cNORM (W. Lenhard, Lenhard & Gary) is a package for the Renvironment for statistical computing that aims at generating continuoustest norms in psychometrics and biometrics and to analyze the model fit.Originally, cNorm exclusively used an approach that makes no assumptionsabout the specific distribution of the raw data (A. Lenhard, Lenhard,Suggate & Segerer, 2016). Since version 3.2 (2024), however, thepackage also offers the option of parametric modeling using thebeta-binomial distribution and since version 3.5 modeling with theSinh-Arcsinh (ShaSh) distribution.
cNORM was developed specifically for achievement tests(e.g. vocabulary development: A. Lenhard, Lenhard, Segerer &Suggate, 2015; written language acquisition: W. Lenhard, Lenhard &Schneider, 2017). However, the package can be used wherever mental(e.g. reaction time), physical (e.g. body weight) or other test scoresdepend on continuous (e.g. age, duration of schooling) or discreteexplanatory variables (e.g. sex, test form). In addition, the packagecan also be used for “conventional” norming based on individual groups,i.e. without including explanatory variables.
The package estimates percentiles as a function of the explanatoryvariable. This is done either parametrically on the basis of thebeta-binomial or the Sinh-Arcsinh (ShaSh) distribution ordistribution-free using Taylor polynomials. For an in-depth tutorial,visit theprojecthomepage, try theonlinedemonstration and have a look at the vignettes.
A quick guide to distribution-free modeling with the essential cNORMfunctions: ```{r example} ## Basic example code for modeling the sampledataset library(cNORM)
cNORM.GUI()
cnorm.elfe <- cnorm(raw = elfe\(raw,group = elfe\)group)
model <- taylorSwift(ppvt\(raw,ppvt\)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)
cnorm.elfe <- cnorm(raw = elfe\(raw,group = elfe\)group, k = 5, t = 3)
plot(cnorm.elfe, “percentiles”)
plot(cnorm.elfe, “norm”) plot(cnorm.elfe, “raw”)
plot(cnorm.elfe, “series”, start=5, end=14)
cnorm.cv(cnorm.elfe$data, max=10, repetitions=3)
cnorm.cv(cnorm.elfe, repetitions=3)
normTable(c(3, 3.25, 3.5), cnorm.elfe)
rawTable(3, cnorm.elfe, CI = .9, reliability = .94)
Modelling norm data using beta-binomial distributions:```{r example}library(cNORM)# cNORM can as well model norm data using the beta-binomial# distribution, which usually performs well on tests with# a fixed number of dichotomous items.model.betabinomial <- cnorm.betabinomial(ppvt$age, ppvt$raw)# Adapt the power parameters for α and β to increase or decrease# the fit:model.betabinomial <- cnorm.betabinomial(ppvt$age, ppvt$raw, alpha = 4)# Plot percentile curves and display manifest and modelled norm scores.plot(model.betabinomial, ppvt$age, ppvt$raw)plotNorm(model.betabinomial, ppvt$age, ppvt$raw, width = 1)# Display fit statistics:summary(model.betabinomial)# Prediction of norm scores for new data and generating norm tablespredict(model.betabinomial, c(8.9, 10.1), c(153, 121))tables <- normTable.betabinomial(model.betabinomial, c(2, 3, 4), reliability=0.9)Modelling norm data using Sinh-Arcsinh (ShaSh) distributions: ```{rexample} library(cNORM) # The Sinh-Arcsinh (ShaSh) distribution is aflexible approach. # It can handle raw score value ranges includingzeros and negative # values, which pose a problem to Box Coxdistributions. # Shape parameters mu, sigma, epsilon and delta can beadjusted as well. model.shash <- cnorm.shash(ppvt
plot(model.shash, ppvt\(age,ppvt\)raw)
summary(model.shash, ppvt\(age,ppvt\)raw)
predict(model.shash, c(8.9, 10.1), c(153, 121)) tables <-normTable.shash(model.shash, c(10, 15), reliability=0.9)
Conventional norming:```{r example}library(cNORM)# cNORM can as well be used for conventional norming:cnorm(raw=elfe$raw)Start vignettes in cNORM: ```{r example} library(cNORM)
vignette(“cNORM-Demo”, package = “cNORM”)vignette(“WeightedRegression”, package = “cNORM”)vignette(“BetaBinomial”, package = “cNORM”) vignette(“sinh”, package =“cNORM”) ```
The package includes data from two large test norming projects,namely ELFE 1-6 (Lenhard & Schneider, 2006) and German adaption ofthe PPVT4 (A. Lenhard, Lenhard, Suggate & Seegerer, 2015), which canbe used to run the analysis. Furthermore, large samples from the Centerof Disease Control (CDC) on growth curves in childhood and adolescence(for computing Body Mass Index ‘BMI’ curves), Type?elfe,?ppvt or?CDC to display information on thedata sets.
cNORM is licensed under GNU Affero General Public License v3(AGPL-3.0). This means that copyrighted parts of cNORM can be used freeof charge for commercial and non-commercial purposes that run under thissame license, retain the copyright notice, provide their source code andcorrectly cite cNORM. Copyright protection includes, for example, thereproduction and distribution of source code or parts of the source codeof cNORM. The integration of the package into a server environment inorder to access the functionality of the software (e.g. for onlinedelivery of norm scores) is also subject to this license. However, aregression function determined with cNORM, the norm tables and plots arenot subject to copyright protection and may be used freely withoutpreconditions. If you want to apply cNORM in a way that is notcompatible with the terms of the AGPL 3.0 license, please do nothesitate to contact us to negotiate individual conditions. If you wantto use cNORM for scientific publications, we would also ask you to quotethe source.
The authors would like to thank WPS (https://www.wpspublish.com/) for providing funding fordeveloping, integrating and evaluating weighting and post stratificationin the cNORM package. The research project was conducted in 2022.