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GlmLink

The link a GeneralizedLinearModel maps its linear predictor to the mean through.

public enum GlmLink

Members — Default is each family's statsmodels default: logit for Binomial, log for Poisson and NegativeBinomial, inverse for Gamma. Log is η = log μ. Inverse is η = 1/μ, statsmodels' InversePower.

Example — the same positive, skewed response through Gamma's two links: the log link's slope reads as a growth rate, and it estimates a smaller dispersion here than the inverse link does.

using Lodestar.Stats.Regression;

double[] x = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0];
double[] cost = [2.1, 1.8, 3.5, 2.9, 4.8, 5.5, 4.9, 7.8, 6.9, 9.4];

GlmSummary inverse = GeneralizedLinearModel.Fit(x, cost, 1, GlmFamily.Gamma);
GlmSummary log = GeneralizedLinearModel.Fit(x, cost, 1, GlmFamily.Gamma, new GlmOptions { Link = GlmLink.Log });

double growth = Math.Round(log.Coefficients[1], 4);            // => 0.1718
double inverseDispersion = Math.Round(inverse.Dispersion, 4);  // => 0.0526
double logDispersion = Math.Round(log.Dispersion, 4);          // => 0.0317

Remarks — not every family takes every link here. Gamma takes Inverse or Log; the two count families take Log, their default; Binomial takes only its logit. Any other pairing is refused by GeneralizedLinearModel.Fit on options, rather than fitted.

Under Inverse, nothing keeps 1/η positive, which is why statsmodels warns when it builds that link. A Gamma fit whose mean reaches zero or below is refused at the iteration it happens, where the reference carries on with |μ| in its variance; Log keeps every mean positive.

Applies to — net10.0, netstandard2.0.

See also — GlmFamily, GlmOptions.