Neuropsycholosia, 1976,Vol. 14,pp. 135-136.PerssmonPress. F’riatedin England.
NOTE HOW TO MEASURE
LATERALIn
JOHN T. E. RICHARDSON* Neuropsychology Unit, Department of Neurology, The Churchill Hospital, Headington, Oxford OX3 7L.I. England (Received 19 September 1974) Abstract-It is suggested that the derivation of measures of laterality is not useful for evaluating theories of cerebral organization. A theory-independent technique is given for ordering the results of experiments in terms of the laterality which they reflect. MARWALL,CAPLAN,and HOW [l] have considered how one might derive a measure of lateralixation of cerebral function from a subject’s performance in a psychological experiment. The basic data from which a measure is to be derived are the percentage of correct responses on trials believed to involve the left hemisphere, Lc, and the percentage of correct responses on trials believed to involve the right hemisphere, Rc. They suggest three fundamental assumptions which they wish to impose upon such a measure. First, if Lc is held constant, then the measure should vary with Rc, and vice veraz. Second, the measure should vary symmetrically about the point where Lc = Rc. Third, the measure should have the same range of et d. [l] propose possible values whatever the total accumcy. On the basis of these assumptions, MARSHALL a measure calculated as follows: when the total accuracy is less than 50x, it is given by (Rc-Lc) divided by the total correct; when the total accuracy is more than SO%, it is given by (Rc-Lc) divided by the total number of errors. If Rc is plotted graphically against Lc, one may identify sets of points which correspond to a particular degree of lateralixation according to this measure. These “isolaterality” functions (as one might call them) consist of two straight-line segments, one from the origin to a point on the minor diagonal of the unit square, the other from that point to the point (100,100). Discussion of measures of laterality might be motivated by one of two concerns. Fit, one might have developed a general theory concerning the nature of cerebral lateralization from which such a measure could be derived. The theory would ascribe certain properties to the measure, and these could be examined in experimental situations. However, in practice this would not be at all a simple matter. The principal ditBculty is that both Lc and Rc are random variables, for each is subject to errors of observation. However, the usual techniques for fitting curves require that one of the variables be determined in advance, and so they could not be used to compare predicted isolaterality functions with the empirical data. A more fruitful way to test such theories would be to isolate their basic assumptions and to submit these to direct test by means of experimental manipulations, in which case a measure of laterality would be redundant. Thus, the derivation of such a measure would be of very little help for the purpose of evaluating a theory of cerebral lateralization. On the other hand, if one lacked a general theory concerning the nature of cerebral lateralixation, one might wish to derive a measure of laterality which was independent of, and neutral between, such theories, in order to investigate other issues; for example, whether a particular function was more lateralixed on the right than on the left. Indeed, all of the theoretical questions which MARSHW er 41. [l] cite in the introduction to their paper could be approached without having a general theory about the nature of lateralixation. Unfortunately, the authors are somewhat muddled over the status of their own discussion. Initially, it is presented as being based upon the rather weak assumptions described above, which, they assert, any measure of Iaterality must satisfy. They then produce a measure which has certain arithmetical properties and which satisfies their assumptions. However, they conclude by pointing out that this is only one of the possible measures which satisfy those assumptions, and by asserting that the isolaterality functions which they derive are “psychological theories of how an S changes Rc and Lc in achieving ditferent overall accuracies”. The arithmetical properties of their measure are logically independent of their initial assumptions, and they receive no theoretical justification in their discussion. Such properties are therefore formal and arbitrary in nature, and they do not increase our understanding of the mechanisms of cerebral lateralixation. Yet, *Present address: Department of Psychology, Brunel University, Uxbridge, Middlesex, England. 135
136
NCKs
it is only because MARSHALL et al. [l] impose this arbitrary arithmetical framework upon their original assumptions that their measure is inconsistent with others. The important point is that no theory of cerebral lateralization has yet been proposed which is sufficiently rich to predict a relationship between Lc and Rc as accuracy varies, on the basis of substantive assumptions about the underlying cerebral structures. The immediate interest must therefore lie in the search for a theory-independent measure of laterality. The imposition of an arbitrary arithmetical framework is symptomatic of a general error in the arguments given by MAIWULL er al. [l]. This is that of assuming that statements about order relations entail statements about quantities. For example, they assume that questions of a “more or less” nature require some numerical measure of laterality. Elsewhere, they drift from an idea of a positive concordance to one of direct proportionality. But this is not simply a case of making one’s statements more precise, but of introducing new properties into one’s measurement system. Ordinal scales have much weaker properties than ratio scales, and the two types should be clearly differentiated. In fact, the basic assumptions given by MARSHALLef al. [I] only detine an ordinal scale, which has no unique representation in the system of real numbers. In other words, without further a priori assumptions, one Cannot arrive at the measurement scale which they wish to adopt, that which would permit one to quantify lateraliitions of cerebral function. Nevertheless, all of the theoretical questions with which MARSHALLet al. [l] introduce their paper could be answered simply by ordering sets of scores with respect to the corresponding laterality, without actually quantifying that laterality. The following procedure defines such an order relation (cf. 121).The set of scores (Lc, Rc) represents a greater degree of (right) lateralization than the set (Lc’, Rc’), if, and only if, Lc is less than Lo’ and Rc is greater than Rc’. This relation is irreflexive, asymmetric, and transitive. It is a partial order, in that if one set has a higher Lc and a higher Rc than the other, the matter is undecided. Nevertheless, thii procedure is sufficient for one to carry out statistical analyses of the results of psychological experiments: the sign test may be used for related samples, the Mann-Whitney U test for independent samples. This is a non-numerical technique for deciding whether one sample reflects more right laterality than another. It requires merely that one rank sets of scores derived from trials associated with each of the two hemispheres. Because of this, it can be extended without modification to situations using different measures of performance. In a reaction-time situation, for example, Lc and Rc might be the speeds of responding. Finally, one can say that a set of scores reilects right laterality, rather than left, if, and only if, Rc is greater than Lc. Acknow&&emenis-This
research was supported
by Medical Research
Council grant G973/144/C.
REFERENCES 1. hrlARsHALL,J. C., C&LAN, D. and HOLMES,J. M. The measure of laterality. Neuropsychologiu 13, 315321,1975. 2. NORMAN,D.A. A comparison of data obtained with diierent false-alarm rates. Psychol. Rev. 71,243-246, 1964.
On sug+ro
que la derivation de mesures de lateralit
pas utile pour Svaluer les theories de l'organisation
n'est
cSr@brale. On
propose une technique independante des theories pour ordonnor 12s x6sultats dcs exp6riences en terme de la iatQralit6 qu'ils r$flSchissent.
Deutschsprachige
Zusammenfassung:
zu bedenken gegeben, daB Theorien zur cerebralen Orgenisation nicht von LateralitSitsmessungenhergeleitet werden diirfen.Es wird eine Theorie-unabhgngige Tec'nnik geboten zur Ordnung der experimentellen Ergehnisse entsprechend der Lateralitlt, auf die sie sich beziehen. Es wird
NOTE HOW TO MEASURE
LATERALIn
JOHN T. E. RICHARDSON* Neuropsychology Unit, Department of Neurology, The Churchill Hospital, Headington, Oxford OX3 7L.I. England (Received 19 September 1974) Abstract-It is suggested that the derivation of measures of laterality is not useful for evaluating theories of cerebral organization. A theory-independent technique is given for ordering the results of experiments in terms of the laterality which they reflect. MARWALL,CAPLAN,and HOW [l] have considered how one might derive a measure of lateralixation of cerebral function from a subject’s performance in a psychological experiment. The basic data from which a measure is to be derived are the percentage of correct responses on trials believed to involve the left hemisphere, Lc, and the percentage of correct responses on trials believed to involve the right hemisphere, Rc. They suggest three fundamental assumptions which they wish to impose upon such a measure. First, if Lc is held constant, then the measure should vary with Rc, and vice veraz. Second, the measure should vary symmetrically about the point where Lc = Rc. Third, the measure should have the same range of et d. [l] propose possible values whatever the total accumcy. On the basis of these assumptions, MARSHALL a measure calculated as follows: when the total accuracy is less than 50x, it is given by (Rc-Lc) divided by the total correct; when the total accuracy is more than SO%, it is given by (Rc-Lc) divided by the total number of errors. If Rc is plotted graphically against Lc, one may identify sets of points which correspond to a particular degree of lateralixation according to this measure. These “isolaterality” functions (as one might call them) consist of two straight-line segments, one from the origin to a point on the minor diagonal of the unit square, the other from that point to the point (100,100). Discussion of measures of laterality might be motivated by one of two concerns. Fit, one might have developed a general theory concerning the nature of cerebral lateralization from which such a measure could be derived. The theory would ascribe certain properties to the measure, and these could be examined in experimental situations. However, in practice this would not be at all a simple matter. The principal ditBculty is that both Lc and Rc are random variables, for each is subject to errors of observation. However, the usual techniques for fitting curves require that one of the variables be determined in advance, and so they could not be used to compare predicted isolaterality functions with the empirical data. A more fruitful way to test such theories would be to isolate their basic assumptions and to submit these to direct test by means of experimental manipulations, in which case a measure of laterality would be redundant. Thus, the derivation of such a measure would be of very little help for the purpose of evaluating a theory of cerebral lateralization. On the other hand, if one lacked a general theory concerning the nature of cerebral lateralixation, one might wish to derive a measure of laterality which was independent of, and neutral between, such theories, in order to investigate other issues; for example, whether a particular function was more lateralixed on the right than on the left. Indeed, all of the theoretical questions which MARSHW er 41. [l] cite in the introduction to their paper could be approached without having a general theory about the nature of lateralixation. Unfortunately, the authors are somewhat muddled over the status of their own discussion. Initially, it is presented as being based upon the rather weak assumptions described above, which, they assert, any measure of Iaterality must satisfy. They then produce a measure which has certain arithmetical properties and which satisfies their assumptions. However, they conclude by pointing out that this is only one of the possible measures which satisfy those assumptions, and by asserting that the isolaterality functions which they derive are “psychological theories of how an S changes Rc and Lc in achieving ditferent overall accuracies”. The arithmetical properties of their measure are logically independent of their initial assumptions, and they receive no theoretical justification in their discussion. Such properties are therefore formal and arbitrary in nature, and they do not increase our understanding of the mechanisms of cerebral lateralixation. Yet, *Present address: Department of Psychology, Brunel University, Uxbridge, Middlesex, England. 135
136
NCKs
it is only because MARSHALL et al. [l] impose this arbitrary arithmetical framework upon their original assumptions that their measure is inconsistent with others. The important point is that no theory of cerebral lateralization has yet been proposed which is sufficiently rich to predict a relationship between Lc and Rc as accuracy varies, on the basis of substantive assumptions about the underlying cerebral structures. The immediate interest must therefore lie in the search for a theory-independent measure of laterality. The imposition of an arbitrary arithmetical framework is symptomatic of a general error in the arguments given by MAIWULL er al. [l]. This is that of assuming that statements about order relations entail statements about quantities. For example, they assume that questions of a “more or less” nature require some numerical measure of laterality. Elsewhere, they drift from an idea of a positive concordance to one of direct proportionality. But this is not simply a case of making one’s statements more precise, but of introducing new properties into one’s measurement system. Ordinal scales have much weaker properties than ratio scales, and the two types should be clearly differentiated. In fact, the basic assumptions given by MARSHALLef al. [I] only detine an ordinal scale, which has no unique representation in the system of real numbers. In other words, without further a priori assumptions, one Cannot arrive at the measurement scale which they wish to adopt, that which would permit one to quantify lateraliitions of cerebral function. Nevertheless, all of the theoretical questions with which MARSHALLet al. [l] introduce their paper could be answered simply by ordering sets of scores with respect to the corresponding laterality, without actually quantifying that laterality. The following procedure defines such an order relation (cf. 121).The set of scores (Lc, Rc) represents a greater degree of (right) lateralization than the set (Lc’, Rc’), if, and only if, Lc is less than Lo’ and Rc is greater than Rc’. This relation is irreflexive, asymmetric, and transitive. It is a partial order, in that if one set has a higher Lc and a higher Rc than the other, the matter is undecided. Nevertheless, thii procedure is sufficient for one to carry out statistical analyses of the results of psychological experiments: the sign test may be used for related samples, the Mann-Whitney U test for independent samples. This is a non-numerical technique for deciding whether one sample reflects more right laterality than another. It requires merely that one rank sets of scores derived from trials associated with each of the two hemispheres. Because of this, it can be extended without modification to situations using different measures of performance. In a reaction-time situation, for example, Lc and Rc might be the speeds of responding. Finally, one can say that a set of scores reilects right laterality, rather than left, if, and only if, Rc is greater than Lc. Acknow&&emenis-This
research was supported
by Medical Research
Council grant G973/144/C.
REFERENCES 1. hrlARsHALL,J. C., C&LAN, D. and HOLMES,J. M. The measure of laterality. Neuropsychologiu 13, 315321,1975. 2. NORMAN,D.A. A comparison of data obtained with diierent false-alarm rates. Psychol. Rev. 71,243-246, 1964.
On sug+ro
que la derivation de mesures de lateralit
pas utile pour Svaluer les theories de l'organisation
n'est
cSr@brale. On
propose une technique independante des theories pour ordonnor 12s x6sultats dcs exp6riences en terme de la iatQralit6 qu'ils r$flSchissent.
Deutschsprachige
Zusammenfassung:
zu bedenken gegeben, daB Theorien zur cerebralen Orgenisation nicht von LateralitSitsmessungenhergeleitet werden diirfen.Es wird eine Theorie-unabhgngige Tec'nnik geboten zur Ordnung der experimentellen Ergehnisse entsprechend der Lateralitlt, auf die sie sich beziehen. Es wird
