Ordinal models to analyze strategy sophistication: Evidence from a learning trajectory efficacy study.

Investigators often rely on the proportion of correct responses in an assessment when describing the impact of early mathematics interventions on child outcomes. Here, we propose a shift in focus to the relative sophistication of problem-solving strategies and offer methodological guidance to researchers interested in working with strategies. We leverage data from a randomized teaching experiment with a kindergarten sample whose details are outlined in Clements et al. (2020). First, we describe our problem-solving strategy data, including how strategies were coded in ways that are amenable to analysis. Second, we explore what kinds of ordinal statistical models best fit the nature of arithmetic strategies, describe what each model implies about problem-solving behavior, and how to interpret model parameters. Third, we discuss the effect of "treatment", operationalized as instruction aligned with an arithmetic Learning Trajectory (LT). We show that arithmetic strategy development is best described as a sequential stepwise process and that children who receive LT instruction use more sophisticated strategies at post-assessment, relative to their peers in a teach-to-target skill condition. We introduce latent strategy sophistication as an analogous metric to traditional Rasch factor scores and demonstrate a moderate correlation them (r = 0.58). Our work suggests strategy sophistication carries information that is unique from, but complimentary to traditional correctness-based Rasch scores, motivating its expanded use in intervention studies.

Modelling the bimodal distribution of indoor gamma-ray dose-rates in Great Britain.

Gamma radiation from naturally occurring sources (including directly ionizing cosmic-rays) is a major component of background radiation. An understanding of the magnitude and variation of doses from these sources is important, and the ability to predict them is required for epidemiological studies. In the present paper, indoor measurements of naturally occurring gamma-rays at representative locations in Great Britain are summarized. It is shown that, although the individual measurement data appear unimodal, the distribution of gamma-ray dose-rates when averaged over relatively small areas, which probably better represents the underlying distribution with inter-house variation reduced, appears bimodal. The dose-rate distributions predicted by three empirical and geostatistical models are also bimodal and compatible with the distributions of the areally averaged dose-rates. The distribution of indoor gamma-ray dose-rates in the UK is compared with those in other countries, which also tend to appear bimodal (or possibly multimodal). The variation of indoor gamma-ray dose-rates with geology, socio-economic status of the area, building type, and period of construction are explored. The factors affecting indoor dose-rates from background gamma radiation are complex and frequently intertwined, but geology, period of construction, and socio-economic status are influential; the first is potentially most influential, perhaps, because it can be used as a general proxy for local building materials. Various statistical models are tested for predicting indoor gamma-ray dose-rates at unmeasured locations. Significant improvements over previous modelling are reported. The dose-rate estimates generated by these models reflect the imputed underlying distribution of dose-rates and provide acceptable predictions at geographical locations without measurements.

Spatial prediction of naturally occurring gamma radiation in Great Britain.

Gamma radiation from natural sources is an important component of background radiation, and correlates with childhood leukaemia risk in Great Britain. The geographic variation of indoor gamma radiation dose-rates in Great Britain is explored using various geo-statistical methods. A multi-resolution Gaussian-process model using radial basis functions with 2, 4, or 8 components, is fitted via maximum likelihood, and a non-spatial model is also used, fitted by ordinary least squares. Because of the dataset size (N = 10,199), four other parametric spatial models are fitted by variogram-fitting. A randomly selected 70:30 split is used for fitting:validation. The models are evaluated based on their predictive performance as measured by Mean Absolute Error, Mean Squared Error, as well as Pearson correlation and rank-correlation between predicted and actual dose-rates. Each of the four parametric models (Matérn, Gaussian, Bessel, Spherical) fitted the empirical variogram well, and yielded similar predictions at >50 km separation, although with more substantial differences in predicted variograms at <50 km. The multi-resolution Gaussian-process model with 8 components had the best predictive accuracy among the models considered. The Spherical, Bessel, Matérn, Gaussian and ordinary least squares models had progressively worse predictive performance, the ordinary least squares model being particularly poor in this respect.