NIX 3931 R3S3oo~O.W c 1983 Prrglmon Prns Ltd.

NOTE

SOME PROBLEMS WITH BRYDEN AND SPROTT’S “STATISTICAL DETERMINATION OF DEGREE OF LATERALIZATION”

BILL JOSEX

Department of Psychology, Carleton University, Ottawa, Ontario. Canada

(Accepted 26 October 1982)

Abstract-BRYDEN and SPROTT (Neuropsychologio 19,571-581, 1982) have proposed an index, i., of lateralization that has the same range of possible values no matter what the overall level of accuracy obtained in the experiment. It is shown by way of example that changes in L over the levels of the

independent variable may be predictable from increases in the level of accuracy for the right and left sides. Specifically, if accuracy increases at a constant rate for the two sides, then i., except for some highly artificial situations, IS constrained to increase or decrease. It is argued that difficulties associated with the difference between correct responses for the two sides as an index of the degree of laterialization may have been exaggerated. It is shown for hypothetical data that the difference measure may have many of the properties of 1..

IN A t_ATFRAt.tTY experiment subjects are required to identify or recall material presented to the right and left sides of perceptual space (visual fields, ears, hands, etc.) on the assumption that information is processed primarily by the contralateral hemisphere. We may represent the proportion ofcorrect responses to items presented on the right and left by PR and P,. respectively with corresponding error proportions I-P, and I-P,. If our assumption is correct, then the observation PR # PL may be taken to mean that processing is relatively lateralized in one hemisphere. Can we go farther and say anything about the degree to which processing is lateralized?

Many authors [e.g. I] have implicitly or explicitly taken the difference, PR - P,., or the absolute difference. [P,, - 5). 3s an index of the degree to which a function or mode of processing is lateralized. Recently MARSHALL, CAPLAS and

H~)LXIES [4] and BRYDEN and SPROTT [I] have suggested alternative measures. They have both argued that PR - PL may be difficult to interpret since the range of values of this measure is a function of the overall accuracy, PR+PL, observed for a subject or a group. Elsewhere [Z] I have discussed some of the difficulties associated with the index proposed by Marshall ef nl. and here I wish to draw attention tosome problems that may arise if Bryden and Sprott’s index is used without sufficient care.

It is useful to note the experimental situation that motivated Bryden and Sprott. They refer to KI~IURA’S [3] well- known study in which children aged from 4 to 9 yr were asked to recall digits dichotically presented to the right and left ears. The difTerence between the two ears declined approximately monotonically as a function of age. If PR - PL reflects the degree of lateralization this must be a surprising result since no theory of hemispheric differences appears to predict decreasing lateralization as a function of increasing age. Bryden and Sprott point out, however, that older children were highly accurate, which restricts the possible values of P,-P,.. They go on to derive an index of lateralization, i., which has the same range, --x to +x, no matter what the value of PR+PL,

The authors give a number of ways of computing L according to the probability model assumed for the experiment. The measure I shall examine is the logarithmic transform of the ratio

P,(I -Pt.)

pL(I-pR).

i.e. the ratio of correct response proportions multiplied by the reciprocal of the ratio of error proportions. This measure would be computed from an experiment in which, for example, subjects were required to identify single stimuli tachistoscopically presented to the right and left visual fields. Bryden and Sprott note that i. is always unbounded and is approximately normally distribured. Since the variance of 1. is easy to derive, standard scores may be constructed and tests of significance performed on the data from individual subjects.

While these are no doubt useful properties we need to determine how L behaves empirically. As Bryden and Sprott

295

pointed out. we need to show whether or not ;. has “consistent and predictable profwttes (p. 580)“. Unfortunately It

will be shoun that some of the properttes of j. may be. if anything, too predictable To begin one might ask “hy be need an 1nde.x that ts independent Im the sense used by Bryden and Sprott) of

PR + PL? To be sure. PR - P, IS wvcrely constrained ifoverall accuracy is too high or too low. However. any task that is

likely to result in either asymptottc correct response or error rates can tell us very little about cerebral laterahzatton no matter what dependent variable we choose as an Index of the degree of laterahzatton. Notice that i. computed for

each group in Ktmura’s experiment according to the formula gt\en her, ---~this Jmounts to assuming that intrusions between the ears are constantPu-ould show the same decrease In the degree of Iaterahzation as PR - PL (Table 1).

Floor andceilingeffects always pose problems ofmterpretation. Houever.eltmtnattngextreme kalues is primartly

a problem of experimental destgn rather than one of finding an appropriate transformation of the raw data. The experimenter must choose a task that will yield a laterality effect and that is not too difficult at some levels of the independent variable and tooeasy at others. Ofcourse. this may often be lery difficult In practice and it is nocriticism

of Kimura’s pioneering experiment to note that it points up certain methodological problems. Some hypothetical data given by Bryden and Sprott in fact show the usefulness of PR - PL when tloor and ceil~nr

elTects have been avoided (Table 2). These data show a very high and positive correlation betueen PR - P,. and >. (r =0.997-t). In other words, for these data both measures preserve the same information Moreover. it is also possible to carry out statistical tests on the raw numbers correct and numbers of errors shown in Table 2. The data are cast

into a 2 (Right-Left) x 2 (Correct Response-Error) contingency table and the x‘ statistic is computed. Values of xz together with the standard score based on i. are also shown in Table 7. It is apparent that both procedures reveal almost precisely the same pattern of statistically significant effects.

Table I. Mean values of P,. and PR (weighted averages of males and females) in KNURA’S [3] experiment together

with PR- P,. and the L measure computed as [P,( I - PL)];[P,_( 1 - P,)]

A_re (yr)

4 0.3450 0.5633 0.21s 0.596 : 0.6250 0.49 17 0.6917 0.8 IS3 0.200 0.193 0.s.t 0.994 I

7 0.7483 0.83 17 0.083 0.508 8 0.8250 0.9138 0.088 0.804 9 0.8767 0.9217 0.045 0.504

Table 7. Number of correct responses (out of 24) for the right and left vtsual fields for 16 hypothetical subjects

(from BRYDEF and SPROTT [I]) together with values PR - PL and L. and Z and zz

Subject Left Right px - s i % L’

I 6 s 7

-4 3 0.406 0.634 0.375

, 5 9 9 13 5 -0.824 0.678 1.256 1.153 I.613 I .3-13

4 6 13 7 1.266 2.027’ 4.269*

5 II I8 7 I.266 2.027* 4.269* 6 10 5 -5 - 0.999 1.534 2.424 7 I8 I4 -4 -0.762 I.214 1.500

8 IS 19 I 0.236 0.342 0.1 I8 9 I8 13 -5 - 0.932 1.492’ x77*

10 I4 18 4 0.767 1.214 1.500 II II 17 6 I .054 1.733* 3.086’ I2 ‘I I3 -8 - 1.779 7.401** 6.454’ 13 II I6 5 0.860 I .443 2.1 I6 14 6 15 9 1.609 2.544** 6.957** 15 IO 12 2 0.337 0.580 0.336 16 13 19 6 I.169 1.x00*’ 3.375+

f/J < 0.05. **p<o.o1.

NOTE 297

Now consider the problem utth which Bryden and Sprott began. Does lateraliration remain constant or vary

imcrease presumably) as a function of age’? In principle we might treat age and side as factors in a p x 2 analysis-of- variance design using proportion correct or some other index ofaccuracy as the dependent variable. A failure to find

a significant Age x Stde interaction would then be evidence for the hypothesis that the degree of lateralization is

roughly constant over development. This is obviously equivalent to treatmg PR - PL as the implicit Index of degree of I3tsralization.

The usual finding in any developmental experiment [e.g. 31 is that overall accuracy increases with age. Within the limits noted PR - P,, may remain constant, increase. or decrease as overall accuracy increases. Thus we are in prrnciple able to use the difference measure to ask whether or not degree of lateralizatton is constant in relation to increasing age. Since. however, i. is a joint function of correct responses and errors. it is easy to show that i. will not in general be constant over trme given an equal rate of increase in PR and PL. Table 3 gives imaginary laterality data for two subjects observed at intervals of 1 yr. For subject 1 values of PR and PL are greater than the respective error rates (i.e. PR and PL less than 0.5). We see that i. increases with increasing age and accuracy for subject I but in fact tiecreases

for subject 2.

Table 3. Imaginary data for two subjects in a laterality experiment where PR and PL increase as a function of

age and PR - PL remains constant

Age PR

I 0.80 2 0.85 3 0.90

Subject 1

PL

0.60 0.65 0.70

I. Pa

0.9808 0.30 1.1156 0.35 1.3499 0.40

Subject Z

PL

0.10 0.15 0.20

i

I .3499 I.1 I56 0.9808

These relations will hold in fact for any index that is not a linear function of differences. Greater weight will be given to differences at extreme ends of the range of PR + PL values. Although L is independent of accuracy in the sense that in principle it is always unbounded, changes in Lover time (the independent variable is merely illustrativ,e) are clearly dependent on overall accuracy.

Of course it is always possible to argue that differences close to the floor or ceiling are in some sense more important than equal differences in the middle ofthe rangeof PR + PL values. If this is theexperimenter’s presumption then the I index is obviously useful. However, any such argument is not about measurement: it is an argument from

some substantive theory of lateralization. As a number of authors (see especially [S]) have noted. we do not have a theory that would allow us to predict how PR and PL should be related. A jbrriori, we do not know how transformations of PR and P,. relate to any theory. It is one thing to identify the formal properties of a measure. Unfortunately without a theory of lateralization we cannot say whether these properties are desirable or not.

A statistician might object to our proposed Age x Side analysis-of-variance that interactions are only meaningful if measurements are on at least an interval scale. This objection has some force and experimenters can rarely give a satisfactory reply. Ordinarily we simply hope that scale properties can take care of themselves since the alternative is the usually impractical task oftesting axioms [6]. We should note, however, that the fact that the scale properties of

the j. are normally or approximately normally distributed has no bearing, numerous assertions to the contrary notwithstanding. since the proposition that a statistic is normally distributed is neither necessary nor sufficient for the conclusion that an attribute is measured on an interval scale (see [6] for a thorough discussion of this issue).

It is perhaps clear that no index of the degree oflateralization will always be completely satisfactory. There are also obvious difiiculties associated with the measure PR - P,.. and where floor and ceiling effects have been avoided the two measures will provide substantially the same information. Bryden and Sprott did point out that if their metric approach failed to yield consistent properties it would be necessary to adopt the non-numerical approach discussed

by RICHARDSON [5]. This may indeed be the case. However. we should also have to abandon the important substantive problem with which Bryden and Sprott began.

rlckno,c/rdgemenrs-The preparation of this manuscript was supported by a grant from the National Science and Engineering Research Council of Canada. I should like to thank anonymous reviewers for their comments.

REFERENCES

I. BRYDEN. M. P. and SPROTT. D.A. Statistical determination ofdegree laterality. Veuropstcholqc/ia 19,571-581. 198’.

2. JOSES. B. Measuring degree ofcerebral lateralization in children as a function of age. Derl Psycho/. In press.

298 NOTE

3. KIMLRL D. Speech laterahatlon m young children as determined by an auditory test. J. camp. Physiol. Ps~chol. 56, 899-902. 1963.

4. MARSHALL. J. C.. CAPLAS. D. and HOLVES. J. M. The measure oflatsrnlity. .Vwropswholoyia 13.315-322. 1975. 5. RICHARDSON.. J. T. E. How to measure laterality. ,Veurops~droloyio 14, 13%-136. 1976. 6. THOMAS. H. 1.Q. interval scales and normal distributions. Psychol. Bull. 91. 19Y~ZO2. 1982.

Bryden et Sprott (Neuropsychologia, 1982, 19 : 571-581) ont oropos6

un index de 1atCralisation (lambda), qui a la m&w gam de valeurs possibles quelque soit le niveau global de prdcision atteint dans 1'expGrience. On montre d l'aide d'un exemple que les variations de lambda selon les niveaux de la variable indipendante peuvent Gtre predits 2 partir de l'auqmentation du deqr6 de pr6cision pour les c6t6s droit et gauche. SpGcifiquement, si la pr6cision augmente de fa$On constante pour les deux c6t6s, lambda, sauf dans des situations tr& artificielles. ne peut qu'augmenter ou diminuer. On avance l'argument que les difficult& li6es i la difference entre rdponses correctes pour les deux c6t6s, comne index du degr6 de latbralisation, hypoth6tiques

ont 6tG exaqerees. On montre 1 partir de donnies que la mesure de la diffsrence Peut avoir beaucoup

des prooriGt6s de lambda.

Jones, E.: Some problems irith Bryd~n...

Zusamrnenfassunp

Bryden und Sprott hnbcn emen Latel.nlisier-ungsindes Lambda vorgeschlagen,

der die yleiche Streuung m6gllcher Werle hat, unabhangig vom generellen

Leistungsnlvew in eincnr Expcrlmen:. I~urci~ tin Beijpiel wird gezeigt, dafi

Ver2nderungen van Lnmbdn iiber verschledene Niveaus der unabhingigen

Variable aus der Zunahme im Niveau der Korrekthelt dcr Leistung fiir die

rechte und Iinke Seite vorausgesagt werden I&xn. Speziell zelgte sich, da(?

Lambda dnnn. wenn die Korrektheit auf beiden Seiten in konstanter Weise

zunimmt, au!’ Zunahme oder Abnahme beschrCinkt bleibr, ausgenommen

einige fizinz artefizielle Situationen.

Es wlrd arcumentlert, da8 die Schwierlgkeiten im Zusammenhang mit

Leistungsunterschieden zwischen den brlden Seiten als Index fiir das Ausrnafl

der Lateralislerung tibersch%tzt worden sind. Fiir hypothetische Daten wird

dargelegt, da0 das Unterschiedsmad vlele der Elgenschaften van Lambda

haben ksnn.