y-hat

Y-hat, denoted as ŷ, is a concept found in statistics and regression analysis, representing the predicted value of the dependent variable (y) based on the constructed linear regression model. It acts as an estimate of y's true value, computed utilizing the independent variable (s) (x) along with the determined coefficients from the regression model.

The Role of y-hat in Regression Analysis

The goal of regression analysis is to depict the relationship between a dependent variable and one or multiple independent variables. In this context, y-hat emerges as the projected value of the dependent variable, offering a comparison point to the actually observed values. Differences between these predicted and actual values enable researchers to gauge the accuracy and efficacy of the regression model.

Calculating y-hat in Simple Linear Regression

In the context of simple linear regression, y-hat calculation follows this equation:
ŷ = b0 + b1 * x

In this formula:

ŷ stands for the predicted value of the dependent variable
b0 signifies the y-intercept of the regression line
b1 represents the slope of the regression line
x denotes the value of the independent variable

Calculating y-hat in Multiple Regression

In multiple regression scenarios, y-hat's calculation involves a comparable formula, yet incorporates multiple independent variables:
ŷ = b0 + b1 * x1 + b2 * x2 +... + bn * xn

Within this equation:

ŷ stands for the predicted value of the dependent variable
b0 represents the y-intercept of the regression plane
b1, b2,..., bn denote coefficients assigned to each independent variable
x1, x2,..., xn signify values attributed to the independent variables

Evaluating Model Performance Using y-hat

Upon computation of y-hat values, these estimates aid in assessing the performance of the regression model. Common techniques for model performance assessment encompass:
Residuals: These represent the discrepancies between observed values (y) and predicted values (ŷ), termed as residuals. Models with smaller residuals generally indicate a superior fit.
Coefficient of Determination (R²) : R² provides a measure of the total variation in the dependent variable explained by the regression model. A higher R² value is indicative of a better fit.