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In this article, we explain and demonstrate how to model norm scores with the cNORM package in R. This package is designed specifically to determine norm scores when the latent ability to be measured covaries with age or other explanatory variables such as grade level. The mathematical method used in this package draws on polynomial regression to model a three-dimensional hyperplane that smoothly and continuously captures the relation between raw scores, norm scores and the explanatory variable. By doing so, it overcomes the typical problems of classical norming methods, such as overly large age intervals, missing norm scores, large amounts of sampling error in the subsamples or huge requirements with regard to the sample size. After a brief introduction to the mathematics of the model, we describe the individual methods of the package. We close the article with a practical example using data from a real reading comprehension test.

The interpretation of psychometric test results is usually based on norm scores. We compared semiparametric continuous norming (SPCN) with conventional norming methods by simulating results for test scales with different item numbers and difficulties via an item response theory approach. Subsequently, we modeled the norm scores based on random samples with varying sizes either with a conventional ranking procedure or SPCN. The norms were then cross-validated by using an entirely representative sample of N = 840,000 for which different measures of norming error were computed. This process was repeated 90,000 times. Both approaches benefitted from an increase in sample size, with SPCN reaching optimal results with much smaller samples. Conventional norming performed worse on data fit, age-related errors, and number of missings in the norm tables. The data fit in conventional norming of fixed subsample sizes varied with the granularity of the age brackets, calling into question general recommendations for sample sizes in test norming. We recommend that test norms should be based on statistical models of the raw score distributions instead of simply compiling norm tables via conventional ranking procedures. [ABSTRACT FROM AUTHOR]

Conventional methods for producing test norms are often plagued with “jumps” or “gaps” (i.e., discontinuities) in norm tables and low confidence for assessing extreme scores. We propose a new approach for producing continuous test norms to address these problems that also has the added advantage of not requiring assumptions about the distribution of the raw data: Norm values are established from raw data by modeling the latter ones as a function of both percentile scores and an explanatory variable (e.g., age). The proposed method appears to minimize bias arising from sampling and measurement error, while handling marked deviations from normality—such as are commonplace in clinical samples. In addition to step-by-step instructions in how to apply this method, we demonstrate its advantages over conventional discrete norming procedures using norming data from two different psychometric tests, employing either age norms (N = 3,555) or grade norms (N = 1,400). [ABSTRACT FROM AUTHOR]