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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)
mdl = GeneralizedLinearModel.fit(tbl) creates a generalized linear model of a table or dataset array tbl.
mdl = GeneralizedLinearModel.fit(X,y) creates a generalized linear model of the responses y to a data matrix X.
mdl = GeneralizedLinearModel.fit(...,modelspec) creates a generalized linear model as specified by modelspec.
mdl = GeneralizedLinearModel.fit(...,Name,Value) or mdl = GeneralizedLinearModel.fit(...,modelspec,Name,Value) creates a generalized linear model with additional options specified by one or more Name,Value pair arguments.
The generalized linear model mdl is a standard linear model unless you specify otherwise with the Distribution name-value pair.
For other methods such as devianceTest, or properties of the GeneralizedLinearModel object, see GeneralizedLinearModel.
A terms matrix is a t-by-(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)
The 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 ds = 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.
ds = 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 two-way interaction terms for the variables, Acceleration and Weight, and a second-order 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 9.7518e-07 7.5389e-07 1.2935 0.19917 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 4.1 R-squared: 0.751, Adjusted R-Squared 0.739 F-statistic vs. constant model: 67, p-value = 4.99e-26
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 = 1.643351e-28 mdl = Linear regression model: y ~ 1 + x2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 49.238 1.6411 30.002 2.7015e-49 x2 -0.0086119 0.0005348 -16.103 1.6434e-28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 R-squared: 0.738, Adjusted R-Squared 0.735 F-statistic vs. constant model: 259, p-value = 1.64e-28
The results of the stepwise regression are consistent with the results of fitlm in the previous step.
A formula for model specification is a string 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 lower-order 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 -1 in the formula. |
For example,
'Y ~ A + B + C' means a three-variable
linear model with intercept.
'Y ~ A + B +
C - 1' is a three-variable linear model without intercept.
'Y ~ A + B + C + B^2' is a three-variable
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 three-way 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 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.
GeneralizedLinearModel | stepwiseglm