Table for Input and Response Data

To create a table from workspace variables:

load carsmall
tbl = table(MPG,Weight);
tbl.Year = ordinal(Model_Year);

Numeric Matrix for Input Data, Numeric Vector for Response

For example, to create numeric arrays from workspace variables:

load carsmall
X = [Weight Horsepower Cylinders Model_Year];
y = MPG;

To create numeric arrays from an Excel^® spreadsheet:

[X, Xnames] = readmatrix('hospital.xls');
y = X(:,4); % response y is systolic pressure
X(:,4) = []; % remove y from the X matrix

Notice that the nonnumeric entries, such as sex, do not appear in X.

Custom Link Function

The link function defines the relationship f(µ) = Xb between the mean response µ and the linear combination Xb = X*b of the predictors. You can choose one of the built-in link functions or define your own by specifying the link function FL, its derivative FD, and its inverse FI:

You can specify a custom link function in either of two equivalent ways. Each way contains function handles that accept a single array of values representing µ or Xb, and returns an array the same size. The function handles are either in a cell array or a structure:

For example, to fit a model using the 'probit' link function:

x = [2100 2300 2500 2700 2900 ...
     3100 3300 3500 3700 3900 4100 4300]';
n = [48 42 31 34 31 21 23 23 21 16 17 21]';
y = [1 2 0 3 8 8 14 17 19 15 17 21]';
g = fitglm(x,[y n],...
    'linear','distr','binomial','link','probit')

g = 

Generalized Linear regression model:
    probit(y) ~ 1 + x1
    Distribution = Binomial

Estimated Coefficients:
                   Estimate     SE            tStat     pValue    
    (Intercept)      -7.3628       0.66815    -11.02    3.0701e-28
    x1             0.0023039    0.00021352     10.79    3.8274e-27

12 observations, 10 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 241, p-value = 2.25e-54

You can perform the same fit using a custom link function that performs identically to the 'probit' link function:

s = {@norminv,@(x)1./normpdf(norminv(x)),@normcdf};
g = fitglm(x,[y n],...
    'linear','distr','binomial','link',s)

g = 

Generalized Linear regression model:
    link(y) ~ 1 + x1
    Distribution = Binomial

Estimated Coefficients:
                   Estimate     SE            tStat     pValue    
    (Intercept)      -7.3628       0.66815    -11.02    3.0701e-28
    x1             0.0023039    0.00021352     10.79    3.8274e-27

12 observations, 10 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 241, p-value = 2.25e-54

The two models are the same.

Equivalently, you can write s as a structure instead of a cell array of function handles:

s.Link = @norminv;
s.Derivative = @(x) 1./normpdf(norminv(x));
s.Inverse = @normcdf;
g = fitglm(x,[y n],...
    'linear','distr','binomial','link',s)

g = 

Generalized Linear regression model:
    link(y) ~ 1 + x1
    Distribution = Binomial

Estimated Coefficients:
                   Estimate     SE            tStat     pValue    
    (Intercept)      -7.3628       0.66815    -11.02    3.0701e-28
    x1             0.0023039    0.00021352     10.79    3.8274e-27

12 observations, 10 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 241, p-value = 2.25e-54

Brief Model Name

Terms Matrix

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 x1, x2, and x3 and the response variable y in the order x1, x2, x3, and y. Each row of T represents one term:

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.

Formula

A formula for a model specification is a character vector or string scalar of the form

Examples:

Model Display

A linear regression model shows several diagnostics when you enter its name or enter disp(mdl). This display gives some of the basic information to check whether the fitted model represents the data adequately.

For example, fit a Poisson model to data constructed with two out of five predictors not affecting the response, and with no intercept term:

rng('default') % for reproducibility
X = randn(100,5);
mu = exp(X(:,[1 4 5])*[.4;.2;.3]);
y = poissrnd(mu);
mdl = fitglm(X,y,...
    'linear','Distribution','poisson')

mdl = 


Generalized Linear regression model:
    log(y) ~ 1 + x1 + x2 + x3 + x4 + x5
    Distribution = Poisson

Estimated Coefficients:
                   Estimate     SE          tStat       pValue    
    (Intercept)     0.039829     0.10793     0.36901       0.71212
    x1               0.38551    0.076116      5.0647    4.0895e-07
    x2             -0.034905    0.086685    -0.40266        0.6872
    x3              -0.17826    0.093552     -1.9054      0.056722
    x4               0.21929     0.09357      2.3436      0.019097
    x5               0.28918      0.1094      2.6432     0.0082126


100 observations, 94 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 44.9, p-value = 1.55e-08

Notice that:

Diagnostic Plots

Diagnostic plots help you identify outliers, and see other problems in your model or fit. To illustrate these plots, consider binomial regression with a logistic link function.

The logistic model is useful for proportion data. It defines the relationship between the proportion p and the weight w by:

This example fits a binomial model to data. The data are derived from carbig.mat, which contains measurements of large cars of various weights. Each weight in w has a corresponding number of cars in total and a corresponding number of poor-mileage cars in poor.

It is reasonable to assume that the values of poor follow Binomial Distribution distributions, with the number of trials given by total and the percentage of successes depending on w. This distribution can be accounted for in the context of a logistic model by using a generalized linear model with link function $$log(\mu /(1–\mu ))=Xb$$. This link function is called "logit".

w = [2100 2300 2500 2700 2900 3100 ...
     3300 3500 3700 3900 4100 4300]';
total = [48 42 31 34 31 21 23 23 21 16 17 21]';
poor = [1 2 0 3 8 8 14 17 19 15 17 21]';
mdl = fitglm(w,[poor total],...
    "linear","Distribution","binomial","link","logit")

mdl = 
Generalized linear regression model:
    logit(y) ~ 1 + x1
    Distribution = Binomial

Estimated Coefficients:
                   Estimate         SE         tStat       pValue  
                   _________    __________    _______    __________

    (Intercept)       -13.38         1.394    -9.5986    8.1019e-22
    x1             0.0041812    0.00044258     9.4474    3.4739e-21


12 observations, 10 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 242, p-value = 1.3e-54

See how well the model fits the data.

plotSlice(mdl)

The fit looks reasonably good, with fairly wide confidence bounds.

To examine further details, create a leverage plot.

plotDiagnostics(mdl)

This is typical of a regression with points ordered by the predictor variable. The leverage of each point on the fit is higher for points with relatively extreme predictor values (in either direction) and low for points with average predictor values. In examples with multiple predictors and with points not ordered by predictor value, this plot can help you identify which observations have high leverage because they are outliers as measured by their predictor values.

Residuals — Model Quality for Training Data

There are several residual plots to help you discover errors, outliers, or correlations in the model or data. The simplest residual plots are the default histogram plot, which shows the range of the residuals and their frequencies, and the probability plot, which shows how the distribution of the residuals compares to a normal distribution with matched variance.

This example shows residual plots for a fitted Poisson model. The data construction has two out of five predictors not affecting the response, and no intercept term:

rng('default') % for reproducibility
X = randn(100,5);
mu = exp(X(:,[1 4 5])*[2;1;.5]);
y = poissrnd(mu);
mdl = fitglm(X,y,...
    'linear','Distribution','poisson');

Examine the residuals:

plotResiduals(mdl)

While most residuals cluster near 0, there are several near ±18. So examine a different residuals plot.

plotResiduals(mdl,'fitted')

The large residuals don’t seem to have much to do with the sizes of the fitted values.

Perhaps a probability plot is more informative.

plotResiduals(mdl,'probability')

Now it is clear. The residuals do not follow a normal distribution. Instead, they have fatter tails, much as an underlying Poisson distribution.

Plots to Understand Predictor Effects and How to Modify a Model

This example shows how to understand the effect each predictor has on a regression model, and how to modify the model to remove unnecessary terms.

Create a model from some predictors in artificial data. The data do not use the second and third columns in X. So you expect the model not to show much dependence on those predictors.

rng("default") % for reproducibility
X = randn(100,5);
mu = exp(X(:,[1 4 5])*[2;1;.5]);
y = poissrnd(mu);
mdl = fitglm(X,y,...
    "linear","Distribution","poisson");

Examine a slice plot of the responses. This displays the effect of each predictor separately.

plotSlice(mdl)

The scale of the first predictor is overwhelming the plot. To disable the first predictor, select only the second through fifth predictors in the Predictors list, and then click Apply.

Now it is clear that predictors 2 and 3 have little to no effect. You can drag the individual predictor values, which are represented by dashed blue vertical lines. You can also choose between simultaneous and non-simultaneous confidence bounds, which are represented by blue shaded area. Dragging the predictor lines confirms that predictors 2 and 3 have little to no effect.

Remove the unnecessary predictors using either removeTerms or step. Using step can be safer, in case there is an unexpected importance to a term that becomes apparent after removing another term. However, sometimes removeTerms can be effective when step does not proceed. In this case, the two give identical results.

mdl1 = removeTerms(mdl,"x2 + x3")

mdl1 = 
Generalized linear regression model:
    log(y) ~ 1 + x1 + x4 + x5
    Distribution = Poisson

Estimated Coefficients:
                   Estimate       SE       tStat       pValue   
                   ________    ________    ______    ___________

    (Intercept)    0.17604     0.062215    2.8295       0.004662
    x1              1.9122     0.024638    77.614              0
    x4             0.98521     0.026393    37.328    5.6696e-305
    x5             0.61321     0.038435    15.955     2.6473e-57


100 observations, 96 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 4.97e+04, p-value = 0

mdl1 = step(mdl,"NSteps",5,"Upper","linear")

1. Removing x3, Deviance = 93.856, Chi2Stat = 0.00075551, PValue = 0.97807
2. Removing x2, Deviance = 96.333, Chi2Stat = 2.4769, PValue = 0.11553

mdl1 = 
Generalized linear regression model:
    log(y) ~ 1 + x1 + x4 + x5
    Distribution = Poisson

Estimated Coefficients:
                   Estimate       SE       tStat       pValue   
                   ________    ________    ______    ___________

    (Intercept)    0.17604     0.062215    2.8295       0.004662
    x1              1.9122     0.024638    77.614              0
    x4             0.98521     0.026393    37.328    5.6696e-305
    x5             0.61321     0.038435    15.955     2.6473e-57


100 observations, 96 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 4.97e+04, p-value = 0

predict

The predict method gives a prediction of the mean responses and, if requested, confidence bounds.

This example shows how to predict and obtain confidence intervals on the predictions using the predict method.

feval

When you construct a model from a table or dataset array, feval is often more convenient for predicting mean responses than predict. However, feval does not provide confidence bounds.

This example shows how to predict mean responses using the feval method.

random

The random method generates new random response values for specified predictor values. The distribution of the response values is the distribution used in the model. random calculates the mean of the distribution from the predictors, estimated coefficients, and link function. For distributions such as normal, the model also provides an estimate of the variance of the response. For the binomial and Poisson distributions, the variance of the response is determined by the mean; random does not use a separate “dispersion” estimate.

This example shows how to simulate responses using the random method.