Question

How do you perform hypothesis testing on multiple regression data from ANOVA table step-by-step? Please provide example.

Answer 1

We know that the Multiple Linear regression equation is

y=a+b1x1+b2x2+...+bnxn

Where y is the dependent variable and x1,x2,...,xn are the independent variables. (Assume there are k data points)

a is the intercept

b1,b2,...,bn are the slope parameters of x1,x2,...,xn respectively

The purpose ANOVA in Multiple linear regression is to test if at least one of the independent variables are significant(i.e. if at least one of the independent variables has an impact on the dependent variable (in other words if one of the independent variables is helpful in predicting the dependent variable). This is equivalent to testing if at least one of the slope parameters namely b1,b2,...,bn   0

Null Hypothesis: Ho: b1=b2=...=bn=0

Alternate Hypothesis H1= At least one of the beta slope parameters 0

Calculation

Compute the following table

Here yi are value of the dependent variable (observed values)

is the average value of the dependent variable
is the predicted value of the dependent variable

(Note: can be found by (y1+y2+...+yk)/k

For calculating substitute the values of the xi's and bi's in the regression equation and you will get

Anova	Degrees of Freedom	Sum fof Squares	Mean Square	F statistic
Regression	n	SSR=	MSR=SSR/n	F=MSR/MSE
Residual(Error)	k-n-1	SSE=	MSE=SSE/(k-n-1)
Total	k-1	SST=

If he calculated "F" value is greater than the F- Table values at F(n,n-k-1), Then we reject Ho and conclude that At least one of the beta slope parameters 0