GLM t-Statistic

Evaluate a GLM contrast using the full already-scaled coefficient covariance; no additional residual variance multiplier is applied.

Description

A fitted GLM contrast estimates a linear combination of coefficients.

A fitted GLM contrast estimates a linear combination of coefficients. Its t statistic divides that estimate by uncertainty derived from the fitted coefficient covariance.

The tool returns contrastEstimate (contrast estimate); contrastStandardError (contrast standard error); tStatistic (t statistic); degreesOfFreedom (degrees of freedom). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use GLM t-Statistic

Use GLM t-Statistic when you already have coefficients, contrast, coefficientCovariance, residualDegreesOfFreedom and want contrastEstimate, contrastStandardError, tStatistic, degreesOfFreedom. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use GLM t-Statistic and understand its results

Enter coefficients, contrast, coefficientCovariance, residualDegreesOfFreedom using the field table below. Match the units, ordering and data boundaries described in the calculation.
Run the tool after checking the stated bounds and matching paired observations or categories. A malformed request gives a tool error rather than a usable measurement.
Read contrastEstimate, contrastStandardError, tStatistic, degreesOfFreedom. Use the calculation and interpretation below to distinguish amounts, ratios, signs and decision flags where applicable.
Review the supplied parameters and assumptions before using the result in a report or comparing it with another observation.

Input field	Meaning	Value type and bounds	Requirement
coefficients	Coefficients	list of number	Required
contrast	Contrast	list of number	Required
coefficientCovariance	Coefficient covariance	list of list of number	Required
residualDegreesOfFreedom	Residual degrees of freedom	integer	Required

Require matching coefficients and contrast, a symmetric positive-definite covariance of at most 32 coefficients and positive integer residual degrees of freedom. Validate covariance by Cholesky factorization.

estimate = cᵀβ; SE = sqrt(cᵀCov(β)c); t = estimate/SE

The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.

json

{
  "input": {
    "coefficients": [
      2,
      3
    ],
    "contrast": [
      1,
      -1
    ],
    "coefficientCovariance": [
      [
        1,
        0
      ],
      [
        0,
        3
      ]
    ],
    "residualDegreesOfFreedom": 10
  },
  "output": {
    "contrastEstimate": -1,
    "contrastStandardError": 2,
    "tStatistic": -0.5,
    "degreesOfFreedom": 10
  }
}

Limitations and assumptions

The covariance must already include the appropriate fitted uncertainty. No model fitting, p value or multiple-comparison correction occurs.

Numerical inputs must be finite. The tool checks its declared input shape and calculation constraints; those checks do not verify the quality, calibration or representativeness of the supplied observations.

Alternative or Complementary methods

Permutation Threshold: Calculate upper-tail max-statistic permutation p-values with the plus-one correction and a separately labeled interpolated reference quantile; decisions use p-values, not an incompatible quantile cutoff. Crawford-Howell t: Compare one case with a small normative control sample using Crawford–Howell t and n−1 degrees of freedom; no p-value, tail choice or clinical interpretation is invented.

Methodological background: Least squares estimation.^[1]

References

Least squares estimation — NIST/SEMATECH

Similar or alternative tools

Permutation Threshold
Calculate upper-tail max-statistic permutation p-values with the plus-one correction and a separately labeled interpolated reference quantile; decisions use p-values, not an incompatible quantile cutoff.
Crawford-Howell t
Compare one case with a small normative control sample using Crawford–Howell t and n−1 degrees of freedom; no p-value, tail choice or clinical interpretation is invented.