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JOURNAL OF CLINICAL AND EXPERIMENTAL NEUROPSYCHOLOGY 2007, 29 (1), 25–35 © 2006 Psychology Press, an imprint of the Taylor & Francis Group, an informa business http://www.psypress.com/jcen DOI: 10.1080/13803390500276859 NCEN Regression-based developmental models exemplified for Wisconsin Card Sorting Test parameters:

Statistics and software for individual predictions Regression-based Developmental Models Christoph Klein, 1 Friedrich Foerster, 2 and Klaus Hartnegg 3 1School of Psychology, University of Wales2Research Group Psychophysiology, University of Freiburg3Brain Research Unit, University of Freiburg The prediction of an individual’s score is relevant in clinical research and requires normative data and a statistical rationale. In the case of developmental research the latter is typically a set of descriptive statistics (e.g., standard scores) for a set of age groups. Here we illustrate a multiple regression approach with a set of 345 Wisconsin Card Sorting Test (WCST) data obtained from subjects aged 6 to 26 years. We modeled linear and curvilinear age effects for each of the 11 WCST variables and, based on this, determined confidence limits for the expected value (mean) and the prediction of individual scores. In these multiple regression models, which accounted for 2% to 26% of the variance, curvilinear age effects clearly dominated linear ones, suggesting the features under scrutiny to be negatively accelerated functions of age. Finally, we developed a statistics program that can be used to apply multiple regression models for individual predictions that are based on normative data with up to 7 predictor var- iables. We discuss the conditions of applicability of the approach, compare it with the conventional standard score approach, discuss its cognitive-developmental implications, and outline the applicability in applied research and practising. INTRODUCTION A “deficit” in neuro-cognitive functions of neuro- logical or psychiatric patients is rarely defined by absolute criteria, but in relation to normative data, which are thus used to make individual predictions about the presence/absence or degree of a deficit in individual patients or groups of patients. “Norma- tive data” in basic research is frequently the aggre- gate statistics of a healthy control group or, if published tests are available, the aggregate statist- ics of a proper normative sample. This holds in principle for investigations both in a develop- mentally rather “static” age range (middle adult- hood) and in developmentally more “dynamic” age ranges (childhood, adolescence). As the latter age ranges are concerned, we suggest and demonstrate here a new approach for making individual predic- tions in clinical trials that are based on normativedata. This approach is based on the following considerations.

Neuropsychological research with children, ado- lescents and seniors can use neuropsychological tests to sensitively track normal developmental or ageing-related changes in cognitive functions and, in consequence of this, their alterations in clinical populations. These tests typically provide quanti- tative scores and, most of them, difficulty ranges that allow for discrimination within given age ranges (e.g., all tests that output reaction time, fre- quency of error, or span scores, or grade test items according to difficulty or complexity criteria).

With these tests development is implicitly assumed to be a step-wise or continuous, but in any case quantitative process rather than a process that includes developmental stages in the sense of quali- tative changes. The usefulness of such tests as assessment or diagnostic instruments for the prediction Address correspondence to PD Dr. Christoph Klein, School of Psychology, University of Wales, Bangor, The Brigantia Building, Penrallt Road, Bangor, Gwynedd LL57 2AS, Wales, UK [E-mail: [email protected]]. 26 KLEIN, FOERSTER, HARTNEGG of individual scores, however, critically depends on the availability of normative data and an appropri- ate statistical modeling of the developmental effects. Unfortunately, some of the neuropsycho- logical tests developed for children and adolescents lack even proper (i.e., large and representative) normative data bases, and most test authors used normative data, if available, to derive means, standard deviations, confidence intervals etc. for the age ranges under scrutiny. The latter situation holds also for some of the commonly used intelli- gence tests for children and adolescents which refer to a neuropsychological theoretical background (like the K-ABC; Melchers & Preuss, 1992) or have provided some of their subtests for neuropsycho- logical assessment (like the WISC; Lezak, 1995).

Although these descriptive statistics obviously allow for correct individual predictions, they are dissatisfying from a theoretical perspective. This is because a chain of age group statistics is not a stat- istical model of (cognitive) development as there is no statistical function that links the different vari- ables across the age groups. Hence, there is a discrepancy between the (implicit) developmental theory underlying these tests conceptualizing development as a continuous quantitative process, and the statistics used for modeling the develop- mental effects and predicting individual scores.

Indeed, evidence from the developmental (neuro-) cognitive literature has shown for a number of fea- tures, that development can be conceived of as a continuous process in which the variable under scrutiny is a negatively accelerated function of age (Fischer, Biscaldi, & Gezeck, 1997; Fry & Hale, 2000; Klein, 2001).

If development is considered a continuous quan- titative process, age should be modeled statistically as a continuous predictor of the quantitative fea- tures of interest. But this is tantamount to requir- ing the application of regression analysis to model developmental effects. Indeed, (multiple) regression functions allow both for individual predictions — as illustrated in this article — and the derivation of statistical models of continuous developmental processes (Klein, 2001). Regression models have three further advantages. First, in the case of ad hoc sampling one or the other age group may be subject to minor sampling biases. Such biases would obviously afflict both analyses that use standard scores for individual age groups, and analyses that use multiple regression statistics.

However, this effect must be considerably dampened for regression functions, because here the estimates at each point of the age continuum are based on the entire sample. As a result, developmental regression functions must result in estimates whichare more stable and precise across samples than group means. Second, the application of regression functions yields one model for an age range, ins- tead of multiple means (but no quantitative model) for various age groups. Given this makes sense the- oretically (see discussion), the regression approach considerably reduces the required sample sizes in normative studies. Finally, the multiple regression model parameters may provide theoretically important insights into the features under scrutiny, which are not obtained by mere comparisons of age group means. This includes direct comparisons of the developmental trajectories of different (sets of) features. In line with this reasoning, Klein (2001) has shown the superposition of (generally strong) curvilinear and (generally weaker) linear developmental effects on different measures of sac- cade control in participants aged 7–28 years. These model parameters differed between measures that are sensitive to prefrontal dysfunctions and mea- sures that are not, thus confirming hypotheses about the protraction of prefrontal cortex develop- ment by way of showing evidence of developmen- tal fractionation.

Using the neuropsychological tests as assess- ment instruments in developmental clinical research, however, additionally and necessarily requires the estimation of confidence limits for individual scores. Here it can be shown that if the regression residues show no evidence of hetero- scedasticity the upper and lower confidence limits are symmetrical with respect to the estimated mean (see appendix).

In this paper we exemplify the suggested regres- sion analytic approach using data for the Wisconsin Card Sorting Test (WCST; Grant & Berg, 1948).

This test has been used both for basic research and for the assessment of cognitive functions in healthy subjects and patients. The WCST has become a popular neuropsychological test in studies with adult psychiatric (Green, Satz, Ganzell, & Vaclav, 1992) or neurological (e.g., Eslinger & Grattan, 1993) patients. A number of cognitive functions such as deductive reasoning, cognitive set shifting, and working memory has been associated with the WCST (Heaton, Grant, & Mathews, 1993). As most of these component functions have been linked with the neuropsychological construct of “executive functions” (EF), the WCST is now widely held as a test of executive functions (Denckla, 1996). These functions, in turn, have been linked to the prefrontal cortex (PFC; Denckla, 1996; Mentzel et al., 1998). Prefrontal regions are considered as higher-order (motor) association areas (Fuster, 1989) and thus operate as the “orchestrating” components in functional REGRESSION-BASED DEVELOPMENTAL MODELS 27 systems that also include more posterior cortical and subcortical brain structures (Alexander, DeLong, & Strick, 1986). Accordingly, functional imaging research has demonstrated metabolic increases in prefrontal and further cortical and subcortical areas during the execution of the WCST (Monchi, Petrides, Petre, Worsley, & Dagher, 2001).

The WCST as a measure of executive functions is potentially useful for developmental cognitive research and assessment. First, both executive functions (e.g., Klein, 2001) and prefrontal cortex (e.g., Jernigan, Trauner, Hesselink, & Tallal, 1991) seem to exhibit protracted development during childhood and adolescence. Second, as WCST and intelligence scores may load on different factors (Ardila, Galeano, & Rosselli, 1998), reflecting their low to moderate correlation (Ardila, Pineda, & Rosselli, 2000), WCST performance may in prin- ciple complement the assessment of cognitive capa- bilities provided by common intelligence tests.

Third, the WCST has a rather play-like nature and is thus suited for also testing younger children. To our knowledge, three studies have examined WCST performance in moderate to large samples of 5–12-year-old children of different cultures (Chelune & Baer, 1986: North-American, N = 105; Rosselli & Ardila, 1993: South-American, N = 233; Shu, Tien, Lung, & Chang, 2000: Asian, N = 219).

Although all studies reported substantial perform- ance improvements for the WCST parameters under scrutiny, age group means (including the age at which an adult level of performance was reached) or correlations between WCST scores and age varied between studies (Rosselli & Ardila, 1993; Shu et al., 2000). In some of these studies, WCST scores did not substantially correlate with teacher ratings of academic achievement (Rosselli & Ardila, 1993), only weakly with the father’s years of education (Shu et al., 2000), the family’s socioeconomic status (Rosselli & Ardila, 1993), and gender (Rosselli & Ardila, 1993; Shu et al., 2000).

Based on the outlined methodological consider- ations, this study aims at demonstrating the useful- ness and limitations of multiple regression models in cognitive (neuro-) developmental research.

Based on a non-representative sample of 345 sub- jects who accomplished a computerized version of the WCST we exemplify the statistical modeling.

This paper, hence, does not provide normative data that could be used in clinical trials for individ- ual predictions. In addition, we offer on our web- site software that allows for the prediction of an individual’s mean and confidence limits. Norma- tive data given, this software can be used byresearchers and practitioners that are interested in making individual predictions in developing or ageing populations.

METHODS Subjects Data from 345 participants were available for stat- istical analyses. Subjects were recruited via news- paper advertisements. Upon the initial phone contact we asked the adult participants or the par- ents of the younger participants about past or cur- rent (a) neurological or (b) psychiatric diseases or (c) viewing problems that could no be corrected. If any of these exclusion criteria was met, a partici- pant candidate was not invited to the lab session.

The participants’ ages ranged from 6 to 26 years (72–312 months), with a mean age of 12.9 ± 3.9 years (159 ± 45 months). 36% of the participants were male, all were German. Gender and age were unrelated, as were IQ and age. The mean T-score in Raven’s Standard Progressive Matrices (SPM; Heller, Kratzmeier, & Lengfelder, 1998) was 51.8 ± 10.9. WCST testing began at 3:15 p.m. on average, and this time was not correlated with the partici- pants’ ages (r = −.07).

Procedures All participants were tested individually with a com- puter-based version of the Wisconsin Card Sorting Test that was tailored following the instructions pro- vided by Grant and Berg (1948; Heaton, Chelune, Talley, Kay, & Curtiss, 1993). Our computerized WCST comprises four stimulus cards presented in the upper part of the screen, and 128 response cards in the lower part. The response cards show figures composed of crosses, circles, triangles, or stars, that differ in the number of elements (one, two, three, and four) and their color (blue, yellow, green, or red).

Response cards are presented one after the other to the participant who is instructed to match it to one of the stimulus cards. No explanation of the sorting principle is given, and the subject is only instructed that s/he will be given feedback (correct or incorrect) after each sort. Matching is initially according to color. After 10 consecutive correct matches the sort- ing principle is changed covertly. Form and number are the subsequent sorting dimensions. This series of sorting dimensions is repeated when the third cate- gory is achieved. Neuropsychological testing took about 10–15 minutes. The WCST was always pre- sented after the SPM. All participants were tested separately, and siblings were tested on the same day 28 KLEIN, FOERSTER, HARTNEGG immediately one after another. At the beginning, the laboratory was shown to the participants (and, in the case of younger children, the accompanying parent).

If a subject agreed to participate, adults or parents (for their children) gave their written consent and testing could begin. Parents could remain in the lab- oratory at the child’s request.

Data analysis The following 11 parameters were defined accord- ing to Grant and Berg (1948) and determined off line by a computer program: (a) number of trials administered (NTA), (b) total number correct (TNC); (c) total number of errors (absolute, TNE); (d) perseverative responses (absolute, PR); (e) per- severative errors (absolute, PE); (f) non-perseverative errors (absolute, NPE); (g) conceptual level responses (absolute, CL); (h) number of categories completed (NCC); (i) trials to complete first category (TCFC); (j) failures to maintain set (FMS). In addition, the “efficiency ratio” was defined as the ratio of NCC and NTA according to Voeller, Edge, Morris, Rao, and Heilman (1993, cf. Riccio et al., 1994).

Following the rationale and procedure described in greater detail in Klein (2001), linear and curvi- linear effects of age on each of these criteria were determined by using age (in months) and its inverse (age −1) as predictors in multiple regression proce- dures. These predictors were found in the follow- ing manner. First, bivariate correlations between the dependent variables on the one hand, and transformations of the age variable were accom- plished. These transformations ranged from age −3 to age −1 in steps of .01. In many cases these corre- lations were greatest for age −1. In most other cases, the correlation between the dependent variable and age −1 was very close (within .02) to the maximum correlation obtained for the set of transformed var- iables. Second, the residues of the dependent varia- ble after partialling out age −1 were examined in order to check for remaining (residual) age effects in the dependent variables’ distributions. As in some cases a linear trend was still discernible, the variables “age −1” and “age” were partialled out, and the residues of the dependent variables were re-examined in order to verify the absence of any systematic relation with chronological age.

The corresponding multiple regressions using least-square estimates of the criteria have the gen- eral form: criterion = a + b 1*age + b 2*age −1 + error, and provide a prediction of the variable under consideration by the two age variables. Pos- sible gender effects on the WCST parameters were assessed with a multivariate analysis of variance(MANOVA). The main effect of factor GENDER as well as its interaction with the WCST scores turned out to be non-significant (Fs ≤ 2). Accord- ingly, male and female participants scored compa- rably in all WCST measures. Hence, our results will be presented for the pooled male and female sub-samples. Finally, we present software (avail- able on our homepage) for the prediction of indi- vidual raw and standardized scores in WCST parameters (described in detail in the appendix).

RESULTS The results of our study are shown in Figures 1–3 and Table 1. Table 1 documents the nonstandard- ized multiple regression parameters (columns “regression equations”), the F-values for the linear (“age”) and curvi-linear (“age −1”) effects of age, and summary statistics like the proportion of vari- ance explained (R 2) and the F-values for the entire model for each of the WCST parameters. The non- standardized regressions weights (b 1, b 2) together with the intercept (a) allow the prediction of individual scores according to equation 1 (see appendix), using age in months and its inverse as predictors.

The corresponding regression functions are graphically shown in Figures 1–3 as dashed lines.

A common feature of these functions is that the WCST scores are, in different degrees, negatively accelerated functions of age. The predominance of curvi-linear over linear age effects is statistically reflected and quantified by the F-values shown in Table 1, which are considerably larger for age −1 than age (except for FMS, which is unrelated to age). The proportions of variance accounted for by the two age variables range between 3% (TNC) and 25–26% (TNE, NCC, E-R). Moderate age effects were found for NTA (17%), PR (18%), PE (20%), and NPE (21%). The regression residues (after partialling out the two age variables) were negligibly correlated with chronological age for most of the WCST variables.

In Figures 1 to 3, the dotted lines close to the estimated mean reflect the boundaries of the 95% confidence limits for the expected value (i.e., the mean), the thin “outer” lines those for the pre- dicted individual scores. As only the latter interval is based on an error variance estimate that includes the mean squared error (see appendix), these boundaries must be considerably broader than those for the expected value. The confidence limits for the predictions of individual scores can be computed for new individual’s scores with the pro- gram CONFIREG (see Appendix). If truly normative REGRESSION-BASED DEVELOPMENTAL MODELS 29 data were available, CONFIREG could be used to assess whether an individual’s score is atypical for the normal population.

The difference between developmental trajecto- ries that are based on regression functions (e.g., Klein, 2001) versus age group means (e.g., Fischer et al., 1997) is illustrated in Figure 3 in comparisonwith Figures 1–2. Figure 3 reveals that at the transition between certain ages the developmental trajectories of some WCST parameters reverse their direction (e.g., from 9 to 10, 13 to 14, and 15 to 16 years). This likely effect of a sampling error is obviously dampened and smoothed in the regres- sion-based trajectories.

Figure 1.Multiple regression functions for criterion variabels TNE, PR, PE, and NPE as predicted by age in months and its inverse.

The dashed line represents the predicted mean, the dotted lines its 95% confidence intervals. The outer solid lines represent the 95% confidence interval for individual scores. 30 KLEIN, FOERSTER, HARTNEGG DISCUSSION In this article we portrayed a multiple regression approach to the modelling of developmental effects using the different parameters of the WCST. The sub- sequent discussion will focus on the following points:

(1) conditions of applicability of the model; (2) cogni- tive-developmental implications; (3) comparison withconventional standard score approaches; (4) applica- bility in applied research and practising.

Ad (1) Statistical models are generally only applicable if their assumptions are met. As the statistical Figure 2.Multiple regression functions for criterion variabels NCC, EFFRATIO, and PCA factors 1 and 2, as predicted by age in months and its inverse. The dashed line represents the predicted mean, the dotted lines its 95% confidence intervals. The outer solid lines represent the 95% confidence interval for individual scores. REGRESSION-BASED DEVELOPMENTAL MODELS 31 assumptions are concerned, a (multiple) regression approach to the modeling of cross-sectional devel- opmental effects makes sense only if both age as the predictor and the criterion variable can be con- sidered as continuous quantitative variables. In this situation, multiple regression models using chronological age and its inverse can model and, hence, quantify superimposed linear and curvilin- ear cross-sectional developmental effects on diff- erent (and, possibly, to-be-compared) criterion variables. From a statistical perspective, these models are appropriate as cross-sectional develop- mental models to the extent that the regression residues (statistical errors) are unrelated to age and the predicted scores (see Klein, 2001). In this situation, the observed response comprises of the predicted response plus error. These assumptions must be verified for each individual data set andvariable, and may obviously not hold (and hence yield awkward functions) for variables with a coarse grading and/or ceiling effects. Concerning the WCST, all number and percentage measures that are based on the entire set of trials are ‘fine- grained’ enough to be treated like continuous vari- ables. Conversely, measures like NCC do not allow for this fine-grained differentiation, and multiple regression models provide at best a rough approxi- mation (see Figure 2).

Another, theoretical, assumption is relevant here as well. This assumption is indirectly related to the application of regression models, but directly related to test construction. This is the assumption of whether the developmental process under scru- tiny is a quantitative or qualitative one. Most cog- nitive, neuropsychological and intelligence tests allow only for the quantitative differentiation Figure 3.Group means in different WCST parameters for the age groups 6–18 years. TABLE 1 Model equations for the WCST task parameters Regression analyses ab 1 b2 Fa pF a pR 2 R2 adj Fp NTA80.80−.0004 3837 <1 n.s. 9.04 .003 .18 .17 37.16 .0001 TNE−38.25 .1330 6954 5.69 .02 35.06 .0001 .25 .25 58.48 .0001 PR−19.18 .0661 3750 2.98 n.s. 21.68 .0001 .19 .18 39.25 .0001 PE−16.16 .0578 3267 3.37 n.s. 24.27 .0001 .20 .20 43.77 .0001 NPE−22.09 .0752 3686 5.39 .02 29.21 .0001 .21 .20 45.39 .0001 NCC10.85−.0119−549 8.71 .003 42.17 .0001 .26 .26 61.27 .0001 E-R.1074−.0008−6.239 1.90 n.s. 26.00 .0001 .26 .26 60.20 .0001 TCFC−17.76 .0824 2744 7.27 .007 18.21 .0001 .09 .08 16.04 .0001 CLR132.9−.1755−5284 13.34 .001 27.27 .0001 .10 .10 19.94 .0001 TNC119.1−.1335−3117 10.26 .002 12.62 .0004 .04 .03 6.38 .002 FMS2.83−.0084−21.43 3.23 n.s. 0.05 n.s. .06 .05 10.42 .0001 Note. CLR = conceputal level responses; E-R = efficiency ratio (NCC/NTA); FMS = failures to maintain set; NPE = non-perseverative errors; NCC = number of categories completed; NTA = number of trials administered; PE = perseverative errors; PR = perseverative responses; TNE = total number of errors; a = F-values based on Type III sum of squares; 32 KLEIN, FOERSTER, HARTNEGG between individuals and would thus be unable to unveil existing qualitative differences. Conversely, a typical Piaget task is designed to test for qualita- tive differences in problem solving between differ- ent stages of cognitive development (and thus “constructs” these developmental stages). How- ever, Pascual-Leone (1970) has nicely shown with his concept of the M space, that the qualitatively different “stages” of cognitive development as sug- gested by Piaget can be theoretically reduced to the quantitative development of M space capacity.

Similarly, intelligence researchers have begun to consider that individual differences in the ability to solve qualitatively different intelligence tasks (which also varies as a function of age) can be a consequence of individual differences in working memory capacity (Kyllonen & Christal, 1990; Schweizer, 1995). It may be possible, hence, to reduce qualitative to quantitative developmental differences in one or more underlying cognitive processes. Conversely, however, quantitative dif- ferences may be the result of using qualitatively different “processing” strategies. Given the quali- tative uniformity of the trials and the purely quan- titative scoring of the WCST this means, that the cognitive developmental processes underlying test performance (such as working memory and cogni- tive set shifting) are considered to be quantitative rather than qualitative in nature.

A limitation of the interpretability of all statist- ical developmental models refers to the fact that age is a global organism variable that may be con- founded with other developmentally powerful fac- tors such as cohort effects. This is probably less of an issue in research with children and adolescents.

Life span developmental research, however, may require the integration of additional information such as cohort effects or other confounds into the model. Unless the developmental trajectories for the confounds are identical (or, perfectly corre- lated) with those of the criteria, the different source of developmental differences can be disentangled.

Ad (2) That the relation between age and most WCST variables was curvilinear reveals that WCST profi- ciency increases as a negatively accelerated func- tion of age. This observation fits into what is known for other features in this age range. For instance, Fischer et al. (1997) described and Klein (2001) quantified the developmental functions for various parameters of saccade control. Similar analyses were accomplished by Fry and Hale (2000) for different cognitive measures such as IQ, work-ing memory, and processing speed, and Klein (2001) for the raw score of Raven’s Standard Pro- gressive Matrices. The diversity of the cognitive functions (WCST, pro- and anti-saccade measures, processing speed, IQ) that seem to follow this pat- tern suggests that a conceptualization of develop- ment as a quantitative-continuous process may be applicable to a broad range of tests of cognitive functions.

Ad (3) An advantage of multiple regression estimates of developmental effects, when compared to estimates based on standardized scores, was illustrated in Figure 3 on the one side and Figures 1–2 on the other. Tracing development via group means results in trajectories that may contain implausible developmental effects. It is, for instance, unlikely that the developmental trajectories of some WCST variables reverse their direction (see Figure 3).

Similar results had been presented by Chelune et al. (1986; failures to maintain set, from 6 to 7 and 10 to 11 years of age), Rosselli and Ardila (1993; e.g., correct responses from 9–10 to 11–12 years of age), and Shu et al. (2000; e.g. perseverative responses and nonperseverative errors from 10 to 11 years of age) for WCST scores, and Fischer et al. (1997) for saccade parameters. Such odd tra- jectories probably do not reflect developmental dif- ferences between the age groups but sampling biases due to non-representative (stratified or ran- dom) sampling. Massive sampling bias at one or the other age can obviously distort the entire regression function to some extent. In this situ- ation, re-sampling or discarding the biased age bin within the traditional approach based on age group means, would obviously be preferable over relying on a seriously distorted regression model.

This should, however, occur in a considerably dampened fashion, because the estimate at each point along the age continuum is based on the entire sample. Nevertheless, both a representative sampling strategy and appropriate statistical tech- niques are required to obtain truly normative data.

Ad (4) With respect to the applicability of the multiple regression model to applied research and practis- ing, it is important to realise that we studied here developmental differences rather than develop- mental changes. Accordingly, the model can be used for status diagnostic / assessment, but not process diagnostic / assessment (Pawlik, 1982). REGRESSION-BASED DEVELOPMENTAL MODELS 33 Status diagnostic, however, is important in many research and practical applications where the pre- diction of individual scores on the basis of norma- tive data is required. Our program CONFIREG which is described in detail in the appendix can, in principle, exploit the information of up to 7 varia- bles to predict an individual’s score in dependent measures such as the WCST scores. Although, for the present purpose, we used only the two age vari- ables (age and age −1) that were necessary to model developmental effects, additional information such as the IQ or other neuropsychological test results that may routinely be available in research studies or clinical trials as well as information relevant to possible cohort effects may provide incremental validity to the prediction and thus enhance its pre- cision. Applied to the data of a truly representative normative sample, this procedure can be used to determine whether an individual’s score lies out- side the confidence intervals of the distribution of normative scores. This may, for instance, be useful in basic research or practical diagnostics of patients with attention-deficit hyperactivity disor- dered (ADHD). Here, it has become clear that only a few cognitive tests or measures can separate patients from controls in group statistics (Klein & von Stralendorff, 2002), and that this is presuma- bly the case because only a portion of ADHD patients show deficient performance at all (Nigg, 2004). The application of our regression-based software CONFIREG could thus help identifying the individual ADHD patients who show the defi- cit and thus determining their proportion in a given sample.

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(a) StErr(predicted value) = √(x iCOVBx i’) and (b) StErr(individual value) = √(MSE+ x iCOVBx i’).

The confidence interval with a level of significance α and the t-statistic t α/2,N-k-1 is symmetric with respect to the predicted value: (b 0 + b 1x1i + ... + b kxki − t α/2,N-k-1 StErr i, b 0 + b 1x1i + ... + b kxki + t α/2,N-k-1 StErr i). Note that the width of a confid- ence interval also depends on the sample size (as reflected by t α/2,N-k-1 StErr i).

CONFIREG and an SAS Macro for statistical analyses is available at our home page at: http:// staff.psychology.bangor.ac.uk/Members/psse0b/ confireg.zip/. yi xx xi1iki’ ,..., ;= () yi REGRESSION-BASED DEVELOPMENTAL MODELS 35 APPLICATION OF “CONFIREG” 1. Start the program under Windows.

(When starting the first time no regression information file is available).

2. Type into the empty sheet the global informa- tion (sample size, the α-level of the confidence interval, mean squared error) and the informa- tion that is specific to the predictors (name, regression weights, generalized inverse covari- ance matrix (lower triangular only)).

3. Type in the predictor values and (for document- ing only) name and value of the dependent variable of the single case for whom you wish to estimate the confidence interval.

4. Click on button APPLY in order to estimate and display the confidence interval and the decision whether the individual’s values lie whithin it.

Also, z-, T-, and stanine scores for the individual are displayed.

5. Chose among the following options: (a) CLEAR:

clear the sheet/formular and restart with 2; (b) STORE: store the information provided in the sheet/formular into a file named *.prf, which can be used for later applications (program termi- nates); (c) enter new predictor scores or alter the α-level, then click APPLY; (d) CANCEL:

program terminates without storing in *.prf.