Elementary Statistics and Computer Application

Testing the significance of regression co-efficient

 
• To test the significance of the regression coefficient we can apply either a t test or analysis of variance (F test).
The ANOVA table for testing the regression coefficient will be as follows:

Sources of variation	d.f.	SS	MS	F
Due to regression	1	SS(b)	Sb2	Sb2 / Se2
Deviation from regression	n-2	SS(Y)-SS(b)	Se2
Total	n-1	SS(Y)


• In case of t test the test statistic is given by
• t = b / SE(b) where SE(b) = se2 / SS(X)
• The regression analysis is useful in predicting the value of one variable from the given values of another variable. Another use of regression analysis is to find out the causal relationship between variables.

Uses of Regression

• The regression analysis is useful in predicting the value of one variable from the given value of another variable. Such predictions are useful when it is very difficult or expensive to measure the dependent variable, Y. The other use of the regression analysis is to find out the causal relationship between variables. Suppose we manipulate the variable X and obtain a significant regression of variables Y on the variable X. Thus we can say that there is a causal relationship between the variable X and Y. The causal relationship between nitrogen content of soil and growth rate in a plant, or the dose of an insecticide and mortality of the insect population may be established in this way

Example 1

• From a paddy field, 36 plants were selected at random. The length of panicles(x) and the number of grains per panicle (y) of the selected plants were recorded. The results are given below. Fit a regression line y on x. Also test the significance (or) regression coefficient.
The length of panicles in cm (x) and the number of grains per panicle (y) of paddy plants.

S.No.	Y	X	S.No.	Y	X	S.No.	Y	X
1	95	22.4	13	143	24.5	25	112	22.9
2	109	23.3	14	127	23.6	26	131	23.9
3	133	24.1	15	92	21.1	27	147	24.8
4	132	24.3	16	88	21.4	28	90	21.2
5	136	23.5	17	99	23.4	29	110	22.2
6	116	22.3	18	129	23.4	30	106	22.7
7	126	23.9	19	91	21.6	31	127	23.0
8	124	24.0	20	103	21.4	32	145	24.0
9	137	24.9	21	114	23.3	33	85	20.6
10	90	20.0	22	124	24.4	34	94	21.0
11	107	19.8	23	143	24.4	35	142	24.0
12	108	22.0	24	108	22.5	36	111	23.1


• Null Hypothesis Ho: regression coefficient is not significant.
• Alternative Hypothesis H1: regression coefficient is significant





The regression line y on x is




• 115.94 = a + (11.5837)(22.86)
• a=115.94-264.8034
• a=-148.8633
• The fitted regression line is y =-148.8633+11.5837x

ANOVA Table

Sources of Variation	d.f.	SS	MSS	F
Regression	1	8950.8841	8950.8841	90.7093
Error	36-2=34	3355.0048	98.6766
Total	35	12305.8889

For t-test



Table Value:

• t(n-2) d.f.=t34 d.f at 5% level=2.032
• t >ttab. we reject Ho.
• Hence t is significant.