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Improvement of Norm Score Quality via Regression-Based Continuous Norming

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The interpretation of psychometric test results is usually based on norm scores. We compared semi-parametric continuous norming (SPCN) with conventional norming methods by simulating results for test scales with different item numbers and difficulties via an Item Response Theory (IRT) approach. Subsequently, we modeled the norm scores based on random samples with varying sizes either with a conventional ranking procedure or SPCN. The norms were then cross-validated by using an entirely representative sample of N = 840,000 for which different measures of norming error, and data fit were computed. This process was repeated 90,000 times. Both approaches benefitted from an increase in sample size, with SPCN reaching optimal results with much smaller samples. Conventional norming also performed worse on the number of missings in the norm tables. The data fit in conventional norming of fixed subsample sizes varied with the granularity of the age brackets, calling into question general recommendations for sample sizes in test norming. We recommend that test norms should be based on statistical models of the raw score distributions instead of simply compiling norm tables via conventional ranking procedures.
Population model: The normally distributed θAge specifies the person location determined with respect to persons of the same age only. By contrast, θPop corresponds to the person location with respect to the total population (c.f. A. Lenhard et al., 2019). The normative samples consisted of N = 700, 1 400, 2 800, 5 600, 11 200 or 22 400 in total and thus covered the complete range of rather small normative samples (100 cases per year with N = 700) up to very large samples, which would probably only be available through large scale assessments. The samples were randomly drawn from the population, that is, the normative samples were afflicted with the regular but unsystematic sampling error. For the cross-validation sample, we generated a completely representative sample of n = 10,000 cases per month, amounting to a total of N = 840,000 cases per simulation cycle. To avoid age bias (as described in the introduction) in the cross-validation, we set the age of each person in this cross-validation sample exactly to the center of the respective months. The person locations in the cross-validation sample followed the same population model as in the normative samples but were not randomly drawn either. Instead, θAge was perfectly normally distributed within each age bracket, with equidistant percentiles ranging from 0.5 í µí±›í µí±› to í µí±›í µí±›−0.5 í µí±›í µí±› . These properties of age and latent ability θAge in the cross-validation sample remained constant in all simulation cycles. The raw scores of the cross-validation sample were however generated anew in each cycle based on the simulated test scales (see b. and c.).
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CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING
Accepted version; reference:
Lenhard, W. & Lenhard A. (in press). Improvement of Norm Score Quality via
Regression-Based Continuous Norming. Educational and Psychological
Measurement.
Improvement of Norm Score Quality via Regression-Based
Continuous Norming
Wolfgang Lenhard
Department of Psychology, University of Wuerzburg
Alexandra Lenhard
Psychometrica
Wolfgang Lenhard, Department of Psychology, University of Wuerzburg, Wuerzburg,
Germany; Alexandra Lenhard, Psychometrica, Dettelbach, Germany.
Declaration of interests: The authors developed the statistical software, which is open
source and free to use under the AGPL-3 license. We do not have financial interests in this
publication. All resources are available as electronic material (R-syntax, data and results) via
https://osf.io/ntydc/. Both authors contributed equally to this manuscript.
Correspondence concerning this article should be addressed to Wolfgang Lenhard,
Department of Psychology, University of Wuerzburg, Wittelsbacherplatz 1, 97074 Wuerzburg,
Germany. E-mail: wolfgang.lenhard@uni-wuerzburg.de
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 2
Abstract
The interpretation of psychometric test results is usually based on norm scores. We compared
semi-parametric continuous norming (SPCN) with conventional norming methods by simulating
results for test scales with different item numbers and difficulties via an Item Response Theory
(IRT) approach. Subsequently, we modeled the norm scores based on random samples with
varying sizes either with a conventional ranking procedure or SPCN. The norms were then cross-
validated by using an entirely representative sample of N = 840,000 for which different measures
of norming error were computed. This process was repeated 90,000 times. Both approaches
benefitted from an increase in sample size, with SPCN reaching optimal results with much
smaller samples. Conventional norming performed worse on data fit, age-related errors, and
number of missings in the norm tables. The data fit in conventional norming of fixed subsample
sizes varied with the granularity of the age brackets, calling into question general
recommendations for sample sizes in test norming. We recommend that test norms should be
based on statistical models of the raw score distributions instead of simply compiling norm tables
via conventional ranking procedures.
Keywords: regression-based norming, continuous norming, inferential norming, data
smoothing, curve fitting, percentile estimation
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 3
The Significance of Norm Scores in Applied Psychometrics
Critical life decisions in education, medicine, and even the judicial system are often based
on the results of psychometric tests. Individual test results are usually interpreted in comparison
to a reference population, i.e. a subset of the target population that serves as a standard (cf. APA,
n.d.). This standard is predominantly reported in the form of norm scores. One of the most drastic
applications of norm scores is, for example, the preclusion of people with mental retardation (i.e.,
with an IQ score < 70) from the death penalty in the U.S. (Duvall & Morris, 2006). Less dramatic
but yet very important decisions based on test norms are, for example, school placement or
advanced placement (American Educational Research Association, 2014), granting of
rehabilitative measures (e.g. diagnosis of learning disorders based on a performance of less than
the 7th percentile; American Psychiatric Association, 2013), job recruitment (Arthur, 2012),
access to elite universities (Duncan & Stevens, 2011) and many more. In all of these examples,
the comparison between an individuals’ test result and the distribution of test results in the
reference population is the basis of the decision. Given that this distribution is usually unknown,
it must be inferred from the test results of a normative sample, that is, a subsample of the
reference population which is much smaller than the reference population but as representative
for it as possible. The statistical procedures used for this inference significantly affect the test
outcome and therefore play a vital role in applied psychometrics and applied psychology (cf. W.
Lenhard, Lenhard & Gary, 2018).
The origins of norm scores in psychometry
The establishment of norm scores in psychometry is tightly linked to intelligence
assessment (e.g., Wassermann, 2018). The first attempts of William Stern (1912) and Lewis
Terman (1916) to measure intelligence in such a way that the test results could be compared
across different psychometric tests and different age groups drew on the definition of an
intelligence quotient (IQ) as mental age divided by chronological age multiplied with 100, thus
trying to take into account the growth of cognitive abilities during childhood. This definition had
multiple drawbacks, most notably the missing linearity of mental growth across age.
It was the distribution-based approach of David Wechsler (1939) in the Bellevue
Intelligence Scale that paved the way for a more statistically sound approach. Wechsler suggested
to z-standardize the subtest raw scores drawing on the age-specific distribution parameters M and
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 4
SD in the reference sample. The resulting z-scores were subsequently transformed into the well-
known Wechsler subtest scores (with M = 10 and SD = 3), summed up and transformed into IQ
scores (with M = 100 and SD = 15).
Wechsler’s approach was not only driven by the motivation to make test results
comparable across different tests and age groups but also to determine the relative location of a
test person among the reference group with which this person was compared. In his seminal work
“The measurement of adult intelligence” (Wechsler, 1939, p. 34) he even stated that “the
important fact about it is this relative standing and not the numerical rating which we may happen
to assign to it.” In fact, determining the relative location of a persons’ latent trait (e.g. in the form
of percentiles) is still the core aspect of psychometric assessment today, as already described in
the introduction section, since the diagnosis of mental disorders and the granting of subsequent
treatment is often based on defined cut-points based on percentiles.
Requirements and limitations of conventional norming procedures
Measurement is generally defined as the assignment of numbers to features in such a way
that the numbers reflect the relationships between the features to be measured (e.g. Brosius,
Koschel, & Haas, 2008, p. 47). Since psychometric tests usually aim at measuring personality
traits or abilities, the numeric test scores must therefore reflect these traits. For example, the raw
scores obtained in an intelligence test must convey the ranking information with regard to the
different intellectual levels of the test persons as precisely as possible. To put it in more technical
terms, tests have to be constructed in such a way that the raw scores are a homomorphous
function of the latent traits or abilities. To achieve this goal, test constructors put major efforts in
item construction, analysis, and selection to establish high test quality in terms of reliability,
validity, unidimensionality, homogeneity, and so on (e.g. Eid & Schmidt, 2014, pp. 360; Kline,
2015; McDonald, 1999). Unfortunately, the calculation of norm scores is often viewed more as a
routine task. There are, however, a number of challenges which must be mastered in the norming
process.
First of all, the normative sample has to be as representative as possible of the reference
population. To establish this representativeness, some test authors rely on random sampling (cf.
Bracken; 1988; Gregory, 1996). In case of unbalanced datasets, the sample should additionally be
stratified according to covariates that influence the test results most significantly, such as sex,
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 5
region or ethnical background. Stratification is especially necessary in the case of small sample
sizes, since they are more prone to imbalances.
If, however, the confounding variable in question does not only affect the raw score but
also defines the reference population, stratification will not even suffice to get to the desired
result. For example, in the case of IQ assessment, the location of a test person is determined with
respect to a reference population of the same age only. (In the following, we will generally refer
to such variables as explanatory variables.) Therefore, the normative samples must usually be
partitioned into smaller subsamples (e.g., age brackets). For conventional norming procedures,
subsample sizes of n = 250 to n = 500 are recommended to generate norms with sufficiently high
precision (depending on the intended diagnostic purpose and the scale properties; Oosterhuis, van
Ark, Sijtsma, 2016; Eid & Schmidt, 2014, pp. 379).
An important question in this context is how the width of the age brackets affects the
accuracy of the norm scores. In the case of intelligence assessment or developmental tests, the
raw scores change rather quickly with age. If, in such cases, the age brackets are chosen too large,
significant jumps between the norm tables of the subsamples will occur (A. Lenhard, Lenhard,
Suggate & Segerer, 2016; Braken, 1998; Voncken, Albers & Timmerman, 2019a; Zachary &
Gorsuch, 1993). These jumps lead to errors because the age of a test person might deviate
considerably from the average age of the respective age bracket. For example, children at the
lower boundary of an age bracket are mainly compared to older children, and vice versa.
Consequently, the performance of children at the lower boundary of the age bracket is
underestimated, while the performance of children at the upper boundary is overestimated. This
age-related norm score bias, which increases with the width of the age brackets and the gradient
of the measured ability across age, is depicted in Figure 1. The data are taken from an unstratified
data pool of a vocabulary test for children (A. Lenhard, Lenhard, Suggate & Segerer, 2015). The
sample was split up into age brackets of one year each to illustrate the effect. The solid lines
represent the assumed continuous development of the raw scores across age. The smileys
represent two children, the first one (light grey) being 5 years and 1 month old with a raw score
of 52, the second one (dark grey) being 6 years and 11 months old with a raw score of 79.
According to the solid lines, both children should be assigned a percentile of 2.5. However, in the
age bracket of the 5-year-olds, children at percentile 2.5 have expected raw scores spanning from
50 to 70 with an average raw score of approximately 60. Therefore, child 1 will be assigned a
percentile below 2.5. In contrast, the opposite effect occurs with the second child (cf. Bracken,
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 6
1988). It might seem that the age bracket of one year is chosen deliberately large in this example
to depict the effect overly dramatic. Yet, some intelligence tests in fact deliver such coarse-
grained norms (e.g., Kubinger & Holocher-Ertl, 2014).
Figure 1. Sources of error with conventional norming. A first source of error originates
from age brackets of finite width. As a consequence, test persons are compared with a reference
group that may be younger or older on average. The second source of error is due to random
sampling.
The age brackets should therefore under no circumstances be chosen too wide. On the other hand,
splitting up into narrow age brackets quickly leads to enormously large sizes of the total sample.
Imagine, for example, an intelligence test spanning ten years. If the age brackets were chosen
with a width of three month and a subsample size of n = 500 each, the total size of the normative
sample would have to be N = 20,000 – a figure that in most cases probably lies far beyond
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 7
feasibility. Therefore, the trade-off between total sample size and width of the age brackets must
be balanced carefully.
Once the subsamples have been established, the next crucial step is to convert the raw
scores into norm scores. As already described above, the numeric test scores assigned to the test
persons must reflect the latent traits and abilities to be measured. This requirement applies not
only to the raw scores, but also to the norm scores. What is even more, the transformation of raw
scores into norm scores must not only maintain the ranking order between any two individuals. In
addition, the norm scores must exactly indicate, how many people in the reference population
have a lower or the same level of the latent trait, i.e., they are supposed to indicate the location of
the test person with respect to the reference population. Technically speaking, norm scores must
represent a specific bijective function of the latent trait. From the well-known law of large
numbers it can be derived that latent traits or abilities must be normally distributed, because they
are usually shaped by a large number of variables (cf. Andersen & Madsen, 1977). This is
especially true for abilities such as intelligence. If the raw scores were perfectly normally
distributed, too, then the norming procedure suggested by Wechsler would actually return norm
scores that could be transformed into percentiles by simply using the cumulative normal
distribution. But perfect normal distribution of the raw scores is rarely given. Instead, raw score
distributions are almost always more or less skewed, at least in some age groups. What remains
constant, though, is the ranking information. Therefore, many test constructors apply inverse
normal transformation (INT), that is, they start by determining the cumulative distribution
function (CDF) of the raw scores and subsequently use the inverse normal CDF to assign norm
scores to the observed percentiles, thereby mapping the raw score distribution onto a normal
distribution. We will refer to this method as conventional norming in the following. It is
important to note that the resulting norm scores do not indicate how many standard deviations the
raw score of an individual is below or above the average raw score of the reference population.
Instead, they are supposed to indicate how many standard deviations the hypothesized latent trait
of an individual is below or above the average latent trait.
Many test constructors consider the norming process to be finished at this point. For
example, we know of only one school achievement test in Germany using advanced modelling
techniques instead of the conventional norming. (It happens to be our own test, W. Lenhard,
Lenhard & Schneider, 2017.) Although it has occasionally been discussed, whether INT should
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 8
be applied in the case of large deviations from the normal distribution (Solomon & Sawilowsky,
2009), other severe drawbacks of this data driven norming procedure received less attention.
A first drawback is that errors resulting from random sampling are either completely
included in the norm tables or they are at best poorly corrected by hand. The sampling errors are
usually of varying magnitude in the different subsamples, which may be even exacerbated by the
fact that sometimes not all subsamples can be stratified accurately. As a consequence, the
sequence of raw scores assigned to a certain percentile across age usually is not smooth but rather
jagged. This effect is also depicted in Figure 1. The dots in this figure represent the CDFs
calculated per age bracket. In some cases, they deviate considerably from the solid lines, which
represent the expected smooth developmental curve. For example, the dot representing the 2.5th
percentile in the age bracket of five-year-olds corresponds to a raw score of 77 whereas the
according dot in the age bracket of six-year-olds corresponds to a raw score of only 51. A 5-year-
old would therefore need a higher raw score than a 6-year-old to attain the same percentile.
Obviously, the numbers in this case do not reliably reflect the relationships between the latent
traits to be measured. Note that the termreliability, as defined in classical test theory, is
supposed to specify the proportion of error variance contained in test scores. Interestingly,
though, the standard methods used to determine reliability indices exclusively draw on raw
scores. Yet the confidence intervals – which are based on these indices – are specified for the
norm scores, not for the raw scores. As a consequence, the error variance contained in norm
scores is not fully covered by conventional confidence intervals, that is, they do not reflect the
error variance resulting from the norming procedure. Unfortunately, this shortcoming has seldom
been the subject of scientific debate (Voncken, Albers & Timmerman, 2019b). Therefore, we
designed our study in such a way that we could determine the error variance caused by the test
items and the error variance caused by the norming method separately. We will later on analyze
the proportion of error variance, the norming procedure adds to the test scores. Furthermore, we
will define a reliability index to evaluate the different norming procedures. This index is an
equivalent to the classical reliability index for test scales. It specifies the proportion of error
variance introduced to test scores by a specific norming procedure and could theoretically be
used to calculate more realistic confidence intervals. Furthermore, it can be used to compare the
error variance the norming process alone contributes to the test scores with the error variance
produced by the items. To the best of our knowledge, this has never been done numerically
before. It is particularly important, though, because some test constructors are of the opinion that
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 9
norming is far from being as important as is generally assumed, specifically if the impact of
norming is compared to the impact of item selection (e.g., Lienert & Raatz, 1998, p272).
A final problem of conventional norming we want to mention here is that the norm tables
can be plagued with gaps because of missing raw scores in some subsamples (cf. Bracken, 1988).
For example, in the dataset depicted in Figure 1, the lowest raw score was 55 in the age bracket
of five-year-olds, but 24 in the age bracket of six-year-olds. The higher the number of items and
the lower the subsample size, the higher is, of course, the number of missings. Moreover, some
achievement tests contain norm tables that apply to specific points in time only, for example, to
the midterm and/or the end of a school year (e.g., Stock, Marx & Schneider, 2017). This means
that sometimes, there are not only missings within norm tables but also large gaps between norm
tables. These gaps are particularly problematic because in diagnostic practice the existing norm
tables are often applied whenever an achievement test is needed, that is, they are used beyond the
specified time slots.
Semi-Parametric Continuous Norming
To overcome some of the problems of conventional norming, different continuous
norming methods have been developed modelling norm scores as functions of the explanatory
variable. Again, the development of these advanced methods was mainly inspired by intelligence
assessment, starting with the WAIS-R (Gorsuch, 1983, quoted from Zachary & Gorsuch, 1985)
in the US and the SON-R 5 ½ – 17 (Snijders, Tellegen & Laros, 1989) in the Netherlands. As far
as we know, the application of these methods is still rather restricted to the field of intelligence
assessment (e.g. WISC-V, Wechsler, 2014; KABC-2, Kaufman & Kaufman, 2004; SON-R 6-40,
Tellegen & Laros, 2012; IDS-2, Grob & Hagman-von Arx, 2018). Clearly, intellectual
assessment strongly profits from continuous norming, since intellectual performance heavily
depends on age (e.g., Horn & Cattell, 1967). Yet, this does not mean that advanced norming
methods can only be applied to intelligence tests. After all, there are many other abilities and
personality traits that develop with age. In addition, quite different continuous covariates are
conceivable, which could be important at least for some diagnostic purposes, e.g. duration of
schooling, socioeconomic level, weight, etc. For example, we ourselves have used continuous
norming to successfully model vocabulary acquisition (A. Lenhard et al., 2015), reading fluency
and reading comprehension (W. Lenhard et al., 2017), competence levels in foreign language
acquisition (A. Lenhard, Bender & Lenhard, in press) but also body mass index and reaction
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 10
times (W. Lenhard et al., 2018). Other examples for the application of advanced norming
procedures have been demonstrated with regard to clinical questionnaires (van Breukelen &
Vlaeyen, 2005) and attention deficits (Stemmler, Lehfeld, Siebert & Horn, 2017).
Although the different continuous norming methods vary in their assumptions about the
raw score distributions and modeling procedures, they all have in common that a statistical model
is established based on the normative data. These models are subsequently used to interpolate
between the age brackets (horizontal interpolation), close the missings between the achieved raw
scores (vertical interpolation), and smooth the norm score distributions horizontally and
vertically. They can as well be used to extrapolate to ability or age levels beyond the ones
contained in the normative data, though extrapolation should be applied cautiously, of course.
The norming methods do not only reduce the required total sample size (e.g., Zhu & Chen, 2011),
but they can also decrease the error variance introduced into the test scores through the norming
procedure (A. Lenhard et al., 2016; Oosterhuis et al., 2016).
In this paper, we focus on a semi-parametric continuous norming approach (SPCN, A.
Lenhard et al., 2016, 2019; W. Lenhard et al., 2018). The essential steps of the method are
illustrated in Figure 2.
Figure 2. Steps in modeling three-dimensional norming functions with semi-parametric
continuous norming (SPCN). The raw scores are modelled as a function of person location and
age via multiple regression, thus fitting a hyperplane to the data that reduces noise and
interpolates missings.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 11
The approach draws on the assumption that the expected raw score r of an individual is a
function f of his or her person location with respect to a latent ability θ and one or several
explanatory variables a. In the following, we will concentrate on the explanatory variable that
probably plays the most important role in norming, namely age. In this specific simulation, we
will use it together with a 1 PL measurement model. SPCN starts the modeling process by using
INT to compute conventional norm scores per age group, which are used as preliminary estimates
l of the age-specific person locations (Fig. 2, Step 1). The aforementioned function f can
subsequently be modeled via the following polynomial (A. Lenhard et al., 2016, 2019):
(1) = 
,
In this equation, k denotes a smoothing parameter which can be chosen as needed with
higher values leading to higher explanation of variance and lower values leading to smoother
curves (Fig. 2, Step 2). Based on our experience, a value of four will in most cases lead to smooth
curves while at the same time fitting the model with a multiple R2 > .99 (A. Lenhard et al., 2016,
2019). The polynomial described in equation (1) contains a total of 2k + k2 variable terms plus the
intercept, which are used to predict the raw scores. The mathematical background of this
approach has been described in more detail elsewhere (A. Lenhard et al., 2016, 2019). Therefore,
in the present article we would like to summarize these descriptions by pointing out that the
principle polynomial approach can be used to model virtually any function with sufficiently high
accuracy as long as the function is smooth and continuous. To meet these requirements, the
discrete raw scores must be underlaid with a continuum. The assumption of an underlying
continuum has, however, been proven to be generally valid for modeling purposes (cf. Hansen,
2004) and applies to all forms of continuous norming (e.g. Oosterhuis et al., 2016; Voncken et
al., 2019a).
In order to avoid overfit, it is advisable not to include the full number of 2k + k2 terms in
the regression function, but to select a minimum number of terms with the highest relevance.
This selection problem can be solved via different approaches as for example stepwise
regression, ridge, lasso and elastic net regularization (e.g., ‘glmnet’ package, Friedman, Hasti &
Tibshirani, 2010) and other methods. In the current version of our approach, we use best subset
regression (Miller, 2002; R package ‘leaps’; Lumley, 2017) to select the respective coefficients
cst (Fig. 2, Step 2; A. Lenhard et al., 2019). The subset regression approach tests all possible
combinations of predictors and returns the models with the highest Radjusted2 for any specified
number of terms. Radjusted2 increases with the number of terms included in the regression equation.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 12
As a consequence, the number of terms is a second, more fine-tuned smoothing parameter, with
low numbers again leading to smooth curves and high numbers leading to a more detailed
mapping of the empirical raw data. In most practical cases, a very small number of terms in the
regression function (e.g., four) is sufficient to model the norming data with a data fit of Radjusted2 >
.99 (A. Lenhard et al., 2019), that is, the regression leads to very efficient models while
minimizing noise and error variance. Graphically speaking, the procedure approximates the
observed data through fitting a hyperplane to the empirical data (Fig. 2, Step 3). In order to
reduce the danger of overfit, it is advisable to keep the number of terms as low as possible and to
cross-validate the retrieved models.
Rationale and goals of the present study
The conditions, under which a new method performs better or worse than other methods
have to be assessed carefully. For example, we have already compared SPCN to another
continuous norming approach, namely parametric continuous norming. SPCN in general
performed very well and its application was particularly favorable when applied on skewed raw
score distributions and when using subsample sizes below n = 250 (A. Lenhard et al., 2019). In
the study reported here, we will compare its performance with respect to conventional norming
under different sample sizes. To this end, we drew normative samples of different sizes from a
simulated population with known population parameters. These normative samples were
subjected to fictitious IRT-based test scales with varying item difficulties and item numbers,
thereby simulating test results with different skewness of the raw score distributions. We
subsequently generated norm scores with SPCN and conventional norming. In a previous study,
we had shown that SPCN is largely invariant to the width of the age brackets (A. Lenhard et al.,
2016). The same does not hold true for conventional norming. We expected that the age bias of
the conventional norms would increase with the width of the age brackets, since the age variance
within each bracket increases. If, on the other hand, the total sample size is fixed, the subsample
size increases with the width of the age bracket, which simultaneously reduces the sampling
error. Therefore, we additionally varied the width of the age brackets in four levels (1 month, 3
months, 6 months, 12 months) when using conventional norming, in order not to disadvantage
this method by selecting a particular unfavorable condition. The real-world analogy would be a
normative dataset with a continuous age distribution where the test constructor has to decide,
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 13
how fine grained the age brackets should be, taking into account the tradeoff between age bias
and subsample size.
The quality of the norming methods was determined by use of a very large and
representative cross-validation sample. On the basis of the “ideal norms” determined in the cross-
validation sample, we calculated different measures of norm score quality. Furthermore, we
compared the proportion of error variance the different norming procedures add to the test scores.
In line with the existing continuous norming literature (e.g., Zhu & Chen, 2011, A. Lenhard et al.,
2016; Oosterhuis et al. 2016), we assumed that conventional norming would lead to a higher
proportion of error variance than SPCN, particularly when used together with small sample sizes,
because the regression draws on the complete normative sample instead of the subsamples only.
In addition, we also expected to find the obvious advantages of continuous norming procedures,
namely, a reduction of missings in the norm tables.
Specifically, we assessed the following hypotheses:
H1: The overall norming error will be lower when SPCN is used as compared to
conventional norming.
H2: The size of the normative sample will have a lower impact on the overall norming
error when SPCN is used as compared to conventional norming.
H3: Conventional norming will lead to a considerably larger number of missings in the
norm tables compared to SPCN. The number of missings will increase as the subsample size
decreases and as the number of items increases.
Method
Procedure
To simulate normative data, we repeatedly drew random samples from a predefined
population and subsequently generated test results for test scales with three different difficulties
(syntax, data, a complete simulation cycle, descriptives of the raw data distributions and
aggregated results are available as via https://osf.io/ntydc/). We delineate only the most important
features below, because this simulation procedure was described in detail in A. Lenhard et al.
(2019).
The simulation process included the following central steps: (a) Simulation of normative
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 14
samples on the basis of the population model, (b) simulation of test scales with n items of varying
average item difficulty, (c) using the test scales to generate raw score distributions for the
normative samples and a representative cross-validation sample by means of a 1 PL measurement
model, (d) retrieving norm scores based on the normative samples by using INT per age group
(conventional norming) or SPCN, (e) cross-validation of the different approaches by computing
conventional norm scores for the large cross-validation sample and comparing them to the norm
scores that would have been assigned to each person based on the much smaller normative
samples.
a. Simulation of normative samples based on the population model.
Each person in the simulated population was assigned an age variable a and a fictitious
latent ability. The uniformly distributed age variable comprised seven years, starting with 0.5 and
ending with 7.5. It is important to note that the latent ability θPop was normally distributed at each
single age level but, but not across the whole age range. Instead, it increased curvilinearly across
age with a slight scissor effect (see Fig. 3). This means, the development across age was not
linear, but slowed down with increasing age, while the standard deviation slightly increased. This
type of development is common for many ability domains in childhood and adolescence and can
be found in our own psychometric tests on vocabulary (A. Lenhard et al., 2015) and reading
comprehension (W. Lenhard et al., 2017), too. The precise numeric description of the population
model is available in A. Lenhard et al. (2019). θPop was z-standardized across the whole age
range, in order to be able to apply it together with the 1 PL model to produce test scores. In
addition, we z-standardized θPop per age level, which lead to a second latent trait score θAge. This
second latent trait score was used to determine the “true” location of the person with respect to
other persons of the same age. Consequently, each person was characterized by three variables:
(a) age (uniformly distributed across the whole age range), (b) the latent ability θPop, which
indicated the person location with respect to the whole population, and (c) θAge, which referred to
the same latent ability, however, this time with regard to persons of the same age only.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 15
Figure 3. Population model: The normally distributed θAge specifies the person location
determined with respect to persons of the same age only. By contrast, θPop corresponds to the
person location with respect to the total population (c.f. A. Lenhard et al., 2019).
The normative samples consisted of N = 700, 1 400, 2 800, 5 600, 11 200 or 22 400 in
total and thus covered the complete range of rather small normative samples (100 cases per year
with N = 700) up to very large samples, which would probably only be available through large
scale assessments. The samples were randomly drawn from the population, that is, the normative
samples were afflicted with the regular but unsystematic sampling error.
For the cross-validation sample, we generated a completely representative sample of n =
10,000 cases per month, amounting to a total of N = 840,000 cases per simulation cycle. To avoid
age bias (as described in the introduction) in the cross-validation, we set the age of each person in
this cross-validation sample exactly to the center of the respective months. The person locations
in the cross-validation sample followed the same population model as in the normative samples
but were not randomly drawn either. Instead, θAge was perfectly normally distributed within each
age bracket, with equidistant percentiles ranging from .
to .
. These properties of age and
latent ability θAge in the cross-validation sample remained constant in all simulation cycles. The
raw scores of the cross-validation sample were however generated anew in each cycle based on
the simulated test scales (see b. and c.).
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 16
b. Simulation of test scales.
We simulated test scales with 10, 20, 30, 40 or 50 items. We used these item numbers to
cover a broad range of possible test scenarios, with 10 items representing a rather small and
unreliable test scale, and 50 items representing a medium to large test scale. For instance, most
test scales of the WISC-V (Wechsler, 2014) and the KABC-2 (Kaufman & Kaufman, 2004)
range between a maximum raw score of 20 to 50. We refrained from including larger item
numbers, since the computational effort for the entire simulation was already enormously high
and we did not expect qualitatively different results with higher item numbers. In each simulation
cycle, the item difficulties δi were drawn randomly from normal distributions with mean item
difficulties of Measy = -1, Mmedium = 0 or Mdifficult = 1 and a standard deviation of SD = 1 for all
scales. Thus, the range of item difficulties was not restricted, but the scales exhibited differences
in the mean scale difficulty. The medium test scale mimicked a test scenario, where most of the
ability levels are covered well across all age levels. The accumulation of item difficulties in the
medium difficulty range for this scale corresponds to real test construction, as items with extreme
difficulties are sorted out more often than items with medium difficulty due to their lower
discriminative power. In contrast to the medium test scale, the easy and difficult scales were
designed such that they differentiated best in the lower or upper age ranges, respectively. The
easy scale resulted in mild to medium ceiling effects at older age levels, and the difficult test
scale resulted in medium to large bottom effects at the younger age levels. Again, our aim was to
cover all kinds of test scenarios as realistically as possible. For example, even a wide-spread
intelligence tests like the KABC-2 (Kaufman & Kaufman, 2004) displays considerable skewness
of the raw score distributions at extreme age levels in more than half of all subtest scales.
c. Generating raw scores for the normative and cross-validation sample
For each person of the normative and the cross-validation sample, we simulated the item
responses in accordance with a 1PL IRT model. We chose this model for three major reasons.
First, the item response theory is one of the most important theoretical models to explain how a
latent ability interacts with items and scale parameters and finally results in a raw score
distribution. It therefore offers a rather simple but plausible way to generate raw score
distributions corresponding to real-world data. More precisely, we do not need to make specific
assumptions about the raw score distributions themselves, apart from the assumption that they
result from an interaction between the latent trait of a certain individual and the test items.
Second, both the normative data and the representative cross-validation data could be generated
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 17
in the exact same way. Third, since we started by assigning a “true” latent ability to each person,
it was not only possible to compare the raw score distributions in the normative sample with the
raw score distributions in the cross-validation sample. Additionally and even more importantly,
we could determine how well the raw scores and the resulting norm scores cover the latent
ability.
To generate the response of person i to item j of a test scale, we computed the probability
of success of that person on each item via the 1PL logistic function (de Ayala, 2009, p. 15;
Embretson & Reise, 2000, p. 50; Rasch, 1980; Wright & Stone, 1979; source code available as
the ‘simRasch’ function in cNORM, W. Lenhard et al., 2018):
(2) = 1|θ ,=θ 
1 + θ 
p(xj=1| θPop i, δj) specifies the probability that a person i with the latent ability θPop i
succeeds on an item j with the given difficulty δj. For each item j and person i, we drew a
uniformly distributed random number ranging from 0 to 1 and compared this number with the
person’s probability to succeed. In case p(xj) was equal or higher than the random number, the
person scored one point, otherwise zero points. Eventually, the item responses were summed up
to a total score, which served as the raw score this person received in the test scale.
On average, the simulated raw score distributions showed a negative skew of gm = -0.506
for the easy, a small positive skew of gm = 0.145 for the medium and a highly positive skew of gm
= 0.859 for the difficult scales. Detailed descriptive data about the raw score distributions per half
year age bracket are available for the normative and cross validation sample in the OSF
repository. As well, the repository includes the data of one complete simulation cycle, including
item difficulties, raw data and cross validation data per condition.
d. Statistical modeling of the raw score distributions and compilation of norm score
tables.
To establish norming models according to the SPCN approach, we used the cNORM
package (W. Lenhard et al., 2018) available for the R platform. The ranking function needed for
the estimation of the location l was set to ‘rankByGroup’ with a width of 6 months, which means
that the percentiles for each person were estimated for each half year bracket of the norm sample
before applying the statistical modeling. The results of the procedure are however relatively
invariant against the bracket width with 6 and 12 month returning almost identical models (A.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 18
Lenhard et al., 2016). The multiple regression was performed with the default setting of k = 4 as
the maximum degree of the polynomial, resulting in a polynomial with 24 terms at most. We
started with four as the default number of terms and only searched for different solutions when
the resulting model showed violations against monotonicity, that is, when the relation between
raw scores and norm scores at a given age was not monotonically increasing throughout the
whole dataset. The monotonicity check was based both on determining the polynomial roots of
the first order derivative of the regression function and on numeric search (method
‘checkConsistency’ of the cNORM package). In case of violations, we repeated the modeling
beginning with three terms and subsequently increased the number of terms in the regression
function until a model without such inconsistencies was found. In 70% of all simulation cycles,
this procedure resulted in a norming model with only six terms or less.
In order to disentangle the effects of sample size and width of the age brackets, we split
up the normative sample in subsamples of the width of 1 month, 3 months, 6 months and 12
months, respectively, when using conventional norming. Table 1 gives an overview over the
average subsample sizes as a function of the total sample size and the width of the age brackets.
Table 1. Subsample sizes as a function of total sample size and width of the age brackets
Total Sample Size N
Age Bracket N = 700 N = 1,400 N = 2,800 N = 5,600 N = 11,200 N = 22,400
1 month 8.3 16.7 33.3 66.7 133.3 266.7
3 months 25 50 100 200 400 800
6 months 50 100 200 400 800 1,200
12 months 100 200 400 800 1,600 3,200
We subsequently performed INT for each age bracket and automatically compiled norm
score tables for the transformation of raw scores into norm scores, which included all raw scores
that had occurred in the respective subsamples.
Since all norm scores will be expressed as T-scores (MT = 50 and SDT = 10) in the
following analyses, we will subsequently refer to norm scores produced with a certain method as
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 19
TMethod. The different conventional norming conditions will be abbreviated to CN1, CN3, CN6
and CN12 for the respective age brackets of 1 month, 3 months, 6 months, and 12 months.
e. Cross-validation.
In the cross-validation sample, we also applied INT to the empirical CDF of the raw
scores in each subsample. Given that the total sample was perfectly representative and included
840,000 cases with n = 10,000 per subsample, we assumed that the resulting norm scores
contained almost no norming error and therefore represented an upper boundary for the quality of
any norming procedure. The fact that these norm scores are nevertheless no perfect predictors of
the latent abilities (R2 < 1) can therefore be completely traced back to limitations of the scales
(distribution of item difficulties; bottom and ceiling effects, limited number of items …), but not
to the norming procedure. We will therefore refer to these norm scores as TIdeal in the analyses.
To be able to compare the different norming methods, we also assigned to each person in
the cross-validation sample the five different norm scores, that had been assigned to the raw score
of this person on the basis of the much smaller normative samples, namely, TSPCN, TCN1, TCN3,
TCN6, and TCN12.
Assessment of model fit and definition of the dependent measures.
We used various measures to determine different aspects of the model fit.
a. Root Mean Square Error (RMSE) and Mean Signed Difference (MSD)
RMSE is the standard measure of model fit. In our simulation, it was defined as follows:
(3)  =
, ×( )
,

The RMSE reflects an overall measure of the error introduced by the norming procedure,
that is, it captures constant as well as variable errors of the norming procedure.
The MSD, by contrast, only reflects constant dislocations of the norm scores within each
simulation cycle. It is defined as
(4)  =
, ×( )
,

Since the average sampling error is expected to be zero in this simulation, the MSD should
also be zero when averaged over multiple simulation cycles, regardless of the total sample size. If
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 20
it is not, this may be due to the fact that a certain method produces constant dislocations of the
norm scores in one specific direction under certain conditions (e.g., for certain scale difficulties).
Therefore, the MSD is a measure of how accurately the method can cover raw score distributions
with specific properties as, for example, skewness.
b. Proportion of error variance
For the following comparisons, the latent variable θAge was also transformed from a z-
score to a T-score for each person in the cross-validation sample so that the variances of the
different scales could directly be compared.
In classical test theory, the reliability of a test scale is usually expressed as the proportion
of variance in the true score that can be predicted from the raw scores. Under the assumption that
raw scores can be converted into norm scores without additional error or loss of information (as
is approximately the case for TIdeal), the coefficient of determination R2 between θAge and TIdeal
therefore equals the reliability RELRaw of the respective scale:
(5)  = =θ
()=θ;
The proportion of error variance (PEV) contained in the ideal norm scores resp. in the raw
scores can subsequently be calculated as follows:
(6)  =  = 1 θ
()= 1 θ;
Note that this proportion of error variance is not due to the norming procedure, but only to
the deficiencies of the test scale.
Since the norm scores are, however, flawed with errors, the proportion of error variance
contained in TSPCN, TCN1, TCN3, TCN6, and TCN12 is higher than the proportion of error variance
contained in the ideal norm scores. We will refer to the proportion of error variance contained in
the norm scores as PEVTotal in the following, because it contains measurement error on the level
of the raw scores as well as additional error due to the norming procedure:
(7)  = 1 θ
()= 1 θ;
The proportion of error variance caused exclusively by the norming procedure can
therefore be calculated by the following equation:
(8)  =   = θ; θ;
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 21
c. Definition of reliability index for norming procedures
We furthermore define the reliability RELNorm of a specific norming procedure as the
proportion of variance in the latent ability the actual norm scores can predict, compared to the
proportion of variance ideal norm scores can predict:
(9)  =θ
():θ
()=()
()=θ;
θ;
Finally, RELTotal, which is the proportion of variance the norm scores share with the true
latent score, can be expressed as the product of RELRaw and RELNorm:
(10)  =   =θ;θ;
θ;=
θ;
RELNorm can be interpreted as a global measure of the precision of a certain norming
procedure that is independent of the specific scale on which the procedure is applied. Moreover,
it can be multiplied with classical reliability indices to retrieve the total reliability of a norm
score, that is, the degree to which this norm score is able to predict the latent ability. If, for
example, the conventionally computed reliability index of a test was RELRaw = .90 and the
norming procedure had a reliability of RELNorm = .95, the de facto reliability of the norm scores
would only be RELTotal = .90 * .95 = .86. RELNorm is easy to interpret, but it only accounts for the
general shape and course of the modeled distributions. It does not capture dislocations or constant
biases that affect all norm scores equally. Therefore, it is only applicable to those norming
methods with an MSD close to zero.
d. Number of missings
Finally, we also counted the missings. They were defined as percentage of persons in the
cross-validation sample for whom no norm score could be determined because the corresponding
raw score had not occured in the normative sample.
Data
The simulation was repeated 1,000 times for each combination of parameters, resulting in
a total number of 6 (size of the normative sample) x 3 (scale difficulty) x 5 (number of items) x
1,000 repetitions = 90,000 cycles.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 22
Statistical analyses
We analyzed RMSE, MSD and the number of missings in the norm tables with repeated
measures ANOVAs, with method (SPCN, CN1, CN3, CN6, and CN12) serving as within factor
and total sample size (N = 700, 1,400, 2,800, 5,600, 11,200, and 22,400), item number (10, 20,
30, 40, and 50) and scale difficulty (easy, medium, and difficult) serving as the between factor.
For specific hypotheses and post hoc analyses, we drew on one-way ANOVAs. Due to the high
number of repetitions for all combinations of the independent variables, the ANOVA was very
robust against violations of homoscedasticity and normality requirements (c.f. Eid, Gollwitzer &
Schmitt, 2017, p. 408). Furthermore, the significance level for all tests was set to α = .001 to
reduce Type I errors. A power analysis with GPower (Faul et al., 2009) showed a β > .9999 for
all analyses on main effects, interactions and post-hoc tests. This means that the ANOVA was
robust yet extremely powerful enough to detect even the smallest main effects and interactions.
Given that the statistical power was extremely high, we will additionally report effect
sizes in the form of partial η2 and interpret only effects with η2 > 3%, which approximately
corresponds to at least small effect sizes (Cohen, 1988, p. 413).
Since PEVNorm and RELNorm are different measures of the proportion of error contained in
test scores, the inferential statistical analysis of these scores does not provide any other results
than the evaluation of the RMSE with regard to the relation between the different methods. What
is different, however, is the fact that the two parameters are not absolute but relative measures of
error, that is, they are independent of the used scaling and therefore provide easy interpretation.
Moreover, the impact of the norming procedure can be compared with the impact of the item
selection. For this reason, we will provide only descriptive data for the two parameters, but we
will add a few fictitious calculation examples using realistic test parameters in order to
demonstrate the importance of a balanced norming procedure.
Results
Precision and accuracy of the norm scores in terms of RMSE and MSD
The RMSE was analyzed as a function of method, total sample size and item number to
investigate the overall quality of the different norming procedures. With regard to hypothesis H1
and H2, the analysis of the RMSE revealed a main effect of method, F(4, 359,980) =
544,560.527, p < .001, ηpart2 = .858, with SPCN delivering on average a much lower RMSE
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 23
(RMSESPCN = .862) than the four conventional norming procedures (RMSECN1 = 2.293, RMSECN3
= 1.445, RMSECN6 = 1.305 and RMSECN12 = 1.665). We also found a main effect of total sample
size (hypothesis H2), F(5, 89,970) = 397,945.050, p < .001, ηpart2 = .957, which was qualified by
a significant Method x Total Sample Size interaction, F(20, 359,880) = 65,863.247, p < .001,
ηpart2 = .785. This interaction is illustrated in Figure 4. The effect of total sample size was only η2
= .184 for SPCN, but much higher for the conventional norming procedures (η2 = .978 for CN1;
η2 = .957 for CN3; η2 = .896 for CN6 and η2 = .600 for CN12). Even with the lowest total sample
size of N = 700 (i.e., 100 persons per year), the RMSE was only 1.25 T-scores for SPCN. In
contrast, the best conventional method required four times the total sample size to approach such
a low value. Moreover, the change in RMSE between the lowest and the highest total sample size
was only ΔRMSESPCN = 0.613. Although the impact of total sample size was similar low in the
CN12 condition (ΔRMSECN12 = 0.572), it must be noted that CN12 showed significantly higher
RMSE at any total sample size. For all other conventional procedures, the impact of the total
sample size was much larger (ΔRMSECN1 = 3.392; ΔRMSECN3 = 2.021 and ΔRMSECN6 = 1.198).
Thus, conventional norming is much more dependent on the total sample size. In general, an
equally low RMSE requires four times the total sample size when conventional norming is
applied as compared to SPCN. Moreover, this result only holds true if an optimal age bracket is
selected. As can be seen from Figure 4, the same subsample size leads to significant differences
in RMSE between the conventional norming conditions. In case no optimum age bracket is
selected, the total sample sizes would have to be even higher when conventional norming is
applied.
The ANOVA also revealed a main effect of the item number, F(4, 89,970) = 13,777.134,
p < .001, ηpart2 = .380, with higher item numbers leading to higher RMSE. This effect is
illustrated in Figure 5.
Due to the huge test power, there were also significant interactions of the item number
with all other independent variables, but the respective effect sizes were too small to be of any
practical relevance (η2 < .02 for all interactions).
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 24
Figure 4. RMSE in the cross-validation as a function of method and sample size. The
numbers indicate the subsample sizes for the conventional norming conditions.
Note. All SE < 0.01. Dashed lines represent conventional norming, the solid line represents
semi-parametric continuous norming (SPCN).
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 25
Figure 5. RMSE in the cross-validation as a function of method and item number.
Note. All SE < 0.01. Dashed lines represent conventional norming, the solid line
represents semi-parametric continuous norming (SPCN).
As a next step, the MSD in the cross-validation was analyzed as a function of method and
scale difficulty to search for constant dislocations. The scale difficulty was chosen as independent
variable in this case, since high skewness of the raw score distributions is most likely to cause a
general dislocation in continuous norming in general. MSD for the different scales is depicted in
Figure 6. There was in fact a significant interaction between method and scale difficulty, F(8,
359,864) = 2,300.986, p < .001, ηpart2 = .049. The interaction was caused by the fact that SPCN
showed a slightly higher MSD than conventional norming at easy scales only. However, with
MSD = 0.054 T-scores, the absolute size of the dislocation was so tiny that it was irrelevant
compared to the other error sources. Moreover, the size of the dislocation is far below the
accuracy with which real test results are actually ever reported.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 26
Figure 6. MSD in the cross-validation as a function of method and scale difficulty (i.e.,
skewness of the raw score distribution).
Note. All SE < 0.01. Dashed lines represent conventional norming, the solid line
represents semi-parametric continuous norming (SPCN).
Analysis of missing norm scores
With regard to the analysis of the missings (H3), the reader should note that SPCN in
most cases allows to compute norm scores to all achievable raw scores, which is the central
advantage of this method, let alone closing gaps between norm tables. However, on some rare
occasions, SPCN cannot return a norm score because of an incoherent model. The analysis of the
missings as a function of method, total sample size and item number confirmed these
expectations. There was a very strong main effect of method, F(4, 359,880) = 3,952,101.067, p <
.001, ηpart2 = .978, with SPCN delivering only 0.258% missings on average. Moreover, in 50% of
all simulation cycles, the number of missings was even below 0.031%. There was, in fact, a small
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 27
main effect of the total sample size in the SPCN condition, F(5, 89,994) = 716.923, p < .001,
ηpart2 = .038, with the smallest total sample size delivering more missings (0,478%) than the
largest one (0,137%). However, conventional norming achieved similarly low numbers only
when the total sample size was four to eight times larger. Moreover, the different conventional
norming procedures showed very large main effects of total sample size and item number, which
were qualified by twofold and threefold interactions (all p’s < .001; all ηpart2 > 0.82). Under
balanced conditions (i.e., age bracket = 6 months, N = 5,600 and item number = 40) the number
of missings was 0.713 %. The effects are depicted in Figures 7 and 8.
Descriptive analysis of PEVNorm and RELNorm
Since the conditions in our study were numerous and covered a wide range of item
numbers and total sample sizes, some of the combinations of conditions would certainly not
occur in reality, for example, conventional norming with N = 700 and an age bracket of 1 month.
Therefore, we will not report PEVNorm and RELNorm for all combinations of conditions in the
following. Instead, we will focus on the conventional norming procedure that performed best
under a specific combination of conditions (= Best CN). That is, we assume that a test constructor
is able to divide the total sample into age brackets with optimal width. Table 2 gives an overview
over PEVNorm and RELNorm under the different combinations of total sample size and item
number analyzed in this study.
First of all, the numbers indicate that the proportion of error variance added to the test
scores through the norming procedure decreases with the total sample size, but increases with the
number of items, as was already reported with regard to RMSE. What we find interesting, though,
is the possibility to interpret the error variance independently of the scaling. For example, when
SPCN was applied, the error variance added to the test scores through the norming procedure,
varied between 0.194% and 1.594% (M = 0.70%). As a consequence, the proportion of true
variance of the latent trait contained in the norm scores, was slightly reduced by a factor ranging
between .980 and .997, with RELNorm = .990 on average. Imagine, for example, a highly reliable
test scale with an indicated reliability of rtt = .90, that is, a proportion of error variance of about
10% in the raw scores. In the worst case, SPCN would increase this variance to a total of 11.6%,
that is, the “effective” reliability of the norm scores would be RELTotal = .90 x .98 = .88, which is
still high. With conventional norming, on the other hand, the introduced error variance varied
between 0.328% and 7.673% (M = 2.53%), corresponding to a factor RELNorm between .994 and
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 28
.901 (M = .966). As a consequence, the reliability of a fictitious test scale of rtt = .90 would
decrease to an effective reliability of RELTotal = .90 x .90 = .81 in the worst case. The error
variance contributed by the norming procedure would be about three quarters as high as the error
variance contributed by the test items in this case. Admittedly, the total sample size was only N =
700 with age brackets of 12 months in this worst-case scenario, that is, the subsample size was
below the generally recommended one for conventional norming. Under more optimal
conditions, that is, with subsample sizes between n = 200 and n = 400, but an attainable total
sample size of up to N = 5,600, the error variance added to the test scores with conventional
norming ranged between 1 and 5%. Again, conventional norming required about four times the
total sample size to achieve the same quality as continuous norming.
Figure 7. Missings in the cross-validation as a function of method and sample size. The
numbers indicate the subsample sizes for the conventional norming conditions.
Note. All SE < 0.01. Dashed lines represent conventional norming, the solid line
represents semi-parametric continuous norming (SPCN).
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 29
Figure 8. Missings in the cross-validation as a function of method and item number.
Note. All SE < 0.01. Dashed lines represent conventional norming, the solid line
represents semi-parametric continuous norming (SPCN).
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 30
Table 2. PEVNorm and RELNorm as functions of approach (continuous vs. conventional), sample
size and item number.
Sample Size
700
1.400
2.800
5.600
11.600
22.400
Total
10 ITEMS
PEVNorm
SPCN
0.97%
0.53%
0.35%
0.37%
0.27%
0.19%
0.45%
Best CN
3.29%
2.07%
1.37%
0.78%
0.49%
0.33%
1.39%
RELNorm
SPCN
.982
.990
.994
.993
.995
.997
.992
Best CN
.941
.963
.975
.986
.991
.994
.975
20 ITEMS
PEVNorm
SPCN
1.27%
0.87%
0.49%
0.41%
0.37%
0.26%
0.61%
Best CN
5.24%
3.32%
2.17%
1.26%
0.82%
0.51%
2.22%
RELNorm SPCN
.982
.988
.993
.994
.995
.996
.991
Best CN
.925
.953
.969
.982
.988
.993
.968
30 ITEMS
PEVNorm
SPCN
1.39%
1.00%
0.71%
0.47%
0.43%
0.33%
0.72%
Best CN
6.32%
4.09%
2.63%
1.55%
1.01%
0.64%
2.71%
RELNorm
SPCN
.982
.987
.991
.994
.994
.996
.991
Best CN
.918
.947
.966
.980
.987
.992
.965
40 ITEMS
PEVNorm
SPCN
1.59%
1.08%
0.75%
0.61%
0.52%
0.49%
0.84%
Best CN
7.12%
4.60%
2.96%
1.74%
1.15%
0.71%
3.05%
RELNorm
SPCN
.980
.987
.991
.992
.994
.994
.990
Best CN
.913
.944
.964
.979
.986
.991
.963
50 ITEMS
PEVNorm
SPCN
1.57%
1.16%
0.82%
0.74%
0.52%
0.49%
0.88%
Best CN
7.67%
5.02%
3.17%
1.90%
1.25%
0.77%
3.30%
RELNorm
SPCN
.981
.986
.990
.991
.994
.994
.989
Best CN
.909
.941
.962
.978
.985
.991
.961
TOTAL
PEVNorm
SPCN
1.36%
0.93%
0.62%
0.52%
0.42%
0.35%
0.70%
Best CN
5.93%
3.82%
2.46%
1.45%
0.94%
0.59%
2.53%
RELNorm
SPCN
.981
.987
.992
.993
.994
.995
.990
Best CN
.921
.949
.967
.981
.988
.992
.966
Note. For the conventional norming procedure, we selected the best fitting model.
Discussion
This simulation study compared continuous to conventional norming, taking into account
that the quality of conventional norms often depends on whether continuous explanatory
variables (e.g., age) are considered in building sufficiently narrow normative subsamples. To
investigate the effects of the different norming procedures under multiple conditions, we
additionally varied the number of items, the difficulty of the used test scales (i.e., the skewness of
the raw score distributions) and the total sample size.
Regardless of the total sample size, the scale difficulty and the number of test items, the
specific continuous norming method we used in this study (SPCN) produced less error variance
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 31
and therefore provided a more reliable estimation of the person location than conventional
norming. Even with an optimal balance between the total sample size and the width of the age
brackets, conventional norming required about four times the total sample size to achieve the
same data fit as continuous norming. In reality, however, it is anything but certain that a test
constructor would always find this optimal balance between the width of the age bracket and the
size of the subsamples. Only with unrealistically high total sample sizes, the quality of both
approaches was similar. At the same time, as indicated in Figure 4, the general recommendations
for sample sizes per age bracket are not sufficient to guarantee optimal results, since the bias
resulting from finite width of the age brackets is neglected.
Since the statistical test power was extremely high in our study, there is the additional
question of whether the size of the effect plays a role for practical applications. On the one hand,
the mean absolute error was mostly within the range of 95%-confidence intervals for both
conventional and continuous norming – even for scales with very high reliability. On the other
hand, many diagnostic decisions are made on the basis of clear cut-points where the confidence
intervals do not play any role at all. Therefore, one T-score more or less can definitely make the
difference here. More importantly, however, the norming errors are never evenly distributed
across all levels of the latent trait. Generally, they are U-shaped, with the slopes on both sides
varying with the average RMSE (c.f. A. Lenhard et al., 2019). Therefore, the difference between
the methods might be irrelevantly small for average test results but significantly high for test
results strongly above or below average. Unfortunately, extreme test results are precisely those
results that are most relevant in diagnostic practice.
Furthermore, conventional norming also lead to a considerably higher number of missings
in the norm tables. Again, conventional norming required about four to eight times the total
sample size to approach the low number of missings of continuous norming. Furthermore, the
missings almost exclusively concern the extreme test results, i.e. the range of diagnostically
relevant cases. Hence, the percentage of test results affected by missings in diagnostic practice is
probably much higher than in our study. Continuous norming can improve the quality of the test
norms because it not only closes these gaps but also smooths the data along the ability dimension.
Finally, we directly compared the error variance contributed to test scores by the different
norming procedures with the error variance produced by the test items. For continuous norming,
the error variance added by the norming procedure was much smaller than the one that is usually
contributed by the test items. Hence, the norming procedure only slightly impaired the
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 32
explanatory power of the test results. By contrast, the proportion of error variance added with
conventional norming considerably increased the total error variance at least under some
conditions. This finding suggests that the selection and accurate implementation of an adequate
norming procedure can be almost as important for the explanatory power of test scores as all item
selection efforts (e.g., increasing the number of items, analyzing the item fit and removing those
that deteriorate homogeneity). In case, relevant covariates like age exist, continuous norming can
help to represent a latent ability more precisely than conventional norming. Our study thus adds
support to the finding of Zhu and Chen (2011) and Oosterhuis, van Ark and Sijtsma (2016) that
with continuous norming, substantially smaller samples are sufficient to reach the same quality of
norm scores or even surpass conventional norming.
Limitations
The advantage of SPCN and other continuous norming procedures is to model the effects
of a covariate on the relation between raw scores and norm scores. This advantage should
increase with the proportion of variance in the raw scores that can be explained by that covariate.
If, on the other hand, a covariate like, for example, age does not exert a relevant influence on the
raw scores, there is no reason to construct norm tables for distinct age brackets and consequently
continuous norming models are unnecessary as well (at least as far as this particular covariate is
concerned). The effects demonstrated in this norming study should therefore increase with an
increasing slope of the age trajectories of the measured trait and decrease with a decreasing slope.
In this simulation study, SPCN returned very robust solutions for the predefined population
model we used, which mimicked the vocabulary development in childhood and adolescence. Yet
it might fail with other application scenarios. In other conceivable scenarios, for example, the
mean raw scores could remain constant across age, while only the variance might change. We
have not modeled such scenarios yet and it is therefore impossible to generalize the results of the
present study to all conceivable conditions. On the other hand, our own experiences with SPCN
showed very good results in many domains other than the originally intended psychometric use
cases. More precisely, we successfully modeled fetal length in dependence of gestation week,
body mass index and body height growth in childhood and adolescence in dependence of age,
world-wide life expectancy at birth and mortality rate on country level in the course of the last 50
years (W. Lenhard et al., 2018). The approach might therefore be useful for other application
scenarios from biometrics to macroeconomics as well.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 33
A second limitation concerns the number of covariates. So far, we have mainly focused on
taking only one covariate into account, namely age. For the application scenarios we mainly
target, such as intelligence tests or developmental tests, other covariates (e.g., sex or ethnical
background) are usually controlled via stratification of the sample. Although from a theoretical
point of view, SPCN is not necessarily restricted to the inclusion of one covariate only, other
approaches (e.g., Van Breukelen & Vlaeyen, 2005; Rigby & Stasinopoulos, 2005) might be better
suited for this demand. Whether it is advisable to compute separate norm scores for covariates
other than age (e.g., sex-specific IQ scores), cannot be solved by statistical procedures, though. It
rather depends on the exact use case.
Conversely, of course, it could also happen that no covariates have to be included at all,
but that the norm score is to be predicted as a function of the raw score only. Equation (1) would
thus be reduced to a simple polynomial regression. At the moment we cannot precisely determine
how significant the advantages of such a regression would be compared to the conventional
calculation of norm scores. While the procedure certainly loses many of its advantages in that
case, it nonetheless closes missings in the tables more precisely than a simple interpolation
between the available data points. Moreover, it smooths the transformation function between raw
scores and norm scores and thus might help to mitigate sampling errors in the normative sample,
though it is not possible to repair completely unrepresentative normative data this way.
Another drawback of the present study might be the use of measures that cannot be
determined in real norming procedures. Since in applied diagnostics, the latent ability is
unknown, the factor RELNorm, specifying the relative amount of error variance added by the
norming procedure, can only be estimated in simulation studies like the present one. Strictly
speaking, the determined values therefore only apply to the conditions established in this
simulation. For SPCN, however, RELNorm remained constant at around .99 even under simulation
conditions other than those implemented in this study, whereas for other methods it obviously
depends more on the actual conditions. We therefore think it is important to keep in mind that in
real life the “effective” reliability of norm scores (as defined by the explained variance of the
latent trait) is usually lower than the effective reliability of raw scores, which is typically
specified in the handbooks.
Of course, a simulation study like the present one is always based on specific
assumptions. For example, we drew the samples from a perfectly representative population. In
real-life scenarios, however, it is not always possible to perfectly stratify each age bracket. As the
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 34
regression-based approach in SPCN always draws on information from the total sample to
determine the mathematical models, such imbalances can at least partially be compensated. The
same does not hold true for conventional norming, though. As a consequence, the differences
between continuous and conventional norming would even increase in real-life scenarios.
Furthermore, we assumed that the latent trait is normally distributed at each level of the
explanatory variable. We believe that this assumption is justified because of the universality of
the law of large numbers. Additionally, the 1PL IRT measurement model we used to produce test
results makes specific assumptions about the interaction between latent traits and test items (e.g.,
constant discriminative power for all test items). We chose to draw on a 1PL IRT model due to
three main reasons. First, the method is suitable to generate sample raw data that align with real
world normative samples. Second, we constructed both the normative and the representative
cross-validation sample from the same population model and subjected them to the same,
randomly generated test scales in order to cross validate different modelling techniques. Third,
the item response theory is one of the most important theoretical models to explain how a latent
ability interacts with items and scale parameters and finally results in a raw score distribution.
That way, we could determine how well the raw scores and consequently the normed raw scores
represent the latent ability. It is important to note, though, that a specific measurement model is
neither required for the application of conventional norming nor for SPCN. The latter is based on
the mathematical principle that any function can be modelled with polynomials as long as it is
smooth and finite. Therefore, it can be applied to all kind of distributions like biometric and
sociometric data. As psychometry most often aims at measuring latent variables, the application
of an IRT model was the most plausible candidate for data simulation purposes in our specific
use case. Furthermore, SPCN is not restricted to 1PL IRT models. For example, we successfully
applied this method to normative data of a test constructed on the basis of a complex
measurement model including power and speed components as well as discrimination parameters
for both (2 x 2 PL model sensu Fox, Klein Entink, & van der Linden, 2007; W. Lenhard et al.,
2017). Moreover, it was also used to model observed biometric data without any measurement
model at all like, for instance, the BMI growth curves in childhood and youth (sample dataset
CDC of the cNORM package). Therefore, we are convinced that the results can in fact be
transferred to various other conditions. We even believe that the universality and simplicity of the
SPCN approach is one of its great strengths.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 35
Finally, we only compared one specific continuous norming method with conventional
norming, namely SPCN. As described above, this particular method makes no specific
assumptions about the raw score distributions and can therefore be used very flexibly. However,
there are other continuous norming methods which may lead to somewhat different results (e.g.
Zachary & Gorsuch, 1985; Cole & Green, 1992; Rigby & Stasinopoulos, 2005 as demonstrated
by Voncken et al., 2019a). When we compared SPCN with parametric continuous norming
methods, it mainly showed superior performance when applied on scales with medium to high
skewness of the raw score distributions (A. Lenhard et al., 2019). When applied on test scales
with very low skewness, both methods performed equally well, with a slight advantage of SPCN
in smaller norming samples and a slightly better performance of parametric models in larger
samples. However, the latter result was only achieved under the condition that an optimal
modelling function was used in the parametric condition. Hence, the search for such an optimal
modelling function is of high importance when applying parametric approaches. If the search is
successful, then parametric and semi-parametric continuous norming deliver comparable results.
Consequently and just like SPCN, parametric continuous norming will very likely outperform
conventional norming in most cases as well.
Conclusion
Our results have clearly shown that test scales contain less error variance and will
therefore be more reliable if continuous models are used to capture the effects of relevant
explanatory variables as, for example, age. We therefore recommend not only to resort to simple
ranking procedures but to complete the analysis by modeling the continuous relation between raw
scores, norm scores, and explanatory variables when necessary. One way to perform this
modeling is the regression-based continuous norming method we have described in this study.
The method not only provides higher norming quality than conventional norming procedures but
also outperforms other continuous norming methods under various conditions (A. Lenhard et al.,
2019). Moreover, the software is available as an R package with a graphical user interface (online
demonstration see https://cnorm.shinyapps.io/cNORM/ ) and online tutorials
(https://www.psychometrica.de/cNorm_en.html ). It can be freely used even with limited prior
knowledge in R. More experienced psychometricians can choose between different norming
methods, for example, semi-parametric or parametric continuous norming. The sample size, the
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 36
impact of covariates and the properties of the raw score distribution are important factors when it
comes to the selection of the most appropriate method.
In recent years, research and published work in the field of norming has increased. We
consider this development to be of paramount importance, because it has the potential not only to
improve the explanatory power of psychometric tests but also to make them more cost-effective
and therefore more wide-spread. We hope that we can significantly contribute to this field with
this simulation study.
CONVENTIONAL VS SEMI-PARAMETRIC CONTINUOUS NORMING 37
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... Moreover, they cannot be interpreted by themselves due to the missing frame of reference. However, what is represented in the eCDF of raw scores in the norm sample is the rank information, that is, information about the raw score that is expected for a certain rank of the latent trait [13]. It should be noted here that just as in IRT, this is a relationship between the latent ability of a person and an expected outcome; however, unlike IRT, it is not the outcome of a single item but of the whole scale. ...
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Background Understanding young people’s subjective well-being, such as how they experience equality, safety, and supportive environments, is particularly important for identifying possible preventive interventions in the school context. Aim The aim of this study was to investigate similarities and differences between 15-year-old girls’ and boys’ subjective well-being, and its associations with psychosocial factors. Methods The study was explorative and cross-sectional. A total of 195 adolescent girls and 116 boys in Grade 9 from four Swedish junior high schools completed a questionnaire on subjective well-being; emotional support opportunities; gender equality; experience of emotional, physical, and sexual violence in close relationships; insecurity due to harassment in the school environment or on social media; and pornography consumption. Results Compared to boys, girls indicated worse subjective well-being and more harassment in school and on social media, and reported that boys received benefits based on gender. Boys reported higher pornography consumption compared with girls. Girls reporting decreased subjective well-being experienced more harassment on social media and in school, and more recent sexual violence. Among boys, with decreased subjective well-being, more psychological risks such as self-harm and suicidal thoughts was reported. Both girls and boys with decreased subjective well-being reported less availability of attachment and insecure close relations. Conclusion The absence of a safe and supportive environment, such as feeling secure at school, and lack of close and trusting relationships are associated with 15-year-olds’ subjective well-being for the worse, regardless of gender. It is suggested that school personnel extend the relational possibilities in the school context.
... (a) urg_p = positive urgence; urg_n = negative urgence; lak_prm = lack of premeditation; lck_prs = lack of perseverance; and sns = sensation seeking. and age, and then, we calculated norms for each factor using procedures from Lenhard et al. (2018Lenhard et al. ( , 2019, Lenhard and Lenhard (2020) . ...
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To compute norms from reference group test scores, continuous norming is preferred over traditional norming. A suitable continuous norming approach for continuous data is the use of the Box-Cox Power Exponential model, which is found in the generalized additive models for location, scale, and shape. Applying the Box-Cox Power Exponential model for test norming requires model selection, but it is unknown how well this can be done with an automatic selection procedure. In a simulation study, we compared the performance of two stepwise model selection procedures combined with four model-fit criteria (Akaike information criterion, Bayesian information criterion, generalized Akaike information criterion (3), cross-validation), varying data complexity, sampling design, and sample size in a fully crossed design. The new procedure combined with one of the generalized Akaike information criterion was the most efficient model selection procedure (i.e., required the smallest sample size). The advocated model selection procedure is illustrated with norming data of an intelligence test.