Significance Test of Regression parameter (ANOVA Test)
SIGNIFICANCE TEST OF REGRESSION PARAMETER
In the significance test, of the regression coefficient, we test whether the given regression coefficient is significant or not. In another word, these tests are performed to know the relation between the dependent and independent variables. There are different tests for regression coefficients which are discussed below.
ANALYSIS OF THE VARIANCE (ANOVA) TEST
ANOVA test is performed to test the linearity of the regression equation as well as the goodness of fit of parameter or overall regression equation. It means this test is performed to test the relation between the dependent and independent variables. It shows whether it is different between the observed or calculated value of a parameter or not also. Hence it measures the goodness of fit of regression parameter or regression line.
To test, we use the F ration test. Ftest is the same as the Z test and Ttest but in the F test, it can test significance for large numbers of variables I.e. more than two variables. In this case, we are dealing with two variables I.e. Y and X hence we use only oneway ANOVA.
We use the following procedure to test the significance of the regression line using the ANOVA test.
Null hypothesis (H_{0}): β = 0
There is no relationship between the dependent and independent variables Y and X. or the test is not significant
Alternative Hypothesis (H_{1}): β ≠ 0 (twotailed test)
There is a linear relationship between the dependent and independent variables. Or test is significant
TEST STAT
F statistics test for simple regression equation Y = α + βX is given by
Where explained variance
where K is the number of variables in the regression. In this case two variables.
And Unexplained variance
where K is the number of variables in regression and n is the sample size or numbers of observations.
CONSTRUCTION OF ANOVA TABLE
ANOVA table for the test of significance of the regression line is given below:
Source of variation  sum of squares  Degree of Freedom  Mean sum of squares  Fstatistics 
Regression  Unexplained  K1  
Error or Residual  Explained  nk  
Total  SST  n1 
DECISION
If the tabulated value of Fration at numerator degree of freedom K1 and denominator degree of freedom nk is greater than the calculated value then H_{0 }is accepted otherwise H_{0} is rejected it means an alternative Hypothesis is accepted.
1.Consider the following data for the supply and price of a commodity for the last seven years.
year  1980  1981  1982  1983  1984  1985  1986 
supply  80  84  86  88  92  96  97 
Price  12  11  15  15  18  16  18 
Estimate the most likely price in 1981 when the supply is 110. Also, calculate the explained and total variation also interpret them using ANOVA test.
Answer:
Let us consider X denotes the supply and Y denotes the price then we have
X  Y  x=XxÌ…  y=YyÌ…  xy  x^{2}  y^{2} 
80  12  9  3  27  81  9 
84  11  5  4  20  25  16 
86  15  3  0  0  9  0 
88  15  1  0  0  1  0 
92  18  3  3  9  9  9 
96  16  7  1  7  49  1 
97  16  7  3  24  64  9 
∑X=623  ∑Y=105  ∑ xy=87  ∑x^{2}=238  ∑y^{2}=44 
Now,
Mean of X i.e. average of supply
Mean of Y i.e. average of price
Let us consider the regression line
Where Y and X are dependent variables and independent variables and a and b are the regression parameter.
Now first of all we try to find the value of a and b which are regression parameters.
Equation (1) can be written as the
Subtracting equation (2) from (1) we get
Multiplying both sides by x and taking summation both side we get
b=0.365
From equation 2 we get
a= 17.48
Now when supplying i.e. X=110, price Y =?
YÌ‚ =22.67 (estimated price)
Sum of a square of the regression
Sum of the square of the total
SST=SSE+SSR
Sum of the square of the error
SSE=SSTSSR=4412.20=31.80
Now,
Source of variation  sum of squares  Degree of Freedom  Mean sum of squares  Fstatistics 
Regression  =12.20  K1=21=1 
12.20/1=12.20 
=6.36/12.20 =0.52 
Error or Residual  =31.80  nk=72=5 
=31.80/5=6.36 

Total  SST=44  n1=6 
Null hypothesis (H_{0}): β = 0
There is no relationship between the dependent and independent variables supply and price X. or the test is not significant
Alternative Hypothesis (H_{1}): β ≠ 0 (twotailed test)
There is a linear relationship between the dependent and independent variables i.e. between supply X and price Y. Or test is significant
TEST STAT
F statistics test for simple regression equation Y = α + βX is given by
DECISION
the tabulated value of Fration at numerator degree of freedom K1=1 and denominator degree of freedom nk =71=5 is 6.61 which greater than the calculated value 0.52 hence H_{0 }is accepted i.e. test is significant.
Note: this table shows F value at a 5% level of significance only. FOr another level of significance use another table.
1. first of all find the value of the denominator degree of freedom. Here we have nk =71=5 where n is the total number of observations (count on the given question) and k is a number of variables. here we have two variables X and Y hence K=2.
2. Now, find the value of the numerator degree of freedom. Here we have K1=21=1
3. we have numerator 1 and denominator 5 so from the table, the value is 6.61.
1. what is Ftest? describe ANOVA test for regression Parameter.
2. How Ftest is different From the Ttest and Ztest?
3. What is the Significance test of regression Parameter?
4. What is the difference between oneway ANOVA and Two way ANOVA?
5. what are SSR and SSE? write down the formula for each.
6. What is the degree of Freedom?
7. if there are only two variable X and Y, does it means K=2?
8. How do you derive Yestimated from Y = α + βX?
9. What is Error or residual?
10. linearity and goodness of fit are related to the ANOVA test?
11. Does ANOVA mean analysis of variance?
12. what is a variance?
 VIA
 Edubomb
 SOURCS
 By G.B. Budhathoki
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