The package cNorm provides methods for generating continuous standard scores, as f. e. for psychometric test development, biometrics (e. g. biological and physiological growth curves), and screenings in the medical domain. It is based on the approach suggested by A. Lenhard et al. (2016, 2019). For an in-depth tutorial please consult the project homepage https://www.psychometrica.de/cNorm_en.html and https://cnorm.shinyapps.io/cNORM/ for an online demonstration.

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. cNORM addresses these problems and also has the added advantage of not requiring assumptions about the distribution of the raw data: The standard scores are established from raw data by modeling the latter ones as a function of both percentile scores and an explanatory variable (e.g., age) through Taylor polynomials. The method minimizes bias arising from sampling and measurement error, while handling marked deviations from normality – such as are commonplace in clinical samples. Contrary to parametric approaches, it does not rely on distribution assumptions of the initial norm data and is thus a very robust approach in generating norm tables.

The rationale of the approach is model the relationship between location / norm score, age and raw score via multiple regression and to fit a 3-dimensional hyperplane. This hyperplane is used to close all gaps and to compute continuous norm scores:

cNORM can be installed via `{r example} install.packages("cNORM", dependencies = TRUE)`

Additionally, you can download a precompiled version or access the github development version via `{r example} install.packages("devtools") devtools::install_github("WLenhard/cNORM") library(cNORM)`

Please report errors. Suggestions for improvement are always welcome!

Conducting the analysis consists of the following steps: 1. Data preparation and establishing the regression model 1. Validating the model 1. Generating norm tables and plotting the results

cNORM offers functions for selecting the best fitting models and in generating the norm tables.

```{r example} ## Basic example code for modeling the sample dataset library(cNORM)

cNORM.GUI()

cnorm.elfe <- cnorm(raw = elfe\(raw, group = elfe\)group)

plot(cnorm.elfe, “subset”, type=0) # plot R2 plot(cnorm.elfe, “subset”, type=3) # plot MSE

cnorm.elfe <- cnorm(raw = elfe\(raw, group = elfe\)group, terms = 4)

plot(cnorm.elfe, “percentiles”)

plot(cnorm.elfe, “norm”) plot(cnorm.elfe, “raw”)

plot(cnorm.elfe, “series”, end=14)

cnorm.cv(cnorm.elfe$data, max=10, repetitions=3)

cnorm.cv(cnorm.elfe, repetitions=3)

normTable(c(3, 3.2, 3.4, 3.6), cnorm.elfe)

rawTable(3, cnorm.elfe, CI = .9, reliability = .94)

d <- rankByGroup(elfe, raw=“raw”, group=FALSE) d <- computePowers(d) m <- bestModel(d) rawTable(0, model = m) # please use an arbitrary value for age when generating the tables

vignette(“cNORM-Demo”, package = “cNORM”) ``` cNORM offers functions to choose the optimal model, both from a visual inspection of the percentiles, as well as by information criteria and model tests:

In this example, a Taylor polynomial with power k = 4 was computed in order to model a sample of the ELFE 1-6 reading comprehension test (sentence completion task; W. Lenhard & Schneider, 2006). In the plot, you can see the share of variance explained by the different models (with progressing number of predictors). Adjusted R2, Mallow’s Cp (an AIC like measure) and BIC is used (BIC is available through the option `type = 2`

). The predefined adjusted R2 value of .99 is already reached with the third model and afterwards we only get minor improvements in adjusted R2. On the other hand, Cp rapidly declines afterwards, so model 3 seems to be a good candidate in terms of the relative information content per predictor and the captured information (adjusted R2). It is advisable to choose a model at the “elbow” in order to avoid over-fitting, but the solution should be tested for violations of model assumptions and the progression of the percentiles should be inspected visually, as well.

The predicted progression over age are displayed as lines and the manifest data as dots. Only three predictors were necessary to almost perfectly model the norm sample data with adjusted R2.

The package includes data from two large test norming projects, namely ELFE 1-6 (Lenhard & Schneider, 2006) and German adaption of the PPVT4 (A. Lenhard, Lenhard, Suggate & Seegerer, 2015), which can be used to run the analysis. Furthermore, large samples from the Center of Disease Control (CDC) on growth curves in childhood and adolescence (for computing Body Mass Index ‘BMI’ curves), life expectancy at birth and mortality per country from 1960 to 2017 (available from The World Bank). Type `?elfe`

, `?ppvt`

, `?CDC`

, ‘?epm’, `?mortality`

or `?life`

to display information on the data sets.

cNORM is licensed under GNU Affero General Public License v3 (AGPL-3.0). This means that copyrighted parts of cNORM can be used free of charge for commercial and non-commercial purposes that run under this same license, retain the copyright notice, provide their source code and correctly cite cNORM. Copyright protection includes, for example, the reproduction and distribution of source code or parts of the source code of cNORM or of graphics created with cNORM. The integration of the package into a server environment in order to access the functionality of the software (e.g. for online delivery of norm scores) is also subject to this license. However, a regression function determined with cNORM is not subject to copyright protection and may be used freely without preconditions. If you want to apply cNORM in a way that is not compatible with the terms of the AGPL 3.0 license, please do not hesitate to contact us to negotiate individual conditions. If you want to use cNORM for scientific publications, we would also ask you to quote the source.

- Gary, S. & Lenhard, W. (2021). In norming we trust. Diagnostica.
- Lenhard, A., Lenhard, W., Segerer, R. & Suggate, S. (2015). Peabody Picture Vocabulary Test - Revision IV (Deutsche Adaption). Frankfurt a. M./Germany: Pearson Assessment.
- Lenhard, A., Lenhard, W., Suggate, S. & Segerer, R. (2016). A continuous solution to the norming problem. Assessment, Online first, 1-14. doi:10.1177/1073191116656437
- Lenhard, A., Lenhard, W., Gary, S. (2018). Continuous Norming (cNORM). The Comprehensive R Network, Package cNORM, available: https://CRAN.R-project.org/package=cNORM
- Lenhard, A., Lenhard, W., Gary, S. (2019). Continuous norming of psychometric tests: A simulation study of parametric and semi-parametric approaches. PLoS ONE, 14(9), e0222279. doi:10.1371/journal.pone.0222279
- Lenhard, W., & Lenhard, A. (2020). Improvement of Norm Score Quality via Regression-Based Continuous Norming. Educational and Psychological Measurement(Online First), 1-33. https://doi.org/10.1177/0013164420928457