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INS Investigators Workshop: Methods for Single-Case Studies in Neuropsychology John R. Crawford School of Psychology College of Life Sciences and Medicine Kings College University of Aberdeen and School of Psychology Flinders University of South Australia Cognitive Neuroscience Research Group

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Collaborators: Prof Paul H Garthwaite The Open University also Prof David C HowellUniversity of Vermont Prof Keith R Laws University of Hertfordshire Prof Annalena VenneriUniversity of Hull Dr Colin D GrayUniversity of Aberdeen Prof Addelchi AzzaliniUniversity of Padova (Dr Sytse Knypstra University of Groningen) Cognitive Neuroscience Research Group

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The Importance of Dissociations Dissociation is the key word of neuropsychology. (Rossetti & Revonsuo, 2000, p. 2)

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The Case for Single Cases Studies in groups of patients which aim at elucidating the neurological and functional architecture of mental processes are useless and harmful, since they provide misleading results. The only appropriate method is to study individual patients (Vallar, 2000, p. 334)

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The need for methodological rigour in single-case studies If advances in theory are to be sustainable they must be based on unimpeachable methodological foundations. (Caramazza & McCloskey, 1988, p.619).

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Evaluating Tests for Deficits in Single-Case Studies Massive revival of interest in single-case studies in neuropsychology and neurology The arguments for single-case studies over group studies are viewed by many as compelling However, it is clear that they present difficulties when it comes to statistical analysis This aspect of single-case studies has been relatively neglected

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Single-case research: The three basic approaches to drawing inferences concerning a patients performance Patient is administered fully standardized neuropsychological tests and performance is compared to large sample normative data At other extreme, patients performance is not referenced to normative data or control performance; i.e., analysis is limited to intra-individual comparisons Patient is compared to a (modestly sized) matched control sample

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Limitations of the fully standardized approach Can only be used in fairly circumscribed situations because: New constructs are constantly emerging in neuropsychology In contrast, collection of norms is a long and arduous process Even where norms are available, may not be applicable to patient

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Dangers of the intra-individual approach Results can be very misleading as performance is not referenced to normal performance Category specificity literature provides a good example of dangers Reports of apparently striking dissociations between naming of living versus non-living things and even within these categories (Broccolis area?) In vast majority of these studies inferences are drawn on the basis of chi-square tests comparing a patients living and non-living naming

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Laws, Gale, Leeson & Crawfords (2005) study on living / non-living naming Laws et al. examined cases of AD who had or had not been classified as exhibiting a dissociation using intra- individual approach (chi-square test) It was found that the performance of some patients with dissociations was not unusual when referenced to control performance Moreover, patients who had not been identified as exhibiting dissociations were identified as such when performance was referenced to controls In one case a patient classified as exhibiting a dissociation in favour of non-living things was found to exhibit a dissociation in the opposite direction

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To recap… Three basic methods of forming inferences Referring patients performance to large scale normative data will often not be possible Intra-individual approach is very problematic Therefore, third approach is commonly employed; i.e. patients performance is referred to that of a matched control sample We will now turn to how such data are analysed starting with the basic question of how we detect a deficit

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Testing for a deficit in single-case studies: the standard method Patients performance is converted to a standard score based on mean and SD of control sample and referred to table of areas under the normal curve The statistics of the control sample are treated as population parameters When sample size is large this is not too much of a problem as the statistics provide sufficiently accurate estimates of the parameters However, large sample sizes are rare in the single-case literature

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Testing for a deficit in single-case studies using Crawford & Howells (1998) proposed method: Uses formula set out by Sokal and Rohlf (1995) Modified t-test: tests hypothesis that patient did not come from the control population (under null hypothesis patient is an observation from this population) Control sample statistics are treated as statistics Crawford & Garthwaite (2002) also developed method of setting confidence limits on abnormality of score (using non-central t distributions)

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Comparison of two methods for testing for a deficit: Type I errors (Crawford & Garthwaite, Neuropsychology,2005) Monte Carlo simulation study 5 control sample sizes (N) were examined: 5, 10, 20, 50 and 100 For each value of N one million observations of N +1 were drawn from a normal distribution The first N observations were taken as the control sample data and the N+1th observation as the control case The alternative tests for deficits were applied to these data and the percentage of Type I errors compared to the specified rate of 5%

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Perform statistical tests comparing control case and Control sample and record if significant, i.e. record if Type I error Step (2) Get machine to repeat this one million times Step (3) Meanwhile go and get yourself a… Monte Carlo simulation: Sampling from the control population

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Comparison of two methods for testing for a deficit: Type I errors (Crawford & Garthwaite, 2005)

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Conclusion Type I error rate is under control when Crawford & Howells test is used to test for a deficit at all values of N Type I error rate is markedly inflated when z is used to detect a deficit with small control samples z will also exaggerate the abnormality of the patients score

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Departures from normality Both z and modified t-test assume control data are drawn from normal distribution However, in single-case studies there is often evidence of negative skew in scores of the control samples (ie control mean=50, SD=10, but max score =55) We have run Monte Carlo simulations to examine control of Type I error rate when control data are non- normal Same method as in previous study except that the N+1 observations were sampled from distributions that were skew and /or leptokurtic

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Perform statistical tests and record if significant, i.e. record if Type I error Sampling from negatively skewed and / or leptokurtic distributions Step (2) Repeat this one million times Step (3) Meanwhile go and get yourself a…

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Results of a Monte Carlo study: Robustness in face of moderate skew

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Effects of departures from normality on methods for testing for a deficit: Conclusions Negative skew and / or leptokurtosis can inflate the Type I error rate for both z and Crawford & Howells test but the effects are not severe (Crawford, Garthwaite, Azzalini, Howell, & Laws, Neuropsychologia, in press) The Type I error rate is markedly inflated for z for small N but this is due more to treating statistics as parameters

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The Internet Most of calculations involved with these methods are simple (exception being CLs) However, still tedious and error prone Therefore, we have written computer programs that implement these methods Freely available on the web Calculations can be performed literally in seconds

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In this example, the patients score is significantly below controls and so we conclude he/she has a deficit. Also, it is estimated that only 1.13% of the control population would exhibit this poor a score; the 95% CI on this estimate of abnormality is from 0.05% to 4.68%

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Modifed T-Test Versus Modified ANOVA Mitchell and colleagues (Mycroft et al, 200; Mitchell et al, 2004) have criticised the foregoing method They argue that (a) a notional patient population will have markedly increased variance relative to the control population, and (b) our method will therefore produce inflated Type I errors Mitchell and colleagues propose an ANOVA that employs more conservative critical values to overcome this perceived problem

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Modifed T-Test Versus Modified ANOVA We believe there are two major problems with Mitchell et als position: The argument over Type I errors is untenable (Crawford et al, Cognitive Neuropsychology, 2004) Statistical power to detect a deficit is very low for Mitchell et als method (Crawford & Garthwaite, Cognitive Neuropsychology, in press)

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A graphic illustrating Mitchell et als. scenario: A notional patient population (gray line) has same mean as controls (dark line) but has greater variability This scenario is not realistic: If the means do not differ but patients are more variable, then scores below control mean must be exactly balanced by scores above control mean

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If the patient mean is lower than control mean (even marginally) then issue of Type I error does NOT arise: a deficit is present and the question is whether it can be detected (i.e., it is a power issue)

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Power to detect a (2 SD ) deficit: comparison of three methods (Crawford & Garthwaite, Cognitive Neuropsychology, in press)

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PART 2 Dissociations in Neuropsychology and Statistical Tests on Differences

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DISSOCIATIONS In neuropsychology, deficits are of limited theoretical interest unless they are accompanied by preserved or less impaired performance on other tasks; i.e. the aim of many single-case studies is to demonstrate dissociations of function

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Conventional Definition of a Classical Dissociation If patient X is impaired on task 1 but performs normally on task 2, then we may claim to have a dissociation between tasks (Ellis and Young, 1996, p. 5)

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A Classical Dissociation (based on Shallice, 1988) Task X Task Y Performance

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The Importance of Dissociations Dissociations play an increasingly crucial role in the methodology of cognitive neuropsychology… they have provided critical support for several influential, almost paradigmatic, models in the field. (Dunn & Kirsner, 2003, p. 2)

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Criteria for Dissociations: Three Problems What constitutes a deficit and being within normal limits is very poorly specified One half of the typical definition essentially involves an attempt to prove the null hypothesis A patients score on the impaired task could lie just below the critical value for defining impairment and the performance on the other test lie just above it (see Caramazza & Shelton, 1998 for similar point)

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Problems with Conventional Criteria for a Classical Dissociation Task X Task Y Performance

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Crawford, Garthwaite & Gray ( 2003):

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Potential Solutions to the Three Problems Crawford et al. (2003) provided fully explicit criteria for a deficit (using Crawford & Howells test) They also introduced a requirement that the patients performance on Task X should be significantly poorer than performance on Task Y This criterion deals with the problem of trivial differences It also provides us with a positive test for a dissociation (thereby lessening reliance on what boils down to an attempt to prove the null hypothesis of no deficit or impairment on Task Y)

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Crawford et als. criteria for a classical dissociation Task X Task Y Performance X significantly below controls on Crawford & Howells test (one- tailed) Y not significantly different from controls on Crawford & Howells test (one-tailed) X significantly different from Y

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How should we test for significant difference between a patients score on Tasks X and Y ? In most single-case studies the two tasks of interest will have different means and SDs For example a patients performance on a ToM task with (mean=35, SD=12) is to be compared with performance on an executive task (mean=22 SD=6) In order to meaningfully compare performance it is necessary to standardize scores on the two tasks Testing for a difference between a patients performance on Tasks X and Y

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A long established method is that of Payne & Jones (1957): The Payne and Jones method

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Testing for a difference between a patients performance on Tasks X and Y: Crawford, Howell & Garthwaite (1998) method

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Revised Standardized Difference Test (Crawford & Garthwaite, 2005b; Garthwaite & Crawford, 2004): Looks nasty but is essentially of familiar form…

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Monte Carlo Evaluation of tests for differences between tasks 5 control sample sizes (N) were examined: 5, 10, 20, 50 and 100 For each value of N and for each of 4 values of r (the correlation between tasks), one million pairs of observations of N +1 were drawn from a bivariate normal distribution The first N pairs were taken as the control sample data and the N+1th pair was as the control case The alternative tests for differences were applied to these data and the percentage of Type I errors compared to the specified rate of 5%

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Perform statistical tests comparing control case with control sample and record if significant, i.e. record if Type I error Simulation study of Type I errors for tests on differences X, Y Meanwhile, have a noodle about on the…

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Monte Carlo simulation: Type I error rate for Revised Standardized Difference Test ( r xy =0.5 in this example)

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Conclusions Payne and Jones method very poor control with modest Ns Although Crawford et als. (1998) test is a marked improvement it is clear that it does not follow a t-distribution and yields inflated Type I error rates In contrast, RSDT controls the Type I error rate

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Evaluating criteria for classical dissociations To recap: we have considered two sets of criteria for detecting dissociations – the conventional criteria and Crawford & Garthwaites (2005b) criteria We now have suitable test for Crawford & Garthwaites third criterion (i.e. it requires a significant difference between a patients scores on X and Y) We (Crawford & Garthwaite, 2005a, Neuropsychology) have examined performance of these two sets of criteria Same approach as used for evaluating foregoing tests; i.e. sample from bivariate distributions but apply the sets of criteria rather than individual tests for components of these criteria

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Type I error rate for Crawford & Garthwaites (2003; 2005b) criteria and conventional criteria for a classical dissociation (in this example r xy = 0.5

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Evaluating criteria for a classical dissociation: Conclusions Conventional criteria for a classical dissociation will misclassify a worryingly high percentage of healthy controls as exhibiting a classical dissociation regardless of the size of the control sample (rate was as high as 18.6% in one of the scenarios) In contrast, Crawford & Garthwaites (2003; 2005b) criteria are conservative; i.e. very low percentage of controls misclassified Results underline importance of testing the difference between patients X and Y scores Crawford & Garthwaites criteria were relatively robust in face of skewed control data

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In this example, the patient is classified as exhibiting a dissociation by Crawford & Garthwaites (2005b) criteria. (Crucially) the RSDT shows that there is a significant difference between the patients scores on the two tasks. Task X is significantly below controls but Task Y is not: therefore it is a classical dissociation.

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Further evaluation of criteria for a classical dissociation Up to this point Type I errors have been defined as incorrectly identifying a control as exhibiting a dissociation However, there is another form of Type I error that should be considered Alternative definition of Type I error: misclassifying a patient who has equivalent deficits on X and Y as exhibiting a classical dissociation

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Perform statistical tests and record if criteria are met, i.e. record if Type I error Lesion study: Type I errors now defined as misclassifying a patient X, Y Lesion the control case, 2 SDs below premorbid Scores on X and Y X-2, Y-2

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Type I error rate for the competing criteria: Type I errors defined as identifying a patient as exhibiting a classical dissociation

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Evaluating criteria for a classical dissociation: Further conclusions Performance of conventional criteria for a classical dissociation is extremely poor: very high percentages of patients will be incorrectly classified as exhibiting a classical dissociation (close to 50% in some scenarios) In contrast, Crawford & Garthwaites (2005) criteria are much more conservative; i.e. percentage is below 5% in most scenarios These results further underline importance of testing the difference between patients X and Y scores

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Power to detect a classical dissociation Up to this point concern has been with Type I errors, i.e. false positives However, we should also be concerned with the statistical power of criteria for dissociations; i.e. the ability of these criteria to avoid false negatives This issue has not previously been examined empirically It does not make sense to examine power unless the Type I error rate is under reasonable control Therefore, power only examined for Crawford & Garthwaites (2005b) criteria

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Why power to detect a dissociation will be low Crawford and colleagues have argued that power is almost inevitably low-to-moderate in single-case studies An individual patient rather than a sample of patients is compared to a control sample The control sample itself is usually modest in size The existence of substantial individual differences in premorbid competencies is a further factor…

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Why power to detect a dissociation will be low Task X Task Y Mean +1 SD -1 SD

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Perform statistical tests and record if correctly identified as a dissociation Lesion study: Power to detect a dissociation X, Y Lesion the control case, 2 SDs below premorbid Scores on X ONLY X-2, Y

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Simulation results: Power to detect a classical dissociation (for control sample N of 20)

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Crawford & Garthwaite (2005a, Neuropsychology) Power to detect a dissociation: Conclusions Results confirm that, if the Type I error rate is to be controlled, power is low in single-case studies Not surprisingly, power is higher with large control samples; therefore, control sample Ns should be larger than is typical currently This is not unreasonable: if a researcher believes single-case studies are more useful than group studies then she/he should be willing to expend the effort More encouragingly, power is higher when tasks are moderately to highly correlated

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Single-Case Methods: Overall Summary Massive revival of interest in single-case studies in neuropsychology and neurology The arguments for single-case studies over group studies are viewed by many as compelling However, it is clear that they present difficulties when it comes to statistical analysis The methods and criteria commonly used in single-case studies are problematic However, many of the problems can be overcome using the methods outlined These latter methods are rigorous but easy to apply using the accompanying computer programs

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Finally… Regression equations can play a useful role in single- case research and clinical practice For example, attempting to detect a deficit by comparing an obtained score with a predicted score based on demographic variables or a measure of premorbid ability In neuropsychology, inferences concerning discrepancies between predicted scores and obtained scores are typically made using the standard error of estimate This method produces inflated Type I error rates (Crawford & Garthwaite, Neuropsychology, in press)

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Finally… Building on earlier work by Crawford & Howell (1998b) we (Crawford & Garthwaite, in press) have recently proposed and evaluated an alternative method that controls Type I error rate The method also produces confidence limits on the abnormality of the discrepancy Methods implemented in accompanying computer program Poster on this topic later today

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THE END Web page summarizing our work on single- case methods: Web page containing the computer programs referred to in this presentation: A reference list for the methods is provided in your handout following the copy of this slide end

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References Crawford, J. R. (2004). Psychometric foundations of neuropsychological assessment. In L. H. Goldstein & J. E. McNeil (Eds.), Clinical neuropsychology: A practical guide to assessment and management for clinicians (pp ). Chichester: Wiley. Crawford, J. R., & Garthwaite, P. H. (2002). Investigation of the single case in neuropsychology: Confidence limits on the abnormality of test scores and test score differences. Neuropsychologia, 40, Crawford, J. R., & Garthwaite, P. H. (2004). Statistical methods for single- case research: Comparing the slope of a patient's regression line with those of a control sample. Cortex, 40, Crawford, J. R., & Garthwaite, P. H. (2005a). Evaluation of criteria for classical dissociations in single-case studies by Monte Carlo simulation. Neuropsychology, 19, Crawford, J. R., & Garthwaite, P. H. (2005b). Testing for suspected impairments and dissociations in single-case studies in neuropsychology: Evaluation of alternatives using Monte Carlo simulations and revised tests for dissociations. Neuropsychology, 19,

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References contd Crawford, J. R., & Garthwaite, P. H. (in press-a). Comparing an individual's predicted test score from a regression equation with an obtained score: a significance test and point estimate of abnormality with accompanying confidence limits. Neuropsychology. Crawford, J. R., & Garthwaite, P. H. (in press-b). Methods of testing for a deficit in single case studies: Evaluation of statistical power by Monte Carlo simulation. Cognitive Neuropsychology. Crawford, J. R., Garthwaite, P. H., Azzalini, A., Howell, D. C., & Laws, K. R. (in press). Testing for a deficit in single case studies: Effects of departures from normality. Neuropsychologia. Crawford, J. R., Garthwaite, P. H., & Gray, C. D. (2003). Wanted: Fully operational definitions of dissociations in single-case studies. Cortex, 39, Crawford, J. R., Garthwaite, P. H., Howell, D. C., & Gray, C. D. (2004). Inferential methods for comparing a single case with a control sample: Modified t- tests versus Mycroft et al's. (2002) modified ANOVA. Cognitive Neuropsychology, 21,

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References contd Crawford, J. R., Garthwaite, P. H., Howell, D. C., & Venneri, A. (2003). Intra-individual measures of association in neuropsychology: Inferential methods for comparing a single case with a control or normative sample. Journal of the International Neuropsychological Society, 9, Crawford, J. R., & Howell, D. C. (1998a). Comparing an individuals test score against norms derived from small samples. The Clinical Neuropsychologist, 12, Crawford, J. R., & Howell, D. C. (1998b). Regression equations in clinical neuropsychology: An evaluation of statistical methods for comparing predicted and obtained scores. Journal of Clinical and Experimental Neuropsychology, 20, Crawford, J. R., Howell, D. C., & Garthwaite, P. H. (1998). Payne and Jones revisited: Estimating the abnormality of test score differences using a modified paired samples t-test. Journal of Clinical and Experimental Neuropsychology, 20, Garthwaite, P. H., & Crawford, J. R. (2004). The distribution of the difference between two t-variates. Biometrika, 91, Laws, K. R., Gale, T. M., Leeson, V. C., & Crawford, J. R. (2005). When is category specific in Alzheimer's disease? Cortex, 44,

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