How to Find the Critical Value of F for Regression ANOVA in R

In regression analysis, the ANOVA F-test is used to evaluate whether the model significantly explains the variability in the dependent variable. The F-statistic measures the ratio of the variance explained by the regression model to the variance not explained by the model (residual variance). To determine whether the F-statistic is significant, we compare it to a critical value of F at a specified significance level (usually 0.05).

• F-Statistic: Measures how much the regression model improves the fit of the data compared to a model with no predictors.
• Critical F-Value: The cutoff point to determine whether the F-statistic is statistically significant at a chosen significance level (usually 0.05).
• Degrees of Freedom: The F-distribution depends on two degrees of freedom: one for the model and one for the residual error.

Let's use a simple linear regression example to explain how to compute the F-statistic and find the critical F-value using R Programming Language.

Step 1: Load and Explore a Sample Dataset

We will use the built-in mtcars dataset, which includes measurements of fuel consumption (miles per gallon) and other car characteristics, to build a simple linear regression model.

# Load the mtcars dataset
data(mtcars)

# View the first few rows of the dataset
head(mtcars)

Output:

                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

• mpg: Miles per gallon (dependent variable).
• wt: Weight of the car (independent variable).

Step 2: Fit a Linear Regression Model

We will fit a simple linear regression model where mpg is the dependent variable and wt (weight) is the independent variable.

# Fit a linear regression model
model <- lm(mpg ~ wt, data = mtcars)

# View the summary of the regression model
summary(model)

Output:

Call:
lm(formula = mpg ~ wt, data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5432 -2.3647 -0.1252  1.4096  6.8727 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
wt           -5.3445     0.5591  -9.559 1.29e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.046 on 30 degrees of freedom
Multiple R-squared:  0.7528,	Adjusted R-squared:  0.7446 
F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

• F-statistic: 91.38
• Degrees of freedom (DF): 1 for the model and 30 for the residuals.
• P-value: 1.294e-10, which indicates that the relationship between the independent variable and the dependent variable is statistically significant.

Step 3: Find the Critical F-Value

To determine whether the F-statistic is significant, we need to find the critical value of F. This value depends on:

• Significance level (alpha): Typically 0.05.
• Degrees of freedom:
  • df1: The degrees of freedom for the model (1).
  • df2: The degrees of freedom for the residuals (30).

We can find the critical F-value using the qf() function in R:

# Set the significance level (alpha)
alpha <- 0.05

# Degrees of freedom for the model and residuals
df1 <- 1  # Model
df2 <- 30  # Residuals

# Find the critical F-value
critical_f_value <- qf(1 - alpha, df1, df2)

# Display the critical F-value
print(paste("Critical F-Value:", critical_f_value))

Output:

[1] "Critical F-Value: 4.17087678576669"

The critical F-value at a significance level of 0.05 with 1 and 30 degrees of freedom is 4.170877.

• F-Statistic: 91.38 (from the model).
• Critical F-Value: 4.17 (calculated using qf()).

Since the F-statistic (91.38) is greater than the critical F-value (4.17), we reject the null hypothesis that the slope of the regression line is zero. This means that the independent variable (car weight) significantly explains the variability in the dependent variable (miles per gallon).

Conclusion

In this guide, we demonstrated how to perform a regression ANOVA in R, calculate the F-statistic, and determine the critical F-value using the qf() function. The critical F-value is essential for comparing against the F-statistic to assess whether the regression model significantly explains the variance in the data. By following these steps, you can determine the significance of your regression models and make informed decisions in your data analysis.