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Generalized Linear Models in Matlab (same results as in R)

Fits Generalized Linear models with one of the following distributions:

'normal' / 'gamma' / 'inv_gsn' / 'poisson' / 'binomial'

And one of the following (if compatible) link functions:

'id' / 'log' / 'sqroot' / 'power' / 'logit' / 'probit' / 'recip' / 'complog'

Estimation procedure has been copied and modified from the package: GLMLAB by Peter Dunn (08 Mar 1999)

This GLM package is essentially the same as Peter Dunn's, but does not recquire the usage of the GLMLAB GUI and outputs results like in R. This package also calculates some additional measure-of-fit statistics.

Some key notes for usage:

  • Supply the X and Y variables in structs, with one variable per element.

    For example:

    • Y.Distance = ...
    • X.Speed = ...
    • X.Time = ...

  • Every supplied variable should be a N x 1 column vector where the rows are the observations.

  • It is also possible to supply weights (in struct W) and an offset (in struct O) to be taken into account in the estimation procedure. As with X and Y, the variables inside the struct should be N x 1.

  • When using the binomial distribution, Y should contain a variable with two columns: (column 1) the responses, (column 2) the sample size per response. If you don't have sample sizes, simply make column 2 a vector of only ones.

Options:

  • GLM.Distribution = 'normal'/'gamma'/'inv_gsn'/'poisson'/'binomial'
  • GLM.Link = 'id'/'log'/'sqroot'/'power'/'logit'/'probit'/'recip'/'complog'
  • GLM.Residual = type of residuals, 'deviance'(default)/'pearson'
  • GLM.Scale = 'mean deviance'(default),'fixed',[positive number]
  • GLM.Intercept = use intercept, 'on'(default)/'off'
  • GLM.ShowEquation = display equation, 'on'(default)/'off'
  • GLM.Display = display output, 'on'(default)/'off'
  • GLM.Recycle = use fit as starting values, 'on'/'off'(default)
  • GLM.illctol = ill-conditioned threshold. default = 1e-10
  • GLM.toler = tolerance. default = 1e-10
  • GLM.maxit = max iterations. default = 30

Estimating a GLM model involves three steps:

  1. Create a GLM model object: mdl = GLM;
  2. Specifying the needed link and distribution: mdl.Distribution = .., mdl.Link = ..
  3. Estimate parameters with the data: mdl.Estimate(Y,X);

After estimation, it is also possible to fit new values to the estimated model as follows: fit = mdl.Fit(Xnew);

The results are almost completely identical with the estimation in R. For comparison, try the Demo.R file (as included in folder) that gives the same output as the Matlab demo examples. See for yourself.

Example usage:

mdl = GLM;
mdl.Distribution = 'gamma';
mdl.Link = 'log';
mdl.Estimate(Y,X);

Example output:

:: convergence in 4 iterations
------------------------------------------------------------------------------------------
   dependent: MilesPerGallon
 independent: (Intercept),Horsepower,Acceleration,Cylinders
------------------------------------------------------------------------------------------
 log(E[MilesPerGallon]) = ß1×(Intercept) + ß2×Horsepower + ß3×Acceleration + ß4×Cylinders
------------------------------------------------------------------------------------------
distribution: GAMMA
        link: LOG
      weight: -
      offset: -
============================================================
    Variable    Estimate     S.E.    z-value    Pr(>|z|)
============================================================
  (Intercept)      4.955     0.510    9.708 	 0.00000
   Horsepower     -0.018     0.004   -4.042 	 0.00005
 Acceleration     -0.026     0.016   -1.680 	 0.09290
    Cylinders      0.093     0.055    1.711 	 0.08706
============================================================
 Residual deviance:     0.0933     Deviance MSE: 0.0085
 Null deviance:         0.3888     Pearson MSE:  0.0088
 Dispersion:            0.0133     Deviance IC:  0.1026
 McFadden R^2:          0.7601     Residual df:  7
============================================================

Disclaimer

The internal estimation of GLM has been directly copied from Peter Dunn. There is no guarantee that this code is correct. Check with other applications (e.g. R) to verify correctness.