psychotropic drug classification based on sleep?wake behaviour of rats

© 2007 Royal Statistical Society 0035–9254/07/56223

Appl. Statist. (2007)56, Part 2, pp. 223–234

Psychotropic drug classification based onsleep–wake behaviour of rats

Kristien Wouters,

Universiteit Hasselt, Diepenbeek, Belgium

Abdellah Ahnaou,

Johnson & Johnson Pharmaceutical Research and Development, Beerse, Belgium

Jose Cortinas Abrahantes and Geert Molenberghs,

Universiteit Hasselt, Diepenbeek, Belgium

Helena Geys

Johnson & Johnson Pharmaceutical Research and Development, Beerse, andUniversiteit Hasselt, Diepenbeek, Belgium

and Luc Bijnens and Wilhelmus H. I. M. Drinkenburg

Johnson & Johnson Pharmaceutical Research and Development, Beerse, Belgium

[Received December 2005. Final revision December 2006]

Summary. The aim of the paper is to present methodology for the classification of potentialpsychotropic drugs on the basis of their activity. We first sketch the background of this classof drugs and then zoom in on so-called pharmacoelectroencephalogram studies. These datapose some statistical challenges. For classification purposes, we propose a flexible hierarchicaldiscriminant analysis tool, allowing us to take the specific nature of the drug class into account,as well as the features of the mixed models, in combination with fractional polynomials, fittedto the electroencephalogram data. The method is evaluated against the background of existingmethods. The method’s performance is studied by using a comprehensive analysis of a largeelectroencephalogram data set.

Keywords: Discriminant analysis; Electroencephalogram; Fractional polynomials;Longitudinal data; Mixed model

1. Introduction

For thousands of years, humans in all known societies have used psychotropic drugs(substances that act on the central nervous system and affect mood, thinking and behaviour).Psychotropic drugs, whatever the substances used, rank from second to tenth among the mostconsumed medicinal products in western nations (Zarifian, 1996). Although there may be somecontroversies about classification for clinical purposes (Zarifian, 1988; American PsychiatricAssociation, 2000), psychotropic drugs can be divided into five major classes: antidepressants,antipsychotics, anxiolytics, hypnotics and stimulants (Deniker, 1982; Oughourlian, 1984; Cohen

Address for correspondence: Kristien Wouters, Center for Statistics, Universiteit Hasselt, Universitaire Cam-pus, B3590 Diepenbeek, Belgium.E-mail: [email protected]

224 K. Wouters et al.

and Cailloux-Cohen, 1995), each typically named according to its main indication in psychia-try. Classifying drugs only on the basis of chemical structure would create numerous categories,which would not necessarily be indicative of their therapeutic use. New chemical entities ideallyshould be classified on the basis of their potential therapeutic activity as early as possible inthe drug discovery process. Availability of an advanced classification model or tool that uses astandardized physiological read-out (e.g. the electroencephalogram (EEG)) would greatly aidefficient determination of psychoactive properties of newly synthesized chemicals.

Pharmacoelectroencephalographical studies aim to characterize psychotropic drug effects,usually on the basis of spectral EEGs, which reflect cortical brain activity. Frequency mea-surements in hertz range from below 3.5 Hz s−1 (so-called δ-activity), 4–7.5 Hz s−1 (θ-activity),8–12 Hz s−1 (α-activity) and above 13 Hz s−1 (β-activity). EEG registrations are reliably carriedout in humans and mammals alike. In rodents the EEG can be used to determine sleep–wakearchitecture, when carried out in conjunction with monitoring movement and a so-called electro-myogram (EMG) that records muscle activity. Typically, six sleep–wake stages are distinguishedregardless of the treatment that is received:

(a) active wake (the AW stage), which is characterized by movement, θ-activity and highEMGs,

(b) passive wake (the PW stage), without movement,(c) light sleep (the SWS1 stage), which is characterized by EEG spindles (short bursts of

phasic brain activity, which are indicative of transitions in neuronal synchronization),(d) deep sleep (the SWS2 stage), with slow waves and prominent δ-activity,(e) intermediate stage sleep (the IS stage), with spindle-like activity against a background of

θ-activity and low EMGs, and(f) rapid eye movement sleep (the RS stage), with θ-activity and very low EMGs.

From a statistical point of view, analysing EEG data poses some important challenges. First,there is the high dimensionality of raw EEG data. Even after the usual initial reduction of dimen-sionality involving a spectral analysis, in which the power spectrum is subdivided into several,predefined frequency bands (e.g. δ and θ), there is a multitude of variables to be analysed.

Secondly, a pharmacoelectroencephalographical study usually consists of subjects that aremeasured repeatedly over time; hence we are dealing with longitudinal data. For this type ofdata, the mixed effects model is a flexible and widely used approach.

Thirdly, there is no generally accepted functional form for the evolution of EEG activity overtime. The longitudinal profiles are usually highly dynamic and unpredictable within a givenshort timeframe; the variability both between and within subjects can be considerable under theinfluence of a certain dosage of the pharmacological treatment, whereas the variance is mostlynon-constant over time. Thus, finding a suitable statistical model is a non-trivial task.

Lastly, given these complexities it is evident that the psychoactivity of a novel drug (i.e. theprediction to which psychoactive class it belongs at a certain dose) cannot easily be discrim-inated. Although conventional discriminant analysis can be used, a fully satisfactory answerrequires appropriately tailored methods.

The aim of the present analysis is therefore to improve classification of psychotropic phar-macological agents into one of the five major classes of psychoactivity or placebo, on the basisof the sleep–wake behaviour of rats as defined by EEG, EMG and locomotor activity.

In the next section, the data will be explained, whereas in Section 3 we turn to methodology,including mixed model methodology, fractional polynomials and discriminant analysis. Thelast is a new, so-called two-step approach, where classification of agents in the second step isbased on appropriate summaries from the model that is built in the first step. Moreover, the

Psychotropic Drug Classification 225

second step proceeds in a hierarchical way. In Section 4, the data are analysed and thereafterthe findings are discussed, in particular in the light of an existing method that was proposed byRuigt et al. (1993).

2. Description of the data

This study includes 26 psychoactive agents at four different doses, including dose 0. To eachcompound, 32 rats are randomly assigned, i.e. eight per dose group. After a wash-out periodof 3 weeks, the same rats can be used in another experiment. In total, 342 rats are includedin the study. Most of the rats are used only once whereas some are used up to eight times.Note that the same compound may belong to different classes at different doses; this is whythe rat level and the compound-by-dose combination will be the levels of interest in ourmethodology.

The brain signals of the rats are monitored during 16 h, which are divided into a light periodof 10 h and a period of darkness of 6 h. The administration of the treatment is done at thebeginning of the light period. For every interval of 30 min the time that is spent in eachof the six sleeping stages is measured. This leads to 32 measurements per sleeping stage perrat.

From these data, a training and a test data set are constructed. In both cases, expert knowledgeis available regarding the class to which a compound-by-dose combination belongs, even thoughthe combinations in the training data set have been used more extensively in clinical practice. Itturns out that there are 64 compound–dose combinations in the training data set: 26 placebos,14 antidepressants, seven antipsychotics, two anxiolytics, five hypnotics and 10 stimulants. Inthe test data set, which is less extensively used in clinical practice, there are four antidepres-sants, two antipsychotics, two hypnotics and three stimulants. For the anxiolytics, only twocompound–dose combinations were available and dividing them over both data sets is thereforenot possible. To solve this problem, 10-fold cross-validation was used, at compound–dose leveland at rat level. This will be explained further in the next section.

3. Methodology

Several flexible modelling techniques can be considered to describe our data. One option is arandom-splines approach (Verbyla et al., 1999; Ruppert et al., 2003). Alternatively, fractionalpolynomial methods can be considered (Royston and Altman, 1994). Both routes were followedbut, although they perform similarly in terms of model fit, the fractional polynomial methodis less computationally demanding and requires fewer parameters. Therefore, for conciseness,only the fractional polynomial results will be presented here.

The fractional polynomial linear mixed model will be described first, to be followed by theclassification methodology. These two methodologies will be used in the first and second phasesrespectively of what will be termed doubly hierarchical discriminant analysis (DHDA).

3.1. Fractional polynomial mixed modelMixed effects models are a widely used approach in the modelling of longitudinal data (Verbekeand Molenberghs, 2000). To capture the irregular trends in the individual profiles, an even moreflexible model is needed. To capture the irregularities that were observed in our mean profiles,we shall use a fractional polynomial mixed model.

https://www.researchgate.net/publication/14921322_Computer-Based_Prediction_of_Psychotropic_Drug_Classes_Based_on_a_Discriminant_Analysis_of_Drug_Effects_on_Rat_Sleep?el=1_x_8&enrichId=rgreq-4fe89960-75ff-4b63-908e-ebd1f5275cba&enrichSource=Y292ZXJQYWdlOzQ3NzI5MjQ7QVM6MTAxMDE2MTU0NDExMDE0QDE0MDEwOTU1MzY0Njg=

https://www.researchgate.net/publication/233621526_Regression_Using_Fractional_Polynomials_of_Continuous_Covariates_Parsimonious_Parametric_Modeling?el=1_x_8&enrichId=rgreq-4fe89960-75ff-4b63-908e-ebd1f5275cba&enrichSource=Y292ZXJQYWdlOzQ3NzI5MjQ7QVM6MTAxMDE2MTU0NDExMDE0QDE0MDEwOTU1MzY0Njg=

https://www.researchgate.net/publication/227390047_Semiparametric_Regression?el=1_x_8&enrichId=rgreq-4fe89960-75ff-4b63-908e-ebd1f5275cba&enrichSource=Y292ZXJQYWdlOzQ3NzI5MjQ7QVM6MTAxMDE2MTU0NDExMDE0QDE0MDEwOTU1MzY0Njg=

https://www.researchgate.net/publication/227623257_The_Analysis_of_Designed_Experiments_and_Longitudinal_Data_by_Using_Smoothing_Splines?el=1_x_8&enrichId=rgreq-4fe89960-75ff-4b63-908e-ebd1f5275cba&enrichSource=Y292ZXJQYWdlOzQ3NzI5MjQ7QVM6MTAxMDE2MTU0NDExMDE0QDE0MDEwOTU1MzY0Njg=


Let us first introduce some notation. Let Yij denote the 32-dimensional vector of measure-ments that are available for rat j in compound–dose combination i. A different linear mixedmodel will be fitted for every compound–dose combination, amounting to 64 models.

A fractional polynomial of degree m over time (Royston and Altman, 1994) is defined by

φm.t;β, p/=m∑

r=0βr Hr.t/, .1/

where m is a positive number, t is the vector of all time periods, t= .1, . . . , 32/′, p is a real-valuedvector of powers with p1 � . . .�pm and β0, β1, . . . , βm are real-valued coefficients. We set p0 =0,H0.t/=1 and, for r =1, . . . , m,

Hr.t/=⎧⎨⎩

ln.t/ if pr =0 and pr �=pr−1,tpr if pr �=pr−1,Hr−1.t/ ln.t/ if pr =pr−1.

.2/

Royston and Altman (1994) argued that fractional polynomials with degree higher than 2 or 3are rarely required in practice and that powers can be restricted to the set Ω={−2, −1:5, −1,−0:5, 0, 0:5, 1, 1:5, 2}. Therefore, second-degree fractional polynomials with random effects willbe used. For all 45 pairwise combinations of p1 and p2 in Ω, a fractional polynomial model wasfitted. To avoid computational difficulties the covariates tp1i

ij and tp2iij are standardized.

The likelihood values, obtained for the 45 models, are sorted and the powers p1 and p2 thatlead to the largest likelihood are retained.

In our particular case, for each compound–dose combination and each sleeping stage, adifferent model is fitted for the light period, the dark period and the first 3 h, resulting in 18models per compound–dose combination. Not only the coefficients but also the powers p1 andp2 can be different for different compound–dose combinations. For example, for the minutesthat were spent in the AW stage in time period k for rat j in compound–dose combination i themodel is given by

.AW min/ijk ={

β0i +b0ij + .β1i +b1ij/tp1ilk −E.tp1il/√

var.tp1il/+ .β2i +b2ij/

tp2ilk −E.tp2il/√

var.tp2il/

}I.tk/

+{γ0i + c0ij + .γ1i + c1ij/

tp1idk −E.tp1id /√

var.tp1id /+ .γ2i + c2ij/

tp2idk −E.tp2id /√

var.tp2id /

}{1−I.tk/}

+ "ijk, .3/

where .AW min/ijk is the number of minutes spent in state AW for rat j in compound–dosecombination i during the kth time period .i = 1, . . . , 64, j = 1, . . . , 8 and k = 1, . . . , 32/. Thethird index l refers to the light period; d to the dark period. The vectors βi = .β0i, β1i, β2i/ andγi = .γ0i, γ1i, γ2i/ are the compound–dose-specific regression coefficients for the light and thedark period respectively, whereas bij = .b0ij, b1ij, b2ij/ and cij = .c0ij, c1ij, c2ij/ are the random-effects or rat-specific coefficients. The random-effects bij and cij are assumed to be independentwith distributions N.0, Db

i / and N.0, Dci / respectively, where Db

i and Dci are unstructured 3×3

matrices. The residual components "ijk are also independent with distribution N.0, σ2i /. The

function I.t/ is an indicator function specified as

I.t/={

1 if t �20,0 otherwise.

This modelling approach allows for a jump between the light and dark period, and a differentbehaviour of the model, in agreement with the biology of the experiment.




Given that the drugs are administered at the beginning of the light period and on thebasis of experts’ belief that the action may be quite different during the initial period, it issensible to allow for a different, perhaps more pronounced, action of the drug during thefirst 3 h after administration. Therefore, we choose to consider a separate model for thefirst 3 h:

.AW min/ijk = δ0i +d0ij + .δ1i +d1ij/tp1ifk −E.tp1if /√

var.tp1if /+ .δ2i +d2ij/

tp2ifk −E.tp2if /√

var.tp2if /+ "ijk: .4/

3.2. Classification procedureLet us first describe a discriminant analysis based on fractional polynomials. To classify psy-choactive compound–dose combinations into one of the five major classes, we need to obtaina good summary of the data, which for each rat is a highly irregular longitudinal profile. Frac-tional polynomials with mixed effects will be used for this. The parameters in the model willthen be used as input in the classification procedure.

To optimize the results, we shall proceed in a stepwise hierarchical way. This approach willbe termed DHDA. In the first step we discriminate, for example, stimulants from the rest byusing a linear discriminant function with the parameters describing the longitudinal profile ofsome sleep–waking stages for the three different periods considered: the first 3 h and the lightand dark periods. Next, we focus on the five remaining classes. This process continues until acomplete decision tree, or classification tree, has been built. In this way, different sleep–wakingstages can be used to discriminate a specific drug class, leading to a more flexible discriminantprocedure.

The term DHDA refers to the hierarchical structure in our data (which is modelled by usinghierarchical models) on the one hand, and the hierarchical character of the classification pro-cedure on the other hand.

Originally, the sleep–waking stages that were used in each step were determined by the resultsof the exploratory analysis of the training and test data sets (the details are not shown), combinedwith prior pharmacological knowledge of the experts. Additionally, we consider two alternativeselection procedures based on 10-fold cross-validation.

The 10-fold cross-validation approaches can be implemented by first using the 512 rats of thedata set, randomly divided into 10 groups (CVrat10). For every parameter combination that isobtained from the fractional polynomial models and for each sleep–waking stage, we use oneof the 10 samples as test data sets, while the remaining nine groups act as a training set. Forthis new training and test data set, the misclassification error and the posterior probabilitiesare calculated. The combination of sleep–waking stages performing best in terms of poster-ior probabilities and misclassification error is retained. This is repeated for every step in theDHDA.

In the second approach, 10-fold cross-validation is performed on the compound–dose com-bination level (CVcomb10). We randomly divide the 64 compound–dose combinations in thedata set into 10 groups and then proceed in the same way as described above.

The posterior probabilities, for a hierarchical process consisting of k steps in which a binarysplit is made, need to be adjusted. For the first step, we calculate the posterior probabilities P1(the probability of belonging to the class that we want to discriminate in step 1) and Q1 =1−P1.The posterior probabilities of the next step are then P2 =Q1P ′

2 and Q2 =Q1Q′2, where P ′

2 andQ′

2 are the unadjusted posterior probabilities for the second step. In general, for the ith split,the adjusted posterior probabilities are Pi =Q.i−1/P

′i and Qi =Q.i−1/Q

′i, P ′

i and Q′i being the

unadjusted posterior probabilities that are obtained in step i.


4. Results and discussion

For each rat and within each time period, the number of minutes that are spent in each of thesix sleep–waking stages are measured in 32 consecutive time intervals of 30 min. We first havea closer look at the fractional polynomial mixed models for all the classes before we use theparameters of these models in our DHDA.

4.1. Fractional polynomial mixed modelAs a prelude to our DHDA, we model, for every compound–dose combination, the number ofminutes that are spent in the six sleep–waking stages by a fractional polynomial model withmixed effects. As explained in Section 3.1 a separate model is fitted for the light period, the darkperiod and the first 3 h, defined by models (3) and (4).

Fig. 1 shows the fitted models in the light and the dark period for one compound–dose com-bination in each of the six classes. The mean fitted profiles are represented by broken curves,with the full curve depicting the mean observed profile over all rats in the compound–dose com-bination. For each of the classes, the fitted profile follows nicely the mean evolution over time.

4.2. Doubly hierarchical discriminant analysisThe training data set is used to build a doubly hierarchical discriminant rule, following the prin-ciple that was laid out in Section 3.2. For each compound–dose combination, we shall derivefive variables, based on the empirical Bayes estimates, obtained from the fractional polynomi-al mixed model. Details on the empirical Bayes estimation method, which are of an entirelystandard linear mixed model type, can be found in Verbeke and Molenberghs (2000).

We shall do so for every response variable in the light period, i.e. β0i +b0ij, β1i +b1ij, β2i +b2ij,p1il and p2il, in the dark period, γ0i + c0ij, γ1i + c1ij, γ2i + c2ij, p1id and p2id, and also for thefirst 3 h of the light period, δ0i + d0ij, δ1i + d1ij, δ2i + d2ij, p1if and p2if , where i = 1, . . . , 64.For each of these new variables, the number of ‘observations’ equals the number of rats in thecompound–dose combination.

We sequentially discriminated for stimulants; then anxiolytics, antipsychotics, antidepres-sants and finally hypnotics are separated from the placebo. This choice was guided by consid-ering the results from the exploratory phase, supplemented with pharmacological informationfrom the experts. Arguably, in general, such a choice will always have a somewhat subjectivecomponent to it and ought to be guided by substantive considerations.

Originally, the sleep–waking stages that were used in each step were determined ad hoc, onthe basis of the results of the exploratory analysis and prior expert’s knowledge. For example, itis known that a stimulant induces the AW stage and reduces light and deep sleep stages SWS1and SWS2. This information is used in the first step. In Table 1 the sleep–waking stages that wereused in the various steps of the DHDA are presented. Parameters for a certain sleep–wakingstage in the light period and the first 3 h are never included in the same step, given that they aredescribing essentially the same period. The fact that a particular sleep–wake stage is retained ina particular step in the DHDA does not mean that it is non-zero in a particular period for theclass that we want to differentiate from the rest. It merely means that the behaviour of the ratsfor this particular compound–dose combination within this particular class is different fromthat of rats not belonging to this class, relative to the sleeping stage being scrutinized.

The results of this method for the training data set, using leave-one-out cross-validation, aredisplayed in Table 2. For every psychotropic class, the number of rats that were classified in eachof the six classes is given. The rows in Table 2 correspond to the true classes; the columns to the


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Table 1. Sleep–waking stages that were used in each step of the DHDA

Step Compound States used in the following periods:

Light period Dark period First 3 h

1 Stimulant AW SWS1 SWS2 RS AW IS PW2 Anxiolytic PW SWS1 SWS1 SWS2 IS3 Antipsychotic PW SWS2 AW SWS2 IS AW IS RS4 Antidepressant AW SWS2 AW SWS1 IS RS5 Hypnotic SWS1 PW AW PW SWS2 RS

Table 2. Summary of the hierarchical discrimination procedure for the training data set (number of obser-vations classified into each drug class and posterior probabilities)

Class Placebo Antidepressant Antipsychotic Anxiolytic Hypnotic Stimulant

Placebo 202 (0.77) 0 (0.13) 0 (0.07) 0 (0.00) 2 (0.00) 0 (0.01)Antidepressant 11 (0.14) 97 (0.78) 0 (0.02) 0 (0.00) 4 (0.05) 0 (0.01)Antipsychotic 1 (0.08) 1 (0.02) 54 (0.86) 0 (0.00) 0 (0.00) 0 (0.04)Anxiolytic 0 (0.00) 0 (0.00) 0 (0.00) 16 (0.96) 0 (0.04) 0 (0.00)Hypnotic 3 (0.10) 0 (0.18) 0 (0.00) 0 (0.06) 29 (0.66) 0 (0.00)Stimulant 0 (0.00) 1 (0.01) 2 (0.04) 4 (0.04) 0 (0.00) 73 (0.91)

Table 3. Summary of the hierarchical discrimination procedure for the validation data set (number of obser-vations classified into each drug class and posterior probabilities)


Antidepressant 2 (0.16) 29 (0.59) 0 (0.09) 0 (0.00) 1 (0.04) 0 (0.12)Antipsychotic 0 (0.00) 0 (0.00) 16 (0.99) 0 (0.00) 0 (0.01) 0 (0.00)Hypnotic 0 (0.00) 0 (0.23) 0 (0.04) 0 (0.00) 12 (0.73) 0 (0.00)Stimulant 0 (0.00) 0 (0.02) 1 (0.03) 0 (0.00) 0 (0.00) 23 (0.95)

predicted classes. In all groups, the number of correctly classified subjects is very high. Theresulting overall error rate for the training data set is about 4.5%.

The adjusted posterior probabilities for the training data set are shown in Table 2, showingposterior probabilities for correct classification exceeding 0.75 for all the classes except hypnotics.

The classification numbers for the validation data set are shown in Table 3. Recall that thereare no compound–dose combinations in the placebo and anxiolytic groups. For the four remain-ing groups, only a few rats are misclassified. The overall error rate for the validation data set isabout 2.3%.

However, the posterior probabilities, that are shown in Table 3, reveal that it can be difficultto classify antidepressants and hypnotics. The posterior probability for hypnotic compoundsto be classified as antidepressants is around 25%, whereas antidepressant compounds can alsopresent placebo and stimulant activities. Antipsychotics and stimulants can be discriminatedfrom the others in an almost error-free way.


Table 4. Sleep–waking stages that were used in each step of the DHDA with 10-foldcross-validation by rat



1 Stimulant PW SWS2 AW SWS1 RS AW SWS12 Anxiolytic PW SWS1 SWS2 IS RS AW SWS13 Antipsychotic PW SWS2 IS AW PW SWS2 AW4 Antidepressant PW IS RS SWS1 IS AW SWS15 Hypnotic AW SWS1 SWS2 IS RS

Table 5. Summary of the hierarchical discrimination procedure for the training data set, using cross-validation by rat (number of observations classified into each drug class and posterior probabilities)



Not all the classes are present in the test data set and the true classification of the compoundsin the test data set is less obvious compared with the training data set. Therefore it would bebetter if we could construct a training and test data set where all classes are present. However,this is not possible, because of the limited amount of compounds. A more realistic alternative isto use 10-fold cross-validation to determine in a more formal way which variables will be usedin each step of the discriminant analysis. In the next sections we shall present the results of thetwo alternative approaches that were described in Section 3.2.

4.2.1 10-fold cross-validation on rat level (CVrat10)The sleep–waking stages that were selected in each of the steps by 10-fold cross-validation usingthe rat level as explained in Section 3.2 are displayed in Table 4. As before, parameters for acertain sleep–waking stage in the light period and the first 3 h are never included in the samestep.

Table 5 shows the number of observations classified in the six classes per class, obtained by10-fold cross-validation using the rat level. The number of correctly classified observations isvery high for all the classes. The overall error rate is about 1.1%.

The adjusted posterior probabilities are shown in Table 5. The posterior probabilities forcorrect classification are all above 90%.

4.2.2 10-fold cross-validation on compound–dose level (CVcomb10)The sleep–waking stages that were selected by 10-fold cross-validation on compound–dose levelare shown in Table 6.


Table 6. Sleep–waking stages used in each step of the DHDA with 10-foldcross-validation by compound–dose combination



1 Stimulant SWS1 SWS1 RS AW PW2 Anxiolytic SWS1 IS PW SWS2 RS3 Antipsychotic PW SWS1 SWS2 IS AW SWS24 Antidepressant IS RS PW SWS2 IS AW SWS15 Hypnotic AW SWS2 AW PW RS

Table 7. Summary of the hierarchical discrimination procedure for the training data set, using 10-foldcross-validation by compound–dose combination (number of observations and proportion classified into eachdrug class)



The number of compound–dose combinations that were classified in each of the six classesis given for each psychotropic class in Table 7. The resulting overall error rate is now 17.6%,which is much higher if we compare it with the results that were presented in the previoussection. First, additional randomness is introduced in CVcomb10 due to random sampling.Second, the data set that was considered contains relatively few compound–dose combinations.Third, and very fundamentally, leaving out rats and leaving out combinations do not assess thesame aspects. Indeed, by removing rats from a combination, we can still estimate, admittedlysomewhat less precisely, all the parameters that are associated with such a combination. This isobviously not true when an entire combination is removed. Therefore, each method focuses ondifferent aspects of variability, associated with rat level or compound–dose combination levelreplication. Thus, there is room for both.

In Table 7 the adjusted posterior probabilities are presented, showing posterior probabilitiesfor correct classification above 80% for placebo, antidepressants and stimulants. The posteriorprobabilities for antipsychotics, hypnotics and anxiolytics are around 60%. Although the pos-terior probabilities and classification percentages are still high, here also we can see that theresults that were obtained by cross-validation using the rat level seem more promising.

5. Conclusion and further research

In this paper we proposed a method to classify compounds with psychotropic potential intothe standard classes that were defined by Deniker (1982) and have been in use for severaldecades.


The data that were used to perform such a classification consist of longitudinal profiles whichwere collected on rats during 16-h periods, over six standard sleep–waking stages.

Owing to the multivariate longitudinal nature of the profiles, classical discriminant analysiscannot be used. Work on discriminant analysis for this type of setting is limited (Verbeke andMolenberghs, 2000). Therefore, a specific longitudinal modelling approach, coupled with whatwe term DHDA, has been proposed.

The performance of the method in the motivating set of data is satisfactory, certainly com-pared with the method that was proposed by Ruigt et al. (1993). In this method, only the first8 h after drug injection are considered. For every sleep–wake stage, the 30-min periods areregrouped in a particular way. For example, for the AW stage the following periods are consid-ered: 0.5–3 h; 3–5 h; 5–8 h. In every period the change in percentage of time spent in a certainsleep–waking stage compared with the placebo group for the compound under investigation isrecorded. These variables are then used in a linear discriminant analysis.

For the training data set, Ruigt et al. (1993) could classify correctly only half of the 12antidepressants, one out of four anxiolytics and one out of four hypnotic compounds. All fourstimulants in the training data set were correctly classified as well as six out of seven antipsy-chotics. For the test data set, nine out of 30 antidepressants, six out of 12 antipsychotics, fourout of nine anxiolytics, one out of two stimulants and all three hypnotics were misclassified.Applying the method of Ruigt et al. (1993) to our data set, we found that only 44% of the ratswere correctly classified, as opposed to the ad hoc 95% that were reached with the method thatis proposed in this paper. The results that were obtained with the ad hoc method are in generalcomparable with those obtained for 10-fold cross-validation when we use the level rat. However,if the compound–dose combination level approach is used, then the results are less promising. Itis also worth noting that the two alternative approaches using cross-validation can be applied toselect the sleep–waking stages that are useful to classify the psychotropic classes. One possibledrawback of this method is the selection of the level to be used in the cross-validation. Anothercould be the fact that several thousand models can be used to discriminate a particular class fromthe rest and we are selecting just the model which performs best in terms of misclassificationerror and posterior probability; then introducing model selection uncertainty.

In general all methods have some difficulty in discriminating between placebo and anti-depressant components and, to a lesser extent, between hypnotic and antidepressant drugs.Although at first sight this is a drawback, we should approach such a conclusion with due cau-tion. First, it is conceivable that a given component at a certain dose has more than one mode ofactivity. Second, the very classification into psychotropic classes, although it is generally used,remains arbitrary and should perhaps be called into question. At least, a revision might beappropriate.

To discriminate a given class from all the others, we have been using estimated parameters,all five for each of the sleeping stage–period combinations, as a summary measure of the com-pound–dose combination. It is to be expected that such combinations, belonging to a particularpsychotropic drug class, behave similarly and thus their summary measure should be similaralso. The reader will note that the sleep–wake stages that are used by the three methods aredifferent. Nevertheless, we point out that a good number of common factors are still presentbetween them. For example, the AW stage is present in the three models to discriminate thestimulant from the rest; this is important from a biological point of view. Another appealingexample is the case of hypnotic compounds, which are known to act primarily on light and deepsleep; the three methods select them to discriminate this class from the others.

Further, note that both longitudinal modelling as well as DHDA should be seen as a generalframework rather than a single technique. Instead of fractional polynomial mixed models, other


longitudinal modelling techniques can be used to describe the data. Also, for the discriminantanalysis, other techniques such as flexible or mixture discriminant analysis can be considered.

An important point is the order in which the steps are conducted. To explore sensitivity, wehave experimented with alternative orders, and the results essentially remained the same. Suchareas of sensitivity, not only with respect to the order, but also regarding model selection, areimportant and are currently being studied, as part of a broader simulation study, addressingboth issues of sensitivity and computational complexity.

To conclude, there is the philosophical question whether having knowledge about the classof the compound, before starting the analysis, is not somehow circular. However, we observethat this is not an uncommon situation in statistics, the simplest and best-known example beingsample size calculations for which knowledge about the quantities to be estimated later is neededat the design stage. However, a key point is that, at such an initial stage, it is often sufficientto have a plausible guess or, put differently, prior knowledge or belief. This applies to our caseas well. Assume, for example, that we are trying to classify a compound in one of the classes,whereas it actually belongs to an entirely different class. It is then realistic to expect that theposterior probabilities will not clearly indicate a ‘winner’ but rather will be distributed fairlyevenly across all classes. At any rate, it is important for the user of the methodology to be awareof this problem.

Acknowledgements

The authors gratefully acknowledge the financial support from the Interuniversity AttractionPoles research network P6/03 ‘Statistical techniques and modeling for complex substantivequestions with complex data’ of the Belgian Government (Belgian science policy).

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psychotropic drug classification based on sleep?wake behaviour of rats

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