EXAMPLES OF NEUROPSYCHOLOGICAL MEASURES WHERE PERFORMANCE IS EXPRESSED AS AN IIMA

The examination of memory for temporal order provides an obvious example of the use of an IIMA. Typically, performance on a temporal order memory task is quantified by computing the intra-individual (rank order) correlation between the actual order in which items (e.g., words or pictures) were presented, and the patient's memory for this order (Shimamura et al., 1990). The correlation computed from the patient's data can be compared with the average correlation obtained from a control or normative sample (a low correlation would suggest impairment); see Mayes et al. (2001) for a recent example.

As another example, in the area of motor control, there is a high intra-individual correlation between maximum grip aperture when reaching for objects and the size of the objects (Carey, 1996). An investigator may wish to determine whether this relationship breaks down in an individual with neurological disease. This could be examined by comparing the slopes of the intra-individual regression lines (relating object size and aperture size) obtained from healthy participants with the slope of the regression line obtained from the patient. Relatedly, there is a positive relationship between the peak velocity of manual reaching and the distance from a target; the further away the target, the faster the peak velocity (Jeannerod, 1984). Again, an investigator may wish to examine whether this relationship breaks down in cases with neurological disease; see Carey et al. (1998) for an example.

As a fourth example, in the area of object recognition, it has been found that there is a robust intra-individual relationship between the latency with which an object is named and the degree to which it is rotated away from its prototypical orientation (Turnbull et al., 1997). Failure to find such an association in a patient would suggest that mental rotation was not being used to achieve object identification; see Turnbull et al. (2002) for a recent single-case study. Distance estimation tasks provide yet another example from the area of visual perception; here the issue would be whether a patient exhibits an attenuation of the expected relationship between visually estimated distance and actual distance (Carey et al., 1998).

IIMAs have also been employed in the area of auditory perception. For example, Steinke et al. (2001) used measures of tonality to assess melody recognition in a patient with right hemisphere damage. Performance on each of four tests of tonality (Steinke et al., 1997) was quantified by computing the Spearman rank order correlation between a participant's ratings of tonality and music-theoretic predicted levels of tonality. Estimation of weight (Jones, 1986) provides another example where IIMAs can be used (to compare estimated against actual weight); estimation of weight has been found to be impaired in a number of neurological conditions (see, e.g., Heindel et al., 1991; Lafargue & Sirigu, 2002).

Finally, in the investigation of time estimation, accuracy can be assessed by examining the strength of the intra-individual relationship between the length of the actual time intervals and the estimated time intervals. There is evidence that, particularly when the time intervals exceed 20 s (Richards, 1973; Sherwin & Effron, 1980), amnesic patients exhibit deficits in time estimation as indicated by a weakening of the association between actual and estimated time (Nichelli et al., 1993; Venneri et al., 1998). Indeed, in severe cases, the direction of the relationship can be reversed (e.g., the correlation, or slope of the regression line, has a negative sign), thereby indicating that shorter intervals are estimated as longer.

In the foregoing examples the breakdown or weakening of an intra-individual association found in healthy participants would suggest impairment. However, in some circumstances an investigator may seek evidence of impairment through demonstrating that there is a strengthening of an intra-individual relationship in a patient. For example, it might be found that a patient's performance on a primary task is differentially sensitive to the degree of working memory load (e.g., number of digits retained) imposed by a secondary task. Were this the case, then one would expect to find a higher (negative) correlation between working memory load and primary task performance in the patient than in the healthy control participants.

TESTING EQUALITY OF PEARSON CORRELATION COEFFICIENTS

When performance on the task of interest is expressed as a Pearson (i.e., parametric) correlation coefficient, we let ρ_i denote the correlation coefficient for the ith person and let r_i denote its sample estimate, based on k trials or items. The sampling distribution of correlation coefficients is not normally distributed and so we apply Fisher's transformation to the coefficients for the individuals in the normative or control sample and the patient:

and,

then approximately,

We assume that ρ₁′ … ρ_n′ are values from a normal distribution,

That is, we accept that the true values of the (transformed) correlations differ between individuals. We denote their mean value in the normative population by R and their variance by λ². Also, we assume these true values (ρ_i′, i = 1… N) follow a normal distribution. We want to test the null hypothesis that ρ_n+1′ (the true correlation coefficient for the patient) is from the same distribution. Assuming k is sufficiently large (i.e., > 10) for the approximation in equation (4) to hold, then r₁′…r_n′ are a simple random sample from the normal distribution,

This last equation (6) shows that r_i′ varies from R both because r_i′ varies about ρ_i′ (with variance 1/(k − 3)), and because ρ_i′ varies about R (with variance λ²). The variance of r_i′ is the sum of these components of variance. If the null hypothesis holds, then r_n+1′ is also from the distribution in (6). We calculate the mean and standard deviation of the transformed correlations in the control sample, i.e.,

and

Then, if r_n+1′ (the patient's observed correlation) comes from the same normal distribution as r_i′…r_n′,

has a t distribution on N − 1 degrees of freedom. If the null hypothesis is rejected at the conventional .05 level, then the patient's correlation is significantly different from that of the controls. Furthermore, if the precise one-tailed probability for t is multiplied by 100 then we have an estimate of the percentage of the population that would obtain a correlation lower than the patient's.

It can readily be appreciated that formula (9) is directly equivalent to the formula employed by Crawford and Howell (1998). In Equation (1) the difference between a patient's score on a neuropsychological test (e.g., number of items passed) and the mean score of the control or normative sample is divided by the standard error of the mean of the control sample. In the present case, where performance on the neuropsychological task is represented not by a conventional score but by a correlation coefficient, the difference between the patient's transformed coefficient and the mean transformed coefficient of the control sample is divided by the standard error of the mean coefficient.

A potential alternative means of testing the difference between the patient's transformed correlation and the mean correlation in the controls would be simply to convert the patient's transformed correlation to z, based on the mean and SD of the controls, and refer this z to a table of the area under the normal curve. (Such a method could also be used with the other forms of correlation coefficients dealt with in the present paper.) However, as when comparing a patient's performance on a conventionally scored test with controls, this method is inappropriate, since it treats the control sample statistics as if they were population parameters. The practical effect of using this alternative method would be to exaggerate the abnormality of the patient's performance and to spuriously inflate the chance of finding statistically significant effects. A comparison of the results from these methods is provided in the worked example (see later section).

Obtaining Confidence Limits on the Abnormality of an IIMA

The above method is designed to yield a point estimate of the rarity or abnormality of a patient's IIMA when the IIMA employed is a correlation coefficient. However, it would also be desirable to obtain confidence limits on the abnormality of the patient's IIMA. That is, using (9) we obtain a point estimate of the percentage of the population that will perform more poorly than the patient but now we wish to obtain confidence limits on this percentage. Such an aim is in keeping with the contemporary emphasis in statistics, psychometrics, and biometrics on the use of confidence limits (American Psychological Association, 2001; Daly et al., 1995; Gardner & Altman, 1989; Zar, 1996). Gardner and Altman (1989), for example, in discussing the general issue of the error associated with sample estimates note that “these quantities will be imprecise estimates of the values in the overall population, but fortunately the imprecision itself can be estimated and incorporated into the findings” (p. 3). Similarly the American Psychological Association (2001) take the view that confidence limits or intervals represent, “in general the best reporting strategy. The use of confidence intervals is therefore strongly recommended” (p. 22). The use of confidence limits is particularly appropriate when results are obtained from small samples.

To generate confidence limits on the abnormality of this form of IIMA we use a result obtained by Crawford and Garthwaite (2002). Let P denote the percentage of the population that will fall below a given individual's transformed correlation (r_n+1′), and suppose we require a 100(1 − α)% confidence interval for P. As in the formula for the point estimate, let (r_n+1′ − r′) represent the difference between the individual's score and the mean score of the normative or control sample, let

be the standard deviation in the normative sample, and let N be the size of the normative sample. We assume scores for the control population are normally distributed. If we put

then c is an observation from a non-central t distribution on N − 1 degrees of freedom. Non-central t-distributions have a non-centrality parameter that affects their shape and skewness. We find a value of this parameter, δ_U, such that the resulting non-central t-distribution has

as its 100α/2 percentile. Then we find the value δ_L such that the resulting distribution has

as its 100(1 − α/2) percentile. From tables for a standard normal distribution we obtain

These probabilities depend upon α, c and N and we denote them by h(α/2;c; N) and h(1 − α/2;c; N), respectively. Then a 100(1 − α)% confidence interval for P may be written as

From equation (3), r_i′ is a monotonically increasing function of r_i, so (h(α/2;c; N), h(1 − α/2;c; N)) is also a 100(1 − α)% confidence interval for the percentage of the population that will fall below a given individual's untransformed correlation, r_n+1. Details of the derivation of h are given in Crawford and Garthwaite (2002) and a worked example of obtaining 95% confidence limits on the rarity of an IIMA is provided in a later section.

TESTING EQUALITY OF RANK CORRELATION COEFFICIENTS

When performance on the task of interest is expressed as a non-parametric correlation coefficient, i.e., Spearman Rank Order Correlation Coefficient or Kendall's Tau, we suggest that, as was done with Pearson correlations, Fisher's transformation is applied to the coefficients of the control or normative sample and the coefficient of the patient. Kraemer (1974), in reviewing inferential methods for rank order coefficients notes that Fisher's transformation “has proved successful as a transformation which, when applied to several forms of correlation coefficients, is approximately normally distributed with variance independent of ρ” (p. 114). Fieller et al. (1957) and Fieller and Pearson (1961) showed that the approximation is good for Spearman's Rank Order correlation coefficient and Kendall's Tau for k > 10 and |ρ| < .9. Therefore, if comparing the non-parametric coefficient obtained for a patient with a control or normative sample, the procedure is identical to that for parametric correlation coefficients. That is, the mean and SD of the transformed coefficients are calculated and entered into formula (9) along with the transformed coefficient for the patient. Similarly, if confidence limits on the abnormality of the patient's score are required, the procedure described for parametric coefficients should be followed.

WORKED EXAMPLES

To illustrate the methods we use the example of the test of temporal order memory developed by Shimamura et al. (1990). In this task participants are read a list of 15 words and are then given the words on individual cards and asked to attempt to put them in the order in which they were originally presented. A 34-year-old female patient with a left frontal lesion was administered this test and the Spearman Rank Order correlation between the actual order of the words and the order produced by the patient was .210; the transformed correlation was .213. It will be noted that k in this example (the number of items used to compute the coefficient) is 15 and is thus sufficiently large for Fisher's transformation to give approximately normal distributions.

In the course of studying the effects of ageing on memory and executive functions, Crawford et al. (2000) obtained data for this task from 111 healthy persons. Because we wish to illustrate the use of this method when the control or normative sample is modest in size, we selected 14 of these participants to match the patient in terms of gender, age and years of education. Mean age in the control sample was 34.2 (SD = 2.69) and mean years of education was 12.0 (SD = 1.51); the patient had 13 years of education. Table 1 presents the Rank Order Correlation Coefficients and the transformed correlations for the controls. The mean of the transformed correlations in the controls is .637 with a SD of .180. Entering these data into formula (9) yields the following result,

Correlation (r) and transformed (r′) correlation between order and reported order on a temporal order memory task in 14 control participants (the correlations are Spearman Rank Order Correlation Coefficients); the final column presents raw scores on a free recall task, used in the example of testing for dissociations.

As the hypothesis tested by the researcher or clinician in this example is directional, that is, that the patient's performance will be significantly lower than matched controls, a one-tailed test is appropriate. The one-tailed critical value for t at the 5% level on 13 degrees of freedom is 1.77. The individual's score is, therefore, significantly different from the controls at the 5% level. The exact one-tailed probability for t in this example is .0202 and so the expectation is that only 2.02% of individuals in the population from which the normative sample was drawn would obtain a score as low as that observed for the patient. To obtain 95% confidence limits on this percentage we proceed as follows:

We want a non-central t distribution on N − 1 = 13 degrees of freedom that has −8.815 as its 0.975 quantile. This determines the non-centrality parameter to be −12.652 so we put δ_L = −12.652. We also want a non-central t distribution on 13 df that has −8.815 as its 0.025 quantile. This gives δ_U = −4.892. Then,

Hence the 95% lower confidence limit for P is 0.04% and the upper limit is 9.56%. To summarize the results for this case: the patient's memory for temporal order was significantly poorer (p < .05) than controls and it is estimated that only 2.02% of the population would exhibit a score poorer than that observed; the 95% confidence interval on this percentage is .04% to 9.56%.

As noted in a previous section, a potential alternative means of testing the difference between the patient and controls would be simply to convert the patient's transformed correlation to z. It is informative to compare this alternative with the proposed method for the present example. The patient's correlation expressed as a z score is −2.36. Referring this z to a table of the normal curve reveals that the estimated percentage of the population that would obtain a correlation lower than this is 0.9%. This exaggeration of the abnormality of the patient's performance would be even more pronounced with smaller control samples. Furthermore, in the present example, the conclusion from application of both the t test and z is that the patient is significantly impaired (p < .05). However, obviously these methods need not be in agreement. For example, if the patient's transformed correlation had been .335, then z (−1.68) would be significant (p < .05). However, this would be a spurious result arising from treating the sample as a population; the t-test would not be significant (t = 1.62, p > .05).

As another example of the use of these methods, Steinke et al. (1997) conducted a single-case study of a patient, C.N., who had suffered bilateral temporal lesions. Steinke et al. administered a series of tests of music perception to C.N. and 6 matched controls. Performance on the tests of music perception was quantified by the rank order correlation between an individual's ratings of the stimulus items (e.g., of tonality) and music-theoretic predicted levels. C.N. performed poorly on these tasks. No inferential method was employed to test whether C.N. differed significantly from controls but this can be readily achieved with the present method. For example, on a test of novel melodies the correlation obtained for C.N. was .46 (hence r′ is .497). The mean of the Fisher transformed correlations (r′) in the 6 controls was 1.29 with a SD of .305. Entering these data into formula (9) yields a t of 2.406 and a one-tailed probability of .031. Therefore, C.N.'s performance was significantly poorer than controls. The confidence interval on the percentage of the population that would obtain a lower score (i.e., a lower correlation) than C.N. was from .008% to 20.3% (the point estimate was 3.1%).

DETECTING DISSOCIATIONS WHEN PERFORMANCE ON ONE OR BOTH OF THE TASKS IS EXPRESSED AS AN IIMA

Up to this point we have been concerned with methods of testing for a significant deficit on a single task. Although the ability to identify a deficit in the individual case is fundamental, the presence of a deficit in a given cognitive function often only acquires theoretical importance when it is accompanied by the absence of a deficit in other related functions. That is, a central aim in many neuropsychological case studies is to fractionate the cognitive system into its constituent parts, and this aim is pursued by attempting to establish the presence of dissociations of function. Typically, if a patient obtains a score in the impaired range on a test of a particular function and is within the normal range on a test of another function, this is regarded as evidence of a dissociation. However, this evidence in isolation may not be at all convincing (Crawford & Garthwaite, 2002). For example, a patient's score on the “impaired” task could lie just below the cut-point for defining impairment and the performance on the other test lie just above it. Therefore, a more stringent test for the presence of a dissociation would also involve a comparison of the difference between tests observed for the patient with the distribution of differences between these same tests in the control sample (Crawford et al., 2003).

As previously noted, Crawford et al. (1998) devised a method that can be used to test whether the difference between an individual's score on two tasks is significantly different from the differences observed in a control sample. This method can, therefore, provide an additional test for the presence of a dissociation. However, it is also useful in the converse situation where a patient's scores are within the impaired range on both tasks. When this pattern is observed, the researcher can still test whether the magnitude of the difference between the two tasks is abnormal; i.e., evidence can be sought for the presence of a differential deficit on the test of one of the functions.

The method was developed for use with tasks in which performance is quantified by conventional means (e.g., number of items correct). For example, Crawford et al. (1998) use the example of testing whether the difference between a patient's performance on a verbal short-term memory task and a spatial short-term memory was significantly larger than the differences in a control sample. However, their method can be just as applicable when performance on one or both of the tasks is expressed as an IIMA. The formula for this test, which is essentially a modified paired samples t test, is

where Z_X and Z_Y are the scores of an individual on Test X and Test Y expressed as z scores formed using the means and SDs of the control sample, r_xy is the correlation between Tests X and Y in the control sample, and N₂ is the number of participants in the control sample. The test statistic follows a t distribution on N₂ − 1 degrees of freedom. Multiplying the one-tailed probability of t by 100 gives the point estimate of the abnormality of the individual's score. A derivation for the formula can be found in Appendix 1 of Crawford et al. (1998).

The use of this method is best illustrated with an example. Let us suppose that the patient whose temporal order memory performance was used to illustrate the method for correlations had also been administered a conventional verbal free recall task involving recall of 15 words. Given the evidence that temporal order memory is differentially impaired following frontal lesions in comparison with free recall (e.g., Shimamura et al., 1991), we could test the directional hypothesis that the patient's temporal order memory will be significantly poorer than her performance on free recall.

The healthy participants (N = 14) selected from Crawford et al.'s (2000) sample to serve as controls for this patient had in fact been administered the verbal free recall task described above. Their mean score (i.e., number of words correctly recalled) was 9.07 with a SD of 2.46 (range = 5–14). The raw scores of the controls on this task are presented in the final column of Table 1 (thereby providing potential users of this method with all the data required to work the example from scratch). Suppose that the patient's score on the free recall task was 8. As recorded earlier, the patient's score on the temporal order task (i.e., the correlation between the actual and reported order of items) was .213 after applying Fisher's transformation. Also as reported earlier, the mean and SD of the transformed correlations in the control sample was .637 with a SD of .180. The only remaining statistic required to test for a dissociation between free recall and temporal order memory is the correlation between performance on the two tasks in the control sample; this (Pearson) correlation was .51 (note that this correlation is computed from the free recall scores and the transformed correlations of the controls).

Using the means and SDs of the controls, the patient's score on the free recall task expressed as a z score is −0.435 and the z score for temporal order is −2.356. We will designate the free recall task as Test X and the temporal order task as Test Y (the choice is arbitrary). Entering these data into formula (12) we obtain t:

The one-tailed probability for a t of 1.87 on 13 degrees of freedom is .042. We would conclude, therefore, that the patient's performance on the temporal order task was significantly poorer than performance on free recall; that is, there is evidence of a dissociation between memory for temporal order and free recall of equivalent stimuli. In addition, by multiplying this p value by 100 we have an estimate of the percentage of the healthy population that would exhibit a discrepancy in favor of recall larger than that observed for the patient (4.2%); that is, discrepancies of this magnitude are fairly rare. A confidence interval on this percentage can be obtained using a method devised by Crawford and Garthwaite (2002). In the interests of brevity we do not provide a worked example here but the 95% confidence interval for this example is from 0.23% to 15.3%.

As another example, we return to Steinke et al.'s (1997) study of impaired music perception in case C.N. An important aim of Steinke et al.'s case study was to test whether music perception could be dissociated from non-musical cognitive abilities. To examine this, C.N. and controls were also administered a series of other tests, including subtests of the WAIS–R (Wechsler, 1981) and the Wisconsin Card Sorting Test (Heaton, 1981). Steinke et al. did not use an inferential method to test whether C.N.'s performance on music perception was significantly poorer than performance on these other tasks but this can readily be achieved using the present method. For example, C.N.'s score on the Wisconsin was 96 and the mean and SD in controls was 89.2 and 3.49, respectively (the correlation between performance on the novel melodies task and the Wisconsin in the controls was 0.83). Entering these data, together with the previously recorded data for the novel melodies task, into formula (12), yields a t of 7.22 and a one-tailed probability of .0004. Therefore, the discrepancy between C.N.'s performance on the music perception and non-musical task was highly significant and there is strong evidence for a dissociation.

The foregoing examples involved comparing the difference between a patient's performance on two tasks with the differences in controls when performance on only one of the tasks was expressed as an IIMA. However, as noted, Crawford et al.'s (1998) method is just as applicable when performance on both tasks is expressed as an IIMA. Therefore, the method can be used whenever there is a need to examine performance under different experimental conditions (i.e., comparison of a patient's correlation obtained under Condition A versus the correlation obtained under Condition B).

For example, Carey et al. (1998) examined the ability of a patient (D.F.) to visually estimate distance; they reported the correlations between actual and estimated distance for D.F. and controls under monocular and binocular conditions. Milner et al. (1991) hypothesized that cases such as D.F. should be markedly more impaired when distance is estimated using monocular versus binocular vision. The present method could be used to test this hypothesis by comparing the difference between the correlations under monocular versus binocular conditions for a patient against the differences between the correlations observed in controls. The indications are that, in the case of D.F., application of such a test would reveal a significant difference. Carey et al. (1998) found that the correlation between actual and estimated distance for D.F. was markedly higher under binocular conditions, whereas the correlations for two controls were very similar under monocular and binocular conditions.

As another example, Venneri et al. (1998) examined time estimation under single and secondary task conditions (the secondary tasks imposed demands on either working memory or attention). The method outlined could be used to examine whether a patient's performance is more sensitive than controls to the effects of these secondary tasks.

Furthermore, memory for temporal order can be tested with verbal and non-verbal stimuli (Mayes et al., 2001); in both cases performance is quantified by computing the correlation between actual and reported order. The method described could be used to test for a significant difference in performance on these two tasks in the individual case. Finally, it was noted that Steinke et al. (2001) have developed four tests of tonality and that performance on each of these tasks is quantified by computing the Spearman rank order correlation between a patient's ratings of tonality and music-theoretic predicted levels of tonality. If a researcher wished to test for possible dissociations between two of these different aspects of tonality the method described above could be used to test for their presence.

Before leaving this topic it should be noted that, in the illustrative examples, the concern was with testing whether there was a statistically significant dissociation between performance on the two tasks. However, in the search for dissociations, the aim is to uncover dissociations between functions, not the tests used as (imperfect) indicators of these functions. Therefore, it can be of some value to demonstrate that tests of a particular putative function provide similar indications of impairment (i.e., that they do not differ significantly). This is perilously close to attempting to prove the null hypothesis and therefore such evidence can play only a supporting role. That is, when there is evidence that was regarded as a unitary function, which can be fractionated into two new putative functions A and B (as indicated by the presence of significant differences between measures of A and B), the lack of significant differences among the tasks assessing each of these putative functions lends support to the view that these new constructs possess internal consistency.

ABNORMALITY AND RELIABILITY OF SCORES AND SCORE DIFFERENCES AND THEIR CORRESPONDING CONFIDENCE INTERVALS

The present methods test whether a patient's score (specifically, an IIMA) or score difference came from the same population as a control or normative sample and are distinct from a test on the reliability of differences. A reliable difference between an individual's test scores is one unlikely to have arisen from measurement error in the tests. However, many healthy individuals will have reliable differences among their abilities in different cognitive domains. Indeed, if the tests concerned have high reliability, reliable differences will be very common and therefore cannot be taken as indicating acquired impairment (see, e.g., Crawford, 2003; Crawford & Allan, 1996). Therefore, methods that quantify the probability that a patient's score difference was drawn from the distribution of score differences in the healthy population are more germane to the needs of clinicians and single-case researchers (Crawford & Garthwaite, 2002; Crawford et al., 1998).

Similarly, the confidence limits presented in the present paper are confidence limits on the estimated rarity or abnormality of a given score (specifically, an IIMA) or difference between scores. As noted, they allow the user to quantify the effects of error arising from using a sample in place of the population: they quantify the fallibility of normative or control sample data. These confidence limits are therefore distinct from confidence limits that quantify the effect of measurement error in a test instrument (or instruments) on an individual's score (or score differences). These latter limits are obtained by multiplying a test's standard error of measurement (or the standard error of measurement of the difference when concerned with score differences) by a standard normal deviate (i.e., 1.96 for 95% limits)

COMPUTER PROGRAM FOR EVALUATING IIMAS IN THE SINGLE CASE

With the exception of the methods for finding confidence limits on the abnormality of a patient's IIMA, the calculations involved in the tests presented in the present paper are relatively straightforward. Armed with a set of basic statistical tables, they could be performed by hand or with the aid of a calculator or spreadsheet. However, as we are aware of the time pressures under which many neuropsychologists operate, we have written a computer program for PCs to accompany this paper. Apart from saving time, use of the program reduces the chance of arithmetic or clerical error. Furthermore, it provides a precise probability value for the tests, whereas tabled values of t only record the t value that must be exceeded to achieve a given level of significance. An exact probability is more useful in many situations as the emphasis will be on the rarity or abnormality of the individual's level of performance. For example, it is still of some value to have an estimate of the percentage of healthy individuals expected to exhibit a coefficient as extreme as the patient's when the t value for a given test does not exceed the most liberal tabled value (e.g., .10 in many tables). It also does away with the need for interpolation when a value falls between two tabled critical values.

The program IIMA.EXE compares the parametric or non-parametric correlation coefficient obtained for a patient with the coefficients of a control or normative sample. The program prompts for the number of cases in the control or normative sample (N), the correlation coefficient for the patient (r_N+1) and the correlation coefficients for each of the controls. The program then applies Fisher's transformation to the coefficients. The output consists of the one and two-tailed significance of the difference between the patient and controls, the abnormality of the patient's coefficient (e.g., the estimated percentage of the population that would obtain a coefficient that was lower than the patient's) and 95% confidence limits on the abnormality. The output can be viewed on the screen, printed, or saved to a file.

As noted, the program incorporates an option of carrying out the test using the mean and SD of the transformed coefficients for controls rather than requiring that the user enters the untransformed correlations individually. This is useful if a researcher or clinician wishes to examine the performance of additional patients against the same controls at a later point (the M and SD of the transformed correlation is included in the output to enable users to use the program in this way). This option also makes it easy for a researcher to use normative data obtained from a third party.¹

If a researcher wishes to test for a dissociation between two tasks when performance on one or both of the tasks is expressed as an IIMA, then a program (DIFFLIMS.EXE) previously provided by Crawford and Garthwaite (2002) can be used.²

Computation of the mean and SDs and the correlation between tasks in the controls could be carried out in a statistics or spreadsheet package. If the package used does not offer Fisher's transformation then the correlations could be entered into the IIMA.EXE program as it provides the transformed correlations for each of the controls and patient as output. (It is likely that these data would be run through the IIMA.EXE program in any case as the researcher would normally wish to test whether the patient's IIMA was significantly different from controls before testing if there is evidence of a dissociation between the IIMA and performance on another task.) The transformed correlations could then be entered into the statistics or spreadsheet package so that the correlation between the IIMA scores and scores on the other task can be computed for the controls. If any doubts remain over how to conduct this test then the data from the worked example for dissociations could be run through the DIFFLIMS.EXE program to verify that the procedure is being followed correctly.