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Shalabh and S. S. Dhar
regression consists of collection of data on input and output variables and finds the
statistical relationship between them.
2.1 Model Description
Let the outcome of an experiment, denoted by dependent or study variable y, depends
upon k variables X 1 , X 2 , . . . , X k , called as covariates or explanatory variables. The
experiment is conducted n times by assigning different values to X 1 , X 2 , . . . , X k as
X 1 = x 1i , X 2 = x 2i , . . . , X k = x ki , i = 1, 2, . . . , n and respective observations on
the outcome y 1 , y 2 , . . . , y n are obtained. So ith set of observation (numerical values)
is represented as (y i , x 1i , x 2i , . . . , x ki ), i = 1, 2, . . . , n. Let the joint relationship
between y and X 1 , X 2 , . . . , X k is linear with respect to the parameters β 1 , β 2 , . . . , β k
in the sense that
∂ E(y)
∂β j
does not depend on any of the β’s and is described as
y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β k X k + ε,
(1)
which is satisfied by each set of observations (y i , x 1i , x 2i , . . . , x ki ), i = 1, 2, . . . , n
as
y i = β 0 + β 1 x 1i + β 2 x 2i + · · · + β k x ki + ε i .
(2)
In practice, many random factors affect the collection of data sets
(y i , x 1i , x 2i , . . . , x ki ), i = 1, 2, . . . , n which may violate the equality sign in the
exact joint relationship between y and X 1 , X 2 , . . . , X k . To take care of the effects
of random factors, a random error or disturbance term ε is introduced in the model
(1), which absorbs the random effects that are present in every set of observation
through (2) and ensures an equality sign in (1). This model (1) is called a multiple
linear regression model with k covariates, and the parameters β j ( j = 0, 1, . . . , k)
are called the regression coefficients. In particular, β 0 is called as an intercept term
and β 1 , β 2 , . . . , β k are called as slope parameters. This model describes a hyperplane
in the k-dimensional space of the explanatory variables X j .
It is more convenient to deal with the multiple regression models when the variables and the observations on them are expressed in matrix notations. This allows a
very compact display of the model, data, and results. In matrix notation, the model
is obtained by combining the n equations with k explanatory variables in (1) as
y = Xβ + ε,
(3)
where y = (y 1 , y 2 , . . . , y n )
T is a n × 1 vector of n observation on dependent or study
variable;
Shalabh and S. S. Dhar
regression consists of collection of data on input and output variables and finds the
statistical relationship between them.
2.1 Model Description
Let the outcome of an experiment, denoted by dependent or study variable y, depends
upon k variables X 1 , X 2 , . . . , X k , called as covariates or explanatory variables. The
experiment is conducted n times by assigning different values to X 1 , X 2 , . . . , X k as
X 1 = x 1i , X 2 = x 2i , . . . , X k = x ki , i = 1, 2, . . . , n and respective observations on
the outcome y 1 , y 2 , . . . , y n are obtained. So ith set of observation (numerical values)
is represented as (y i , x 1i , x 2i , . . . , x ki ), i = 1, 2, . . . , n. Let the joint relationship
between y and X 1 , X 2 , . . . , X k is linear with respect to the parameters β 1 , β 2 , . . . , β k
in the sense that
∂ E(y)
∂β j
does not depend on any of the β’s and is described as
y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β k X k + ε,
(1)
which is satisfied by each set of observations (y i , x 1i , x 2i , . . . , x ki ), i = 1, 2, . . . , n
as
y i = β 0 + β 1 x 1i + β 2 x 2i + · · · + β k x ki + ε i .
(2)
In practice, many random factors affect the collection of data sets
(y i , x 1i , x 2i , . . . , x ki ), i = 1, 2, . . . , n which may violate the equality sign in the
exact joint relationship between y and X 1 , X 2 , . . . , X k . To take care of the effects
of random factors, a random error or disturbance term ε is introduced in the model
(1), which absorbs the random effects that are present in every set of observation
through (2) and ensures an equality sign in (1). This model (1) is called a multiple
linear regression model with k covariates, and the parameters β j ( j = 0, 1, . . . , k)
are called the regression coefficients. In particular, β 0 is called as an intercept term
and β 1 , β 2 , . . . , β k are called as slope parameters. This model describes a hyperplane
in the k-dimensional space of the explanatory variables X j .
It is more convenient to deal with the multiple regression models when the variables and the observations on them are expressed in matrix notations. This allows a
very compact display of the model, data, and results. In matrix notation, the model
is obtained by combining the n equations with k explanatory variables in (1) as
y = Xβ + ε,
(3)
where y = (y 1 , y 2 , . . . , y n )
T is a n × 1 vector of n observation on dependent or study
variable;