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Multiple linear regression is a statistical technique used to analyze the relationship between multiple independent variables and a single dependent variable. It is commonly used in various fields, such as economics, psychology, and social sciences, to predict and explain the behavior of a dependent variable based on the values of the independent variables.
Performing multiple linear regression by hand using a step-by-step approach involves several key steps. First, the data must be collected and organized into a matrix format. Then, the regression coefficients can be calculated using the least squares method, which involves minimizing the sum of the squared differences between the actual and predicted values of the dependent variable. This process can be done manually using mathematical equations and calculations.
Next, the significance of the regression coefficients can be determined by calculating the t-statistic and comparing it to a critical value. This helps to identify which independent variables are statistically significant in predicting the dependent variable.
Furthermore, the overall fit of the regression model can be assessed by calculating the coefficient of determination (R-squared) and the adjusted R-squared. These measures indicate the proportion of variation in the dependent variable that can be explained by the independent variables.
Finally, the results of the analysis can be interpreted and used to make predictions and draw conclusions about the relationship between the variables.
In summary, performing multiple linear regression by hand using a step-by-step approach requires careful data collection and organization, as well as the use of mathematical calculations and statistical tests to determine the significance and fit of the regression model. This process can provide valuable insights and predictions in various research and practical applications.
Multiple Linear Regression by Hand (Step-by-Step)
Multiple linear regression is a method we can use to quantify the relationship between two or more predictor variables and a response variable.
This tutorial explains how to perform multiple linear regression by hand.
Example: Multiple Linear Regression by Hand
Suppose we have the following dataset with one response variable y and two predictor variables X1 and X2:
Use the following steps to fit a multiple linear regression model to this dataset.
Step 1: Calculate X12, X22, X1y, X2y and X1X2.
Step 2: Calculate Regression Sums.
Next, make the following regression sum calculations:
- Σx12 = ΣX12 – (ΣX1)2 / n = 38,767 – (555)2 / 8 = 263.875
- Σx22 = ΣX22 – (ΣX2)2 / n = 2,823 – (145)2 / 8 = 194.875
- Σx1y = ΣX1y – (ΣX1Σy) / n = 101,895 – (555*1,452) / 8 = 1,162.5
- Σx2y = ΣX2y – (ΣX2Σy) / n = 25,364 – (145*1,452) / 8 = -953.5
- Σx1x2 = ΣX1X2 – (ΣX1ΣX2) / n = 9,859 – (555*145) / 8 = -200.375
Step 3: Calculate b0, b1, and b2.
The formula to calculate b1 is: [(Σx22)(Σx1y) – (Σx1x2)(Σx2y)] / [(Σx12) (Σx22) – (Σx1x2)2]
Thus, b1 = [(194.875)(1162.5) – (-200.375)(-953.5)] / [(263.875) (194.875) – (-200.375)2] = 3.148
The formula to calculate b2 is: [(Σx12)(Σx2y) – (Σx1x2)(Σx1y)] / [(Σx12) (Σx22) – (Σx1x2)2]
Thus, b2 = [(263.875)(-953.5) – (-200.375)(1152.5)] / [(263.875) (194.875) – (-200.375)2] = -1.656
Thus, b0 = 181.5 – 3.148(69.375) – (-1.656)(18.125) = -6.867
Step 5: Place b0, b1, and b2 in the estimated linear regression equation.
The estimated linear regression equation is: ŷ = b0 + b1*x1 + b2*x2
In our example, it is ŷ = -6.867 + 3.148x1 – 1.656x2
How to Interpret a Multiple Linear Regression Equation
Here is how to interpret this estimated linear regression equation: ŷ = -6.867 + 3.148x1 – 1.656x2
b0 = -6.867. When both predictor variables are equal to zero, the mean value for y is -6.867.
b1 = 3.148. A one unit increase in x1 is associated with a 3.148 unit increase in y, on average, assuming x2 is held constant.
b2 = -1.656. A one unit increase in x2 is associated with a 1.656 unit decrease in y, on average, assuming x1 is held constant.
Additional Resources
An Introduction to Multiple Linear Regression
How to Perform Simple Linear Regression by Hand