Nonparametric scaling: A descriptive index

June 5, 2017 | Autor: Alessandro Giuliani | Categoria: Pharmacology, Nonparametric Statistics, Placebos, Drug evaluation, Indexation
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Nonparametric Scaling: A Descriptive Index O. Ghirardi, A. Giuliani, and R. Cozzolino

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Research oh Sewsceace, Signa-Tau Pomezìa, Rofta,

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A des.riptive index (CI) based on the U statistic is pr€sented. The CI permits to quantify the differences beiwe€n experimental samples takìng into a.count bolh the localion and variability features of the data without any disEibutional assumplion. This index can be useful to build activity scal€s for senes of conpounds.

Keyworats: Scales: Ranhng, Mann Whihey's U; Statistical methods

Intrcduction Many colleagues working in experimental situations have often expressed the need to quantify the difference between two groups using a synthetic index that can provide an immediate perception of the size of the observed differcnces. Two main approaches can be followed to solve this problem: one involves the use of inferential statisticsi the other is based on the calculation of descriptive indexes. The fi.st strategy involves calculating the level of signifrcance of the observed differences by means of a statistical test, for exarnple, typically, the Mann wtitney U test in nonparametric cases and the Student's , test in the other cases. This strategy allows an evaluation to be made of the statistrcal reliability of the resùlt, although it does not reFesent a relevant fteasure of the size of the observed difference (K$emer 1992). The value of p is actually crucially dependent on the number of cases analyzed and on the variability. Consequently, for the same size differcnces between two groups, we note that one can obtair different significance values for samples differing in number of cases and vaiability. FuthenDore, the cases in u/hich the small size of the sample makes it very difficult to attain statistical signi6cance can be hardly quantified in this way. Above all, it is important to keep the problem of the statistical reli-

Atldress rcprint req@sts 10 Dr, O. Ghirsrdi, Insdtute for Reseùch on Senescencè, via Po.rina Xm.30.,100, 00040 Pomezia, Roma, Italy. Reeived Novenbd L 1993: rcvisèd dd ecepted May 1, 1994. J@md of

Phmolosical ùd

@ l99t El.èvièr Scitue ù8, 655 Avaù. of ln. Amri6,

Toxicological Methods 32, 105-107 (1994)

NN Yoit, NY 1ml0

ability of a difference sepamte from that of its size (Kraemer, 1992).

The descriptive indexes most commonly used are on percentage calculations. Depending on the

based

case, these percentages are calculated on the difference between rhe means of the two groups (parametdc case)

exceeding a given endpoint (nonparametric case). This strategy, when applied conectly, provides a measure of the size of the mean difference between the two groups, but does not

or on rhe number of individuals

give us any information about the distribution of the difference (percent variation) or else leads to a loss of what often amounts to a considerable quantity of information contained in the data (number of individuals exceeding an endpoint). The index presented in this article is a descriptive one and was constructed for the puryose of remedying some

above limitations, at least partially, of being applicable io nonparametric cases with only a small

of the

number of observations and of haviIlg a simple relation with validated statistical tests. The typical experimental situation for which this index is intended, is the activity scaling of a series of molecules in which each molecule is assayed in a separate e\periment versus a control or reference standard of its own (Angelucci et al., 1993). This is a very common experimental ftame, especially in b€havioral pharmacological screening (Buder et al., 1984). The raw data we face in this kind of experiment arc characterized by different (and generally small) sample sizes for each drug-placebo pair, nonparametric measures, and heterogeneity of the

con[ol groups due to

106

JPM Vol. 32. No, 2 odober !94rlo5 t0?

the intrinsic variability of the methods. These features resemble the melaanalyses frames (Riedemann et al.,

ofp values and the comparisons of median differences or percentage counts based on statistics have obvious limitations. What l,e need is an effective melhod to compute sone 1993) wherc such solurions as the order

sort of generalized distance between drug-placebo pairs simultaneously taking inro account the different centers of gravity of the two sets (differences in locations) and thei va.dability (degrees of discrimination). These lwo types of informadon must converge in a single ìndex in

order to have an activity scale useful to perform a QSAR procedure or any other kind of decisional process.

ical groups A and B with N" = 3 and Nb = 4. The individual values

a.re as

follows:

A: 9, 11, 15; B: E, 10, 6,

13.

After making 3*4 comparisons, we realize that Cij = (x, > xj and then the A values arc grcater than lhe B values) in 9 occasions. There are no cases of eqùality, and so io the remaining three situations, xi < xj and so Ci; = -1. When applyinS Eq. 1 we 8et the following: CIab = (9+(1) + 3*(-l )/(344) = 6tr2 = 0.5

I

Construction Using U Statistics Also U calculation is based on the reciprocal ordering samples. The value of U is obtained from the number of times a group B value is greater than a group A value. U' statistics is the complement of U, as given

of two CalculÀtion Method The proposed comparability index (CI) can be derived in two different ways: by intuitive construclion and a relation between it and Mann Whitney U sratis, lics.

by the followidg:

U' = N,*Nb

-

The numerato in Eq. 1 can be thus expressed follows:

!', c,, = g' Eq.

lntuitiye Construction Let us assume that we have to compare two groups, A and B, with reference to a measured parametet X- The lwo groups A and B have a numerosity of Ni and Nb. respectively. Let us call the individual values of the rwo groups xi and xj, rcspecrively. From what has been said above we shall. therefore, have N, xi elements and Nb xj elements. From this, we obtain the existence of N,*Nb s€pamte comparisoos between pairs having one element

from g1oup A and one from group B. Each of these compaisons can have three possible outcomes, and to each of these outcomes the procedure assigns a value C, in accordance wilh lhe following pàrameters.

l) if xi > xj. then Cii = l; 2) if xi < xj, then C,i = -l; 3) if xi = xj, thenC,J=0 The algebraic sum of all the Cir, afrer normaÌization for the total number of comparisons, gives us the CI for the lwo groups A and B according lo the expression:

(>

(l) Y(N,*Nb) It is clear from Eq. I ùat rhe value of C[ varies

CI"b

=

C,

between I (maximum separation of the two groups: all the groups A values are higher than those of group B) and -1 (maximum separation: all the group A values are lower than those of group B); CI will take on a value

approximating zero

if

rhe two groups have similar

To give a practical example, let us take two hypotheÈ

U (Siegel, 1956).

cr,b =

g.

1 thus becomes as

expressing 2 becomes

CI,h

follows:

(u'-u/(N.*Nh)

'When

=I

as

U'

as a function of

- (2*U/N.*NJ

(2)

U, we note Eq. G)

Convendonally, it is assumed that U < U'. Thereif CI is calculated by means of Eq. 3, then it will always be positive. Il can be brought back to ahe condi-

fore,

tion of the sign in Eq.

I

simply by referring to the

medians of the two groups.

Discussion The construction of CI is affected both by lhe mean intergroup size difference and the variability. The index provides an indirect measure of the quantitative aspect of the intergoup differences: a comparatively high observed value will "win" numerous comparisons for its group by increasing the conesponding value of C. This way of keeping track of lhe size of the differences is based on an ordinal scale rhat depends on ranking nther than on the numerical value. This means lhat CI is particularly suitable for formalizing all those cases in which a nonparametric approach must be used (nonassumability of a normal disribulion, presence of an endpoint, semiquantitative measures). In such cases, the only possible quantitative approach ro oblain a descriptive index of differcnces belween two groups is ro count rhe observalions exceeding a given endpoint in the two groups. This approach has two obvious drawbacks: 1) the possible arbitmry nature of

to7

O, CHIRARDI ET AL,

NONPARAMEIRIC SCALINC

lhe endpoint itself and 2) the loss of information about differences between values below the endpoint. The use of percent differences between medians should be considered an incoffect procedure as it consideN a measure which actually has an ordinal basis as being based on intervals. In such cases, which are panicularly ftequent in the behavioral sciences, CI has a definitive edge on the other quantificatioo methods. ln drug screeniog, in which each product is compared with a corresponding control or a rcference standard, the superiority of CI is even more evident given the possibility of this index to be used for consrucdng a scale of activity of the goducls under investigalion. Each molecule will have a C[ value that is independent of the number of cases analyzed in each test and can be easily used as a dependent variable for structure/activity relation procedures. IÌ is wonh noting rhat, in lhe case of an equal number of cases for each dmg-placebo pair, lhe U statistics is cornpletely interchangeable with CI: but this is rarcly the case in drug screening. In other words, CI can be interpreted as a normalized version of U slatistics for a different number of cases.

'We have successfully lested the usefirlness

of CI in many different situations, including the one reported (Angelucci et al., 1993) with reference to the investigarion of the nootropic activities of a series of sùbstances. References Crini l. C@mho U, Coalino R, D. Wni P. OhiErdi O. Oilnrcsi F, Giuliani A, Cwldi D, Misiù D. R.m@i MT, Scolasico C, Tini MO (1993) Synlhesn ùd mcsia-rcvesal adirny of À s.ries of 7- and 5-DembeEd 3-acylamìho lactams. r M.d CÀ.r 36(l I):l5l l-1519. BuÙ.r DE, Nordin lC, L I.alien YJ. Z*enl.l L, P@hel PH. Mùiolr ,G (1984) Al@sia-Evesal &ivny of r *ries of N [disubsrùùtcd .mino) dryll-2-orGl [email protected]@id.s, including pmn&.ran. I Med Chéu 2115)16a4 @t , Kreher HC (1992) Reponing ùe size ol eiLcrs in Esqrch sudies to facilita& d*stm. of p@tical or clini.al significdce P$rlorc! AnSelucci L, Calvisi P.

r@ndrritulotr

11

16):521 576.

(l9l)

Ri.d.mann PJ. B.6inic S, Cudd, U, Torance GW, Tu8well PX A iudy to d.rcrmrne ùe effi(ac) dnd \J'elJ ot kno\icrm !er\u\

pioxican, diclotenrc and indomeùaci. in pltiens with ostee

fihnù

A E.la-anaìysis. J

rR

reuùotol 2U\2):2@5 2t03.

Si.e.l S (1956) Non?arunctrit Stu,isti.s lÒt Ncw Yoik: M.c Graw-Hill

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