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Sequential Generalized Likelihood Ratio Tests for Online Item Monitoring

Abstract

The study presents statistical procedures that monitor functioning of items over time. We propose generalized likelihood ratio tests that surveil multiple item parameters and implement with various sampling techniques to perform continuous or intermittent monitoring. The procedures examine stability of item parameters across time and inform compromise as soon as they identify significant parameter shift. The performance of the monitoring procedures was validated using simulated and real-assessment data. The empirical evaluation suggests that the proposed procedures perform adequately well in identifying the parameter drift. They showed satisfactory detection power and gave timely signals while regulating error rates reasonably low. The procedures also showed superior performance when compared with the existent methods. The empirical findings suggest that multivariate parametric monitoring can provide an efficient and powerful control tool for maintaining the quality of items. The procedures allow joint monitoring of multiple item parameters and achieve sufficient power using powerful likelihood-ratio tests. Based on the findings from the empirical experimentation, we suggest some practical strategies for performing online item monitoring.

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Notes

  1. Previous studies defined the reference sample as \({\mathcal {R}} = \{i: \, i = 1 , \, \ldots \, , \,t - m \}\) such that it increases as the monitoring progresses. This study uses a fixed reference sample to alleviate the probable impact of false negatives in the expanding reference sample.

  2. Under the simulation design in Sect. 4, sequential testing based on \({\mathcal {T}}^2\) showed average false positive rate of 16.88%. The procedure also tended to flag drift items prematurely before the actual parameter shift, exhibiting early detection rate of 9.70%. The chart seemed overly sensitive to small fluctuations that occur from the sampling and calibration error. Note that, unlike standard Shewhart control charts, which examine manifest variables, the Shewhart chart based on \({\mathcal {T}}^2\) examines estimable parameters and can be influenced by the sampling and estimation error.

  3. We note that there are other ways of constructing a multivariate chart (e.g., Healy, 1987, Pignatiello & Runger, 1990, Woodall & Ncube, 1985). These procedures, however, make impractical assumptions (e.g., known directions or multiple univariate charts) or make little difference in the monitoring statistics in the present setting because only one observation is evaluated each time.

  4. In sequential testing, Type I error can be defined in three ways–across the event times, the items, and across both the events and items.

  5. Recall that the charting statistics are obtained by subtracting the reference values (k). The larger the k, the smaller the null charting statistics, and thus, the smaller the decision limit.

  6. We also contemplated simulation for attaining the threshold values. The resulting values, however, did not generally accord with the statistics in the real data possibly due to disparity in sampling (e.g., content-balancing and item exposure control in real testing.)

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Kang, HA. Sequential Generalized Likelihood Ratio Tests for Online Item Monitoring. Psychometrika (2022). https://doi.org/10.1007/s11336-022-09871-9

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Keywords

  • item parameter drift
  • online monitoring
  • sequential generalized likelihood ratio test
  • cumulative sum control chart
  • response time