GeneralizedLinearModel.fit
will be removed
in a future release. Use fitglm
instead.
mdl = GeneralizedLinearModel.fit(tbl)
mdl = GeneralizedLinearModel.fit(X,y)
mdl = GeneralizedLinearModel.fit(...,modelspec)
mdl = GeneralizedLinearModel.fit(...,Name,Value)
mdl =
GeneralizedLinearModel.fit(...,modelspec,Name,Value)
creates
a generalized linear model of a table or dataset array mdl
= GeneralizedLinearModel.fit(tbl
)tbl
.
creates
a generalized linear model of the responses mdl
= GeneralizedLinearModel.fit(X
,y
)y
to
a data matrix X
.
creates
a generalized linear model as specified by mdl
= GeneralizedLinearModel.fit(...,modelspec
)modelspec
.
or mdl
= GeneralizedLinearModel.fit(...,Name,Value
)
creates
a generalized linear model with additional options specified by one
or more mdl
=
GeneralizedLinearModel.fit(...,modelspec
,Name,Value
)Name,Value
pair arguments.
The generalized linear model mdl
is
a standard linear model unless you specify otherwise with the Distribution
namevalue
pair.
For other methods such as devianceTest
,
or properties of the GeneralizedLinearModel
object,
see GeneralizedLinearModel
.
A terms matrix is a tby(p + 1) matrix specifying terms in a model, where t is the number of terms, p is the number of predictor variables, and plus one is for the response variable.
The value of T(i,j)
is the exponent of variable j
in
term i
. Suppose there are three predictor variables A
, B
,
and C
:
[0 0 0 0] % Constant term or intercept [0 1 0 0] % B; equivalently, A^0 * B^1 * C^0 [1 0 1 0] % A*C [2 0 0 0] % A^2 [0 1 2 0] % B*(C^2)
0
at
the end of each term represents the response variable. In general,If you have the variables in a table or dataset array,
then 0
must represent the response variable depending
on the position of the response variable. The following example illustrates
this.
Load the sample data and define the dataset array.
load hospital dsa = dataset(hospital.Sex,hospital.BloodPressure(:,1),hospital.Age,... hospital.Smoker,'VarNames',{'Sex','BloodPressure','Age','Smoker'});
Represent the linear model 'BloodPressure ~ 1 + Sex
+ Age + Smoker'
in a terms matrix. The response variable
is in the second column of the dataset array, so there must be a column
of 0s for the response variable in the second column of the terms
matrix.
T = [0 0 0 0;1 0 0 0;0 0 1 0;0 0 0 1]
T = 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1
Redefine the dataset array.
dsa = dataset(hospital.BloodPressure(:,1),hospital.Sex,hospital.Age,... hospital.Smoker,'VarNames',{'BloodPressure','Sex','Age','Smoker'});
Now, the response variable is the first term in the dataset
array. Specify the same linear model, 'BloodPressure ~ 1
+ Sex + Age + Smoker'
, using a terms matrix.
T = [0 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]
T = 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
If you have the predictor and response variables in
a matrix and column vector, then you must include 0
for
the response variable at the end of each term. The following example
illustrates this.
Load the sample data and define the matrix of predictors.
load carsmall
X = [Acceleration,Weight];
Specify the model 'MPG ~ Acceleration + Weight + Acceleration:Weight
+ Weight^2'
using a term matrix and fit the model to the
data. This model includes the main effect and twoway interaction
terms for the variables, Acceleration
and Weight
,
and a secondorder term for the variable, Weight
.
T = [0 0 0;1 0 0;0 1 0;1 1 0;0 2 0]
T = 0 0 0 1 0 0 0 1 0 1 1 0 0 2 0
Fit a linear model.
mdl = fitlm(X,MPG,T)
mdl = Linear regression model: y ~ 1 + x1*x2 + x2^2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 48.906 12.589 3.8847 0.00019665 x1 0.54418 0.57125 0.95261 0.34337 x2 0.012781 0.0060312 2.1192 0.036857 x1:x2 0.00010892 0.00017925 0.6076 0.545 x2^2 9.7518e07 7.5389e07 1.2935 0.19917 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 4.1 Rsquared: 0.751, Adjusted RSquared 0.739 Fstatistic vs. constant model: 67, pvalue = 4.99e26
Only the intercept and x2
term, which correspond
to the Weight
variable, are significant at the
5% significance level.
Now, perform a stepwise regression with a constant model as the starting model and a linear model with interactions as the upper model.
T = [0 0 0;1 0 0;0 1 0;1 1 0];
mdl = stepwiselm(X,MPG,[0 0 0],'upper',T)
1. Adding x2, FStat = 259.3087, pValue = 1.643351e28 mdl = Linear regression model: y ~ 1 + x2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 49.238 1.6411 30.002 2.7015e49 x2 0.0086119 0.0005348 16.103 1.6434e28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 Rsquared: 0.738, Adjusted RSquared 0.735 Fstatistic vs. constant model: 259, pvalue = 1.64e28
The results of the stepwise regression are consistent with the
results of fitlm
in the previous step.
A formula for model specification is a character
vector of the form '
Y
~ terms
'
where
Y
is the response name.
terms
contains
Variable names
+
means include the next variable

means do not include the next
variable
:
defines an interaction, a product
of terms
*
defines an interaction and all lowerorder terms
^
raises the predictor to a power,
exactly as in *
repeated, so ^
includes
lower order terms as well
()
groups terms
Note:
Formulas include a constant (intercept) term by default. To
exclude a constant term from the model, include 
For example,
'Y ~ A + B + C'
means a threevariable
linear model with intercept.'Y ~ A + B +
C  1'
is a threevariable linear model without intercept.'Y ~ A + B + C + B^2'
is a threevariable
model with intercept and a B^2
term.'Y
~ A + B^2 + C'
is the same as the previous example because B^2
includes
a B
term.'Y ~ A + B +
C + A:B'
includes an A*B
term.'Y
~ A*B + C'
is the same as the previous example because A*B
= A + B + A:B
.'Y ~ A*B*C  A:B:C'
has
all interactions among A
, B
,
and C
, except the threeway interaction.'Y
~ A*(B + C + D)'
has all linear terms, plus products of A
with
each of the other variables.
Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.
Wilkinson Notation  Factors in Standard Notation 

1  Constant (intercept) term 
A^k , where k is a positive
integer  A , A^{2} ,
..., A^{k} 
A + B  A , B 
A*B  A , B , A*B 
A:B  A*B only 
B  Do not include B 
A*B + C  A , B , C , A*B 
A + B + C + A:B  A , B , C , A*B 
A*B*C  A:B:C  A , B , C , A*B , A*C , B*C 
A*(B + C)  A , B , C , A*B , A*C 
Statistics and Machine Learning Toolbox™ notation always includes a constant term
unless you explicitly remove the term using 1
.
The default link function for a generalized linear model is the canonical link function.
Canonical Link Functions for Generalized Linear Models
Distribution  Link Function Name  Link Function  Mean (Inverse) Function 

'normal'  'identity'  f(μ) = μ  μ = Xb 
'binomial'  'logit'  f(μ) = log(μ/(1–μ))  μ = exp(Xb) / (1 + exp(Xb)) 
'poisson'  'log'  f(μ) = log(μ)  μ = exp(Xb) 
'gamma'  1  f(μ) = 1/μ  μ = 1/(Xb) 
'inverse gaussian'  2  f(μ) = 1/μ^{2}  μ = (Xb)^{–1/2} 
You can also construct a generalized linear model using fitglm
.
Use stepwiseglm
to select
a model specification automatically. Use step
, addTerms
,
or removeTerms
to adjust a fitted model.
[1] Collett, D. Modeling Binary Data. New York: Chapman & Hall, 2002.
[2] Dobson, A. J. An Introduction to Generalized Linear Models. New York: Chapman & Hall, 1990.
[3] McCullagh, P., and J. A. Nelder. Generalized Linear Models. New York: Chapman & Hall, 1990.