DNA Fingerprinting

A DNA "fingerprint" provides a profile of an individual's genes. Researchers use this information to estimate how closely related individuals are. Geyer and Thompson (G&T, 1992) remark that previous statistical models have omitted important parameters "because without a stochastic model to guide inference, it is difficult to know what to do with them." G&T see the value of sufficient statistics for this problem and construct an autologistic model for DNA data that nearly gives Rasch measures. Here is a revision of their model that does provide Rasch measurement.

G&T's simulated DNA data for 13 individuals is illustrative (see Table). A "-" indicates the presence of the gene in the individual. Some genes are more widespread in the sample than others. Some individuals have more different genes and so are more genetically diverse than others. Individuals who share more genes are hypothesized to be more closely related. This is the basis of G&T's model, which can be expressed in Rasch form as:

Pnb(present) is the probability that individual n exhibits gene b.
Wb is how Widespread gene b is in the sample.
Dn is the genetic Diversity of individual n.
Cnm is the Closeness of the relationship between individuals n and m among the S individuals.

This model, as it stands, does not provide unambiguous measures because the individual's own genetic diversity cannot be disentangled from that due to relationships with others. This obstacle is overcome with a two-stage approach. First, as much genetic variation as possible is explained by the main effects - the measures of the prevalence of each gene and of the genetic diversity of each individual. Thus,

Then, these main-effect, Wb and Dn, measures are fixed (anchored) at their estimates, and the remaining variance is modelled as interactions, i.e., paired relationships among the individuals in the sample:

This approach produces the plotted results. Gene 15 is exhibited by all but one individual. Individual 7 has the fewest different genes. Individuals 7 and 9 are the most closely related, because 9 exhibits all of 7's genes.

Linacre JM (1993) DNA Fingerprinting. Rasch Measurement Transactions, 7:1, p. 274.

Geyer C.J. & Thompson E.A. 1992 Constrained Monte Carlo maximum likelihood for dependent data. Journal Royal Statistical Society, B, 54(3), 657-699

Logit      Gene Band   Individual  Pairing
           Prevalence  Gene Mix    Relation
           Widespread  Diverse     Close
 3                                 7-9
           15 ***

           ********     *******    **************
 0         ****         8 ***      ***************
           ******       9          ******
-1         *****                   8-9 *
           **           7

           19 **                   (* = 2 pairs)
-3         Rare        Inbred      Distant

DNA fingerprinting. Linacre JM. … Rasch Measurement Transactions, 1993, 7:1 p.274

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
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Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
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Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
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