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Multiple regression analysis
The earliest form of regression was
the method of least squares, which was
published by Legendre in 1805, and by Gauss in
The term "regression" was used by British
biometrician sir Francis Galton in the (1822-
1911), to describe a biological phenomenon.
Sir Galton's work on inherited characteristics
of sweet peas led to the initial conception of
Regression is a statistical technique for investigating
and modeling the relationship between variables.
Applications of regression are numerous and occur
in almost every field, including engineering, the
physical and the social sciences, and the biological
Usually, the investigator seeks to ascertain the causal
effect of one variable upon another—the effect of a
price increase upon demand, for example, or the effect
of changes in the money supply upon the inflation rate.
Regression is the measure of the average
relationship between two or more variables in terms of
the original units of the data. It is unquestionably the
most widely used statistical technique in social
sciences. It is also widely used in biological and
Regression equation is (y) =a + b x
Slope (b) = (NΣXY-(ΣX)( ΣY)) / (NΣX2 – (ΣX)2)
Intercept (a) = (ΣY-b(ΣX)) / N
Review of Simple linear regression.
A simple linear regression is carried out to
estimate the relationship between a dependent variable, Y
and a single explanatory variable, x given a set of data
that includes observations for both of these variables for a
•For ex: A real estate agent wishes to examine the
relationship between the selling price of a home and its size
(measured in square feet)
•A random sample of 10 houses is selected
Dependent variable (Y) = house price
Independent variable (X) = square feet
Simple Linear Regression Model
The simple linear regression equation provides
an estimate of the population regression line
Estimate of the
Y value for
Value of X for
The individual random error terms ei have a mean of zero
Prediction equation is given by:
Estimation of coefficients:
Measures of Variation
Total variation is made up of two parts:
Total Sum of
Error Sum of
= Average value of the dependent variable
Yi = Observed values of the dependent variable
i = Predicted value of Y for the given Xi valueYˆ
Measures of Variation
SST = (Yi - Y)2
SSE = (Yi - Yi )2
SSR = (Yi - Y)2
Coefficient of Determination, r2
• The coefficient of determination is the
portion of the total variation in the
dependent variable that is explained by
variation in the independent variable
• The coefficient of determination is also
called r-squared and is denoted as r2
Multiple linear regression
The general purpose of multiple regression (the
term was first used by Pearson, 1908) is to learn more
about the relationship between several independent or
predictor variables and a dependent or criterion
A regression model that involves the relationship
between two or more explanatory variables and a response
variable by fitting a linear equation to observed data (more
than one regressor variable) is called a multiple regression
model. Every value of the independent variable x is
associated with a value of the dependent variable y.
Suppose that the yield in the pounds of conversation in a
chemical process depends on temperature and the catalyst
concentration. A multiple regression model that might
describe the relationship is
where y denotes the yield,x1denotes the temperature,x2
denotes the catalyst concentration. This is multiple linear
regression model with two regressor variables.
The term linear is used because equation is a linear function
of the known parameters β0,β1& β2 and ε is error term.
The parameter β1 indicates that the expected
change in response (y) per unit change in x1 when x2 is held
constant. Similarly β2 measures the expected change in (y)
per unit change in x2 when x1 held constant.
In general, the response y may be related to k regressor (or)
predictor variables. The model
y= β0+β1x1+β2x2+……………+ βkxk+ε
is a multiple linear regression with k regressors. The parameters
βj, j=0,1,…….k. are called regression coefficients.
The parameter βj represents the expected change in the response (y)
per unit change in xj when all of the remaining regressor variables xi
(i≠j) are held constant. For this reason the parameters βj, j=1,…….k are
often called partial regression coefficients.
Assumptions of Regression
• For any given set of values of x1, x2, … , xk, the random
error has a probability distribution with the following
• 1. Mean equal to 0
• 2. Variance equal to 2
• 3. Normal distribution
• 4. Random errors are independent
Regression Analysis: Model Building
• General Linear Model
• Determining When to Add or Delete Variables
• Analysis of a Larger Problem
• Multiple Regression Approach
to Analysis of Variance
General Linear Model
Models in which the parameters (β0, β1, . . . , βp)
all have exponents of one are called linear
• First-Order Model with One Predictor
y x0 1 1
y x0 1 1
Variable Selection Procedures
• Stepwise Regression
• Forward Selection
• Backward Elimination
variable at a time
is added or
the F statistic
Variable Selection Procedures
• F Test
• To test whether the addition of x2 to a model
involving x1 (or the deletion of x2 from a model
involving x1and x2) is statistically significant
The p-value corresponding to the F statistic is the
criterion used to determine if a variable should be added or
(SSE(reduced)-SSE(full))/number of extra terms
• This procedure is similar to stepwise-
regression, but does not permit a variable to
• This forward-selection procedure starts with
no independent variables.
• It adds variables one at a time as long as a
significant reduction in the error sum of
squares (SSE) can be achieved.
• This procedure begins with a model that
includes all the independent variables the
modeler wants considered.
• It then attempts to delete one variable at a
time by determining whether the least
significant variable currently in the model
can be removed because its p-value is less
than the user-specified or default value.
• Once a variable has been removed from the
model it cannot re enter at a subsequent step.
Procedure of simultaneous forward and backward
selection also available
In a stepwise regression, predictor variables are
entered into the regression equation one at a time
based upon statistical criteria.
At each step in the analysis the predictor variable that
contributes the most to the prediction equation in
terms of increasing the multiple correlation, R, is
entered first. This process is continued only if
additional variables add anything statistically to the
The choosing is done according to following
i.e.) delete x.i if ^i
2 E( 2 )(Z1
enter x.j if (n-r-2)cjq
2) > Fin=F1,n-r-2
here either pin or pout are specify the stepwise
procedure is terminated when either of the two
following points happens
We can’t enter or delete the variables according to the above criteria
i.e.) this includes the case where enter all regressor & can’t delete any.
The processor dictates that the same regressor be enter and deleted in
successive operations the stepwise selection procedure is an attempt to
achieve to insert variables in terms until the regression equation is
When additional predictor variables add anything statistically
meaningful to the regression equation, the analysis stops. Thus, not all
predictor variables may enter the equation in stepwise regression.
There are a number of multiple regression variants. Stepwise is usually
a good choice though one can enter all variables simultaneously as an
alternative. Similarly, one can enter all of the variables simultaneously
and gradually eliminate predictors one by one if elimination does little to
change the overall prediction.
Stepwise regression procedure is the best procedure when
compared to the all procedures we have see earlier.
Uses of Regression Analysis:
1.Regression analysis helps in establishing a functional
Relationship between two or more variables.
2. Since most of the problems of economic analysis are based
on cause and effect relationships, the regression analysis is a
highly valuable tool in economic and business research.
3. Regression analysis predicts the values of dependent
variables from the values of independent variables.
4. We can calculate coefficient of correlation (r) and
coefficient of determination (R2) with the help of regression
Source Degrees of freedom Sum of squares Mean Square F
Regression 2 5550.8166 2775.4083 4.7*10-16
Residual 22 233.7260 10.6239
Total 24 5784.5426
R2 = 0.9596 Adjusted R2 = 0.9559
Scatter plot for cases and