formula of correlation coefficient of multiple regression
The formula for the correlation coefficient (r) of multiple regression can be calculated using the following steps:
1. Calculate the predicted values (Y') for each observation using the multiple regression equation:
Y' = b0 + b1*X1 + b2*X2 + ... + bn*Xn
Where b0, b1, b2, ..., bn are the regression coefficients and X1, X2, ..., Xn are the predictor variables.
2. Calculate the sum of squares total (SST) using the formula:
SST = Σ((Y - Ȳ)^2)
Where Y is the observed dependent variable and Ȳ is the mean of the observed dependent variable.
3. Calculate the sum of squares regression (SSR) using the formula:
SSR = Σ((Y' - Ȳ)^2)
4. Calculate the sum of squares error (SSE) using the formula:
SSE = Σ((Y - Y')^2)
5. Calculate the correlation coefficient (r) of the multiple regression using the formula:
r = √((SSR / SST))
Note: The correlation coefficient (r) ranges from -1 to +1, where -1 indicates a perfect negative relationship, +1 indicates a perfect positive relationship, and 0 indicates no relationship between the variables.
The formula for the correlation coefficient (R) of multiple regression is derived from the coefficient of determination (R²). The formula for R is the square root of R². Here's the step-by-step process to calculate R:
Step 1: Calculate the coefficient of determination (R²)
R² is calculated by dividing the explained sum of squares (ESS) by the total sum of squares (TSS), and it represents the proportion of variance in the dependent variable that is explained by the independent variables.
R² = ESS / TSS
Step 2: Calculate the explained sum of squares (ESS)
The ESS is the sum of the squared differences between the predicted values (ŷ) and the average of the dependent variable (ȳ).
ESS = Σ(ŷ - ȳ)²
Step 3: Calculate the total sum of squares (TSS)
The TSS is the sum of the squared differences between the actual values (y) and the average of the dependent variable (ȳ).
TSS = Σ(y - ȳ)²
Step 4: Calculate the correlation coefficient (R)
The correlation coefficient (R) is calculated as the square root of the coefficient of determination (R²).
R = √(R²)
Once you have calculated R, it represents the strength and direction of the linear relationship between the independent variables and the dependent variable in a multiple regression model.
The formula for calculating the correlation coefficient (R) of a multiple regression model involves several steps. Here's a general outline of the process:
1. Obtain the estimated regression coefficients (β) for each independent variable from the multiple regression model. These coefficients represent the slopes of the regression lines for each independent variable.
2. Calculate the predicted values (ŷ) for the dependent variable using the estimated regression coefficients and the observed values of the independent variables. This can be done using the formula:
ŷ = β₁ * x₁ + β₂ * x₂ + ... + βₚ * xₚ
where ŷ is the predicted value of the dependent variable, β₁, β₂, ..., βₚ are the estimated regression coefficients, and x₁, x₂, ..., xₚ are the observed values of the independent variables.
3. Calculate the residuals (e) by subtracting the predicted values (ŷ) from the observed values of the dependent variable (y):
e = y - ŷ
4. Calculate the residual sum of squares (RSS) by summing the squared residuals:
RSS = e₁² + e₂² + ... + eₙ²
where e₁, e₂, ..., eₙ are the individual residuals.
5. Calculate the total sum of squares (TSS) by summing the squared differences between the observed values of the dependent variable (y) and its mean (ȳ):
TSS = (y₁ - ȳ)² + (y₂ - ȳ)² + ... + (yₙ - ȳ)²
where y₁, y₂, ..., yₙ are the observed values of the dependent variable, and ȳ is the mean of y.
6. Calculate the coefficient of determination (R²) by taking the ratio of the explained variation (TSS - RSS) to the total variation (TSS):
R² = (TSS - RSS) / TSS
R² represents the proportion of the variance in the dependent variable that is explained by the independent variables.
7. Finally, take the square root of R² to obtain the correlation coefficient (R) of the multiple regression model.
It's important to note that the correlation coefficient (R) in multiple regression is the multiple correlation coefficient, which measures the strength and direction of the linear relationship between the dependent variable and a set of independent variables. It represents the overall goodness-of-fit of the multiple regression model.