 # Least Squares in Multiple Regressions

## Question

Discuss the method of least squares in multiple regressions, including:
1. Summarize what the method of least squares means.
2. What are the assumptions necessary for hypothesis testing?
3. Discuss lack of fit – how do we test for lack of fit.
4. What does it mean to have “orthogonal” predictor variables and why do we want that?
5. Discuss the “extra sum of squares” idea and how we use it.
What are sequential sums of squares? What is a partial F test?
6. Explain what Minitab does in backward elimination and in stepwise (forward and backward) automatic regression procedures.
7. Discuss the statistics shown in Minitab’s “best subsets” procedure.
8. Discuss the role of residual plots.
9. Discuss how you can code dummy variables when you have categorical predictors.
10. When is a transformation in order (how would you know)? How might you go about deciding on a transformation?

## Solution Preview

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Discuss the method of least squares in multiple regressions, including:
1. Summarize what the method of least squares means.
The multiple regression models contain more than one independent variables and it takes the form;
y = bjxi + c,………………………………..(i)
Where;
y is the dependent variable
xi is the independent variables for i=1,2,3,……….n.
bi are the coefficients for j=1,2,3,……..,n
c is the constant.
The method of least squares is used to determine the values of coefficients (bj) in a regression model. The selection is based on selecting coefficients that provide us with the smallest sum of squares. It is a method of finding lines that fit the data. These sum of squares are sums of differences between the predicted and observed values of the dependent variable of the model. The y are obtained from the vertical differences between the lines from the data

2. What are the assumptions necessary for hypothesis testing?
When testing hypothesis about a multiple regression model stated in (i) with many independent variables (xi) we make a slight modification on the assumptions of independence, distributions to have same variance, linear relationships between two and the dependent variables follows a normal distribution for any value of the independent variable.

We extend these assumptions for multiple regressions which contains multiple independent variables. These assumptions will be modified to become;
a) The relationship between the dependent variable and independent variables is linear.
b) The relationship must be linear for each set of values of the independent variable.
c) The distribution of the dependent variable is normal with a constant variable.
d) Data is drawn from an independence sample....
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