Details

The statistics are computed given the following assumptions about the data distribution:

  • Responses y ij , i = 1, ..., n, are independent and have a constant variance σ j2, j = 1, ..., k
  • Conditional expectation of responses y .j , j = 1, ..., k, is linear in input variables x. = (x .1, ... , x .p )
  • Deviations of y ij , i = 1, ..., n, around the mean of expected responses ERM j , j = 1, ..., k, are additive and Gaussian.

Testing Insignificance of a Single Beta

The library uses the following quality metrics:

Quality Metric

Definition

Root Mean Square (RMS) Error

Vector of variances

A set of variance-covariance matrices C = C 1, ..., C k for vectors of betas β jt , j = 1, ..., k

Z-score statistics used in testing of insignificance of a single coefficient β jt



σ i is the j-th element of the vector of variance σ 2 and ν t is the t-th diagonal element of the matrix (X TX)-1

Confidence interval for β jt



pc 1 - α is the (1 - α) percentile of the Gaussian distribution, σ i is the j-th element of the vector of variance σ 2, and ν t is the t-th diagonal element of the matrix (X TX)-1

Testing Insignificance of a Group of Betas

The library uses the following quality metrics:

Quality Metric

Definition

Mean of expected responses, ERM = (ERM 1, ..., ERM k )

Variance of expected responses, ERV = (ERV 1, ..., ERV k )

Regression Sum of Squares RegSS = (RegSS 1, ..., RegSS k )

Sum of Squares of Residuals ResSS = (ResSS 1, ..., ResSS k )

Total Sum of Squares TSS = (TSS 1, ..., TSS k )

Determination Coefficient



F-statistics used in testing insignificance of a group of betas F = (F 1, ..., F k )



where ResSS j are computed for a model with p +1 betas and ResSS 0j are computed for a reduced model with p 0+1 betas (p - p 0 betas are set to zero)

For more complete information about compiler optimizations, see our Optimization Notice.
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