Standardized IQ scales are established through large-scale normative sampling and statistical transformation.
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Peer-reviewed literature on psychometric testing confirms that standardized intelligence scales like the Stanford-Binet are developed using large-scale normative sampling and advanced statistical transformation methods.
Norm scores are an essential source of information in individual diagnostics. Given the scope of the decisions this information may entail, establishing high-quality, representative norms is of tremendous importance in test construction. Representativeness is difficult to establish, though, especially with limited resources and when multiple stratification variables and their joint probabilities come into play. Sample stratification requires knowing which stratum an individual belongs to prior to data collection, but the required variables for the individual's classification, such as socio-economic status or demographic characteristics, are often collected within the survey or test data. Therefore, post-stratification techniques, like iterative proportional fitting (= raking), aim at simulating representativeness of normative samples and can thus enhance the overall quality of the norm scores. This tutorial describes the application of raking to normative samples, the calculation of weights, the application of these weights in percentile estimation, and the retrieval of continuous, regression-based norm models with the cNORM package on the R platform. We demonstrate this procedure using a large, non-representative dataset of vocabulary development in childhood and adolescence (N = 4542), using sex and ethnical background as stratification variables.
Instead, the norm scores must be derived from a normative sample, that is, a much smaller, representative subsample of the reference population. To this end, statistical methods designed for the norm score computation must be applied to the raw scores (Cole, 1988 ; Gary et al., 2021 ). In recent years, advanced norming approaches have been developed and evaluated with respect to their influence on the norm score quality (for an overview, see Gary & Lenhard, 2021 ). In many psychometric tests (e.g., intelligence scales), the norm scores refer only to individuals of the same age. Therefore, conventional approaches usually split the normative sample into several distinct age groups.
In particular, ensuring that the normative sample is representative of the reference population might be difficult. A sample is representative with respect to the relevant stratification variables (SVs) if the proportions of the various subgroups in the sample match the proportions of the respective strata in the reference population (Kruskal & Mosteller, 1979 ; Moosbrugger & Kelava, 2012 ). In other words, representativeness is established when the marginal and joint probabilities of a set of relevant SVs equal the corresponding proportions in the reference population.
In the following section, we will describe how to correct or at least mitigate the biases of norm scores introduced by non-representative samples. Countering non-representativeness in normative samples Probability sampling and sample stratification Probability or random sampling is probably the best-known strategy for establishing representativeness of normative samples. The data are drawn such that every individual in the population has the same chance to be included in the normative sample (Lumley, 2011 ).
Weights are then iteratively adjusted until the marginal proportions of the dataset match the composition of the target population Limitations and recommendations for the usage of weighting While weighting has the potential to reduce the adverse effects of non-representative normative samples on norm score quality, we strongly recommend its thoughtful application and limiting its use to cases where random sampling is not feasible, such as due to logistical issues, clustered data collection, or restrictions in accessing all strata.
The statistical modelling of cNORM then (c) uses polynomial regression to continuously model raw scores as a function of the norm scores and age or other explanatory variables. Once this norm score model is established, arbitrarily fine-grained norms can be determined. The method reduces the necessity for large norm samples, as it relies on the complete sample rather than distinct groups. It only requires continuity of the dependent variable and does not make distribution assumptions. Thus, it can effectively handle skewed distributions, which frequently arise in test construction due to ceiling and floor effects (Lenhard et al., 2019 ).
Step 2: Ranking of the test raw scores using the standardized raking weights To apply the generated weights, they must be passed via the parameter ‘weights’ in the ‘cnorm()’ function. This function ranks the data groupwise and converts the ranks into percentiles. It subsequently applies inverse normal transformation (INT) to convert percentiles to preliminary norm scores. Finally, multiple regression is performed to establish a continuous norm model. Hence, the ‘cnorm()’ function performs step 2 and step 3 in one single process.
The resulting vector ‘weights’ contains a weight for each individual case in the normative sample obtained by raking and subsequent standardization of the weights ( Appendix , Step 1b). Ranking with standardized raking weights In our example, both the ranking and the best-subset regression are conducted with the ‘cnorm()’-function. This function returns an object containing the original data, the weights, the group-specific ranks, the preliminary norm scores, and powers of the norm scores, of the grouping variable and all interactions between them (cf. Gary et al., 2021 ).
The object also contains the final statistical model describing the functional relation between raw scores, norm scores, and the explanatory variable, which is age in the example. The norm scores are returned as T scores (M = 50, SD = 10) by default, but other types of scales such as z scores or IQ scores can also be used. By default, cNORM calculates unweighted percentiles but automatically switches to weighted percentiles, if a vector with weights is provided (see description above and Appendix , Step 2).
Developmental domains, such as cognitive, language, and motor, are key concepts of interest in longitudinal studies of intellectual and developmental disabilities (IDD). Normative scores (e.g., IQ) are often used to operationalize performance on standardized tests of these concepts, but it is the interval-distributed person-ability scores that are intended for the assessment of within-individual change. Here we illustrate the use and interpretation of several Stanford Binet, 5th Edition score types (IQ, extended IQ, Z-normalized raw score, developmental quotient, raw sum score, age equivalent, and ability score) using data from two longitudinal studies of rare genetic conditions associated with IDD. We found that, although normality assumptions were tenuous for all score types, floor effects led to model unsuitability for longitudinal analysis of most types of norm-referenced scores, and that the validity of interpretation with respect to individual change was best for ability scores.
Normative scores (e.g., IQ) are often used to operationalize performance on standardized tests of these concepts, but it is the interval-distributed person-ability scores that are intended for the assessment of within-individual change. Here we illustrate the use and interpretation of several Stanford Binet, 5 th Edition score types (IQ, extended IQ, Z-normalized raw score, developmental quotient, raw sum score, age equivalent, and ability score) using data from two longitudinal studies of rare genetic conditions associated with IDD.
Because directly estimating a score at the <1 st percentile requires a prohibitively large sample per normative group, scores more than about 3 SD below average are extrapolations (i.e., actual standardization data in this range may not be present; see Timmerman et al. (2021) for a tutorial on one type of regression-based norming procedures). Even after borrowing statistical information from adjacent age groups, the precision of the extrapolated values is low. Thus, by convention, scores more than 3-to-4 SD below average are usually censored. This lowest standard score offered by the publisher is the third type of floor effect, referred to here as the standard score floor. Table 1.
Summary of available methods for operationalizing performance on the SB5 Full Scale Composite. Feature Relative Absolute Hybrid Intelligence Quotient (IQ) Extended IQ (EXIQ) Deviation Z Score (Z) Developmental Quotient (DQ) Raw Sum Score Age Equivalent (AE) Change Sensitive Score (CSS) Projected Retained Ability Score (PRAS) Possible Range on SB5 40 – 160 10 – 225 −17.5 – 231.7 0 – 1050 0 – 358 24 – 252 376 – 592 40 – 160 Floor types Test, normative Test, normative Test Test, age equivalent Test Test, age equivalent Test Test, normative Derivation Sums of normalized scaled scores are tabulated and smoothed within age groups.
First, it rests on the assumption that raw scores are normally distributed within each normative age group. Skewness, which is often observed especially at the youngest and oldest ages, compromises this. When test developers do base standard scores on raw scores, this skewness is addressed by first normalizing, or converting to percentiles, the raw scores. Second, the Z-score method creates discontinuity at age breaks, such that one could observe a dramatic difference in Z-score for the similar performance across two adjacent age groups. For standard scores based on raw scores this is addressed with the statistical procedure of smoothing growth curves.
Interval-level measurement means that a given difference in ability score has the same meaning at all points in the scale – this property is an essential assumption for most statistical models common to longitudinal data analysis and is required for the valid interpretation of the magnitude of resulting parameters. Because all Rasch-based and most IRT-based ability scores have a monotonic relationship with the raw sum score (the raw sum score is an ordered approximation of the ability score; Sijtsma et al., 2024 ), valid interpretation of the direction of change is possible.
Participants in the dataset were included in the analysis if they had at least one assessment with the Stanford Binet, 5 th edition. Measures The Stanford Binet, 5 th edition (SB5) was refined using Rasch analysis and normed on a nationally representative sample of N=4,800 aged 2 to 85 years ( Roid, 2003 ). There are 10 subtests which feed into the full-scale (FS) composite used in this study. The available score types for the SB5 FS are described in detail in Table 1 ; in the current study we evaluated the IQ, extended IQ (EXIQ), developmental quotient (DQ), Z-score (Z), raw sum score (RAW), age equivalent
However, for individuals with IDD, norm-referenced scores like IQ are often significantly limited by floor effects, which the EXIQ is intended to address. As illustrated in both samples in this paper, almost no intermediate values were assigned between the original floor of 40 and the EXIQ floor of 10. While the EXIQ method only moved the standard score floor, the Z-score method did successfully remove it, revealing variability that was censored by the IQ and EXIQ. However, we observed that despite the normative metrics all putatively measuring relative standing of an individual’s cognitive ability, the results of statistical analysis did not always lead to the same interpretation.
Projected Retained Ability Score (PRAS): A New Methodology for Quantifying Absolute Change in Norm-Based Psychological Test Scores Over Time. Assessment, 28(2), 367–379. 10.1177/1073191119872250 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Kwok E, Feiner H, Grauzer J, Kaat A, & Roberts MY (2022). Measuring Change During Intervention Using Norm-Referenced, Standardized Measures: A Comparison of Raw Scores, Standard Scores, Age Equivalents, and Growth Scale Values From the Preschool Language Scales-Fifth Edition. J Speech Lang Hear Res, 65(11), 4268–4279.
This report presents an open-source dataset investigating neurodevelopmental profiles in children. The dataset consists of EEG, ERP, and cognitive assessments from 100 Iranian non-clinical participants (age range 6-11 years, Mean = 8.52 ± 1.5 SD). Notably, this is a smaller group drawn from a larger longitudinal ongoing study. The research aligns with the Research Domain Criteria (RDoC) framework, aiming to enhance diagnostic precision and intervention efficacy for specific learning disabilities (SLD) using EEG/ERP measures and machine learning. Cognitive assessments included non-verbal intelligence (Raven Test), attention (IVA-2), and working memory tasks. EEG recordings captured resting-state (eyes closed/open) and brain activity during working memory tasks with numerical and non-numerical stimuli (ERPs). Additionally, demographic information such as age, gender, education, handedness, parental history of learning difficulties, and child symptom inventory-4 (CSI-4) were collected. This dataset provides a valuable resource for exploring the neurophysiological correlates of cognitive functions in typically developing children, which can advance our understanding of the neural foundations of cognitive development in children.
See Appendix 2 for details (Supplementary Appendices). Language of test 1 34207 Persian Input Device 1 34208 USB Test Validity Checks 4 34209-34212 Valid = 1 Valid (Interpret with caution: excessive idiopathic errors) = 2 Invalid (excessive idiopathic errors) = 3 Quotient Scale Scores 45 34213-34257 See Table S2 for details (Supplementary Tables). Note that the variable is assigned a value of -9 when the information is missing or unavailable CSI-4 97 34258-34354 See Table S3 for details (Supplementary Tables).
Parent’s Reading History 25 34390-34414 Questions are in a Likert scale format in which the participant could specify the degree of struggle ratings along a continuum from 0 to 4 (no difficulty = 0, extreme difficulty = 4). Working Memory Task 39 34415-34453 See Table S4 for details (Supplementary Tables). Ravan Test 3 34454-34456 See Table S5 for details (Supplementary Tables). Please note that in Table 2 , the column(s) number(s) are not Python indices; they start from 1 instead of 0. Additionally, both the start and stop of the ranges are inclusive.
This transformation allowed for a comprehensive spectral analysis, enabling the decomposition of EEG signals into distinct frequency components. The extracted QEEG values were then categorized across several frequency bands, including Delta (1.0–4.0 Hz), Theta (4.0–8.0 Hz), Alpha (8.0–12.0 Hz), Beta (12.0–25.0 Hz), High Beta (25.0–30.0 Hz), Gamma (30.0–40.0 Hz), High Gamma (40.0–50.0 Hz), and sub-bands such as Alpha 1 (8.0–10.0 Hz), Alpha 2 (10.0–12.0 Hz), Beta 1 (12.0–15.0 Hz), Beta 2 (15.0–18.0 Hz), Beta 3 (18.0–25.0 Hz), Gamma 1 (30.0–35.0 Hz), and Gamma 2 (35.0–40.0 Hz).
To aid in the interpretation and clinical relevance of these features, z-scores were calculated using the NeuroGuide software’s normative database. These z-scores allowed for a standardized comparison of an individual’s QEEG values against age-matched norms, helping to identify deviations from typical brain activity patterns. The use of these normative comparisons facilitated a more objective and comprehensive understanding of the brain’s functional state Raw data for all indicators are reported alongside their age-specific z-scores (see Table 2 and Table S1 in Supplementary Tables).
Research indicates that children from families with a history of reading difficulties may exhibit poorer letter-word knowledge and phonological awareness 41 , 42 . These foundational skills, crucial for future reading success, are typically established before formal education begins 43 . Additionally, children with a family history of reading difficulties often struggle with word recognition by the time they enter elementary school, impacting their ability to benefit from explicit reading instruction 44 , 45 . The ARHQ 46 is a reliable screening tool used to assess the risk of dyslexia in
All data were collected by trained experts under the supervision of a board-certified clinical psychologist. After input, data were organized by a research and development team at the Imâge Brain Institute using statistical summary tools to ensure quality control. All tests and questionnaires used in this study, except for our novel working memory task, are validated and standardized. Previous studies 51 have shown that working memory can mediate the relationship between fluid intelligence, as measured by Raven’s Matrices. As expected, a significant positive correlation was found between performance on our working memory task (i.e.
The experimental environment was carefully controlled to reduce the impact of external factors, such as noise, temperature, and lighting, on the EEG recordings. The technicians documented ocular movements and other relevant events during the recording sessions. All technicians were trained experts, qualified in EEG recording procedures. Statistical Quality Control: Alpha power suppression is a well-established EEG phenomenon, particularly noticeable at posterior sites during the transition from an eyes-closed to an eyes-open condition 52 . To visualize these changes, we created a topographic map of alpha power (Fig. 6 ).
R2023a), following established protocols and Makoto’s pipelines (Miyakoshi, 2018). EDF + files containing 20 channels (19 EEG, 1 event) were imported. A band-pass filter (0.5-30 Hz) and notch filters (45-55 Hz, 95-105 Hz) were applied to remove noise. Artifact Subspace Reconstruction (ASR) was used to eliminate large artifacts, and Independent Component Analysis (ICA) was employed to remove non-brain sources. Components with a brain source probability exceeding 70% were retained. ERP Computation ERPs were computed for each block (numerical and non-numerical). Data were locked to stimulus 1 and averaged across all trials.
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