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Neuropsychologia 47 (2009) 2687–2689

Contents lists available at ScienceDirect

Neuropsychologia

journa l homepage: www.e lsev ier .com/ locate /neuropsychologia

eviews and perspectives

omparing a single case with a control sample: Refinements and extensions

ichael C. Corballisepartment of Psychology, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

r t i c l e i n f o

rticle history:eceived 9 September 2008eceived in revised form 14 January 2009ccepted 12 April 2009vailable online 19 April 2009

a b s t r a c t

Crawford and Garthwaite [Crawford, J. R. & Garthwaite, P. H. (2002). Investigation of the single case inneuropsychology: Confidence limits on the abnormality and test score differences. Neuropsychologia, 40,1196–1208] have proposed an adjusted t-test, widely used in experimental neuropsychology, for compar-ing a single case with a control sample. This test does not assess whether the single-case score belongs

eywords:nalysis of varianceingle-case studiestatistics

in the population from which the control sample is drawn, but rather whether the mean of the distribu-tion from which the case was drawn differs significantly from the mean of the control population. Thisapproach is readily extended to more complex designs in which the analysis of variance is appropriate,and the single case is treated as belonging to a group of size one. The main qualification in using eitherthis or Crawford and Howell’s approach is that it makes the untestable assumption of homogeneity ofvariance between the two populations, but a simple adjustment either to the t-test or to the analysis of

variance allows one to draw conclusions about the relation of the case itself to the control population.

© 2009 Elsevier Ltd. All rights reserved.

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26872. Alternative t-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26873. Extension to analysis of variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26884. Adjusted analysis of variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26885. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2688

Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2689References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2689

. Introduction

In experimental studies in neuropsychology, there is often theeed to relate a score arising from a single case to scores from theeneral population. Traditionally, the way to do this is to relate thecore, say X1, to norms based on large-sample norms, where theean � and standard deviation � can be taken as population values.z-score can then be computed from the formula

2. Alternative t-tests

As Crawford and Howell (1998) point out, this procedure isvalid only if the sample is large. In neuropsychological research,tests are often improvised to focus on particular deficits, andlarge-scale norms are not available. Researchers may then test con-trol participants, typically in relatively small samples, since theestablishment of large-scale norms is generally impracticable, and

= X1 − �

�(1)

he z-score can then be referred to the normal distribution to deter-ine where the case stands in relation to the normal population at

arge.

E-mail address: m.corballis@auckland.ac.nz.

028-3932/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.oi:10.1016/j.neuropsychologia.2009.04.007

indeed unnecessary. In such cases � and � are replaced by sampleestimates X̄2 and S2, respectively, and the corresponding formulabecomes

t = X1 − X̄2 . (2)

S2

The computed value can then be referred to the t distribu-tion with (N2 − 1) degrees of freedom, where N2 is the samplesize.

Crawford and Howell (1998), following Sokal and Rohlf (1995),also suggest that Eq. (2) should be amended as follows:

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688 M.C. Corballis / Neuropsy

= X1 − X̄2

S2

√N2

N2 + 1. (3)

This version is in widespread use, and unless N2 is very smallhis adjustment makes little difference. Crawford and Garthwaite2002) give procedures for computing confidence intervals for thebnormality of a test score based on this formula, and offer a com-uter program for this purpose.

There is nevertheless a case for using Eq. (2) rather than Eq. (3),nd the choice of equation depends on the question that is asked.here are correspondingly two such questions. First, one might askhether the score can be considered part of the population fromhich the control sample is drawn. Here the null hypothesis is that

1 − �2 = 0. In this case Eq. (2), the small-sample equivalent of Eq.1), is appropriate. The second question is whether the single case1 and the control sample come from populations with the sameean; the null hypothesis is that �1 − �2 = 0. What Eq. (3) does,

n effect, is compute the independent-samples t-test for testing theifference between two means, one represented by a sample of sizene and the other on a sample of size N2 (see Appendix A). Crawfordnd Howell therefore implicitly assume that the investigator wisheso make inferences about the population to which the case belongs,hereas the aim may be simply to determine whether the case itself

elongs in the control population.The choice of Eqs. (2) and (3) may therefore depend on the aims

f the investigator. A clinician, for example, may be primarily inter-sted in the single case alone, and whether his or her scores fallutside the normal range, in which case question (2) is appropri-te. Experimental neuropsychologists, on the other hand, may beore concerned to generalize the results of the single-case study

o the hypothetical population of those with the same neurologicalondition. In this case qualified use of Eq. (3) is appropriate.

A disadvantage of using Eq. (3) and drawing inferences beyondhe single case is that it assumes homogeneity of variance betweenhe two hypothetical populations, and there is no way to test thisrom the given data. The standard error against which the differences measured is based entirely on the control sample. Imagine, forxample, that one is confronted with a single visitor from Mars, whoappens to be very tall. One might readily conclude that she is tooall to belong to the earthly human population, but we cannot reallyell how the mean Martian compares in height to the mean humannless we have some indication of variation. Even so, neuropsycho-

ogical investigators may be willing to make the assumption that apecific brain lesion or neurological disorder does not appreciablylter the variance among those afflicted, so the t-test may well beeasonable.

. Extension to analysis of variance

The use of Eq. (3) is equivalent to using a simple t-test, and iseadily generalized to analysis of variance, allowing the researchero include different treatments, and so assess individual cases in

ultifactorial experiments. This involves including the between-ubjects factor “group” in the analysis of variance and testing forts main effect and interactions. This analysis is easily run usingtatistical packages such as SPSS (e.g., SPSS Inc., 1999). As with the-test, the main problem is that homogeneity of variance betweenroups must be assumed, and all error terms are based only on theariation within the control sample.

An experiment by Corballis, Boyd, Schulze, and Rutherford1998) provides an example of analysis of variance used in this

ay. A commissurotomized man, L.B., and 20 control participantsere tested for the ability to discriminate whether pairs of spatially

eparated lights were simultaneous or successive. (For illustrativeurposes, the experiment is here slightly simplified.) Each pairould be presented wholly in the left visual field, wholly in the right

ia 47 (2009) 2687–2689

visual field, or bilaterally, with one light in each visual field. Thestimulus onset asynchronies (SOA) were 0, 17, 33, 50, and 67 ms.The critical question was whether the commissurotomized manwas impaired in making temporal discriminations when the lightswere presented in opposite visual fields, and therefore to oppo-site hemispheres, relative to discrimination when the lights werewithin the same visual field, and relative to control participantswith intact commissures.

In the experiment, the dependent variable was the proportion oftrials on which the lights were judged successive, and was analyzedwithin participants as a function of location and SOA. Group wasthen added as a between-participants factor, with one “group” rep-resenting the commissurotomized man and the other the 24 controlparticipants. Analysis of variance was carried out using SPSS. Ofinterest were the variable group and its interactions with locationand SOA. It transpired that the interaction between group and SOAwas significant, as was the triple interaction between group, SOA,and location. Analyzing the simple group-by-SOA interaction sep-arately for each location showed temporal discrimination in thecommissurotomized participant to be significantly impaired in theleft visual field and when the pairs were presented bilaterally, butnot when they were in the right visual field. Technically, these con-clusions apply to the populations from which L.B. and the controlsample were drawn, and require assumptions of homogeneity ofvariance and covariance.

4. Adjusted analysis of variance

The t-test adjustment proposed by Crawford and Howell (1998)can be reversed to provide a test of whether a single case belongsin the population defined by the control sample. That is, Eq. (2) canbe derived from Eq. (3) by multiplying by

√(N2 + 1)/N2.

This can be extended to any contrast where the aim is to comparethe single-case value with the sample value (see Appendix A). Sinceanalysis of variance can be reduced to orthogonal contrasts, thesum of squares for any effect or interaction involving group, whereone “group” is the single case and the other the control samplecan be multiplied by (N2 + 1)/N2, so the resulting F-tests no longertest hypotheses about two populations, but rather test whether thesingle case belongs in the sample population with respect to theeffect in question.

This provides relief from assumptions about homogeneity ofvariance, but by the same token means that inferences about thesingle case refer to that case alone, and not to the population fromwhich it is drawn. With respect to the example given above, theinteractions between group and SOA, and between group, location,and SOA could have been adjusted to test the relation of L.B. aloneto the control sample, without any necessary implications for theeffects of commissurotomy in general.

The adjustment is slight, and since it always results in a largersum of squares the original analysis can be regarded as conservativewith respect to the comparison of the single case with the controls.Comparison between the populations represented by the case andthe control group may well be suspect due to lack of homogeneityof variance, but investigators may nevertheless feel secure that anysignificant effect relating to group applies to the comparison of thesingle case with the control population.

5. Conclusions

This article has two main aims. One is to draw attention tothe two questions that can be asked with respect to comparisonsbetween a single case and a control sample. The first question iswhether the case can be considered to belong in the populationfrom which the control sample was drawn. Here, Eq. (2) is the

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M.C. Corballis / Neuropsy

ppropriate test for a simple comparison. The second question ishether the population to which the case belongs differs from that

rom which the control sample was drawn. In this case Eq. (3),hich also corresponds to a regular independent-samples t-test,

s appropriate, but requires the additional untestable assumptionf homogeneity of variance.

The second and more important aim is to spell out the impli-ations of extending single-case studies to more complex designs,here multifactorial analysis of variance can be applied. Thus

nvolves entering the factor group into the analysis, where the sin-le case represents a “group” with a single member. As with Eq. (3),omparisons between the single case and the control group requirehe untestable assumption of homogeneity of variance between theopulations from which single case and sample are drawn. Nev-rtheless, just as Eq. (2) can be derived from question (3), so theroup effects can be adjusted by multiplying their sums of squaresy (N2 + 1)/N2. Analysis of variance can therefore be recommended,ith the proviso that investigators either accept the assumption of

omogeneity of variance and draw population-level conclusions, orccept that the results, whether adjusted or not, apply only to theomparison of the single case with the control population.

ppendix A.

The formula for comparing two sample means is

= X̄1 − X̄2

S√

(1/N1) + (1/N2)(4)

here

=

√∑(X1 − X̄1)2 + (X2 − X̄2)

2

N1 + N2 − 2. (5)

If N1 = 1 then Eq. (4) reduces to Eqs. (3) and (5) reduces to theormula for the standard deviation of the control sample, namely

2.

This can be extended to comparison between contrasts. Let’s saycontrast is defined by the coefficients c1, c2, ... ck. In the case of

wo groups, sum of squares for the interaction between group andontrast is given by

ia 47 (2009) 2687–2689 2689

SS =

[∑kj=1(cjX̄1 − cjX̄2)

]2

∑kj=1[(c2

j/N1j) + (c2

j/N2j)]

.

If the first group is a single case and the second group is of sizeN2, then this equation reduces to

SS =

[∑kj=1(cjX1j − cjX̄2j)

]2

(N2 + 1)/N2∑k

j=1c2j

. (6)

Dividing this by the error term gives a test of whether the twopopulations differ with respect to the contrast. But if we wish todetermine whether the contrast for the single case belongs in thatfor the control sample, the appropriate sum of squares is simply

SS =

[∑kj=1(cjX1j − cjX̄2j)

]2

∑kj=1c2

j

. (7)

Thus to obtain Eq. (7) from Eq. (6), we multiply the sum ofsquares by (N2 + 1)/N2. Note that this is essentially the same adjust-ment as is required to derive Eq. (2) from Eq. (3).

Since an entire analysis of variance can be broken down intoorthogonal contrasts, each contrast can be so adjusted. It followsthat any comparisons or interactions involving the factor group,where one “group” is the single case and the other the controlgroup, can be adjusted to test whether the single case belongs inthe population from which the control sample is drawn by simplymultiplying the sum of squares by (N2 + 1)/N2.

References

Corballis, M. C., Boyd, L., Schulze, A., & Rutherford, B. J. (1998). Role of the commis-sures in interhemispheric temporal judgments. Neuropsychology, 12, 519–525.

Crawford, J. R., & Garthwaite, P. H. (2002). Investigation of the single case in neu-

Neuropsychologia, 40, 1196–1208.Crawford, J. R., & Howell, D. C. (1998). Comparing an individual’s test score against

norms derived from small samples. The Clinical Neuropsychologist, 12, 482–486.Sokal, R. R., & Rohlf, J. F. (1995). Biometry. San Francisco, CA: W.H. Freeman.SPSS Inc. (1999). SPSS Base 10 for Windows User’s Guide. Chicago, IL: SPSS Inc.