(\beta_0\)): Insert Interception Coefficient Here
  • Category Coefficient (\(\beta_1\)): Insert Category Coefficient Here
  • QuantVar Coefficient (\(\beta_2\)): Insert QuantVar Coefficient Here

These coefficients represent the expected change in the response variable for a one-unit change in the respective predictor variable, holding other variables constant.

Statistical Significance

The p-values associated with each coefficient are essential for assessing their statistical significance.

  • Category P-value: Insert Category P-value Here
  • QuantVar P-value: Insert QuantVar P-value Here

A small p-value (typically < 0.05) suggests that the corresponding predictor variable is statistically significant in predicting the response variable.

Model Fit

The overall model fit is evaluated through the R-squared value:

  • R-squared: Insert R-squared Value Here

This value represents the proportion of variance in the response variable explained by the model. A higher R-squared indicates a better fit.

Conclusion

In conclusion, the linear regression analysis reveals the impact of both categorical and quantitative variables on the response variable. The coefficients and p-values provide insights into the strength and significance of these relationships. Further interpretation and exploration may be needed for a comprehensive understanding of the dataset.

Thank you for reading! ```