Scatter Plot and Properties of Residual Lecture 03

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Scatter Diagram (Plot)

The scatter diagram in the regression shows the relationship between a response variable (usually Y) and an explanatory variable (usually X). The scatter diagram is built using the explanatory variable on the x-axis and the response variable on the y-axis. Plot bi-variate data on (X, Y) on graphic paper. The relationship between response variable and explanatory variable will be linear if the plotted points portray a relationship represented by straight line otherwise the relationship between the response variable and explanatory variable will be nonlinear.

Practice Question – 1.2

Construct scatter plot for the data given in practice question – 1.1 (Lecture 02)

Solution: The data of practice question – 1 is given below:

No	X	Y	XY	square of X
1 2 3 4 5 6 7 8 9	5 6 8 10 12 13 15 16 17	16 19 23 28 36 41 44 45 50	80 114 184 280 432 533 660 720 850	25 36 64 100 144 169 225 256 289
	102	302	3853	1308

The estimated regression equation of Y on X is

Using SPSS

Residual

Th residual is measuring the amount of deviation of systematic pattern from actual. Th residual is measuring the amount of deviation of systematic pattern from actual or true value. the residual for sample is denoted by e and can be defined as follows:

The difference between observed or true value of response variable and estimated value of response variable by least squares regression model is called residual denoted by e.

The presence of a significant residual value does not imply that the study is flawed; rather, it indicates that there is still something unexplained that should be addressed by including more influential predictors in the regression model.

When the regression model is correctly and accurately defined by including all predictors in the model for a specific phenomenon or scenario, the true value of the response variable is equal to the systematic pattern, and the deviation of the systematic pattern from the true value of the response variable is zero. It signifies that the true value of the response is the same as the estimated value of the response variable.

 That’s

Keeping these considerations in mind, the residual has the following properties.

i.                    The sum of residual is zero.

proof:

ii.                    The sum of the product of regressor and residual is zero.

proof:

iii.          The sum of the product of estimated values of Y  and residual is zero.

proof 

iv.                    The sum of the product of residual and predictor is equal to the sum of square of residual.

 

proof:

Practice Question 1.3

Consider a hypothetical data given below:

X	2	4	6	8
Y	3	7	5	10

Find the least squares regression line and show that the sum of residual is equal to zero.

Solution: The least squares method is used to estimate the parameters of the simple linear regression model.

X	Y	XY	square of X
2	3	6	4	3.4	- 0.4
4	7	28	16	5.3	1.7
6	5	30	36	7.2	- 2.2
8	10	80	64	9.1	0.9
20	25	144	120	25	0

The estimated regression model:

Y^ = 1.50 + 0.95 X

• Read More: Standard Error of the Estimate