Generalized Linear Models 2

Fit Model to Data

Create a fitted model using fitglm or stepwiseglm. Choose between them as in Choose Fitting Method and Model. For generalized linear models other than those with a normal distribution, give a Distribution name-value pair as in Choose Generalized Linear Model and Link Function. For example,

mdl = fitglm(X,y,'linear','Distribution','poisson')
% or
mdl = fitglm(X,y,'quadratic',...
         'Distribution','binomial')

Examine Quality and Adjust the Fitted Model

After fitting a model, examine the result.

• Model Display

• Diagnostic Plots

• Residuals — Model Quality for Training Data

• Plots to Understand Predictor Effects and How to Modify a Model

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 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(µ/(1 – µ)) = 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.

Predict or Simulate Responses to New Data

There are three ways to use a linear model to predict the response to new data:

• predict

• feval

• random

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.

Share Fitted Models

The model display contains enough information to enable someone else to recreate the model in a theoretical sense. For example,

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

1. Adding x1, Deviance = 2515.02869, Chi2Stat = 47242.9622, PValue = 0
2. Adding x4, Deviance = 328.39679, Chi2Stat = 2186.6319, PValue = 0
3. Adding x5, Deviance = 96.3326, Chi2Stat = 232.0642, PValue = 2.114384e-52

mdl = 


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

You can access the model description programmatically, too. For example,

mdl.Coefficients.Estimate

ans =

    0.1760
    1.9122
    0.9852
    0.6132

mdl.Formula

ans = 

log(y) ~ 1 + x1 + x4 + x5