cNORM (W. Lenhard, Lenhard & Gary) is a package for the R environment for statistical computing that aims at generating continuous test norms in psychometrics and biometrics and to analyze the model fit. Originally, cNorm exclusively used an approach that makes no assumptions about the specific distribution of the raw data (A. Lenhard, Lenhard, Suggate & Segerer, 2016). Since version 3.2 (2024), however, the package also offers the option of parametric modeling using the beta-binomial 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 scores depend on continuous (e.g. age, duration of schooling) or discrete explanatory variables (e.g. sex, test form). In addition, the package can 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 explanatory variable. This is done either parametrically on the basis of the beta-binomial distribution or distribution-free using Taylor polynomials. For an in-depth tutorial, visit the project homepage, try the online demonstration and have a look at the vignettes.

A quick guide to distribution-free modeling with the essential cNORM functions: ```{r example} ## Basic example code for modeling the sample dataset 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 tables
predict(model.betabinomial, c(8.9, 10.1), c(153, 121))
tables <- normTable.betabinomial(model.betabinomial, c(2, 3, 4),
reliability=0.9)
```

Conventional norming: ```{r example} library(cNORM)

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")
```

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), Type `?elfe`

,
`?ppvt`

or `?CDC`

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, the norm tables … are 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.

The authors would like to thank WPS (https://www.wpspublish.com/) for providing funding for developing, integrating and evaluating weighting and post stratification in the cNORM package. The research project was conducted in 2022.

- Gary, S., Lenhard, W., Lenhard, A. et al. A tutorial on automatic post-stratification and weighting in conventional and regression-based norming of psychometric tests. Behav Res (2023a). https://doi.org/10.3758/s13428-023-02207-0
- Gary, S., Lenhard, A., Lenhard, W., & Herzberg, D. S. (2023b). Reducing the bias of norm scores in non-representative samples: Weighting as an adjunct to continuous norming methods. Assessment, 10731911231153832. https://doi.org/10.1177/10731911231153832
- 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. https://doi.org/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. https://doi.org/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