Logistic regression

fitglm

Create generalized linear regression model

Syntax

mdl = fitglm(tbl)

mdl = fitglm(X,y)

mdl = fitglm(___,modelspec)

mdl = fitglm(___,Name,Value)

Description

mdl = fitglm(tbl) returns a generalized linear model fit to variables in the table or dataset array tbl. By default, fitglm takes the last variable as the response variable.

mdl = fitglm(X,y) returns a generalized linear model of the responses y, fit to the data matrix X.

mdl = fitglm(___,modelspec) returns a generalized linear model of the type you specify in modelspec.

mdl = fitglm(___,Name,Value) returns a generalized linear model with additional options specified by one or more Name,Value pair arguments.

For example, you can specify which variables are categorical, the distribution of the response variable, and the link function to use.

Examples

Fit a Logistic Regression Model

Make a logistic binomial model of the probability of smoking as a function of age, weight, and sex, using a two-way interactions model.

Load the hospital dataset array.

load hospital
dsa = hospital;

Specify the model using a formula that allows up to two-way interactions between the variables age, weight, and sex. Smoker is the response variable.

modelspec = 'Smoker ~ Age*Weight*Sex - Age:Weight:Sex';

Fit a logistic binomial model.

mdl = fitglm(dsa,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

All of the p-values (under pValue) are large. This means none of the coefficients are significant. The large p-value for the test of the model, 0.535, indicates that this model might not differ statistically from a constant model.

GLM for Poisson Response

Create sample data with 20 predictors, and Poisson response using just three of the predictors, plus a constant.

rng('default') % for reproducibility
X = randn(100,7);
mu = exp(X(:,[1 3 6])*[.4;.2;.3] + 1);
y = poissrnd(mu);

Fit a generalized linear model using the Poisson distribution.

mdl =  fitglm(X,y,'linear','Distribution','poisson')

mdl = 
Generalized linear regression model:
    log(y) ~ 1 + x1 + x2 + x3 + x4 + x5 + x6 + x7
    Distribution = Poisson

Estimated Coefficients:
                   Estimate        SE        tStat        pValue  
                   _________    ________    ________    __________

    (Intercept)      0.88723    0.070969      12.502    7.3149e-36
    x1               0.44413    0.052337      8.4858    2.1416e-17
    x2             0.0083388    0.056527     0.14752       0.88272
    x3               0.21518    0.063416      3.3932    0.00069087
    x4             -0.058386    0.065503    -0.89135       0.37274
    x5             -0.060824    0.073441     -0.8282       0.40756
    x6               0.34267    0.056778      6.0352    1.5878e-09
    x7               0.04316     0.06146     0.70225       0.48252


100 observations, 92 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 119, p-value = 1.55e-22

The p-values of 2.14e-17, 0.00069, and 1.58e-09 indicate that the coefficients of the variables x1, x3, and x6 are statistically significant.

Input Arguments

`tbl` — Input data
table | dataset array

Input 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, fitglm 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. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

`X` — Predictor variables
matrix

Predictor 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 variable
vector | matrix

Response 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

`modelspec` — Model specification
`'linear'` (default) | character vector or string scalar naming the model | t-by-(p + 1) terms matrix | character vector or string scalar formula in the form `'y ~ terms'`

Model specification, specified as one of these values.

A character vector or string scalar naming the model.

Value	Model Type
`'constant'`	Model contains only a constant (intercept) term.
`'linear'`	Model contains an intercept and linear term for each predictor.
`'interactions'`	Model contains an intercept, linear term for each predictor, and all products of pairs of distinct predictors (no squared terms).
`'purequadratic'`	Model contains an intercept term and linear and squared terms for each predictor.
`'quadratic'`	Model contains an intercept term, linear and squared terms for each predictor, and all products of pairs of distinct predictors.
`'polyijk'`	Model is a polynomial with all terms up to degree `i` in the first predictor, degree `j` in the second predictor, and so on. Specify the maximum degree for each predictor by using numerals 0 though 9. The model contains interaction terms, but the degree of each interaction term does not exceed the maximum value of the specified degrees. For example, `'poly13'` has an intercept and x₁, x₂, x₂², x₂³, x₁x₂, and x₁x₂² terms, where x₁ and x₂ are the first and second predictors, respectively.

A t-by-(p + 1) matrix, or a Terms Matrix, specifying terms in the model, where t is the number of terms and p is the number of predictor variables, and +1 accounts for the response variable. A terms matrix is convenient when the number of predictors is large and you want to generate the terms programmatically.
A character vector or string scalar Formula in the form

'y ~ terms',

where the terms are in Wilkinson Notation. The variable names in the formula must be variable names in tbl or variable names specified by Varnames. Also, the variable names must be valid MATLAB identifiers.

The software determines the order of terms in a fitted model by using the order of terms in tbl or X. Therefore, the order of terms in the model can be different from the order of terms in the specified formula.

Example: 'quadratic'

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Distribution','normal','link','probit','Exclude',[23,59] specifies that the distribution of the response is normal, and instructs fitglm to use the probit link function and exclude the 23rd and 59th observations from the fit.

`BinomialSize` — Number of trials for binomial distribution
1 (default) | numeric scalar | numeric vector | character vector | string scalar

Number of trials for binomial distribution, that is the sample size, specified as the comma-separated pair consisting of 'BinomialSize' and the variable name in tbl, a numeric scalar, or a numeric vector of the same length as the response. This is the parameter n for the fitted binomial distribution. BinomialSize applies only when the Distribution parameter is 'binomial'.

If BinomialSize is a scalar value, that means all observations have the same number of trials.

As an alternative to BinomialSize, you can specify the response as a two-column matrix with counts in column 1 and BinomialSize in column 2.

Data Types: single | double | char | string

`B0` — Initial values for coefficient estimates
numeric vector

Initial values for the coefficient estimates, specified as a numeric vector. The default values are initial fitted values derived from the input data.

Data Types: single | double

`CategoricalVars` — Categorical variable list
string array | cell array of character vectors | logical or numeric index vector

Categorical variable list, specified as the comma-separated pair consisting of 'CategoricalVars' and either a string array or cell array of character vectors containing categorical variable names in the table or dataset array tbl, or a logical or numeric index vector indicating which columns are categorical.

If data is in a table or dataset array tbl, then, by default, fitglm treats all categorical values, logical values, character arrays, string arrays, and cell arrays of character vectors as categorical variables.
If data is in matrix X, then the default value of 'CategoricalVars' is an empty matrix []. That is, no variable is categorical unless you specify it as categorical.

For example, you can specify the second and third variables out of six as categorical using either of the following:

Example: 'CategoricalVars',[2,3]

Example: 'CategoricalVars',logical([0 1 1 0 0 0])

Data Types: single | double | logical | string | cell

`DispersionFlag` — Indicator to compute dispersion parameter
`false` for `'binomial'` and `'poisson'` distributions (default) | `true`

Indicator to compute dispersion parameter for 'binomial' and 'poisson' distributions, specified as the comma-separated pair consisting of 'DispersionFlag' and one of the following.

`true`	Estimate a dispersion parameter when computing standard errors. The estimated dispersion parameter value is the sum of squared Pearson residuals divided by the degrees of freedom for error (DFE).
`false`	Default. Use the theoretical value of 1 when computing standard errors.

The fitting function always estimates the dispersion for other distributions.

Example: 'DispersionFlag',true

`Distribution` — Distribution of the response variable

`'normal'` (default) | `'binomial'` | `'poisson'` | `'gamma'` | `'inverse gaussian'`

Distribution of the response variable, specified as the comma-separated pair consisting of 'Distribution' and one of the following.

`'normal'`	Normal distribution
`'binomial'`	Binomial distribution
`'poisson'`	Poisson distribution
`'gamma'`	Gamma distribution
`'inverse gaussian'`	Inverse Gaussian distribution

Example: 'Distribution','gamma'

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