cNORM — Continuous Norming

This application generates continuous norms for psychometric and biometric test scores. Instead of computing norms separately for each age or grade group, cNORM models the raw score as a function of both the norm score and the explanatory variable (e. g. age). The app implements the distribution-free approach based on regression with Taylor polynomials (Lenhard et al., 2018, 2019; Lenhard & Lenhard, 2021), as provided by the cNORM package for R (version 3.7).

Quick start

1. Data – load a built-in example or upload your own file
2. Modeling – select the raw-score variable and, if the scores depend on age or grade, the grouping variable. Choose the norm scale (T, IQ or z) and adjust the polynomial degrees, then click Run model. Inspect the percentile plot, the consistency check and the model-fit curve to decide on the number of terms.
3. Prediction – convert single raw scores to norm scores (and vice versa) or generate and download complete norm tables. Diagnostic plots are available in the Visualization tab; a stratified Cross-validation can be run separately.

Data requirements

• Raw-score variable – the test score to be normed. Ascending scales are supported in this app (descending scales require the R console, see below).
• Grouping variable (optional) – age group, grade or year (the group mean or midpoint is appropriate). The groups define the subsamples in which ranks and norm scores are computed before the model is fitted. If the test is not expected to depend on age or grade, select (none).
• Cases with missing values in the selected variables are removed automatically; the number of removed cases is reported in the model summary.
• If age is recorded continuously, use the parametric modelling app, or the sliding-window ranking (age and width arguments) on the R console.

Built-in example data sets

• elfe – reading comprehension in primary school (sentence-completion task, raw score 0–28), grouped by grade.
• ppvt – Peabody Picture Vocabulary Test IV norming sample, ages 2.5–17.5, raw score 0–228.
• CDC – epidemiological growth data (BMI, weight, height), ages 2–18, from the CDC.

Modelling without a grouping variable

When the grouping variable is set to (none), the whole sample is treated as a single group. Norm scores are derived from the ranks in the total sample and the raw score is modelled as a polynomial of the norm score only (t = 0; the degree t is ignored). The resulting norm table is valid for one population only. Please note that some of the other functions are unavailbale in this case.

Model parameters

Evaluating the model

A high R² alone does not establish a good norming model. Please check the following before using the norms:

1. Percentile plot – the model-implied curves should follow the empirical percentiles (points) closely and run smoothly across the full age range.
2. Consistency check – percentile curves must not intersect, and raw scores must increase monotonically with the norm score at every age. Violations, especially at the extremes or at the edges of the age range, indicate overfitting or a too complex model.
3. Model-fit curve – compare the fit across models of different size and choose the smallest model beyond which the gain is negligible.
4. Cross-validation – assess the stability of the model on held-out data and compare the candidate models.
5. Range of validity – norm scores are interpolated within the observed age range and should not be extrapolated beyond it. Extreme norm scores (beyond approximately ±2.5 SD) depend strongly on the model and on the sample size and should be interpreted with caution.

In case of overfit (e. g. wavy percentile lines) reduce the degree of age (t) and/or location (k).

Advanced options on the R console

The GUI covers the most important functionality. The following require the R console: weighting and post-stratification (e. g. raking via iterative proportional fitting) to correct for unrepresentative samples, descending scores, the choice of ranking method and handling of ties, sliding-window ranking for continuous age, extended cross-validation, model-implied moments (predictMoments()) and predicting norm scores for new data based on an existing model.

Privacy

When launched locally via cNORM.GUI(), all data stays on your computer. The online application is hosted on ISO 27001-certified IONOS servers located in the European Union (Frankfurt, Germany), in compliance with the General Data Protection Regulation (GDPR). cNORM does not store uploaded data; nothing you upload is retained.

Citation and references

Please cite the package (see citation("cNORM")) and the methodological papers relevant to your analysis.

• cNORM on CRAN · Tutorial & manual · GitHub (issues & source)
• Lenhard, A., Lenhard, W., Suggate, S. & Segerer, R. (2018). A continuous solution to the norming problem. Assessment, 25(1), 112–125. doi:10.1177/1073191116656437
• 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.
• Lenhard, W. & Lenhard, A. (2021). Improvement of norm score quality via regression-based continuous norming. Educational and Psychological Measurement, 81(2), 229–261. doi:10.1177/0013164420928457
• Gary, S., Lenhard, W. & Lenhard, A. (2021). Modelling norm scores with the cNORM package in R. Psych, 3(3), 501–521. doi:10.3390/psych3030033