# Discrete-time Survival Analysis: A Rasch-Model Logistic Regression

Singer and Willett (S&W, 1993) use a logistic regression model for discrete-time survival analysis that works because it is a Rasch model. They study the drop-out rate for 3,941 special education teachers hired in Michigan between 1972 to 1978.

Their technique and rationale provide support and insight for Rasch practitioners. S&W Model A, reparameterized, is where
Pni = the probability that teacher n drops out in year i, -Di = the baseline hazard level in year i, B = the "fixed-effect" proclivity to drop-out, common to all teachers. Set at B=0.

Years are encountered sequentially until a teacher drops out or data collection ceases causing the teacher's data to become missing. The S&W teacher survival data for 12 years is depicted in the Figure. The years are rescaled quadratically to clarify the trend. Manual calculation and a Facets Rasch analysis confirm S&W's results.

S&W can take their analysis further by using the measurement properties of their models. Linearity is assured for fitting data because their models incorporate the necessary and sufficient conditions for constructing linear measures. What is not assured is the extent to which their data cooperate in constructing this linearity, i.e., fit their model.

S&W suggest two methods for checking linearity. The first is to partition data and check for parameter invariance. This method is standard in Rasch work. Rasch (1960/1980) uses grouping to look for systematic deviations. His method is widely used to check for differential item functioning.

S&W's second method is to investigate whether better global fit is obtained by adding non-linear predictors. This strategy is disastrous because it sacrifices linearity for better model-to-data fit, degrading the model into a transient description of local data.

Linearity, however, is vital for comprehensibility and generality. Only with linearity does one unit mean the same amount everywhere - only with linearity are graphs readable. Introducing non-linear components produces uneven and hence incomparable units. This disaster is also incurred when a "guessing" parameter is added to a model to "explain" misfitting low ability performances.

Instead of introducing interaction terms to account for misfit, a measurement- oriented strategy examines residual patterns and quality-control fit statistics for useful data partitions. Local failures of the data to support linear measurement construction are detected by local misfits, rather than by the increase in global fit that occurs when local misfits are modelled non-linearly. Identification of local misfit motivates the analyst either to edit the data toward coherent subsets or to reconceptualize the linear model. In the S&W data where "very high-scoring teachers [on the National Teacher's Examination] are disproportionately likely to leave," it makes sense to stratify teachers by NTE score group, just as Rasch (1980 p.96-100) stratified test takers by their speed. Then, since the hazard measures for each score group are in a linear frame of reference, these score group measures can be compared by standard graphical and statistical techniques. Benjamin D. Wright

Singer J. D. & Willett J. B. (1993) It's about time: Using discrete-time survival analysis to study duration and the timing of events. Journal of Educational Statistics. Summer. 18(2) p.155-195.

Discrete-time survival analysis. Wright BD. Rasch Measurement Transactions 1993 7:3 p.307

Discrete-time survival analysis: a Rasch-model logistic regression. Wright BD. … Rasch Measurement Transactions, 1993, 7:3 p.307

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
Rating Scale Analysis - free, Wright & Masters
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
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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