Superclasses: CompactGeneralizedLinearModel
Generalized linear regression model class
An object comprising training data, model description, diagnostic
information, and fitted coefficients for a generalized linear regression.
Predict model responses with the predict
or feval
methods.
or mdl
=
fitglm(tbl
)
creates
a generalized linear model of a table or dataset array mdl
=
fitglm(X
,y
)tbl
,
or of the responses y
to a data matrix X
.
For details, see fitglm
.
or mdl
= stepwiseglm(tbl
)
creates
a generalized linear model of a table or dataset array mdl
=
stepwiseglm(X
,y
)tbl
,
or of the responses y
to a data matrix X
,
with unimportant predictors excluded. For details, see stepwiseglm
.
tbl
— Input dataInput data including predictor and response variables, specified as a table or dataset array.
The predictor variables and response variable can be numeric, logical, categorical,
character, or string. The response variable can have a data type other than numeric only
if 'Distribution'
is 'binomial'
.
By default, GeneralizedLinearModel
takes the last variable as
the response variable and the others as the predictor variables.
To set a different column as the response variable, use the
ResponseVar
name-value pair argument.
To use a subset of the columns as predictors, use the
PredictorVars
name-value pair argument.
To define a model specification, set the modelspec
argument using a formula or terms matrix. The formula or terms matrix
specifies which columns to use as the predictor or response
variables.
The variable names in a table do not have to be valid MATLAB® identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:
You cannot specify modelspec
using a formula.
You cannot use a formula to specify the terms to add or remove when you
use the addTerms
function or the
removeTerms
function,
respectively.
You cannot use a formula to specify the lower and upper bounds of the
model when you use the step
or stepwiseglm
function with the
name-value pair arguments 'Lower'
and
'Upper'
, respectively.
You can verify the variable names in tbl
by using the isvarname
function. The following code returns logical 1
(true
) for each variable that has a valid variable name.
cellfun(@isvarname,tbl.Properties.VariableNames)
tbl
are not valid, then convert them by using the matlab.lang.makeValidName
function.tbl.Properties.VariableNames = matlab.lang.makeValidName(tbl.Properties.VariableNames);
X
— Predictor variablesPredictor variables, specified as an n-by-p matrix,
where n is the number of observations and p is
the number of predictor variables. Each column of X
represents
one variable, and each row represents one observation.
By default, there is a constant term in the model, unless you
explicitly remove it, so do not include a column of 1s in X
.
Data Types: single
| double
y
— Response variableResponse variable, specified as a vector or matrix.
If 'Distribution'
is not
'binomial'
, then y
must be an
n-by-1 vector, where n is the
number of observations. Each entry in y
is the response
for the corresponding row of X
. The data type must be
single or double.
If 'Distribution'
is 'binomial'
,
then y
can be an n-by-1 vector or
n-by-2 matrix with counts in column 1 and
BinomialSize
in column 2.
Data Types: single
| double
| logical
| categorical
CoefficientCovariance
— Covariance matrix of coefficient estimatesThis property is read-only.
Covariance matrix of coefficient estimates, specified as a p-by-p matrix of numeric values. p is the number of coefficients in the fitted model.
For details, see Coefficient Standard Errors and Confidence Intervals.
Data Types: single
| double
CoefficientNames
— Coefficient namesThis property is read-only.
Coefficient names, specified as a cell array of character vectors, each containing the name of the corresponding term.
Data Types: cell
Coefficients
— Coefficient valuesThis property is read-only.
Coefficient values, specified as a table.
Coefficients
contains one row for each coefficient and these
columns:
Estimate
— Estimated
coefficient value
SE
— Standard error
of the estimate
tStat
— t-statistic for a test that the
coefficient is zero
pValue
— p-value for the
t-statistic
Use anova
(only for a linear regression model) or
coefTest
to perform other tests on the coefficients. Use
coefCI
to find the confidence intervals of the coefficient
estimates.
To obtain any of these columns as a vector, index into the property
using dot notation. For example, obtain the estimated coefficient vector in the model
mdl
:
beta = mdl.Coefficients.Estimate
Data Types: table
Deviance
— Deviance of the fitThis property is read-only.
Deviance of the fit, specified as a numeric value. Deviance is useful for comparing two models when one model is a special case of the other model. The difference between the deviance of the two models has a chi-square distribution with degrees of freedom equal to the difference in the number of estimated parameters between the two models. For more information on deviance, see Deviance.
Data Types: single
| double
DFE
— Degrees of freedom for errorThis property is read-only.
Degrees of freedom for the error (residuals), equal to the number of observations minus the number of estimated coefficients, specified as a positive integer.
Data Types: double
Diagnostics
— Diagnostic informationThis property is read-only.
Diagnostic information for the model, specified as a table. Diagnostics
can help identify outliers and influential observations.
Diagnostics
contains the following fields.
Field | Meaning | Utility |
---|---|---|
Leverage | Diagonal elements of HatMatrix | Leverage indicates to what extent the predicted value for
an observation is determined by the observed value for that
observation. A value close to 1 indicates
that the prediction is largely determined by that
observation, with little contribution from the other
observations. A value close to 0
indicates the fit is largely determined by the other
observations. For a model with p
coefficients and n observations, the
average value of Leverage is
p/n. An
observation with Leverage larger than
2*p/n can be an
outlier. |
CooksDistance | Cook's measure of scaled change in fitted values | CooksDistance is a measure of scaled
change in fitted values. An observation with
CooksDistance larger than three times
the mean Cook's distance can be an outlier. |
HatMatrix | Projection matrix to compute fitted from observed responses | HatMatrix is an
n-by-n matrix such
that Fitted = HatMatrix*Y ,
where Y is the response vector and
Fitted is the vector of fitted
response values. |
All of these quantities are computed on the scale of the linear predictor. For example, in the equation that defines the hat matrix:
Yfit = glm.Fitted.LinearPredictor Y = glm.Fitted.LinearPredictor + glm.Residuals.LinearPredictor
Data Types: table
Dispersion
— Scale factor of the variance of the responseThis property is read-only.
Scale factor of the variance of the response, specified as a numeric
value. Dispersion
multiplies the variance function for
the distribution.
For example, the variance function for the binomial distribution is
p(1–p)/n, where
p is the probability parameter and
n is the sample size parameter. If
Dispersion
is near 1
, the variance
of the data appears to agree with the theoretical variance of the binomial
distribution. If Dispersion
is larger than
1
, the data set is “overdispersed”
relative to the binomial distribution.
Data Types: double
DispersionEstimated
— Flag to indicate use of dispersion scale factorThis property is read-only.
Flag to indicate whether fitglm
used the
Dispersion
scale factor to compute standard errors
for the coefficients in Coefficients.SE
, specified as a
logical value. If DispersionEstimated
is
false
, fitglm
used the theoretical
value of the variance.
DispersionEstimated
can be
false
only for 'binomial'
or 'poisson'
distributions.
Set DispersionEstimated
by setting the
DispersionFlag
name-value pair in
fitglm
.
Data Types: logical
Distribution
— Generalized distribution informationThis property is read-only.
Generalized distribution information, specified as a structure with the following fields relating to the generalized distribution.
Field | Description |
---|---|
Name | Name of the distribution, one of
'normal' ,
'binomial' ,
'poisson' ,
'gamma' , or 'inverse
gaussian' . |
DevianceFunction | Function that computes the components of the deviance as a function of the fitted parameter values and the response values. |
VarianceFunction | Function that computes the theoretical variance for the
distribution as a function of the fitted parameter values.
When DispersionEstimated is
true , Dispersion
multiplies the variance function in the computation of the
coefficient standard errors. |
Data Types: struct
Fitted
— Fitted response values based on input dataThis property is read-only.
Fitted (predicted) values based on the input data, specified as a table with one row for each observation and the following columns.
Field | Description |
---|---|
Response | Predicted values on the scale of the response. |
LinearPredictor | Predicted values on the scale of the linear predictor.
These are the same as the link function applied to the
Response fitted values. |
Probability | Fitted probabilities (this column is included only with the binomial distribution). |
To obtain any of the columns as a vector, index into the property using
dot notation. For example, in the model mdl
, the vector
f
of fitted values on the response scale is
f = mdl.Fitted.Response
Use predict
to compute predictions
for other predictor values, or to compute confidence bounds on
Fitted
.
Data Types: table
Formula
— Model informationLinearFormula
objectThis property is read-only.
Model information, specified as a LinearFormula
object.
Display the formula of the fitted model mdl
using dot
notation:
mdl.Formula
Link
— Link functionThis property is read-only.
Link function, specified as a structure with the following fields.
Field | Description |
---|---|
Name | Name of the link function, or '' if
you specified the link as a function handle rather than a
character vector. |
LinkFunction | The function that defines f, a function handle. |
DevianceFunction | Derivative of f, a function handle. |
VarianceFunction | Inverse of f, a function handle. |
The link is a function f that links the distribution parameter μ to the fitted linear combination Xb of the predictors:
f(μ) = Xb.
Data Types: struct
LogLikelihood
— Log likelihoodThis property is read-only.
Log likelihood of the model distribution at the response values, specified as a numeric value. The mean is fitted from the model, and other parameters are estimated as part of the model fit.
Data Types: single
| double
ModelCriterion
— Criterion for model comparisonThis property is read-only.
Criterion for model comparison, specified as a structure with these fields:
AIC
— Akaike information criterion.
AIC = –2*logL + 2*m
, where logL
is the
loglikelihood and m
is the number of estimated
parameters.
AICc
— Akaike information criterion corrected for
the sample size. AICc = AIC + (2*m*(m+1))/(n–m–1)
, where
n
is the number of observations.
BIC
— Bayesian information criterion.
BIC = –2*logL + m*log(n)
.
CAIC
— Consistent Akaike information criterion.
CAIC = –2*logL + m*(log(n)+1)
.
Information criteria are model selection tools that you can use to compare multiple models fit to the same data. These criteria are likelihood-based measures of model fit that include a penalty for complexity (specifically, the number of parameters). Different information criteria are distinguished by the form of the penalty.
When you compare multiple models, the model with the lowest information criterion value is the best-fitting model. The best-fitting model can vary depending on the criterion used for model comparison.
To obtain any of the criterion values as a scalar, index into the property using dot
notation. For example, obtain the AIC value aic
in the model
mdl
:
aic = mdl.ModelCriterion.AIC
Data Types: struct
NumCoefficients
— Number of model coefficientsThis property is read-only.
Number of model coefficients, specified as a positive integer.
NumCoefficients
includes coefficients that are set to zero when
the model terms are rank deficient.
Data Types: double
NumEstimatedCoefficients
— Number of estimated coefficientsThis property is read-only.
Number of estimated coefficients in the model, specified as a positive integer.
NumEstimatedCoefficients
does not include coefficients that are
set to zero when the model terms are rank deficient.
NumEstimatedCoefficients
is the degrees of freedom for
regression.
Data Types: double
NumObservations
— Number of observationsThis property is read-only.
Number of observations the fitting function used in fitting, specified
as a positive integer. NumObservations
is the
number of observations supplied in the original table, dataset,
or matrix, minus any excluded rows (set with the
'Exclude'
name-value pair
argument) or rows with missing values.
Data Types: double
NumPredictors
— Number of predictor variablesThis property is read-only.
Number of predictor variables used to fit the model, specified as a positive integer.
Data Types: double
NumVariables
— Number of variablesThis property is read-only.
Number of variables in the input data, specified as a positive integer.
NumVariables
is the number of variables in the original table or
dataset, or the total number of columns in the predictor matrix and response
vector.
NumVariables
also includes any variables that are not used to fit
the model as predictors or as the response.
Data Types: double
ObservationInfo
— Observation informationThis property is read-only.
Observation information, specified as an n-by-4 table, where
n is equal to the number of rows of input data. The
ObservationInfo
contains the columns described in this
table.
Column | Description |
---|---|
Weights | Observation weight, specified as a numeric value. The default value
is 1 . |
Excluded | Indicator of excluded observation, specified as a logical value. The
value is true if you exclude the observation from the
fit by using the 'Exclude' name-value pair
argument. |
Missing | Indicator of missing observation, specified as a logical value. The
value is true if the observation is missing. |
Subset | Indicator of whether or not a fitting function uses the observation,
specified as a logical value. The value is true if
the observation is not excluded or missing, meaning that fitting
function uses the observation. |
To obtain any of these columns as a vector, index into the property using dot
notation. For example, obtain the weight vector w
of the model
mdl
:
w = mdl.ObservationInfo.Weights
Data Types: table
ObservationNames
— Observation namesThis property is read-only.
Observation names, specified as a cell array of character vectors containing the names of the observations used in the fit.
If the fit is based on a table or dataset
containing observation names,
ObservationNames
uses those
names.
Otherwise, ObservationNames
is an empty cell array.
Data Types: cell
Offset
— Offset variableThis property is read-only.
Offset variable, specified as a numeric vector with the same length as the
number of rows in the data. Offset
is passed from
fitglm
or stepwiseglm
in the
Offset
name-value pair. The fitting function used
Offset
as a predictor variable, but with the
coefficient set to exactly 1
. In other words, the formula
for fitting was
μ ~ Offset + (terms involving real
predictors)
with the Offset
predictor having coefficient
1
.
For example, consider a Poisson regression model. Suppose the number of
counts is known for theoretical reasons to be proportional to a predictor
A
. By using the log link function and by specifying
log(A)
as an offset, you can force the model to
satisfy this theoretical constraint.
Data Types: double
PredictorNames
— Names of predictors used to fit modelThis property is read-only.
Names of predictors used to fit the model, specified as a cell array of character vectors.
Data Types: cell
Residuals
— Residuals for fitted modelThis property is read-only.
Residuals for the fitted model, specified as a table with one row for each observation and the following columns.
Field | Description |
---|---|
Raw | Observed minus fitted values. |
LinearPredictor | Residuals on the linear predictor scale, equal to the adjusted response value minus the fitted linear combination of the predictors. |
Pearson | Raw residuals divided by the estimated standard deviation of the response. |
Anscombe | Residuals defined on transformed data with the transformation chosen to remove skewness. |
Deviance | Residuals based on the contribution of each observation to the deviance. |
To obtain any of these columns as a vector, index into the property using
dot notation. For example, in a model mdl
, the ordinary
raw residual vector r
is:
r = mdl.Residuals.Raw
Rows not used in the fit because of missing values (in
ObservationInfo.Missing
) contain
NaN
values.
Rows not used in the fit because of excluded values (in
ObservationInfo.Excluded
) contain
NaN
values, with the following exceptions:
raw
contains the difference between the
observed and predicted values.
standardized
is the residual, standardized in
the usual way.
studentized
matches the standardized values
because this residual is not used in the estimate of the residual
standard deviation.
Data Types: table
ResponseName
— Response variable nameThis property is read-only.
Response variable name, specified as a character vector.
Data Types: char
Rsquared
— R-squared value for the modelThis property is read-only.
R-squared value for the model, specified as a structure with five fields:
Ordinary
— Ordinary (unadjusted)
R-squared
Adjusted
— R-squared adjusted for the number of
coefficients
LLR
— Log-likelihood ratio
Deviance
— Deviance
AdjGeneralized
— Adjusted generalized
R-squared
The R-squared value is the proportion of total sum of squares explained by the model.
The ordinary R-squared value relates to the SSR
and
SST
properties:
Rsquared = SSR/SST = 1 - SSE/SST
.
To obtain any of these values as a scalar, index into the property using dot notation.
For example, the adjusted R-squared value in mdl
is
r2 = mdl.Rsquared.Adjusted
Data Types: struct
SSE
— Sum of squared errorsThis property is read-only.
Sum of squared errors (residuals), specified as a numeric value.
The Pythagorean theorem implies
SST = SSE + SSR
,
where SST
is the total sum of squares,
SSE
is the sum of squared errors, and SSR
is
the regression sum of squares.
Data Types: single
| double
SSR
— Regression sum of squaresThis property is read-only.
Regression sum of squares, specified as a numeric value. The regression sum of squares is equal to the sum of squared deviations of the fitted values from their mean.
The Pythagorean theorem implies
SST = SSE +
SSR
,
where SST
is the total sum
of squares, SSE
is the sum of squared errors,
and SSR
is the regression sum of
squares.
Data Types: single
| double
SST
— Total sum of squaresThis property is read-only.
Total sum of squares, specified as a numeric value. The total sum of squares is equal
to the sum of squared deviations of the response vector y
from the
mean(y)
.
The Pythagorean theorem implies
SST = SSE + SSR
,
where SST
is the total sum of squares,
SSE
is the sum of squared errors, and SSR
is
the regression sum of squares.
Data Types: single
| double
Steps
— Stepwise fitting informationThis property is read-only.
Stepwise fitting information, specified as a structure with the fields described in this table.
Field | Description |
---|---|
Start | Formula representing the starting model |
Lower | Formula representing the lower bound model. The terms in
Lower must remain in the model. |
Upper | Formula representing the upper bound model. The model cannot contain
more terms than Upper . |
Criterion | Criterion used for the stepwise algorithm, such as
'sse' |
PEnter | Threshold for Criterion to add a term |
PRemove | Threshold for Criterion to remove a term |
History | Table representing the steps taken in the fit |
The History
table contains one row for each step, including the
initial fit, and the columns described in this table.
Column | Description |
---|---|
Action | Action taken during the step:
|
TermName |
|
Terms | Model specification in a Terms Matrix |
DF | Regression degrees of freedom after the step |
delDF | Change in regression degrees of freedom from the previous step (negative for steps that remove a term) |
Deviance | Deviance (residual sum of squares) at the step (only for a generalized linear regression model) |
FStat | F-statistic that leads to the step |
PValue | p-value of the F-statistic |
The structure is empty unless you fit the model using stepwise regression.
Data Types: struct
VariableInfo
— Information about variablesThis property is read-only.
Information about variables contained in Variables
, specified as a
table with one row for each variable and the columns described in this table.
Column | Description |
---|---|
Class | Variable class, specified as a cell array of character vectors, such
as 'double' and
'categorical' |
Range | Variable range, specified as a cell array of vectors
|
InModel | Indicator of which variables are in the fitted model, specified as a
logical vector. The value is true if the model
includes the variable. |
IsCategorical | Indicator of categorical variables, specified as a logical vector.
The value is true if the variable is
categorical. |
VariableInfo
also includes any variables that are not used to fit
the model as predictors or as the response.
Data Types: table
VariableNames
— Names of variablesThis property is read-only.
Names of variables, specified as a cell array of character vectors.
If the fit is based on a table or dataset, this property provides the names of the variables in the table or dataset.
If the fit is based on a predictor matrix and response vector,
VariableNames
contains the values specified by the
'VarNames'
name-value pair argument of the fitting
method. The default value of 'VarNames'
is
{'x1','x2',...,'xn','y'}
.
VariableNames
also includes any variables that are not used to fit
the model as predictors or as the response.
Data Types: cell
Variables
— Input dataThis property is read-only.
Input data, specified as a table. Variables
contains both predictor
and response values. If the fit is based on a table or dataset array,
Variables
contains all the data from the table or dataset array.
Otherwise, Variables
is a table created from the input data matrix
X
and response the vector y
.
Variables
also includes any variables that are not used to fit the
model as predictors or as the response.
Data Types: table
addTerms | Add terms to generalized linear model |
compact | Compact generalized linear regression model |
fit | (Not Recommended) Create generalized linear regression model |
plotDiagnostics | Plot diagnostics of generalized linear regression model |
plotResiduals | Plot residuals of generalized linear regression model |
removeTerms | Remove terms from generalized linear model |
step | Improve generalized linear regression model by adding or removing terms |
stepwise | (Not Recommended) Create generalized linear regression model by stepwise regression |
coefCI | Confidence intervals of coefficient estimates of generalized linear model |
coefTest | Linear hypothesis test on generalized linear regression model coefficients |
devianceTest | Analysis of deviance |
disp | Display generalized linear regression model |
feval | Evaluate generalized linear regression model prediction |
plotSlice | Plot of slices through fitted generalized linear regression surface |
predict | Predict response of generalized linear regression model |
random | Simulate responses for generalized linear regression model |
Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).
Fit a logistic regression model of probability of smoking as a function of age, weight, and sex, using a two-way interactions model.
Load the hospital
dataset array.
load hospital ds = hospital; % just to use the ds name
Specify the model using a formula that allows up to two-way interactions.
modelspec = 'Smoker ~ Age*Weight*Sex - Age:Weight:Sex';
Create the generalized linear model.
mdl = fitglm(ds,modelspec,'Distribution','binomial')
mdl = Generalized linear regression model: logit(Smoker) ~ 1 + Sex*Age + Sex*Weight + Age*Weight Distribution = Binomial Estimated Coefficients: Estimate SE tStat pValue ___________ _________ ________ _______ (Intercept) -6.0492 19.749 -0.3063 0.75938 Sex_Male -2.2859 12.424 -0.18399 0.85402 Age 0.11691 0.50977 0.22934 0.81861 Weight 0.031109 0.15208 0.20455 0.83792 Sex_Male:Age 0.020734 0.20681 0.10025 0.92014 Sex_Male:Weight 0.01216 0.053168 0.22871 0.8191 Age:Weight -0.00071959 0.0038964 -0.18468 0.85348 100 observations, 93 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 5.07, p-value = 0.535
The large -value indicates the model might not differ statistically from a constant.
Create response data using just three of 20 predictors, and create a generalized linear model stepwise to see if it uses just the correct predictors.
Create data with 20 predictors, and Poisson response using just three of the predictors, plus a constant.
rng default % for reproducibility X = randn(100,20); mu = exp(X(:,[5 10 15])*[.4;.2;.3] + 1); y = poissrnd(mu);
Fit a generalized linear model using the Poisson distribution.
mdl = stepwiseglm(X,y,... 'constant','upper','linear','Distribution','poisson')
1. Adding x5, Deviance = 134.439, Chi2Stat = 52.24814, PValue = 4.891229e-13 2. Adding x15, Deviance = 106.285, Chi2Stat = 28.15393, PValue = 1.1204e-07 3. Adding x10, Deviance = 95.0207, Chi2Stat = 11.2644, PValue = 0.000790094
mdl = Generalized linear regression model: log(y) ~ 1 + x5 + x10 + x15 Distribution = Poisson Estimated Coefficients: Estimate SE tStat pValue ________ ________ ______ __________ (Intercept) 1.0115 0.064275 15.737 8.4217e-56 x5 0.39508 0.066665 5.9263 3.0977e-09 x10 0.18863 0.05534 3.4085 0.0006532 x15 0.29295 0.053269 5.4995 3.8089e-08 100 observations, 96 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 91.7, p-value = 9.61e-20
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/μ2 | μ = (Xb)–1/2 |
The hat matrix H is defined in terms of the data matrix X and a diagonal weight matrix W:
H = X(XTWX)–1XTWT.
W has diagonal elements wi:
where
g is the link function mapping yi to xib.
is the derivative of the link function g.
V is the variance function.
μi is the ith mean.
The diagonal elements Hii satisfy
where n is the number of observations (rows of X), and p is the number of coefficients in the regression model.
The leverage of observation i is the value of the ith diagonal term hii of the hat matrix H. Because the sum of the leverage values is p (the number of coefficients in the regression model), an observation i can be considered an outlier if its leverage substantially exceeds p/n, where n is the number of observations.
The Cook’s distance Di of observation i is
where
is the dispersion parameter (estimated or theoretical).
ei is the linear predictor residual, , where
g is the link function.
yi is the observed response.
xi is the observation.
is the estimated coefficient vector.
p is the number of coefficients in the regression model.
hii is the ith diagonal element of the Hat Matrix H.
Deviance of a model M1 is twice the difference between the loglikelihood of that model and the saturated model, MS. The saturated model is the model with the maximum number of parameters that can be estimated. For example, if there are n observations yi, i = 1, 2, ..., n, with potentially different values for XiTβ, then you can define a saturated model with n parameters. Let L(b,y) denote the maximum value of the likelihood function for a model. Then the deviance of model M1 is
where b1 are the estimated parameters for model M1 and bS are the estimated parameters for the saturated model. The deviance has a chi-square distribution with n – p degrees of freedom, where n is the number of parameters in the saturated model and p is the number of parameters in model M1.
If M1 and M2 are two different generalized linear models, then the fit of the models can be assessed by comparing the deviances D1 and D2 of these models. The difference of the deviances is
Asymptotically, this difference has a chi-square distribution with degrees of freedom
v equal to the number of parameters that are estimated in one model
but fixed (typically at 0) in the other. It is equal to the difference in the number of
parameters estimated in M1 and M2. You can get
the p-value for this test using
1 - chi2cdf(D,V)
, where D =
D2 –
D1.
A terms matrix T
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 +1 accounts for the response variable. The value of
T(i,j)
is the exponent of variable j
in term
i
.
For example, suppose that an input includes three predictor variables A
,
B
, and C
and the response variable
Y
in the order A
, B
,
C
, and Y
. Each row of T
represents one term:
[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, a column vector of zeros in a terms matrix represents the position of the response
variable. If you have the predictor and response variables in a matrix and column vector,
then you must include 0
for the response variable in the last column of
each row.
Usage notes and limitations:
When you fit a model by using fitglm
or stepwiseglm
, the following restrictions apply.
Code generation does not support categorical predictors. You cannot
supply training data in a table that contains a logical vector,
character array, categorical array, string array, or cell array of
character vectors. Also, you cannot use the 'CategoricalVars'
name-value pair argument. To include categorical predictors
in a model, preprocess the categorical predictors by using dummyvar
before
fitting the model.
The Link
, Derivative
, and
Inverse
fields of the 'Link'
name-value pair argument cannot be anonymous functions. That is, you
cannot generate code using a generalized linear model that was created
using anonymous functions for links. Instead, define functions for link
components.
For more information, see Introduction to Code Generation.
LinearModel
| NonLinearModel
| fitglm
| plotPartialDependence
| stepwiseglm
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