Applied Statistics Using SPSS, STATISTICA, MATLAB and R

The Yi can also be linearly modelled as:
i = β 0 + β1u
i1
+ β 2u
i2
+ i with
Y ε
7.2 Multiple Regression 301
2
i1
xi
; ui
2 xi
u = = .
As a matter of fact, many complex dependency models can be transformed into
the general linear model after suitable transformation of the variables. The general
linear model encompasses also the interaction effects, as in the following example:
i = β 0 + β1x
i1
+ β 2 xi2
+ β 3 xi1x
i2
+ i , 7.47
Y ε
which can be transformed into the linear model, using the extra
variable x i3
= xi1x
i2
for the cross-term x i1x i2
.
Frequently, when dealing with polynomial models, the predictor variables are
previously centred, replacing xi by xi − x . The reason is that, for instance, X and
X 2 will often be highly correlated. Using centred variables reduces multicollinearity
and tends to avoid computational difficulties.
Note that in all the previous examples, the model is linear in the parameters βk.
When this condition is not satisfied, we are dealing with a non-linear model, as in
the following example of the so-called exponential regression:
Y = β ) + ε
i
0 exp( β1x
i i . 7.48
Unlike linear models, it is not generally possible to find analytical expressions
for the estimates of the coefficients of non-linear models, similar to the normal
equations 7.3. These have to be found using standard numerical search procedures.
The statistical analysis of these models is also a lot more complex. For instance, if
we linearise the model 7.48 using a logarithmic transformation, the errors will no
longer be normal and with equal variance.
Commands 7.3. SPSS, STATISTICA, MATLAB and R commands used to
perform polynomial and non-linear regression.
SPSS
STATISTICA
MATLAB
R
Analyze; Regression; Curve Estimation
Analyze; Regression; Nonlinear
Statistics; Advanced Linear/Nonlinear
Models; General Linear Models; Polynomial
Regression
Statistics; Advanced Linear/Nonlinear
Models; Non-Linear Estimation
[p,S] = polyfit(X,y,n)
[y,delta] = polyconf(p,X,S)
[beta,r,J]= nlinfit(X,y,FUN,beta0)
lm(formula) | glm(formula)
nls(formula, start, algorithm, trace)