BUSINESS STATISTICS NOTES
B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)
REGRESSION ANALYSIS
Meaning of Regression Analysis
Regression is the measure of the
average relationship between two or more variable in terms of the original
units of the data. It is a statistical tool with the help of which the unknown
values of one variable can be estimated from known values of another variable.
In the words of Ya Lum Chou,
“Regression analysis attempts to establish the nature of the relationship
between variables – that is, to study the functional relationship between the
variable and thereby provide a mechanism for prediction, or forecasting.”
It is clear from above definitions
that regression analysis is a statistical device with the help of which we are
in a position to estimate the unknown values of one variable from known values
of another variable.
Table of
Contents |
1. Meaning and Properties of Regression
Analysis 2. Significance and Limitations of Correlation
Analysis 3. Various Types of Regression
analysis 4. Difference between Correlation
and Regression 5. Regression lines. Why there are
two regression lines? 6. Standard error of estimates ALSO READ: REGRESSION ANALYSIS
COMPLETE FORMULA (COMING SOON) ALSO READ: CORRELATION AND
REGRESSION ANALYSIS MCQs |
Characteristics of regression coefficients:
1. Both regression co-efficients will
have the same sign.
2. If one regression co-efficient is
above unity, then the other regression co-efficient should be below unity.
3. If both the regression co-efficient
are negative, correlation co-efficient should be negative
4. Regression co-efficients are
independent of change of origin but not of scale.
Uses and Significance of Regression Analysis
The following are main Advantages of
regression analysis:
(1) Helpful to statisticians: The
study of regression helps the statisticians to estimate the most probable value
of one variable of a series for the given values of the other related variables
of the series.
(2) Nature of relationship: Regression
is useful in describing the nature of the relationship between two variables.
(3) Estimation of relationship: Regression
analysis is widely used for the measurement and estimation of relationship
among economic variables.
(4) Predictions: Regression analysis
is helpful in making quantitative predictions on the basis of estimated
relationship among variables.
(5) Policy formulation: The
predictions made on the basis of estimated relationship are used in policy
making.
Limitations of Regression analysis
The following are the main limitation
of regression:
1) No change in relationship: Regression
analysis is based on the assumption that while computing regression equation;
the relationship between variables will not change.
(2) Conditions: The application of
regression analysis is based on certain conditions like, for existence of
linear relationship between the variables; exact values are needed for the
independent variable.
(3) Spurious relationships: There may
be nonsense and spurious regression relationships. In such case, the regression
analysis is of no use.
Kinds of Regression Analysis
Kinds of regression may be studied on
the basis of:
I. Change in proportions.
II. Number of variation.
(I) Basis of change in proportion: There
are two important regressions on the basis of change in proportion. They are:
(a) Linear regression: Regression is
said to be linear when one variable move with the other variable in fixed
proportion
(b) Non-linear regression: Regression
is said to be non-linear when one variable move with the other variable in
changing proportion.
(II) On the basis of number of
variables: On the basis of number of variables, regression may be:
(a) Simple regression: When only two
variables are studied it is a simple regression.
(b) Partial regression: When more than
two variables are studied keeping other variables constant, it is called
partial regression.
(c) Multiple regressions: When at
least three variables are studied and their relationships are simultaneously
worked out, it is a case of multiple regressions.
Distinguish between correlation and regression.
There are some basis difference
between correlation and regression:
(1) Nature of relationship: Correlation
explains the degree of relationship, whereas regression explains the nature of
the relationship.
(2) Causal relationship: Correlation
does not explain the cause behind the relationship whereas regression studies
the cause and effect relationship.
(3) Prediction: Correlation does not
help in making prediction whereas regression enables us to make prediction.
(4) Origin and scale: Correlation
coefficient is independent of the change of origin and scale, whereas
regression coefficient is independent of change of origin but not of scale.
(5) Nature of variables: Correlation
analysis does make any difference between dependent and independent variable.
On the other hand, regression analysis makes difference between dependent and
independent variable.
Regression lines:
A line of regression by the method of
“least square” shows an average relationship between variables under study.
This regression line can be drawn graphically or derived algebraically. A line
fitted by method of least square is known as the line of best fit. There are
two regression lines:-
Regression line of x on y: Regression
line of x on y is used to predict x for a given value of y. The regression
equation of x on y is x=a+by.
Regression line of y on x: Regression
line of y on x is used to predict y for a given value of x. The regression
equation of y on x is y=a+bx
Why do we
generally have two regression equations?
Two regression lines: We know that
there are two lines of regression: - x on y and y on x. For these lines, the
sum of the square of the deviations between the given values and their
corresponding estimated values obtained from the line is least as compared to
other line. One regression line cannot minimise the sum of squares for both the
variables that is why we are getting two regression lines. (We get one
regression line when r = +1 and Two regression lines will be at right angles
when r = 0.)
Standard
error of estimate
With the help of regression equations, perfect prediction of values is not possible. In order to measure the accuracy of estimated figures, a statistical tool is used which is known as standard error of estimate. Calculation of standard error of estimate, symbolized as S_{xy }similar to standard deviation. Standard deviation measures the dispersion about an average, such as mean. The standard measure of estimate measures the dispersion about an average line, called the regression line. The formula for calculating the standard error of estimate is:
The standard error of estimate measures the accuracy of the estimated figures. The smaller the values of standard error of estimate, the closer will the actual value and estimated value. If standard error of estimate is zero, then there is no variation.
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