/// <summary>
/// Gets the Wald Test for all coefficients.
/// </summary>
///
/// <remarks>
/// The Wald statistical test is a test for a model parameter in which
/// the estimated parameter θ is compared with another proposed parameter
/// under the assumption that the difference between them will be approximately
/// normal. There are several problems with the use of the Wald test. Please
/// take a look on substitute tests based on the log-likelihood if possible.
/// </remarks>
///
/// <param name="category">The category's index.</param>
///
public WaldTest[] GetWaldTest(int category)
{
var tests = new WaldTest[NumberOfInputs + 1];
for (int i = 0; i < tests.Length; i++)
{
tests[i] = GetWaldTest(category, i);
}
return(tests);
}
public void ComputeTest1()
{
// Consider the following example data, adapted from John C. Pezzullo's
// example for his great Cox's proportional hazards model example in
// JavaScript (http://www.sph.emory.edu/~cdckms/CoxPH/prophaz2.html).
// In this data, we have three columns. The first column denotes the
// input variables for the problem. The second column, the survival
// times. And the last one is the output of the experiment (if the
// subject has died [1] or has survived [0]).
double[,] example =
{
// input time censor
{ 50, 1, 0 },
{ 70, 2, 1 },
{ 45, 3, 0 },
{ 35, 5, 0 },
{ 62, 7, 1 },
{ 50, 11, 0 },
{ 45, 4, 0 },
{ 57, 6, 0 },
{ 32, 8, 0 },
{ 57, 9, 1 },
{ 60, 10, 1 },
};
// First we will extract the input, times and outputs
double[,] inputs = example.GetColumns(0);
double[] times = example.GetColumn(1);
int[] output = example.GetColumn(2).ToInt32();
// Now we can proceed and create the analysis
var cox = new ProportionalHazardsAnalysis(inputs, times, output);
cox.Compute(); // compute the analysis
// Now we can show an analysis summary
// DataGridBox.Show(cox.Coefficients);
// We can also investigate all parameters individually. For
// example the coefficients values will be available at
double[] coef = cox.CoefficientValues;
double[] stde = cox.StandardErrors;
// We can also obtain the hazards ratios
double[] ratios = cox.HazardRatios;
// And other information such as the partial
// likelihood, the deviance and also make
// hypothesis tests on the parameters
double partial = cox.LogLikelihood;
double deviance = cox.Deviance;
// Chi-Square for whole model
ChiSquareTest chi = cox.ChiSquare;
// Wald tests for individual parameters
WaldTest wald = cox.Coefficients[0].Wald;
// Finally, we can also use the model to predict
// scores for new observations (without considering time)
double y1 = cox.Regression.Compute(new double[] { 63 });
double y2 = cox.Regression.Compute(new double[] { 32 });
// Those scores can be interpreted by comparing then
// to 1. If they are greater than one, the odds are
// the patient will not survive. If the value is less
// than one, the patient is likely to survive.
// The first value, y1, gives approximately 86.138,
// while the second value, y2, gives about 0.00072.
// We can also consider instant estimates for a given time:
double p1 = cox.Regression.Compute(new double[] { 63 }, 2);
double p2 = cox.Regression.Compute(new double[] { 63 }, 10);
// Here, p1 is the score after 2 time instants, with a
// value of 0.0656. The second value, p2, is the time
// after 10 time instants, with a value of 6.2907.
Assert.AreEqual(86.138421225296526, y1);
Assert.AreEqual(0.00072281400325299814, y2);
Assert.AreEqual(0.065660458190496693, p1);
Assert.AreEqual(6.2907511049223928, p2);
Assert.AreEqual(1, coef.Length);
Assert.AreEqual(0.37704239281490765, coef[0]);
Assert.AreEqual(0.25415746361167235, stde[0]);
Assert.AreEqual(1.4579661153488215, ratios[0]);
Assert.AreEqual(-2.0252666205735466, partial);
Assert.AreEqual(4.0505332411470931, deviance);
Assert.AreEqual(0.13794183001851756, wald.PValue, 1e-4);
Assert.AreEqual(1, chi.DegreesOfFreedom);
Assert.AreEqual(7.3570, chi.Statistic, 1e-4);
Assert.AreEqual(0.0067, chi.PValue, 1e-3);
}
public void learn_test()
{
#region doc_learn_part1
// Consider the following example data, adapted from John C. Pezzullo's
// example for his great Cox's proportional hazards model example in
// JavaScript (http://statpages.org/prophaz2.html).
// In this data, we have three columns. The first column denotes the
// input variables for the problem. The second column, the survival
// times. And the last one is the output of the experiment (if the
// subject has died [1] or has survived [0]).
double[][] example =
{
// input time censor
new double[] { 50, 1, 0 },
new double[] { 70, 2, 1 },
new double[] { 45, 3, 0 },
new double[] { 35, 5, 0 },
new double[] { 62, 7, 1 },
new double[] { 50, 11, 0 },
new double[] { 45, 4, 0 },
new double[] { 57, 6, 0 },
new double[] { 32, 8, 0 },
new double[] { 57, 9, 1 },
new double[] { 60, 10, 1 },
};
// First we will extract the input, times and outputs
double[][] inputs = example.Get(null, 0, 1);
double[] times = example.GetColumn(1);
SurvivalOutcome[] output = example.GetColumn(2).To <SurvivalOutcome[]>();
// Now we can proceed and create the analysis (giving optional variable names)
var cox = new ProportionalHazardsAnalysis(new[] { "input" }, "time", "censor");
// Then compute the analysis, learning a regression in the process:
ProportionalHazards regression = cox.Learn(inputs, times, output);
// Now we can show an analysis summary
// Accord.Controls.DataGridBox.Show(cox.Coefficients);
#endregion
#region doc_learn_part2
// We can also investigate all parameters individually. For
// example the coefficients values will be available at
double[] coef = cox.CoefficientValues; // should be { 0.37704239281490765 }
double[] stde = cox.StandardErrors; // should be { 0.25415746361167235 }
// We can also obtain the hazards ratios
double[] ratios = cox.HazardRatios; // should be { 1.4579661153488215 }
// And other information such as the partial
// likelihood, the deviance and also make
// hypothesis tests on the parameters
double partialL = cox.LogLikelihood; // should be -2.0252666205735466
double deviance = cox.Deviance; // should be 4.0505332411470931
// Chi-Square for whole model
ChiSquareTest chi = cox.ChiSquare; // should be 7.3570 (p=0.0067)
// Wald tests for individual parameters
WaldTest wald = cox.Coefficients[0].Wald; // should be 1.4834 (p=0.1379)
// Finally, we can also use the model to predict
// scores for new observations (without considering time)
double y1 = cox.Regression.Probability(new double[] { 63 }); // should be 86.138421225296526
double y2 = cox.Regression.Probability(new double[] { 32 }); // should be 0.00072281400325299814
// Those scores can be interpreted by comparing then
// to 1. If they are greater than one, the odds are
// the patient will not survive. If the value is less
// than one, the patient is likely to survive.
// The first value, y1, gives approximately 86.138,
// while the second value, y2, gives about 0.00072.
// We can also consider instant estimates for a given time:
double p1 = cox.Regression.Probability(new double[] { 63 }, 2); // should be 0.17989138010770425
double p2 = cox.Regression.Probability(new double[] { 63 }, 10); // should be 15.950244161356357
// Here, p1 is the score after 2 time instants, with a
// value of 0.0656. The second value, p2, is the time
// after 10 time instants, with a value of 6.2907.
// In addition, if we would like a higher precision when
// computing very small probabilities using the methods
// above, we can use the LogLikelihood methods instead:
double log_y1 = cox.Regression.LogLikelihood(new double[] { 63 }); // should be 4.4559555514489091
double log_y2 = cox.Regression.LogLikelihood(new double[] { 32 }); // should be -7.2323586258132284
double log_p1 = cox.Regression.LogLikelihood(new double[] { 63 }, 2); // should be -1.7154020540835324
double log_p2 = cox.Regression.LogLikelihood(new double[] { 63 }, 10); // should be 2.7694741370357177
#endregion
Assert.AreEqual(86.138421225296526, y1, 1e-10);
Assert.AreEqual(0.00072281400325299814, y2, 1e-10);
Assert.AreEqual(0.17989138010770425, p1, 1e-10);
Assert.AreEqual(15.950244161356357, p2, 1e-10);
Assert.AreEqual(4.4559555514489091, log_y1, 1e-10);
Assert.AreEqual(-7.2323586258132284, log_y2, 1e-10);
Assert.AreEqual(-1.7154020540835324, log_p1, 1e-10);
Assert.AreEqual(2.7694741370357177, log_p2, 1e-10);
Assert.AreEqual(System.Math.Log(y1), log_y1, 1e-10);
Assert.AreEqual(System.Math.Log(y2), log_y2, 1e-10);
Assert.AreEqual(System.Math.Log(p1), log_p1, 1e-10);
Assert.AreEqual(System.Math.Log(p2), log_p2, 1e-10);
Assert.AreEqual(1, coef.Length);
Assert.AreEqual(0.37704239281490765, coef[0]);
Assert.AreEqual(0.25415746361167235, stde[0]);
Assert.AreEqual(1.4579661153488215, ratios[0]);
Assert.AreEqual(-2.0252666205735466, partialL, 1e-6);
Assert.AreEqual(4.0505332411470931, deviance, 1e-6);
Assert.AreEqual(1.4834991955655938, wald.Statistic, 1e-4);
Assert.AreEqual(0.13794183001851756, wald.PValue, 1e-4);
Assert.AreEqual(1, chi.DegreesOfFreedom);
Assert.AreEqual(7.3570, chi.Statistic, 1e-4);
Assert.AreEqual(0.0067, chi.PValue, 1e-3);
}
