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ParametricSummary

A fitted parametric model: its parameters with their inference, and its curves.

public sealed class ParametricSummary

Properties — the per-parameter lists are parallel and in lifelines' order:

• ParameterNames are lifelines' names: lambda_ and rho_ for the Weibull, mu_ and sigma_ for the log-normal, and so on; ParametricModel lists them.
• Parameters are the fitted values.
• StandardErrors are the square roots of the inverse observed information's diagonal.
• ZStatistics is each parameter less lifelines' comparison value, over its standard error.
• PValues are two-sided, from the normal tail.
• ConfidenceLower and ConfidenceUpper are each parameter's interval at ConfidenceLevel.

For the model:

• Model is the ParametricModel fitted.
• LogLikelihood is the log-likelihood at the fit, summed over the subjects with their weights.
• Akaike is Akaike's criterion, −2 · LogLikelihood + 2k over the k parameters.
• MedianSurvivalTime is the time the fitted survival falls to one half, lifelines' median_survival_time_.
• ConfidenceLevel is the level the intervals were built at.

Example — the Weibull fit of the ParametricSurvival page, its shape read against one.

using Lodestar.Survival;

double[] months = [5, 8, 12, 3, 15, 9, 20, 6, 11, 14];
bool[] died = [true, true, false, true, true, true, false, true, true, false];

ParametricSummary fit = ParametricSurvival.Fit(months, died, ParametricModel.Weibull);

string shapeName = fit.ParameterNames[1];            // => rho_
double shapeZ = Math.Round(fit.ZStatistics[1], 6);   // => 1.306226
double shapeP = Math.Round(fit.PValues[1], 6);       // => 0.191476
double aic = Math.Round(fit.Akaike, 6);              // => 53.400484

Remarks — each z statistic is measured from lifelines' _compare_to_values, not from zero. A Weibull rho_ of one is the exponential model, so its z statistic above tests whether the hazard changes with time at all, and a p-value of 0.19 says ten patients cannot tell. The values are one for the Weibull's and the log-logistic's two parameters and for the log-normal's sigma_, zero for the log-normal's mu_, the exponential's lambda_ and the generalized gamma's mu_ and ln_sigma_, one for its lambda_, and lifelines' starting values for the piecewise exponential's rates. AftSummary measures its coefficients from zero.

The generalized gamma's shape derivatives are numerical on both sides. Its likelihood holds the regularised incomplete gamma function, differentiated in its shape parameter; lifelines' autograd_gamma takes that derivative by finite differences, which puts lifelines' own standard errors up to about 2e-5 relative off. These take central differences extrapolated by Richardson's method. Within 0.35 of a zero lambda_ the likelihood here follows Temme's uniform expansion about the log-normal instead, with exact derivatives, short of the tail where lambda_ times the log time standardized through the expansion passes 2.2 in magnitude: through the gamma at shape 1/lambda_² they lost their digits, and a log-normal sample's fit did not converge. Below about 1e-10 lifelines' own clip of that gamma's variate at 1e20 holds at every point; this follows the smooth limit there.

A class rather than a record, as CoxSummary is: it carries the model it evaluates its curves with.

Applies to — net10.0, netstandard2.0.

See also — ParametricSurvival.Fit, ParametricModel, ParametricOptions.

Members

Member	What it does
ParametricSummary.CumulativeHazard	The fitted cumulative hazard at given times.
ParametricSummary.CumulativeHazardBounds	The delta-method interval of the cumulative hazard.
ParametricSummary.Hazard	The fitted hazard at given times.
ParametricSummary.Percentile	The time by which survival falls to a level.
ParametricSummary.Survival	The fitted survival function at given times.
ParametricSummary.SurvivalBounds	The delta-method interval of the survival function.