Analysis of longitudinal “time series” data in toxicology

FUNDAMENTALANDAPPLIEDTOXICOLOGY8,159-169(1987)

Analysis of Longitudinal CHRISTOPHER

“Time Series” Data in Toxicology’

Cox* AND DEBORAH A. CORY-SLECHTA~

*Division of Biostatistics. and TDivision of Toxicology and Environmental Health Sciences Center, Department of Biophysics, School ofMedicine and Dentistry, University of Rochester, Rochester, New York 14642

Analysis of Longitudinal “Time Series” Data in Toxicology. Cox, C., AND CORY-SLECHTA, ( 1987). Fundam. Appl. Toxicol. 8, 159- 169. Studies focusing on chronic toxicity or on the time course of toxicant effect often involve repeated measurements or longitudinal observations of endpoints of interest. Experimental design considerations frequently necessitate between-group comparisons of the resulting trends. Typically, procedures such as the repeatedmeasures analysis of variance have been used for statistical analysis, even though the required assumptions may not be satisfied in some circumstances. This paper describes an alternative analytical approach which summarizes curvilinear trends by fitting cubic orthogonal polynomials to individual profiles of effect. The resulting regression coe5cients serve as quantitative descriptors which can be subjected to group significance testing. Randomization tests based on medians are proposed to provide a comparison of treatment and control groups. Examples from the behavioral toxicology literature are considered, and the results are compared to more traditional approaches, such as repeated-measures analysis of variance. 0 1987 Society ofToxicology. D. A.

quently involve the longitudinal assessment of effect(s), and, thus, may be considered prototypical examples. A full characterization of toxicant-induced changes in a particular behavior often requires the repeated examination of the performance over a prolonged period. Given the sustained nature of the evaluation and/or exposure in such experiments, the use of an animal as its own control may not be feasible. Furthermore, some compounds may produce irreversible effects, preventing a crossover design. Thus, the examples presented here consider an experiment with both a treatment group and a control group having, of necessity, relatively small numbers (e.g., between 5 and 15 animals) in each group. In these examples, the data from a single animal over the measurement interval will be referred to as a response profile, and the experimental data consist of a response profile for each animal in both the control and treated groups. The goal of the analysis (ultimately to be translated into null

Human exposure to toxicants, whether environmental or occupational, frequently involves sustained low level contact and gradual onset of effect. Examples include environmental exposure to inorganic lead and methyl mercury. In other cases, acute administration of compounds such as trimethyltin (e.g., Bushnell and Evans, 1985), organic lead (Tilson et al., 1982), or organophosphates produce effects that are not immediately discernible, but unfold only over subsequent intervals of time. Evaluation of the toxicological effects of such exposures clearly necessitates repeated observations and longitudinal assessment of the experimental endpoints of interest. Studies in behavioral toxicology fre’ This work was supported by Grants ES-01247, ES0 1248, ES-03054, and ES-03079 from the National Institutes of Environmental Health Sciences and under Contract DE-AC02-76EV03490 with the U.S. Department of Energy at the University of Rochester, Department of Biophysics, and has been assigned Report No. DGE/EV/ 3490-25 15.

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0272-0590187 $3.00 Copyright 0 1987 by the Society of Toxicology. All rights of reproduction in any form reserved.

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and alternative hypotheses in the context of significance testing) is the discovery and characterization of systematic differences in performance between the two groups. Such differences may conceivably take several forms. For example, the two groups may attain the same level of performance over time, but do so at different (linear) rates and/or exhibit differential curvature (quadratic or higher order) in attaining this level. Another alternative is that the groups may exhibit similar curvature without attaining comparable levels of performance. An analytical approach is required that can describe such longitudinal trends using a reasonable number of parameters or descriptors which can then be used for statistical comparison of the two groups. In this paper an approach based on orthogonal polynomial fits of the response profiles is presented. The regression coefficients from these fits are treated as derived descriptors, and analyzed by median-based randomization tests. The proposed analysis is also compared to other procedures including a more traditional repeated-measures analysis of variance based on medians of consecutive data points. METHODS

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In considering the appropriate statistical analysis of response profile data, it should be pointed out that even when serial measurements on the same animal are made at widely different times, they do not constitute truly independent measurements, because the single animal is the smallest biological entity in which treatment effects can be observed. Consequently, for purposes of statistical comparison, repeated measurements on the same animal are assumed to be correlated, as in a repeated-measures analysis of variance (RMAOV), in contrast to indepcndent measurements on different animals. Greater variability would be expected in the latter case, as compared to repeated measurements on the same animal. One approach to the complexity of longitudinal data is provided by traditional “time series” analysis. The type of studies described here focus on different aspects of the data than does classical time series analysis, however. First, the primary interest is centered on long-term trends, traditionally regarded in time series analysis as

nuisance variation to be eliminated, so that the study of periodic fluctuations (short-term trend) can proceed. Second, in these between-groups experiments, the effect of the treatment is exhibited on a group, rather than an individual, basis. Most importantly, the statistical regularity of the data may not be sufficiently great, nor the observation period sufficiently long for the usual time series methods. Thus, it is improbable that traditional time series methods, applied to such response profiles, will play a significant analytical role. An alternative strategy, when the number of measurement intervals is relatively small, has been the RMAOV. For larger numbers of observations, means or medians of consecutive blocks of data are often used. The advantage of the RMAOV approach is that it tests for differences in means; no specific type of trend is assumed, and this component of the analysis is perfectly general. In addition, standard statistical tests can bc used subsequently for post hoc analysis of differences between the two groups. The disadvantages of the RMAOV include its dependence on the assumption that random variation is normally distributed and that the correlation structure assumes a particularly simple form. Furthermore, it is conceivable that trends over time in response or effects assume a smooth, curvilinear form. If such trends can be adequately modeled, a more sensitive analysis is possible. The simplest model for a smooth trend is a polynomial. These curves have the advantage of producing interpretable parameters (intercept, linear trend, quadratic trend, etc.) and can be easily fitted to the data by least squares. The degree of the polynomial must be specified, however, since any subsequent analysis will require the same number of coefficients for each animal. The degree must also be reasonably low if a simple description of the data is to result. Our experience with data from a number of experiments (Cory-Slechta et al., 1981, 1983, 1985) is that third-degree polynomials can often adequately describe long-term response trends. Orthogonal polynomials can be used in order to isolate pure trends of each degree; addition or deletion of one term will not change any of the other coefficients. Use of third-degree polynomials results in four descriptors of performance: a constant (intercept) which is the average of the series of observations, a linear coefficient, a quadratic coefficient, and a cubic coefficient. The choice of a subsequent statistical evaluation of the descriptors was influenced by the observation that a frequent complication of such toxicology experiments is the presence of “nonresponders” in the treated group. Additionally, there may be a small number of spontaneous responders in the control group. This is a dose-response phenomenon in which the dose may not exceed the “threshold” for behavioral impairment in every treated animal, as well as one in which background response

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may occur in controls. Increasing the sample size or the toxicant dose have been two approaches commonly used to overcome such complications. However, when interest is focused on low-level exposure, or on chronic toxicity, increasing the dose is not an alternative. Additionally, sample size may be constrained both by equipment availability and timing factors. Conventional significance tests, whether parametric t tests or nonparametric rank tests such as the MannWhitney, are unlikely to be useful for detecting differences on a group basis when such dose-response phenomena are evident. As an alternative, Good (1979) has recommended randomization tests. These are completely nonparametric tests: thep value is obtained by rerandomizing the data into two groups of the appropriate size and determining how unusual the observed difference (as measured by the chosen statistic, see below) is against the background provided by all possible rearrangements of the actual numbers. The advantage over standard nonparametric procedures such as the Mann-Whitney is that the actual data, rather than the ranks, are used in the analysis, i.e., in generating the underlying distribution ofthe test statistic. Such procedures are best carried out on a computer, and Edgington (1980) has provided short FORTRAN programs which can easily be adapted to perform arbitrary tests comparing two groups. The randomization procedure may be used to generate a p value for any choice of test statistic, i.e., any method of measuring the difference between the two groups. Good ( 1979) proposed a version of the t statistic modified for sensitivity to differences in variability and skewness caused by the presence of nonresponders. We have found that tests based on the mean may lack sensitivity in a dose-response situation; tests based instead on the median are more likely to be sensitive to group differences (Hays, 1973). In the data sets presented here, a simple statistic was chosen for a two-sided comparison of the group median coefficients: the square of the difference between them as a percentage of the control median. This evolved from our experience with a number of similar data sets. An alternative statistic would be, e.g., the square of the difference in the medians. A multivariate approach, e.g., one using all the regression coefficients in one test, is also possible for the between-group comparison. Zerbe and Walker (1977) proposed one such test, also based on polynomial fits, which assessesthe distance between a pair of response profiles within a group by averaging the squared difference between the two curves. This distance depends only on the regression coefficients. The test statistic is based on the sum of the within-group distances for all pairs of curves in each group. A slightly simpler version of this test was used with our examples as a comparison to the medianbased univariate randomization tests.

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RESULTS The examples presented use data from experiments investigating the behavioral toxicity of chronic low-level lead exposure. In the first experiment, 12 randomly selected LongEvans male rats were chronically exposed to 25 ppm sodium acetate (controls) and another 12 to 25 ppm lead acetate (treated) in drinking water from weaning. At 55 days of age, evaluation of behavioral performance on a fixed-interval (FI) schedule of food reinforcement began. (Details of the methods are reported in Cot-y-Slechta et al., 1985.) The FI schedule designates that a specified response (in this case, a lever press) be reinforced (here by food delivery) only after a fixed interval of time has elapsed since the preceding delivery of food (Ferster and Skinner, 1957). One of the traditional measures of FI performance is overall response rate, or simply the number of specified responses (lever depressions) divided by the total session time. Figure 1 shows the response profiles, in this case the overall response rates, for all individual control (top panel) and lead-treated animals (bottom panel) over the first 40 sessions of the experiment. One striking difference between the two groups of response profiles emerged across sessions. Response rates of 10 of the 12 control animals were less than 15 responses/minute over the first 20 sessions, and below 25 responses/minute over the entire course of the experiment. In contrast, response rates of half of the treated animals exceeded 15 responses/min by session 20, and 9 of the 12 exceeded 25 responses/min by session 40, replicating effects previously observed at a higher concentration, 50 ppm (Cot-y-Slechta and Thompson, 1979; Cory-Slechta et al., 1983). Three lead-treated animals (E3, closed triangle; ElO, open triangle; and El 1, open diamond) exhibit “control” type performance, while two control animals (C4, closed diamond; and C6, asterisk) display response

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SESSIONS FIG. 1. Overall response rates of individual control (top panel) and 25 ppm lead acetate-treated rats (bottom panel) on a fixed interval 1-min schedule of food reinforcement over the first 40 experimental sessions. Each animal is represented by a unique symbol as follows: 0 1; n 2; A 3; * 4; v 5; * 6; X 7; 0 8; o 9; A I 0; 0 11; V 12. Data are from Cory-Slechta et al. ( 1985).

profiles resembling those characteristic of most treated animals. Figure 2 displays the constant (intercept) and quadratic regression coefficients resulting from the third-degree orthogonal polynomial fits to the individual response profiles shown in Fig. 1. The constant coefficient (intercept) can be interpreted as the average response rate level throughout the experiment. Intercept values of the control animals (circled numbers) generally cluster toward the top left comer of the plot. In accord with Fig. 1, two notable exceptions are controls C4 and C6, with constant coefficients to the extreme right of all but one treated animal, and smaller quadratic coefficients than all other animals. Experimental animals E3 (closed triangles), E 10 (open triangles), and E 11 (open diamonds) show response rate profiles (Fig. 1) similar to those of typical controls and the resulting coefficients (Fig. 2) corre-

spondingly fall within the cluster of controls. Those rats showing the highest response rates in the treated group likewise show the highest constant coefficients of the experimental animals (E8, open circles; E4, closed diamonds; E6, asterisk; E9, open square; E12, inverted open triangle; and E2, closed square). The quadratic coefficient, representing first-order curvature or deviation from a linear trend, produces some clustering of treated animals. Although the separation of the two groups is not as apparent as with the constant term, treated animals tend to have more negative curvature than controls. Linear vs cubic regression coefficients from the orthogonal polynominal fits are presented in Fig. 3. As might be expected, the linear term was found to be positively correlated with the constant term, as evidenced by the similarity of the distribution of animals along these two parameters (cf. Figs. 2 and 3). Lin-

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0

163

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FIG. 2. Plot of the quadratic vs constant (intercept) regression coefficients resulting from the cubic orthogonal polynomial fit to each individual animal’s overall response rate profile in Fig. 1. Individual rats are numbered in correspondence with Fig. 1. Control animals are circled; experimental points are enclosed by squares.

ear coefficients of 10 of the 12 control animals cluster toward the upper right central portion of the plot. Again, the two remaining controls (C4 and C6) exhibit linear terms quite distinct from the majority of control as well as treated animals. (This deviant performance is typically observed in about 15-20% of control animals in our studies (CorySlechta and Thompson, 1979; Cory-Slechta et al., 1983) and its basis is currently unclear.) In accord with the agreement between the linear and constant coefficients, lead-treated animals with the highest response rates also exhibit the largest linear coe5cients (E6, E8, E9, E4, El, E2, E5, E12). For this data set, the cubic coefficient produces little obvious clustering. To assess the adequacy of the regression modeling for individual animals, residuals were examined for evidence of any systematic patterns remaining after the three low-degree trends had been removed. Values of R2 (coefficient of determination) were generally high (70-900/o); lower values were observed only in cases where response rates remained low throughout the experiment. Further-

more, plots of the first two serial correlations for the raw residuals (not shown) revealed no apparent differences in remaining oscillation between the two groups. For further assurance, the coefficients for a polynomial of degree 10 were computed and displayed in a biplot (Gabriel, 197 I), which indicated little, if any, need for terms beyond the third degree. Qualitatively, the effects of lead treatment on the coefficients, especially the constant and linear terms, can be seen clearly in Figs. 2 and 3. The animals respond as individuals, but, nevertheless show obvious indications of differences on a group basis. The four coefficients from the cubic orthogonal polynomial fit were then subjected to tests of significance using the median-based randomization (permutation) test already described. Examination of the regression coefficient plots (Figs. 2 and 3) argued for maximal group differences based on the constant and linear coefficients. To avoid calculating the large number of possible permutations of the data, a random sample of 10,000 was selected and the p value was estimated. The result for the constant co-

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COX AND CORY-SLECHTA

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FIG. 3. Plot of the cubic vs linear regression coefficients resulting from the orthogonal polynomial fit to each individual animal’s response profile in Fig. 1. Data as in Fig. 2.

efficient was a two-sided p value (p = 0.04) confirming that the median value for the treated animals exceeded that of controls more than expected by chance if the intercept were unrelated to treatment. As predicted by the correlation between the constant and linear terms, the p value for the latter was also low (0.07), suggesting that, in addition to the higher average rates of responding, leadtreated animals exhibited larger linear trends. In contrast, the p value for the multivariate randomization test (Zerbe and Walker, 1977) was only 0.30, probably reflecting the fact that this test averages across all four regression coefficients. The median-based randomization test proposed here is a simple test, and certainly alternatives are possible. It has proven adequate, however, in a number of studies and appears to be an important component of the approach (e.g., Cory-Slechta et al., 1983, 1985). For comparison, both an approximate unequal variance t test and a Mann-Whitney

test were used with the constant coefficient. Neither proved sensitive (p = 0.26 and 0.11, respectively). In Table 1, the results of another alternative test, a repeated-measures analysis of variance based on medians of five consecutive sessions, are presented. A three-way RMAOV model was used: the treatment and block (session) effects were fixed, while the animal effect was nested within treatments. The expected mean squares were computed on the basis of this model, and from these, the appropriate F ratios were determined. This model is discussed fully in Winer ( 197 1; the analysis of variance table is given in Table 7.2-l). In this model, differences in trend between the two groups would be indicated by a significant Treatment X Block interaction, and a difference in the overall level of response by a significant treatment effect. In such cases, it is customary to partition the interactions sum of squares into single degreeof-freedom contrasts to represent differences

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TABLE 1 ANALYSISOFVARIANCETABLEFOROVERALLRATES Source of variation

Sum of squares

df

MS

F

p value

Tmt Animal (Tmt) Block Tmt X Block Error corrected total

935.89 16,133 11,313 237.95 6,886.1 35,506

1 22 7 7 154 191

935.9 733.3 1616 33.99 44.7 I

1.28” 36.14 0.76

n.s.
’ The denominator mean square is Animal (Tmt).

in linear, quadratic, etc., trends. This proved fits were all above 80%, with the exception of difficult in the present case because the Treatanimal E2 (68%; right panel: solid squares). ment X Block means were not statistically in- Residual plots for this animal showed a clear dependent within each treatment. This trend remaining in the data, indicating unmakes calculation of a standard error for the usual behavior as compared with the other desired contrasts more complicated. Furtheranimals in the experiment. more, the Treatment X Block interaction The constant and quadratic regression coturned out to be highly nonsignificant. The efficients from the polynomial fits are plotted in Fig. 5. The constant term provides a treatment effect was similarly not significant, mainly due to the large amount of variation marked separation of the two groups: coeffibetween animals. In this respect, the results cients of the control animals were, in general, are similar to those from the multivariate test lower than values for lead-treated animals, and obviously less sensitive than the randomindicating lower overall levels of perforization tests using the constant and linear co- mance. The exceptions are animal E2 with a more negative (downward) quadratic curvaefficients. In another experiment (Cory-Slechta, ture than any control animal, and Cl, with 1986), changes in a different schedule of food the largest constant coefficient of the controls. As before, the plot of the first two autoreinforcement, the fixed ratio, were studied. The effects of chronic postweaning exposure correlations on the least-squares residuals showed no remaining difference in trend beto 500 ppm lead acetate on this performance tween the two groups. The simple medianare shown in Fig. 4, which plots the median test on the constant cointerresponse times (IRT: time between suc- based randomization cessive responses) for the 6 control (left panel) efficient resulted in the two-sided p value of randomization proand lead-treated rats (right panel) over the 0.03. The multivariate first 10 sessions when five responses were re- duced a similar value in this case (p = 0.03); quired for each delivery of food. the corresponding value for the t test was yielded a Again, the two families of curves are visu- 0.06, while the Mann-Whitney ally different; the treated animals appear to p = 0.05. exhibit longer IRTs. These data provided an For comparison, a repeated-measures even more severe test of this approach, as analysis of variance on the data using all 10 both the number of sessions and animals was sessions was performed. The original analysis small, making it more difficult to estimate indicated that the data were somewhat skewed, so the logarithms of the raw data trends. The values of R* for the polynomial

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1

5

10

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SESSIONS

FIG. 4. Median times between responses (interresponse times: IRTS) of individual control (left panel) and 500 ppm lead acetate-treated rats over the first 10 sessions of a fixed ratio 5 schedule of food reinforcement. Shorter interresponse times indicate higher overall response rates. Each animal is identified by a unique symbol as follows: l 1; n 2; A 3; 4 4; v 5; * 6. Data are from Cory-Slechta (1986).

were used (Table 2). This time the variation between animals was less, and a significant treatment effect was obtained (p = 0.04). The Treatment X Block interaction was, however, nonsignificant as was the median-based randomization test using the linear coefficient. In both of the examples presented, the interaction sum of squares is relatively small. Thus, the treatment effect appeared primarily as a difference in the overall level of response as was seen with the randomization test, although the difference in linear trend approached significance in the first data set. DISCUSSION The statistical approach described here is designed to describe longitudinal trends occurring over the duration of an experimental observation period. In both of the examples considered, the median-based randomization test performed well, proving more sensitive than either a repeated-measures analysis of variance or the multivariate randomization test, which provided results similar to

RMAOV. Clear differences between control and treated animals in the constant coefficient (intercept) of the orthogonal polynomials confirmed the differences in the response profiles. It should be emphasized that this type of approach is by no means limited to behavioral data. It is applicable to other cases of repeated measurements of effect when a sufficient number of data points are generated for an adequate polynomial fit. More specifically, the cubic orthogonal polynomials are best suited for modeling smooth or gradual onset trends. In cases involving both a gradual onset trend and its subsequent reversal, the cubic and quadratic coefficients may delineate effects. The cubic polynomial, however, is not amenable to curves with sharp elbows or angles and those that plateau. In such cases, other types of curves might be considered. The median-based randomization test appears to be an important component of this approach as evidenced by a comparison of its performance with an approximate unequal

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FIG. 5. Plot of the quadratic vs constant term regression coefficients resulting from the cubic orthogonal polynomial fit to each individual animal’s IRT protile in Fig. 4. Data as in Fig. 2.

variance t test and the Mann-Whitney test. This greater sensitivity may be related to the fact that the choice of the test statistic for the randomization test is not dictated by statistical theory, but by experience with similar data. The randomization test may not prove suitable for all applications, however. Plots of the regression coefficients sometimes depict extreme displacements of the higher order coefficients of certain animals. In such cases, multivariate tests, which are responsive to such deviation, may be more sensitive than

univariate tests. This proved not to be the case with the data sets used here, however, as a procedure similar to the multivariate approach of Zerbe and Walker (1977) gave results similar to those produced by RMAOV. An advantage of the approach outlined here relative to other procedures, including the repeated-measures analysis of variance and multivariate tests, lies in the extent of the information provided by a significant treatment effect. A significant F value, such as provided by the RMAOV, specifies only that groups differ; a significant interaction term

TABLE 2 ANALYSIS

Source of variation

Sum of squares

Tmt Animal (Tmt) Block Tmt X Block Error Corrected total

28.309

OF VARMNCE

52.243

22.549 1.113 6.851

111.07

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TABLE

FOR LOG IRT

p value

df

MS

F

1 10 9 9 90 119

28.31 5.22

5.42"

0.04

31.38 1.62


2.51 0.12 0.08

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COX AND CORY-SLECHTA

further indicates lack of parallelism in the effects. In contrast, a significant betweengroups difference in the various coefficients denotes as well the source of the group differences. For example, a change in the constant coefficients specifies a difference in average response rates over the experiment, i.e., in the level of performance attained. Significantly higher linear coefficients provide evidence of an alteration in the linear trend or slope; quadratic coefficient differences suggest specific changes in the curvature of the response rate profiles. Thus, a composite description of treatment effects on the response rate profile can be generated. Thus, for example, low-level lead exposure (example 1) not only increased the overall rate of responding (as indicated by the constant coefficient), but also increased the rate at which peak level of performance occurred (linear term). This occurred despite the fact that the FI schedule actually requires 0nIy a single response for each food delivery. The sustained nature of the increased rates may result from the delivery of food right at the time when these high response rates were occurring, thus reinforcing them. Significant effects in the two examples provided were restricted primarily to the constant and linear coefficients. It might be argued that an effect only on the overall mean response (constant coefficient) does not warrant an analysis by orthogonal polynomials. However, the significance of the constant and linear coefficients in differentiating the groups was only revealed by examination of the regression coefficient plots. Furthermore, for other data sets or types of curves, it is certainly possible that the cubic or quadratic coefficients may additionally or even solely differentiate groups, as described above. Examination of the regression coefficient plots was informative for several reasons. They frequently highlighted the dose-response aspect of the data. Additionally, they disclosed that certain animals whose response profiles were not obviously atypical,

nonetheless, were quite different in some respects. For example, constant, linear, and quadratic coefficients of C4 and C6 in experiment 1 (Figs. l-3) differed not only from other controls, but also from almost all treated animals. Another example was E2 in the second experiment (Figs. 4 and 5). The coefficient plots also emphasize the multidimensional nature of treatment effects, in contrast to differences based only on a single coefficient. Finally, in many experiments little may be known about the anticipated pattern of group differences a priori. The coefficient plots are informative in this regard. As the nature of the effect becomes more fully characterized, test statistics can be designed which are sensitive to predicted changes. The median-based tests which we have described are designed for the comparison of two groups. If more than two groups are involved, the test may be applied to various pairs. To adjust for the number of statistical comparisons, a Bonferroni correction (dividing the overall significance level by the number of tests) may be used. Another alternative would be some analog of the Kruskal-Wallis test as an overall procedure. In summary, the proposed analysis consists as much of a point of view, as a specific application of available methodologies. These ideas should not be construed as a recipe, but as a stimulus for further thought and discussion. In using such an approach, careful consideration should be given to various aspects of the analysis, such as the selection of coefficient(s) for statistical tests, and the choice of the test statistic itself. If the polynomial fits appear to be reasonable, and the coefficient plots do not indicate clear separation of treatment groups, adding a fourth- or fifth-degree polynomial is unlikely to be helpful. Finally, the mechanics of these procedures should be simple enough to anyone with the appropriate background, but if modifications seem necessary, statistical expertise may be beneficial.

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ACKNOWLEDGMENTS We thank Christopher Newland and Ronald W. Wood for their comments on this manuscript and Pam Arnold for statistical programming.

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CORY-SLECHTA, D. A., WEISS, B., AND Cox, C. (1985). Performance and exposure indices of rats exposed to

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low concentrations of lead. Toxicol. Appl. Pharmacol. 78,29

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CORY-SLECHTA, D. A. (1986). Prolonged lead exposure and tied-ratio performance. Neurobehav. Toxicol. Teratol. 8,237-244. EDGINGTON, E. S. (1980). Randomization Tests. Dekker, New York. FERSTER, C. B., AND SKINNER, B. F. (1957). Schedules of Reinforcement, pp. 133-325. Prentice-Hall, New York. GABRIEL, K. R. ( 197 1). The biplot graphic display of matrices with application to principal component analysis. Biometrika S&458-467. GOOD, P. (1979). Detectors of treatment effect when not all experimental subjects will respond to treatment. Biometrics 35,483-489. HAYS, W. L. ( 1973). Statistics for the Social Sciences, p. 223. Holt, Rinehart & Winston, New York. TILSON, H. A., MACTUTUS, C. F., MCLAMB, R. L., AND BURNE, T. A. ( 1982). Characterization of triethyl lead chloride neurotoxicity in adult rats. Neurobehav. Toxicol. Teratol. 4,67 l-68 1. WINER, B. J. ( 197 1). Statistical Principles in Experimental Design, 2nd ed., McGraw-Hill, New York. ZERBE, G. O., AND WALKER, S. H. ( 1977). A randomization test ofgroups ofgrowth curves with different polynomial design matrices. Biometrics 33,653-667.