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Least Squares Regression is a popular method for finding the best-fitting line through a set of data points. It does this by minimizing the sum of the squared differences between each data point and the line itself. In simple terms, it tries to make the data points as close to the line as possible. The equation resulting from least squares regression is usually in the form \[ \hat{y} = a + b_1x_1 + b_2x_2 + \ldots + b_nx_n \],where \(\hat{y}\) is the predicted value, \(a\) is the y-intercept, and \(b_1, b_2, \ldots, b_n\) are the coefficients for the independent variables \(x_1, x_2, \ldots, x_n\).
These coefficients are calculated so that the line best represents the data, balancing any over-predictions with under-predictions across all points collected.

The Correlation Coefficient, denoted as \(r\), is a measure of the strength and direction of a linear relationship between two variables.
It can range from -1 to 1, where:

• -1 means a perfect negative linear relationship,
• 0 indicates no linear relationship, and
• 1 represents a perfect positive linear relationship.

In the context of regression, the correlation coefficient helps to describe how well the data points fit along the predicted line. Larger absolute values of \(r\) indicate stronger relationships. In multiple regressions, although individual pairwise correlation values are less relevant, understanding the relationship between predictors and the dependent variable is key for interpreting the regression model.

When dealing with multiple regression, the concept of Multiple Correlation comes into play. Here, we talk about \(R\), the multiple correlation coefficient, which measures the strength of the relationship between the dependent variable and multiple independent variables.
Unlike the simple correlation coefficient, \(R\) takes into account how these independent variables, in combination, affect the dependent variable. The square of \(R\) is known as \(R^2\), which indicates the proportion of the variance in the dependent variable that is predictable from the independent variables. Adding a new variable into a multiple regression usually increases or maintains the \(R^2\), never decreasing it, since it allows the model to possibly capture more variance.

Variance Explained, often expressed as \(R^2\), is a principle gauge of the effectiveness of a regression model.
When we mention that a model "explains" variance, it implies that the model can account for a certain proportion of the variability seen in the data set. For example, an \(R^2\) of 0.70 suggests that 70% of the variance in the output variable can be explained by the inputs used in the model.
This is a powerful concept because it provides insight into how well the regression model predicts the outcome. When a new predictor is added, the sum of squared deviations decreases, meaning that either more variance is explained or the model remains the same, ensuring that the Variance Explained (\(R^2\)) does not decrease.