404 TiPS - October 1990 [Vol. 11]

blood pressure, therefore, may be a validation of the principle that everything is connected to every- thing e l s e - that, in this model at least, the immune system may be related to blood pressure regu- lation; i.e. hypertension may be an autoimmune disease. It may also explain why aspirin, a cyclo- oxygenase inhibit.or that releases IL-2 from thymus T cells, is anti- hypertensive in the immature spontaneously hypertensive rat, but not in human essential hyper- tension. Further support for the role of the thymus comes from the observation that thymus extracts from normotensive WKY Wistar rats normalize blood pressure in spontaneously hypertensive rats.

The bad news in all this is that the spontaneously hypertensive

rat may have outlived its usetul- ness as a model of essential hyper- tension. It is inevitable that all models will fail eventually; if they did not, it would imply that the model and the process being modeled were one and the same. However, scientific inertia will doubtless ensure that papers on the spontaneously hypertensive rat will continue to appear, albeit at a diminish_~ng rate.

B. MAX

R e f e r e n c e s

1 Guideri, G., Barletta, M. A. and Lehr, D. (1974) Cardiovasc. Res. 8, 775-786

2 Pavlov, I. P. (1927) Conditioned Reflexes, Oxford University Press

3 Baker, T. B. and Tiffany, S. T. (1990) Psychol. Rev. 92, 78--108

4 Siegel, S (1975) J. Comp. Physiol. Psychol. 89, 498-506

5 Kayan, S., Woods, L. A. and Mitchell, C. L. (1969) Eur. J. Pharmacol. 6, 333-339

6 Siegel, S., Hinson, R. E., Krank, M. D. and McCully, J. (1982) Science 216, 436-437

7 Siegel, S., Hinson, R. E. and Krank, M. D. (1978) J. Exp. Psychol. Anita. F~ehav. Processes 4, 188-196

8 Mucha, R. F., Volkovsk..'s, C. and Kalant, H. (1981) ~. Comp. Physiol. Psychol. 95, 351-362

9 Solomon, R. L. (1980) Am. Psychol. 35, 691-712

10 Siegel, S. and Sdao-Jarvie, K. (1986) Psychopharmacology 88, 258-261

11 Raffa, R. B. and Porreca, F. (1986) Neurosci. Lett. 67, 229-232

12 Okamoto, K. et al. (1972) in Spontaneous Hypertension: Its Pathogenesis and Complications (Okamoto, K., ed.), pp. 1-8, Igaku Shoin, Tokyo

13 Freis, E. D. (1972) in Spontaneous Hyper- tension: Its Pathogenesis and Complica- tions (Okamoto, K., ed.), pp. 231-237, Igaku Shoin, Tokyo

14 Tuttle, R. S. and Boppana, D. P. (1990) Hypertension 15, 89-94

' pili :f r,a ,,,,. .,,, 1 !. i I [ J- I I " " ' ' " " ' i l - " T " I " .... i i , i i l l .1_

Cumulative frequency curves in population analysis Richard Barlow

In this article Dick Barlow explains how you can easily obtain answers to the questions: 'Do the results come from one normal distribution or from two populations?'; and in the more complex situation when your results consist of sets of values X and Y which should lie on a straight line: 'Do the points fit a single line or are they normally distributed about two straight lines?'

When examining results to see how they are distributed it is common to construct a series of histograms showing how many lie within a particular range. If the results were normally distributed it would be expected that: 2.3% would be less than the mean - 2.0 SD (standard deviation) or greater than the mean + 2.0 SD; 19.2% would lie between the mean - 0.5 SD and the mean, or between the mean and the mean + 0.5 SD, etc. Itis then possible to compare the observed and expected distributions to see how different these are.

R. B. Barlow is Reader in Chemical Pharma- cology at the Department of Pharmacology, University of Bristol, The Medical School, University Walk, Bristol BS8 ITD, UK.

An alternative method of deal- ing with the data: which uses individual results and avoids the loss of information associated with grouping results in ranges, is to construct a cumulative ('integrated') frequency curve. To do this the result, R, is plotted on the X-axis; the proportion of results which have values less than or equal to R is plotted on the Y-axis. If the distribution is normal, then the value of R corresponding to a Y-axis value of 50% is the mean; the value of R corresponding to 2 . 3 % should be the mean - 2 SD; the value ef R corresponding to 97.7% should be the mean + 2 SD, and so on. The curve is S-shaped and is the integrated noi:~al fre- quency curve, but it is not poss-

19q0. Elsevier Science Publishers Ltd. (UK) 0165 - 6147/90/$02.00

ible to express this as an equat ion because the integral is intractable. An acceptable approximation 1, however, is provided by the logis- tic equation:

M . X P y - XP+KP

where p is an exponent of steep- ness, M is the maximum and K is the value of X when Y is half- maximal. This equation gives an S-shaped curve when Y is plotted against log X. This may be familiar to pharmacologists as a model for the log dose-response curve 2 and for a cumulative frequency curve, M = I .

Finney 3 writes: "the logistic is scarcely distinguishable from the normal between response rates of 0.01 and 0.99', and in a direct comparison the difference is less than 1% (Ref. 4). Programs are available for fitting cumulative frequencies to the logistic equation to calculate the mean and standard deviation, and to check the distri- bution to see if it appears normal 4.

If the results do not belong to the same population, the cumu- lative frequency curve will be stepped, as with a b inding curve when there is more than one b inding site, and it may be poss- ible to fit the data to the sum of two logistic curves that represent two normal distributions. If the first populat ion is a fraction Q T of the total, and the parameters K1

TiPS - October 1990 [Vol. 11]

2 . 5 0 0

0 0

,

lJ o O O @

0.0 O @ 5. l , ,

o o

@

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Fig. 1 (above). Data from Gualtieri s fitted to a single straight line. Results (obtained from his figure 5 using a digitizing tablet) ShGw log of relative potency of enantiomers (Y-axis)

plotted against log of affinity of the stronger enantiomer (X-axis). Open circles ~'Tdicate 1,3-oxathiolane-3-oxides; filled c, ircles indicate 1,3-oxathiolanes. Fig. 2 ~'left). Distribution of aifferences betwee~ observed and fitted values of Y from Fig. 1, the mean is 0 and the width

-2 of each block in the histogram is 0.5 so. - 1 0 1 2

and p, determine the first mean and s tandard deviation, the second population will be a frac- tion (1 -QT) of the total; par- ameters K2 and P2 determine the second mean and standard devi- ation. The values of Y and X are therefore fitted to the equation:

extended to the situation where each result consists of a pair of values, X and Y, which should fit a straight line. Do the values fall along a single line or should there be two lines? An example may be taken from a recent note in TiPS

405

by Professor Gualtieri 5 in which values of the 'eudismic index' (logarithm of the ratio of the ac- tivities of enantiomeric pairs), Y, was plotted against the log of the affinity of the stronger enan- tiomer, X. If the compounds obey Pfeiffer's rule the results should lie along a single straight line, but Professor Gualtieri argued that the results for one type of compound (1,3-oxathiolanes) lie along one line and the results for another type (1,3-oxathiolane-3-oxides) lie along a second line. Are there statistical reasons which justify this division based on structure?

The null hypothesis is that there is only one relation and, when the data are fitted to a single line (Fig. 1), the correlation coef- ficient r is 0.52 (with 16 degrees of freedom); the Spearman rank co- efficient is 0.68; and the Kendall rank coefficient is 0.51, all indi- cating that there is a significant correlation (p<0.05). The null hypothesis predicts that the values of ( Y o b s e r v e d - - Ycalculated) for each point should be normally distrib- uted about zero. The distri- bution, plotted as a histogram (Fig. 2) suggests that there are two populations. It looks skew and the coefficient of skewness is 0.78. This should be O, but the estimate of standard error is 0.58, making it difficult to reject the null hypothesis.

y _ QT. X P1 21- (1-QT)" X p2 X pl + K1 pl X p2 Jr- K2 p2

and should give the values of the parameters K1, Pl, K2, P2 and QT, which give the min imum value of the sum of (Yobsen, ed -- Ycalculated) 2. The fitting process requires start- ing guesses and, because it is very sensitive to changes in QT (which must lie between 0 and 1), it is in practice often necessary to set QT by inspection of the fit to a single logistic curve, and to obtain starting estimates of the values of the other four parameters before allowing QT to vary. The values of K and p obtained with QT fixed are used to start the fit; then the value of QT is allowed to 'float' and all five parameters are calcu- lated. A program of this type (for Amiga or IBM machines) can be obtained from the author.

The main aim of this article is to show how the analysis can be

100%

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Fig. 3. Cumulative frequency curve, with differences (Yob ...... d - Yca,cu,ated from Fig. 1) plotted on the X-axis and percentage of group with values less than or equal to this plotted on the Y-axis. Broken line shows logistic approximation to the normal distribution, with vertical lines marking mean (50%), mean - so (16%), mean + so (84%) and mean +2 SO (97.7%). Unbroken line shows fit to two populations with the means (-0.35 and +0.79) marked on the X-axis.

406 TiPS - October 1990 [Vol. 11]

The cumulative frequency curve is shown in Fig. 3; the distance (Yob ....... d -- YcalculaLed) of the points from the fitted single line from Fig. 1 is plotted on the X-axis, and the percentage of the total (18) points which are less than or equal to this distance from the line is plotted on the Y-axis. The broken line indicates the normal distribution (using the logistic approximation), and the full line shows the fit for two normally distributed populations with the two means indicated by vertical rules at -0.35 and +0.79. The fitted value of QT is 0.749, which is nearer to the value (0.722) ex- pected for groups of 13 and 5 points than for groups of 12 and 6. Moreover, as can also be seen

from the histogram (Fig. 2), the smaller group consists of positive values, so the statistical test has identified the points above the line in Fig. i as forming or.~ group, with the larger number o~ points below the line forming the second group.

This is not the same as Professor Gualtieri 's division based on structure. If there is chemical evidence that differences in struc- ture affect b inding then it is reas- onable to divide the groups up as he proposed - perhaps there are differences in the temperature coefficients of affinity which indi- cate different enthalpies of bind- ing. If, however, the inference is merely supposition and depends upon the idea that particular

structures b ind in a particular way, then the division is more arbitrary. It does not fit the stat- istical analysis.

The method described is an analytical tool which is easy to use and has a wide range of appli- cations.

References 1 Berkson, J. (1944) J. Am. Stat. Assoc. 39,

357-365 2 Parker, R. P. and Waud, D. R. (1971) J.

Pharmacol. Exp. Ther. 177, 1-12 3 Finney, D. J. (1978) Statistical Methods in

Biological Assays, p. 358, Griffin 4 Barlow, R. B. (1983) Biodata Handling with

Microcomputers, pp. 156-162, Elsevier Biosoft

5 Gualtieri, F. (1990) Trends Pharmacol. Sci. 11, 315-316

m]mmm!mmNmn ,vp i

Dopamine receptor J i

antagonism in schizophrenia: is there reduced risk of extrapyramidal side-effects? Jarmo Hietala, Jaakko Lappalainen, Markku Koulu and Erkka Syv&lahti

The first selective D1 dopamine receptor antagonist, SCH23390, has been reported to be active in preclinical tests that predict antipsychotic activity in schizophrenic patients. This is particularly exciting because it has been claimed that this compound is "atypical', in that it has a reduced propensity to induce extrapyramidal side-effects. However, in considering the evidence from preclinical screening tests for antipsychotic activity and extrapyramidal side- effects of potential neuroleptic drugs, Jarmo Hietala and colleagues conclude that the majority of available data is not compatible with the postulated atypical profile of SCH23390.

The mechanism of action of classical neuroleptic drugs such as haloperidol is generally attribu- ted to antagonism of central dopamine D2 receptors. Although

]. Hietala is acting Associate Professor and ]. Lappalainen is a Research Associate in the Department of Pharmacology, University of Turku; M. Koulu is a Researcher and E. Syviilahti is a Psychiatrist and Senior Resear- cher with the Academy of Finland, Medical Research Council, Kiinamyllynkatu 10, SF- 20520 Turku 52, Finland.

effective in a high proportion of patients, these drugs are often limited by their intrinsic ability to induce extrapyramidal side- effects such as parkinsonism, aka- thisia and, in the worst cases, tardive dyskinesia, which can be irreversible even after withdrawal of the drug. These disturbing side-effects have generally been associated with antagonism of striatal dopamine D2 receptors. Because of this, drug companies

lqg0. Elsevier Science Publishers Ltd. (UK) 0105 - 6147/90/$02.00

have been investigating other ap- proaches to the therapy of schizo- phrenia. One of these approaches has been the development of Dz receptor antagonists.

The role of Dz receptors in schizophrenia is uncertain z'2, but the first D 1 receptor antagonist, SCH23390 (Ref. 3), has recently been reported to be active in pre- clinical tests predictive of anti- psychotic action in schizophrenic patients. Moreover, it has been claimed that preclinical evidence suggests a reduced propensity of D1 antagonists to induce extra- pyramidal side-effects 4-6. This property would place D1 antag- onists in the category of 'atypical ' neuroleptics.

Atypical neuroleptics are those drugs that are effective in the control of schizophrenic symp- toms, but do not tend to induce extrapyramidal side-effects 7. The prototypic drug in this group is clozapine. Atypical neuroleptics generally have a weaker affinity for dopamine receptors than do classical neuroleptics, but at least some have a greater potency to displace [3H]SCH23390 in vivo, with a rank order of potency roughly comparable with their clinical potencies 8, suggesting the involvement of D1 receptor occu- pancy in their therapeutic actions. However, direct or indirect inter- actions with non-dopaminergic systems, e.g. muscarinic, 0c-adren- ergic, 5-HT or GABA, may also contribute to the atypical profile of 'dirty ' drugs such as clozapine.