Br. J. clin. Pharmac. (1991), 32, 159-166
A D 0 N I S 030652519100162Q
A new, sensitive graphical method for detecting deviations from the normal distribution of drug responses: the NTV plot LASZLO ENDRENYI & MAYANK PATEL University of Toronto, Department of Pharmacology, Toronto, Ontario M5S 1A8, Canada
1 A new graphical method was developed for the detection of deviations from the normal distribution. The approach took advantage of the similarity of graphical features of a graded dose-response relationship and a cumulative normal distribution. 2 The behaviour of the new normal test variable (NTV) plot was evaluated, in comparison with that of the probit plot and probability density functions (the generalization of histograms), for various assumed distributions. These included skewed distributions and composites of normal distributions with a variety of separations, ratios of peak sizes and widths. 3 The NTV approach generally detected deviations from the normal distribution more sensitively than the probit plot. 4 The NTV and probit plots may be able to identify bimodality by complementary approaches. 5 The characteristics of the three graphical representations were illustrated by a simulated sample from a composite of normal distributions and by an example of sparteine metabolism in 142 Cuna Amerindians.
Keywords drug polymorphism graphical method
bimodal distributions
normal distribution
Introduction
Methods
Jackson et al. (1989a,b,c) have recently called attention both to the importance of identifying the possible multiplicity of population distributions and to the related difficulties. Knowledge of the features of observed distributions is essential, e.g. for recognizing if in vivo measurements of drug metabolism suggest the polymorphism of metabolic activity (Kalow 1982, 1986; Vesell, 1984; Weinshilboum, 1984). Jackson et al. (1989a,b,c) considered the behaviour of graphical methods such as histograms and probit plots customarily applied for indentifying deviations from the normal distribution, and also demonstrated features of statistical procedures. Both approaches were shown to be distressingly insensitive to detect deviations from the normal distribution and, in particular, to reveal the multiplicity of distributions. Evans et al. (1983) and Jack (1983) described additional statistical considerations for the analysis of data evaluating polymorphic drug metabolism. We shall describe a new graphical method for the identification of deviations from the normal distribution. The procedure is simple and very sensitively detects such deviations including the multiplicity of distributions.
The normal test variable (NTV) plot
Procedure for detecting deviations from the normal distribution Convert the doses or concentrations (c) to their logarithmic form: 1 = lnc or L = logloc Obtain the median (ml or mL) and standard deviation (sl or sL) of the logarithmic concentrations. Calculate the standardized form of each logarithmic concentration: x = (1-m)lsl = (L-mL)/sL (1) Record the cumulative number (n) of subjects, all of whom show a defined response up to a given dose or concentration of a drug. Calculate the corresponding cumulative frequency (F) by dividing the cumulative number by the total number (N) of the subjects: F= n/N. Note that the value of F is between 0 and 1. Evaluate also the secondary measure of: Z = F-(1-F)el 6x
Correspondence: Dr L. Endrenyi, University of Toronto, Department of Pharmacology, Toronto, Ontario M5S 1A8, Canada
159
160
L. Endrenyi & M. Patel
Calculate the normal test variable (NTV) as: NTV = -Z if F 0.5 NTV = Zel .6x, and if F 0. 5 Plot NTV against F.
(2)
Rationale for the method The features of two kinds of responses will be compared. On the one hand, let us assume that a rectangular hyperbola (the Michaelis-Menten equation) describes the dependence of a graded response (y) on concentration. The same reponse shows a symmetrical and sigmoidal (S-shaped) relationship when plotted against the logarithm of the concentration. At the same time, the cumulative frequency (F) of a normally-distributed response also has a sigmoidal shape when plotted against the logarithmic concentration. It is therefore assumed in the development of this method that the responses follow a normal distribution with respect to the logarithm of the concentration. This well-known similarity can be demonstrated further. The cumulative normal distribution of a response yields, by definition, a straight line displayed in a probit diagram. The hyperbolic graded response shown in the same plot against the logarithm of concentration is almost straight. The parallelism between the behaviours of the two kinds of responses will be exploited in order to demonstrate deviations from the normal distribution. Thus, among the linearizations of the Michaelis-Menten equation, the Scatchard plot is known to detect deviations from hyperbolic behaviour particularly sensitively. The plot contrasts ykc with y. The corresponding diagram for normally distributed responses would plot F/exp(x) against F. Figure la shows that the Scatchard plot applied directly to normally distributed responses does not yield a straight line. However, if the denominator of the ordinate is empirically changed to exp(1.6x) then the right-hand side of the relationship, in the range of higher concentrations, is almost straight (Figure lb). The advantageous feature of approximate linearity can be extended to the low end of concentrations by reflection. (Recall the symmetry around the midpoint in the sigmoidal relationships). The reflection involves substituting 1/x for x, and 1-F for F, in the region when F 0.5. Figure lc illustrates the approximate linearity of the modified function and its good resemblance to a Scatchard plot. It was considered to be useful to lift, by adding F, the diagonal relationship to horizontal, and to lower it, by subtracting 1, to the level of zero (Figure ld). The results of these transformations are expressed in Equation 2 which defines the measure of NTV. -
Results
Effect of the separation between distributions Figure ld illustrates that, for a normally distributed response, NTV deviates only very slightly from a horizontal line. Furthermore, the deviations are all positive,
have maxima of 0.0185 at F = 0.05 and 0.95, and are symmetrical around F = 0.5. NTV for the normal distribution is reproduced in Figure 2. The figure shows also the distribution itself (Figure 2a) and the corresponding probit plot (Figure 2b) which, for the normal distribution, is defined to be straight. The corresponding NTV plots are shown in Figure 2c. Features are illustrated also for composite distributions. These contain two normally distributed components having identical heights and identical widths which reflect that the 2 standard deviations are the same ((cl = Co2 = (J). However, the two peaks, defined by the respective means, ,ti and R2, are separated. The two components are resolved in all representations when the true means differ by 3cF. The distributions themselves are bimodal by showing two distinct peaks. The probit plot has a fairly clear inflection point. The NTV plot descends conspicuously and takes negative values. - The three graphical representations offer particularly interesting contrasts when the true means are separated by 2or. The components of the distributions are no longer resolved. Instead, the composite distribution has a single, flat peak. The probit plot still has a theoretical inflection point but, in practice, this can not be detected clearly. By contrast, the descent of the NTV plot into negative territory is substantial and meaningful.
Effect of the size ratio of components Figure 3 shows graphical representations for combined normal distributions with various relative contributions of the two components. The illustrations assume that the true standard deviations, i.e. the widths of the two components are identical, and that their means are separated by 3 standard deviations (w1-2 = 3cr). In the frequency distributions (Figure 3a), the lesser component becomes, of course, decreasingly noticeable as its contribution is reduced. This feature is reflected by the NTV plot (Figure 3c) in which the descent into the negative region declines when the importance of the smaller component is moderated. On the other hand, the breadth of the inflection range increases in the probit plot with growing discrepancy between the two components (Figure 3b). However, this range also rises toward larger probit values which occur closer to the terminal region of the distribution. In this region, the variability of probits increases substantially and, consequently, the inflection range cannot be easily detected. Effect of relative width Figure 4 illustrates graphical representations of composite normal distributions for various relative widths of the two components. Thus, the diagram shows the effect of varying the ratio of the 2 standard deviations. The illustrations assume that the two components have, with respect to area, true relative magnitudes of 4 to 1. The modes are separated by 3 standard deviations belonging to the larger peak. Detection of the lesser component becomes more sensitive as its relative standard deviation decreases (Figure 4a). This is intuitively reasonable since the smaller component appears more sharply when its relative width
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Detection of deviations from normal distribution
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A D 0 N I S 030652519100162Q
A new, sensitive graphical method for detecting deviations from the normal distribution of drug responses: the NTV plot LASZLO ENDRENYI & MAYANK PATEL University of Toronto, Department of Pharmacology, Toronto, Ontario M5S 1A8, Canada
1 A new graphical method was developed for the detection of deviations from the normal distribution. The approach took advantage of the similarity of graphical features of a graded dose-response relationship and a cumulative normal distribution. 2 The behaviour of the new normal test variable (NTV) plot was evaluated, in comparison with that of the probit plot and probability density functions (the generalization of histograms), for various assumed distributions. These included skewed distributions and composites of normal distributions with a variety of separations, ratios of peak sizes and widths. 3 The NTV approach generally detected deviations from the normal distribution more sensitively than the probit plot. 4 The NTV and probit plots may be able to identify bimodality by complementary approaches. 5 The characteristics of the three graphical representations were illustrated by a simulated sample from a composite of normal distributions and by an example of sparteine metabolism in 142 Cuna Amerindians.
Keywords drug polymorphism graphical method
bimodal distributions
normal distribution
Introduction
Methods
Jackson et al. (1989a,b,c) have recently called attention both to the importance of identifying the possible multiplicity of population distributions and to the related difficulties. Knowledge of the features of observed distributions is essential, e.g. for recognizing if in vivo measurements of drug metabolism suggest the polymorphism of metabolic activity (Kalow 1982, 1986; Vesell, 1984; Weinshilboum, 1984). Jackson et al. (1989a,b,c) considered the behaviour of graphical methods such as histograms and probit plots customarily applied for indentifying deviations from the normal distribution, and also demonstrated features of statistical procedures. Both approaches were shown to be distressingly insensitive to detect deviations from the normal distribution and, in particular, to reveal the multiplicity of distributions. Evans et al. (1983) and Jack (1983) described additional statistical considerations for the analysis of data evaluating polymorphic drug metabolism. We shall describe a new graphical method for the identification of deviations from the normal distribution. The procedure is simple and very sensitively detects such deviations including the multiplicity of distributions.
The normal test variable (NTV) plot
Procedure for detecting deviations from the normal distribution Convert the doses or concentrations (c) to their logarithmic form: 1 = lnc or L = logloc Obtain the median (ml or mL) and standard deviation (sl or sL) of the logarithmic concentrations. Calculate the standardized form of each logarithmic concentration: x = (1-m)lsl = (L-mL)/sL (1) Record the cumulative number (n) of subjects, all of whom show a defined response up to a given dose or concentration of a drug. Calculate the corresponding cumulative frequency (F) by dividing the cumulative number by the total number (N) of the subjects: F= n/N. Note that the value of F is between 0 and 1. Evaluate also the secondary measure of: Z = F-(1-F)el 6x
Correspondence: Dr L. Endrenyi, University of Toronto, Department of Pharmacology, Toronto, Ontario M5S 1A8, Canada
159
160
L. Endrenyi & M. Patel
Calculate the normal test variable (NTV) as: NTV = -Z if F 0.5 NTV = Zel .6x, and if F 0. 5 Plot NTV against F.
(2)
Rationale for the method The features of two kinds of responses will be compared. On the one hand, let us assume that a rectangular hyperbola (the Michaelis-Menten equation) describes the dependence of a graded response (y) on concentration. The same reponse shows a symmetrical and sigmoidal (S-shaped) relationship when plotted against the logarithm of the concentration. At the same time, the cumulative frequency (F) of a normally-distributed response also has a sigmoidal shape when plotted against the logarithmic concentration. It is therefore assumed in the development of this method that the responses follow a normal distribution with respect to the logarithm of the concentration. This well-known similarity can be demonstrated further. The cumulative normal distribution of a response yields, by definition, a straight line displayed in a probit diagram. The hyperbolic graded response shown in the same plot against the logarithm of concentration is almost straight. The parallelism between the behaviours of the two kinds of responses will be exploited in order to demonstrate deviations from the normal distribution. Thus, among the linearizations of the Michaelis-Menten equation, the Scatchard plot is known to detect deviations from hyperbolic behaviour particularly sensitively. The plot contrasts ykc with y. The corresponding diagram for normally distributed responses would plot F/exp(x) against F. Figure la shows that the Scatchard plot applied directly to normally distributed responses does not yield a straight line. However, if the denominator of the ordinate is empirically changed to exp(1.6x) then the right-hand side of the relationship, in the range of higher concentrations, is almost straight (Figure lb). The advantageous feature of approximate linearity can be extended to the low end of concentrations by reflection. (Recall the symmetry around the midpoint in the sigmoidal relationships). The reflection involves substituting 1/x for x, and 1-F for F, in the region when F 0.5. Figure lc illustrates the approximate linearity of the modified function and its good resemblance to a Scatchard plot. It was considered to be useful to lift, by adding F, the diagonal relationship to horizontal, and to lower it, by subtracting 1, to the level of zero (Figure ld). The results of these transformations are expressed in Equation 2 which defines the measure of NTV. -
Results
Effect of the separation between distributions Figure ld illustrates that, for a normally distributed response, NTV deviates only very slightly from a horizontal line. Furthermore, the deviations are all positive,
have maxima of 0.0185 at F = 0.05 and 0.95, and are symmetrical around F = 0.5. NTV for the normal distribution is reproduced in Figure 2. The figure shows also the distribution itself (Figure 2a) and the corresponding probit plot (Figure 2b) which, for the normal distribution, is defined to be straight. The corresponding NTV plots are shown in Figure 2c. Features are illustrated also for composite distributions. These contain two normally distributed components having identical heights and identical widths which reflect that the 2 standard deviations are the same ((cl = Co2 = (J). However, the two peaks, defined by the respective means, ,ti and R2, are separated. The two components are resolved in all representations when the true means differ by 3cF. The distributions themselves are bimodal by showing two distinct peaks. The probit plot has a fairly clear inflection point. The NTV plot descends conspicuously and takes negative values. - The three graphical representations offer particularly interesting contrasts when the true means are separated by 2or. The components of the distributions are no longer resolved. Instead, the composite distribution has a single, flat peak. The probit plot still has a theoretical inflection point but, in practice, this can not be detected clearly. By contrast, the descent of the NTV plot into negative territory is substantial and meaningful.
Effect of the size ratio of components Figure 3 shows graphical representations for combined normal distributions with various relative contributions of the two components. The illustrations assume that the true standard deviations, i.e. the widths of the two components are identical, and that their means are separated by 3 standard deviations (w1-2 = 3cr). In the frequency distributions (Figure 3a), the lesser component becomes, of course, decreasingly noticeable as its contribution is reduced. This feature is reflected by the NTV plot (Figure 3c) in which the descent into the negative region declines when the importance of the smaller component is moderated. On the other hand, the breadth of the inflection range increases in the probit plot with growing discrepancy between the two components (Figure 3b). However, this range also rises toward larger probit values which occur closer to the terminal region of the distribution. In this region, the variability of probits increases substantially and, consequently, the inflection range cannot be easily detected. Effect of relative width Figure 4 illustrates graphical representations of composite normal distributions for various relative widths of the two components. Thus, the diagram shows the effect of varying the ratio of the 2 standard deviations. The illustrations assume that the two components have, with respect to area, true relative magnitudes of 4 to 1. The modes are separated by 3 standard deviations belonging to the larger peak. Detection of the lesser component becomes more sensitive as its relative standard deviation decreases (Figure 4a). This is intuitively reasonable since the smaller component appears more sharply when its relative width
ta3¢i>Ar~ ~ 40Y
Detection of deviations from normal distribution
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Cumulative frequency (F) Figure 1 Illustration of rationale for the development of the NTV plot based on the similar features of graded dose-response curves and cumulative normal distributions. a) Cumulative normal distribution in the analogue of the Scatchard plot and b) in an empirical modification of the plot. c) The lower half of Figure lb is reflected across the midpoint. d) The curve is then rotated in the horizontal direction. Details are presented in the text. 1:1
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