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Unit 9: Regression Analysis

9.4 Keywords Notes

t
Exponential trend: The general form of an exponential trend is Y = a.b , where a and b are
constants.
Least square methods: This is one of the most popular methods of fitting a mathematical trend.
The fitted trend is termed as the best in the sense that the sum of squares of deviations of
observations, from it, are minimized.
Line of Regression Y on X: The general form of the line of regression of Y on X is Y = a + bX ,
Ci i
where Y denotes the average or predicted or calculated value of Y for a given value of X = X .
Ci i
This line has two constants, a and b.
Line of Regression of X on Y: The general form of the line of regression of X on Y is X = c + dY,
Ci i
where X denotes the predicted or calculated or estimated value of X for a given value of Y = Y
Ci i
and c and d are constants. d is known as the regression coefficient of regression of X on Y.
Linear Trend: The linear trend equation is given by relation Y = a + bt. where t denotes time
t
period such as year, month, day, etc., and a, b are the constants.
2
Parabolic Trend: The mathematical form of a parabolic trend is given by Y = a + bt + ct or
t
2
Y = a + bt + ct Where a, b and c are constants.
Regression equation: If the coefficient of correlation calculated for bivariate data (X , Y ), i =
i i
1,2, ...... n, is reasonably high and a cause and effect type of relation is also believed to be existing
between them, the next logical step is to obtain a functional relation between these variables.
This functional relation is known as regression equation in statistics.
9.5 Review Questions

1. Distinguish between correlation and regression. Discuss least square method of fitting
regression.

2. What do you understand by linear regression? Why there are two lines of regression?
Under what condition(s) can there be only one line?

3. Define the regression of Y on X and of X on Y for a bivariate data (X , Y ), i = 1, 2, ...... n. What
i i
would be the values of the coefficient of correlation if the two regression lines (a) intersect
at right angle and (b) coincide?
4. (a) Show that the proportion of variations explained by a regression equation is r .
2
(b) What is the relation between Total Sum of Squares (TSS), Explained Sum of Squares
(ESS) and Residual Sum of squares (RSS)? Use this relationship to prove that the
coefficient of correlation has a value between –1 and +1.
5. Write a note on the standard error of the estimate.

6. “The regression line gives only a ‘best estimate’ of the quantity in question. We may
assess the degree of uncertainty in this estimate by calculating its standard error “. Explain.

7. Show that the coefficient of correlation is the geometric mean of the two regression
coefficients.

8. What is the method of least squares? Show that the two lines of regression obtained by this
method are irreversible except when r = ± 1. Explain.

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