Hotelling's T2
Definition
The multivariate generalisation of the t-test used to compare mean element-concentration vectors between questioned and known glass populations; accounts for the full covariance structure of the element data and is specified by ASTM E2927.
- Type
- Multivariate generalisation of the t-test
- Application
- Compares mean element-concentration vectors between glass populations
- Standard
- ASTM E2927
- Key strength
- Accounts for the full covariance structure of the element data
Common questions
Why not just run separate t-tests on each trace element?+
Running one univariate test per element inflates the overall false-positive rate and ignores correlations between elements, for example elements that co-vary because they come from the same raw material batch, which Hotelling's T2 accounts for by testing the full vector jointly.
What does a non-significant Hotelling's T2 result mean for a glass comparison?+
It means the questioned and known samples' mean element vectors are not statistically distinguishable given the measured variability, which supports but does not prove common origin, since it cannot rule out that two different sources happen to share a similar elemental signature.
Does the test require a minimum number of measurements per sample?+
Yes, LA-ICP-MS glass comparisons under ASTM E2927 specify a minimum number of replicate ablations per fragment so the within-sample covariance can be estimated reliably before the between-sample test is applied.
Related terms
- ASTM E1967
- The ASTM International standard for forensic glass RI measurement; specifies three acceptable methods (automated, semi-automated, manual), minimum precision of ±0.0001 RI units,...
- ASTM E2927
- Standard test method for the determination of trace elements in glass samples using LA-ICP-MS; specifies the element suite, calibration standards (NIST SRM...
- Bayesian Likelihood Ratio (LR)
- The statistical framework that evaluates glass-fragment evidence by comparing the probability of the observed fragment population assuming contact with the source glass...
- ENFSI ENG3-2013
- The ENFSI Glass Expert Working Group guideline for glass comparison casework; specifies the examination sequence, within-source variation requirements, statistical comparison framework, and...
- GRIM-3
- Glass Refractive Index Measurement, third-generation instrument by Foster + Freeman (UK); an automated phase-contrast hot-stage system that identifies the RI match point...
- LA-ICP-MS
- Laser ablation inductively coupled plasma mass spectrometry; a technique that fires a laser pulse at a sample surface, vaporises a tiny amount,...
- Mahalanobis Distance
- A multivariate distance measure that expresses the separation between two observations in units of the population standard deviation, accounting for covariance; used...
- NIST SRM 612
- National Institute of Standards and Technology Standard Reference Material 612 (Trace Elements in Glass); the primary calibration material for LA-ICP-MS glass comparison,...
- Secondary Transfer
- The movement of a glass fragment from a primary surface (which contacted the broken glass) to a secondary surface (which contacted the...
- Sequential Gate Design
- The casework workflow structure in which analytical methods are applied in order of increasing cost and complexity, with an exclusion possible at...
- Within-Source Variation
- The natural variation observed among multiple samples collected from a single source location. Used to estimate the probability of the observed difference...