Unformatted text preview: Lessons in Business Statistics
Prepared By
P.K. Viswanathan Chapter 10: Correlation and Regression Introduction Managers very often have to assess the nature and degree
of relationship between variables. For example, a
marketing manager would like to know the degree of
relationship between advertising expenditure and the sales
volume. Normally, you expect a positive relationship
between sales and advertising expenditure. The manager
would like to know whether money spent on advertising is
justified in terms of sales generated; flat 10 percent
increase in advertisement expenditure will result in how
much extra sales volume? This type of question could be
answered by Correlation and Regression. This Chapter
covers the nittygritty of correlation and regression. 1) What is Correlation?
The manager of the business environment of today is very
often interested in finding out whether there is any
association between two or more variables and if it is true,
he would like to know the strength of relationship between
the variables. The strength of relationship is also known
as the degree of relationship. In the previous Chapter, we
have provided a conceptual framework of the Chi Square
distribution that does try to provide some answer to the
question of finding out whether the two attributes in a
contingency table or associated or not. The degree of
relationship between two variables can be elegantly
worked out by correlation coefficient when the variables
are intervally scaled. 1) What is Correlation?Continues
What is the correlation between demand and price of a
product? For all normal commodities we know that when
price increases, the demand decreases and when price
decreases, the demand increases. Economists call this
inverse relationship between demand and price as the price
elasticity of demand. So, logically speaking the correlation
coefficient between demand and price must be negative. 2) Insights into Correlation 2) Insights into Correlation
Positive Correlation: As the value of one variable increases,
the value of the other variable also increases. For example
you normally expect a positive correlation between
advertising and sales. As you increase the amount spent on
advertising, the sales volume will also increase. 2) Insights into Correlation
Negative Correlation: As the value of one variable increases, the
value of the other variable decreases. For example the correlation
between demand and price is negative for all normal commodities.
The economists say the price elasticity is negative meaning the
relationship between demand and price is negative. 2) Insights into Correlation
No Correlation: At times, we may not be able to find any
correlation pattern. It may be a case of absence of correlation.
We say that no linear correlation is observed. This is because the
correlation coefficient that we apply in practice is based on a
linear relationship. 2) Insights into Correlation 2) Insights into Correlation
For a sample of n observations selected on two variables X and Y, the
sample correlation coefficient of Karl Pearson is defined as follows:
r (X X)(Y Y)
(X
(Y X) Y)
2 2 This is also known as Product Moment Correlation.
Here r represents the sample correlation coefficient. 2) Insights into Correlation Properties of Correlation Coefficient
The correlation coefficient is a pure number independent
of unit of measurement and scale. The value of r will not
change if X and Y are converted into U and V by
transformation of scale. The correlation coefficient always lies between –1 and +1 The three extreme positional values of r are shown below: 2) Insights into Correlation
Example: The following data refer to two variablespromotional expenses (Rs. Lakhs) and sales (1000
units) collected in the context of a promotional study.
Calculate the correlation coefficient and comment.
Promotional
Expenses
7
10
9
4
11
5
3 Sales
12
14
13
5
15
7
4 2) Insights into Correlation
Solution: The basic calculations are shown in the
following spreadsheet. 2) Insights into Correlation
In the spreadsheet calculations
shown above, in the first two
columns, the numbers 7 and 10
in the bottom row are the mean
of X and Y. That is
X = 7 and Y = 10 . Likewise,
(X X )(Y Y ) = 83 ,
(X X ) 2 = 58 and (Y Y ) 2 = 124. 2) Insights into Correlation
r (X X)(Y Y)
=
(X X ) (Y Y)
2 2 83
(58)(124 ) = 0.9787. Comments: The promotional expense is
strongly associated with sales and the
correlation is very close to 1. 3) Basics of Regression
Need for Regression
The Pearson’s correlation coefficient gives you just the
degree of relationship or association. It cannot help you
estimate or predict the response variable for a given
independent variable. The response variable is called the
dependent variable. In the example we have taken for the
correlation coefficient, ‘promotional expense’ is the
independent variable and ‘sales’ is the dependent variable.
Sales depend on promotional expense. Using regression
analysis, it is possible to predict sales for a given promotion
expense. For business planning and forecasting, regression is
much more useful than correlation. 3) Basics of Regression 4) Regression Model
Simple Linear Regression Model: In this model, dependent variable is a linear function of one independent variable. For example, demand
may be structured as a linear function of price. Based on sample
data collected for the dependent and independent variable, a model
is postulated connecting the dependent variable with the
independent variable in a linear equation form. Symbolically, we
write the sample regression line as follows:
Y = a+bx
where
Y is the dependent variable
X is the independent variable
a and b are constants.
a and b are determined by statistical least square method. b is
called the regression coefficient(slope) and a is the constant term
(intercept). 4) Regression Model
Historical Perspective
Just for knowledge sake, it is worth pointing out here that
the estimates for a and b obtained by least square method
are called ‘Best Linear Unbiased Estimates’ (BLUE) first
pioneered by Gauss and Markoff in the context of
General Linear Models that take care of Multiple Linear
Regression as well. 4) Regression Model
Values of a and b in the case of simple linear regression model
The values of a and b are obtained by solving the normal equations that are given below: Y na b X YX a X b X 2 Here Y is the dependent variable, X is the indepe ndent variable, and n is the sample size.
Solving these two normal equations,
You will find b= (X X )(Y Y )
( X X ) b
a= Y X 2 4) Regression Model
Multiple Linear Regression Model: Whenever we
are interested in the combined influence of several
independent variables upon one dependent variable,
our model is that of multiple regression. Demand for
example, may be a function of price, income of the
consumer, advertising expense, industrial growth,
and competitor’s price. When all these independent
variables change, what happens to the demand is a
study of multiple linear regression. 4) Regression Model
Example: The following data refer to two
variablespromotional expenses(Rs. Lakhs)
and sales(1000 units) collected in the context
of a promotional study. Set up the simple
linear regression model and predict sales
when promotional expense is Rs.13 lakhs. 4) Regression Model
Promotional
Expenses
7
10
9
4
11
5
3 Sales
12
14
13
5
15
7
4 4) Regression Model 4) Regression Model
You postulate the model in the standard form as follows:
Y = a+bx
where
Y is the dependent variable
X is the independent variable
a and b are constants.
As already worked out by solving the two normal equations,
b= Y
(X X)(Y )
( X X )
2 = (83/58) = 1.4310 b
a = Y X = 101.4310(7) = .017 So the fitted equation is
Y = 0.017+1.4310X. This is the line of best fit. 4) Regression Model To predict the sales when promotional expense
=13, put X =13 in the fitted equation, you will
get the answer.
Y = 0.017+1.4310(13) = 18.59. The estimated
sales when promotional expense is Rs. 13 lakhs
is = 18.59(1000) units =18590. 4) Regression Model
The concept of Coefficient of Determination for
Statistical Validity
R2 is called the coefficient of determination. This
gives the contribution made by regression in
explaining the variations in the dependent variable.
This is worked out as a ratio between the regression
sum of square and the total sum of square. In other
words, R2 measures the % variation in the dependent
variable as explained by the independent variable.
Closer the value of R2 to 1, greater is the veracity of
the model. To calculate , you need the following
terms. 4) Regression Model
(Ye )2
Y
Regression Sum of Squares = Error Sum of Squares = Total Sum of Squares = (Y e ) 2
Y (Y ) 2
Y Where Ye is the estimated value of Y for a given X. This is obtained from the
fitted line of regression.
Please note
Total Sum of Squares = Regression Sum of Squares+ Error Sum of Squares 4) Regression Model
Basic Calculations for the Example 4) Regression Model
From the spreadsheet, we have the following:
Total Sum of Squares = (Y ) 2 =
Y Regression Sum of Squares =
Error Sum of Squares = (Ye )2 =
Y (Y e ) 2 =
Y 124.00
118.78
5.22 2 R = (Regression Sum of squares/Error Sum of Squares) = (118.72/124) =
0.9579
The interpretation is 95.79% of the variations in sales is explained by
promotional expense and only about 4.21% is explained by the error
or residual term. So, the model fitted is fairly accurate. 4) Regression Model
Things to do in a Simple Linear Regression Model Postulate the model Y =a+bX. Enter the sample data for X and Y in Microsoft Excel. Perform the Regression Analysis and get the summary
output from Excel Write the Regression Equation using the intercept and coefficient of X from Excel summary output. Predict Y for a
given X Validate the model statistically by looking at R2 as well as F
statistic in the ANOVA that tests the null hypothesis of no
linear relationship. After statistical validation use the model for estimation and
prediction 4) Regression Model
Multiple Linear Regression Model
Multiple linear regression is a logical extension of the simple
linear regression. The number of independent variables will
be more than one. The same procedure of setting up the
model is followed as in the case of simple linear regression.
When the number of independent variables increases,
Microsoft Excel is the only way out. Doing the calculations
using a calculator is not only very tedious but also error
prone. If you want to do a multiple regression model
involving 10 independent variables using a calculator, you
must be crazy! The best way to understand how multiple
regression works in practice is through an example. 4) Regression Model
Example: Eight patients underwent an operation in a hospital.
Measurements of weight(kg), duration of operation(minutes),
and blood loss(ml) were taken. The hospital authorities would
like to know whether the blood loss was related to weight and
duration of operation. The data are as follows:
Weight(X1) 44
42
70
45
50
51
36
53 Duration of
Operation
(X2)
108
85
88
114
110
101
97
121 Blood
Loss(Y)
505
492
472
506
484
492
515
466 4) Regression Model 4) Regression Model
1) Regression equation is Y = 584.4716 1.35887X1 0.25783X2
2) From regression Statistics on top R2 =0.6551. This means that 65.51%
of variations in blood loss is explained by weight and duration. About
35.49% are accounted by error. The R2 value suggests that the model is
not robust and more factors will have to be added. Let us see what
ANOVA concludes.
3) In ANOVA, calculated F value is 4.75 and F significance is 0.0699(Pvalue). Since the P value is more than the level of significance 0.05,
accept the null hypothesis of no linear relationship between Blood loss
and weight and duration. You get the same conclusion by working out F
critical using the paste function. F or F table in Appendix G. Critical for
F(2,5) for 5% is = 5.79. Calculated F is less than critical F. So, accept
the null hypothesis. 4) Regression Model
Limitations of Multiple Regression Model The most crucial assumption made is that the independent variabl es are not
correlated with each other. If they are correlated, then the reg ression coefficients
cannot be estimated. This problem is called multicollinearity. The procedure
followed for resolving multicollinearity is to drop the independent variable that
has the highest standard deviation and then rework the model again. You may also
like to use twostage least square method that is part of econometrics. The other
way is to transform a set of correlated independent variables into an uncorrelated
set of variables by the technique called principal component analysis. This is an
advanced technique requiring the help of advanced statistical software like SPSS. When there are wild fluctuations in one or more of the independe nt variables,
multiple regression model crumbles and will be highly unreliable. In order to use the multiple regression model for prediction, you have to first
predict the values of the independent variables using some other prediction
method. In forecasting problems, multiple regression at best can work fo r short and
medium term only. It cannot be successfully used for long term forecasting. ...
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 Spring '12
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