2.-Statistics/F-value.md

@@ -8,41 +8,41 @@ The **F-value** is a test statistic used in ANOVA and regression models to asses
 
 To calculate the F-value, you need to partition the total variance in the outcome variable. This total variance is known as the **Total Sum of Squares (SST)**, and it is calculated by taking the squared differences between the observed data points and the overall mean of the outcome:
 
-\[
-SST = \sum (Y_i - \bar{Y})^2
-\]
+$$
+SST = \sum (y_i - \bar{y})^2
+$$
 
 Where:
-- $Y_i$: The observed value for each data point
-- $\bar{Y}$: The mean of the outcome variable
+- $y_i$: The observed value for each data point
+- $\bar{y}$: The mean of the outcome variable
 
 The total variance is then split into two components:
 
 1. **Explained Variance (Regression Sum of Squares - SSR)**: This is the portion of variance explained by the model. It is the difference between the predicted values from the model and the overall mean of the outcome variable:
 
-\[
-SSR = \sum (\hat{Y}_i - \bar{Y})^2
-\]
+$$
+SSR = \sum (\hat{y}_i - \bar{y})^2
+$$
 
 Where:
-- $\hat{Y}_i$: The predicted value from the model for each data point
-- $\bar{Y}$: The mean of the outcome variable
+- $\hat{y}_i$: The predicted value from the model for each data point
+- $\bar{y}$: The mean of the outcome variable
 
 2. **Unexplained Variance (Residual Sum of Squares - SSE)**: This is the variance that remains unexplained by the model, which measures how far the observed values differ from the predicted values:
 
-\[
-SSE = \sum (Y_i - \hat{Y}_i)^2
-\]
+$$
+SSE = \sum (y_i - \hat{y}_i)^2
+$$
 
 Where:
-- $Y_i$: The observed value for each data point
-- $\hat{Y}_i$: The predicted value from the model
+- $y_i$: The observed value for each data point
+- $\hat{y}_i$: The predicted value from the model
 
 Next, you calculate the **Mean Squares** for both the explained and unexplained variances. These values account for the number of predictors and observations in the model. The **Mean Square Between (MSB)**, representing the explained variance, is calculated by dividing the **SSR** by the number of predictors ($p$):
 
-\[
+$$
 MSB = \frac{SSR}{p}
-\]
+$$
 
 Where:
 - $SSR$: The regression sum of squares (explained variance)
@@ -50,9 +50,9 @@ Where:
 
 The **Mean Square Within (MSW)**, representing the unexplained variance, is calculated by dividing the **SSE** by the degrees of freedom (the number of observations $n$, minus the number of predictors $p$, and minus one):
 
-\[
+$$
 MSW = \frac{SSE}{n - p - 1}
-\]
+$$
 
 Where:
 - $SSE$: The residual sum of squares (unexplained variance)
@@ -61,9 +61,9 @@ Where:
 
 Finally, the **F-value** is calculated by taking the ratio of the explained variance (MSB) to the unexplained variance (MSW):
 
-\[
+$$
 F = \frac{MSB}{MSW}
-\]
+$$
 
 This ratio indicates how much more variance is explained by the model compared to the variance that remains unexplained. A large F-value suggests that the model explains much more variance than what is left unexplained, meaning the model is statistically significant. Conversely, a small F-value indicates that the model is not significantly better than using the mean to predict the outcome.