### Introduction

Interindividual variation in drug response among patients is well known and poses a serious problem in medicine. The variation could be because of multiple factors such as disease phenotypes, genetic and environmental factors and the variability in drug target response (pharmacodynamic response) or allergic response, all factors that affect drug absorption, distribution, metabolism and excretion, side effects or efficacy.^{1, 2, 3} However, at present, few biomarkers can predict, which group of patients will respond positively, which patients are nonresponders and who might experience adverse reactions for the same medication and dose. To realize personalized medicine, it is critically important to observe individual differences in drug response and the role of genetic polymorphisms that are relevant to the pathways of drug metabolism and the biology of drug responses in the pharmacogenomics of common diseases.^{4}

With this background, the Food and Drug Administration (FDA) recognizes the importance of pharmacogenomics and has issued a guidance that encourages pharmacogenomics during drug development.^{5} Many pharmacogenomics studies have been launched worldwide, such as a combination of a pharmacokinetic (PK) study and analyses of single nucleotide polymorphisms (SNPs) in a candidate gene or in a genome-wide approach. Following the completion of the HapMap project,^{6} the advent of the powerful array-based SNP typing platforms has heralded an era in which a genome-wide approach is a popular or standard strategy for identifying disease susceptibility or drug response genes for common diseases.^{7, 8, 9}

To identify the SNPs, which relate to the pathways of drug metabolism, biomedical researchers usually test the null hypothesis (H_{0}) that there is no difference of the population mean of PK parameters (that is, area under the curve (AUC), maximum drug concentration (*C*_{max}), half-life period (*t*_{1/2}) and so on) between genotypes by using mainly the nonparametric analysis of variance, that is, the Kruskal–Wallis test.^{10, 11} On the basis of the statistical significance of the results from that test, researchers check the PK-genotype response patterns by sight, and then detect an additive, recessive or dominant model. The PK-related genes indicate a monotone increasing pattern in the number of alleles such as genetic models (additive, recessive and dominant pattern) as shown in Figure 1. Because a genome-wide association study is often designed as a multistage process, a relatively relaxed type I error rate is adopted in the first stage screening to assure the overall power of the study. If the Kruskal–Wallis test is applied to a significance level of *P*=0.05 in the first stage of a genome-wide association study, it is expected that as many as 5000–50 000 SNPs should be visually inspected for the current standard genome scan typing platform. Development of more objective and efficient screening method than the current subjective procedure is needed.

As a statistical method for detecting a monotonic dose–response relationship, a maximum contrast method has been proposed in clinical trials and toxicological trials.^{12, 13, 14, 15, 16, 17} This method, formed by taking the maximum over multiple contrast statistics for detecting a monotonic increase with the dose levels, is useful. The dose–response curves are modeled as the response patterns, and the set of contrast statistics is determined based on the set of contrast coefficients that correspond to these patterns. The contrast statistics should consist of contrast coefficients that are highly correlated with the population means for realistic dose–response relationships. The maximum contrast method is then applied to this set, and the pattern of the contrast coefficients for the contrast statistic that takes the maximum value is then selected as the true response pattern (see Materials and methods section).

In a typical pharmacogenomics study, the association of between PK parameter and genotype is modeled as the response patterns in Figure 1, and then the maximum contrast method is applied to this study on three contrast statistics with the following coefficients: ** c_{1}**=(−1 0 1)

^{t},

**=(−2 1 1)**

*c*_{2}^{t},

**=(−1 −1 2)**

*c*_{3}^{t}. The first contrast statistics corresponds to an additive model, the second to a recessive model and the third to a dominant model. As a result, the pattern of the coefficients for the contrast statistic that takes the maximum value is then selected as the true response pattern. However, the sample size of each genotype was enormously unbalanced, because the minor allele frequency (MAF) is less than 0.5, and the population is in Hardy–Weinberg equilibrium (HWE). Here we propose a modified maximum contrast method to detect an association between a PK parameter and genotype that accounts for the imbalance in genotype groups (see Materials and methods section).

In this paper, we proposed a modified maximum contrast method for detecting an association between PK parameter and genotype. Further, using simulation studies, we compare the Kruskal–Wallis test, the maximum contrast method and the modified maximum contrast method for pharmacogenomics data. We also suggest rules of thumb for choosing a powerful maximum contrast statistic for such data. Finally, the discussed methodology is illustrated by its application to a pharmacogenomics study to antitumor drugs on Japanese cancer patients.

### Results

#### Simulation study

To assess the power of the Kruskal–Wallis test, the maximum contrast method and the modified maximum contrast method, we first performed a Monte-Carlo simulation. Summary results of the simulation for each method are shown in Table 1 for various values of Δ and MAF when sample size (*n*) is set at 300. Note first that the diagonal box shows the positive predictive value (*R*_{TP}) for detection of true response patterns, whereas the row at the far right is the power (*R*_{p}) for detection of PK-related SNPs and includes the misidentification of true response patterns, where the result of *R*_{TP} by the Kruskal–Wallis test is blank because that test is an overall test and rejects the null hypothesis that there is no difference of the population mean of PK parameters. In a case where Δ=0, the *R*_{p} corresponds to a type I error rate of 5%, and is mostly controlled below 5%, but is inflated slightly for some models. The *R*_{p} and *R*_{TP} increase with increasing Δ; on the other hand, both *R*_{p} and *R*_{TP} decrease as the MAF decreases. We evaluated each method for detecting true response patterns. As a result, when the MAF is less than or equal to 0.25, in the additive and dominant pattern, the modified maximum contrast method is about 0.1–0.4 higher than the maximum contrast method in the *R*_{TP}. However, in the recessive model, the modified maximum contrast method is about 0.4 lower than the maximum contrast method in the *R*_{TP}. Therefore, under unbalanced sample size, the modified maximum contrast method is more powerful for detecting true response patterns than the maximum contrast method in the additive and dominant model, whereas the maximum contrast method is more powerful than the modified maximum contrast method in the recessive model.

For detecting PK-related genes, the Kruskal–Wallis test is lower in power than both maximum contrast methods in the *R*_{p}, under equivalent sample size (MAF=0.5), whereas the Kruskal–Wallis test is about 0.05–0.10 higher in power than both maximum contrast methods under the unbalanced sample size (MAF0.25). On the other hand, the *R*_{p} of the Kruskal–Wallis test is about 0.10–0.50 higher than both maximum contrast methods under the simulation condition without the PK-related SNPs in Table 2. Therefore the simulation studies suggest that the Kruskal–Wallis test detects many PK-nonrelated SNPs because of not considering the response patterns.

#### Questionnaire survey on judgment

We next conducted a questionnaire survey to compare the judgment by each statistical method (modified maximum contrast method and maximum contrast method) with the judgment by experts. The results are shown in Table 3. For the recessive and no-response pattern, the judgment of both statistical methods well accords with the judgment by expert. For the additive and dominant pattern, the maximum contrast method detects as a recessive pattern by mistake, and the modified maximum contrast method well accords with the judgment by expert. The Kendall’s rank correlation coefficient between modified maximum contrast method and expert’s judgment is 0.731; on the other hand, the correlation coefficient between the maximum contrast method and expert’s judgment is 0.423. The conclusion is that the judgment of modified maximum contrast method is closer to the judgment by expert than is the maximum contrast method.

#### Application to actual genome-wide pharmacogenomics study

In this paper, we focus on the elimination rate constant (*K*_{el}), which is the first order rate constant describing drug elimination from the body, and report the results of PK-related SNPs associated with *K*_{el}. The histogram for the *K*_{el} is shown in Figure 2. The PK data seem to be skewed to the left tail of the distribution; therefore we transformed data before applying both maximum contrast methods by taking the natural logarithm of the observed values.

##### Figure 2.

Histogram of elimination rate constant (*K*_{el}) for the gemcitabine pharmacogenomic (PGx) study.

Table 4 summarizes the number of significant SNPs for various *P*-value cutoffs by each method. For the *P*-value cutoff of 0.001, 84 SNPs are significant by the Kruskal–Wallis test, 77 SNPs by the maximum contrast method and 84 SNPs by the modified maximum contrast method. Two SNPs (GLT25D1: rs3848643 and BCNP1: rs6512201) are detected as additive model by all three methods in Table 5. In considering a multiple testing problem, we assume the existence of about 10 000 linkage disequilibrium blocks within 100 000 gene-centric SNPs, which are concentrated in about 2% of the human genome (that is, average interval of two SNPs is 600 bp). It follows that the *P*-value cutoff is set at 5.0 × 10^{−6} as Bonferroni correlation. As a result, two SNPs (rs2148582, rs699) in angiotensinogen (*AGT*) gene are detected as a dominant model by both the maximum contrast method and modified maximum contrast method, but no SNPs are detected by the Kruskal–Wallis test. We show all of the results for association between PK parameters and SNPs at Genome Medicine Database of Japan (http://gemdbj.nibio.go.jp).

### Discussion

We reviewed the current methodology for analyzing PK-SNP data in pharmacogenomics studies. We proposed a modified maximum contrast method for detecting a PK-genotype response pattern under unbalanced sample size. The simulation study suggested that both the maximum contrast method and the modified maximum contrast method were more useful for screening PK-related SNPs in consideration of genetic models than is the Kruskal–Wallis test, and that under unbalanced sample size, the modified maximum contrast method was more powerful for detecting true response patterns than the maximum contrast method in the additive and dominant pattern, whereas the maximum contrast method is more powerful than the modified maximum contrast method in the recessive model.

In addition, the survey on judgment for PK-related SNPs makes evident that the modified maximum contrast method gave a judgment closer to the judgment of experts than did the maximum contrast method. The expert’s judgment was subjective, and also different by each expert. The modified maximum contrast method and the maximum contrast method were able to objectively detect the response pattern and the PK-related SNPs under unbalanced sample size.

The application of the maximum contrast method, modified maximum contrast method and Kruskal–Wallis test to a pharmacogenomics study supported these considerations. When the *P*-value cutoff was set at 0.05 in this application, about 5000 SNPs were detected as statistical significance, and both maximum contrast methods were able to show each genetic model to the biomedical researcher. We showed the list of identified SNPs by each method. The two SNPs in the *AGT* gene showed a statistical significant difference in *K*_{el} by both the maximum contrast method and modified maximum contrast method, but the Kruskal–Wallis test was not able to detect the gene after multiple testing adjustment. Polymorphic variations in the human *AGT* gene have been shown to be associated with increased circulating AGT concentrations,^{18} an increased risk of essential hypertension,^{18, 19, 20, 21, 22} and a decline in renal function.^{23} Although it is possible that the *AGT* gene polymorphisms influence gemcitabine PK through yet unknown mechanisms, the result and the suggested hypothesis in our study should be evaluated both in replication study using another sample set and also in biological functional analyses.

Many biomedical researchers have used the Kruskal–Wallis test for finding significant SNPs, and visually checked PK-genotype response patterns. Recently, powerful array-based SNP typing platforms have heralded an era in which a genome-wide association study is a popular or standard strategy for identifying disease associated genes or drug response genes for common diseases, and genotype data on 100 000–1 000 000 SNPs are increasingly available to researchers. It is virtually impossible for biomedical researchers to visually check the response patterns on such a genome scan data. Our study has proposed an alternative by showing that either a modified maximum contrast method or a maximum contrast method can readily be applied to genome scans as a statistical screening method.

### Materials and methods

#### Maximum contrast method and modified maximum contrast method

We first describe a general form for a maximum contrast method as discussed in Yoshimura *et al.*,^{14} and Wakana *et al.*,^{17} in a pharmacogenomics study. We assume *Y*_{ij} (*i*=1, 2, 3; *j*=1, 2, …, *n*_{i}) as an observed response for *j*th individual in *i*th genotype group (*AA*, *Aa* and *aa*, where ‘*A*’ is major allele, and ‘*a*’ is minor allele), and *Y*_{ij}s are independently and normally distributed with *E*(*Y*_{ij})=*μ*_{i} and Var(*Y*_{ij})=*σ*^{2}. Under these conditions, **=(**_{1}, _{2}, _{3})^{t} follows the trivariate normal distribution *N*(*μ*, *σ*^{2}*D*) where

diag(•) being a diagonal matrix, and the superscript t indicates the transpose of a matrix or vector. In the maximum contrast method, some contrast statistics are set according to the approach. Let contrast statistic *T*_{k} with a coefficient ** c_{k}**=(

*c*

_{k1},

*c*

_{k2},

*c*

_{k3})

^{t}where

*c*

_{k1}+

*c*

_{k2}+

*c*

_{k3}=0, and

*k*=1, 2 and 3 correspond to an additive, recessive and dominant response pattern, respectively, as shown in Figure 1. The following coefficients were used for each genetic model:

The contrast statistics is defined in terms of the contrast vector *c*_{k},

where ^{2} is and is degree of freedom of ^{2}. Then the statistic is defined as:

The statistics can be used to test over null hypotheses, H_{0}: *μ*_{1}=*μ*_{2}=*μ*_{3}, on the other hand, in pharmacogenomics studies, alternative hypotheses such as H_{1}: *μ*_{1}<*μ*_{2}<*μ*_{3,} *μ*_{1}=*μ*_{2}<*μ*_{3,} *μ*_{1}<*μ*_{2}=*μ*_{3}. The *P*-value for the probability distribution of *T*_{max} under the over null hypotheses is calculated by using the complex property of integration in the multivariate *t*-distribution with singular correlation matrix,^{24} and the *P*-value is defined as in the following formula:

where *t _{max}^{*}* is the observed value of the test statistics.

This method can select the response pattern, which best fits the observed data among a set of patterns. In clinical and toxicological trials, the sample size of each group is almost equivalent; therefore the maximum contrast method assumes an equivalent sample size of each group. However, the sample size of each group is not equivalent in pharmacogenomics studies, because the MAF is less than 0.5 and is generally around 0.2. For this reason, under the unbalanced sample size, the adjoining denominator of contrast statistic is overestimated although the studentized statistics by this variance estimate is robust. Therefore, the maximum contrast method cannot be applied for detecting a true response pattern in pharmacogenomics studies.

Here we propose a modified maximum contrast method for unbalanced sample size. First, the modified contrast statistic is given by

thus the adjoining denominator of the modified contrast statistic is and it is not influenced by unbalanced sample size. The modified maximum contrast statistic is given by

In addition, the multiplicity adjusted *P*-value for the probability distribution of *T*′_{max} under H_{0} was calculated by the permutation procedure.^{25} Simple random sampling without replacement method generates the sampling distribution of the statistic by drawing repeated samples from the observed sample itself. Consequently, the permutation procedure is used to simulate *P*-values; the lowest *P*-value is calculated for each of 100 000 permutations for the entire dataset, and from this distribution.

#### Simulation study

We assess the power of the Kruskal–Wallis test, the maximum contrast method (Equations (1) and (2)) and the modified maximum contrast method (Equations (4) and (5)) by simulation studies. The scenarios discussed here are motivated by the anticancer drug’s pharmacogenomics data that will be introduced below. In particular, we are interested in the performance of all three tests when the MAF decreases. To examine these effects, we simulate PK parameter data from transformed log-normal distribution by each genotype. The distributions of PK parameters are often unknown, but empirically are modeled under the assumption of a log-normal distribution because (1) PK parameter values must be nonnegative and the normal assumption does not enforce this assumption; (2) distribution of estimated PK parameters is often left skewed, which is compatible with a log-normal distribution.^{26} Here, for applying an intravenous anticancer drug in actual data, we focus on the elimination rate constant (*K*_{el}), which is the first order rate constant describing drug elimination from the body. We assume that a sample of patients with any genotype (*AA*, *Aa* or *aa*) and the proportion of MAF is given, and that this population is in HWE. For each patient, the PK parameter, *K*_{el} is given. Under these assumptions, we generated *K*_{el} by each genotype from the log-normal distribution with mean *μ*_{i}, that is, _{i}~LN(*μ*_{i}, 1), *i*=1, 2, 3 in *i*th genotype group (*AA*, *Aa* and *aa*), where *μ*_{i}=Δ·*c*_{ki} and Δ is given coefficient and *c*_{ki} corresponded to element of contrast statistics vector ** c_{k}**=(

*c*

_{k1}

*c*

_{k2}

*c*

_{k3})

^{t}. Here, Δ is set at 0.0, 0.5 and 1.0, respectively, and the contrast statistics vectors are set as the following coefficients corresponding to the genetic models (

*k*=1, 2 and 3) in Figure 1 (i)–(iii):

In addition, the MAF was set at 0.12, 0.25 or 0.50, respectively, where total sample size (*n*) was 300 or 600 subjects. The criteria to evaluate the performance of each method were two indicators, *R*_{P} and *R*_{TP}, defined by Equations (6) and (7).

*R*_{p} is a proportion of detected PK-related SNPs (power), whereas *R*_{TP} is the proportion of detected true response patterns among PK-related SNPs (positive predictive value). The Monte-Carlo simulation to calculate *R*_{p} and *R*_{TP} was repeated 10 000 times and the mean values of indicators were calculated. Note that *N* was a constant fixed as the repetition number of the simulation, whereas *N*_{R} was a random variable realized as the number of rejections by the hypothesis test, and *N*_{TP} was the number of detected true response patterns.

#### Questionnaire survey on judgment

To identify PK-related genes, biological experts who are pharmacokinetists, molecular biologists and geneticists used to draw box-and-whisker plots on PK parameters by each genotype, and check these response patterns by sight. This judgment involves much experience, but it is subjective and may differ by expert. We conducted a questionnaire survey to compare statistical judgment with judgment by experts. First, we showed summary statistics and box–whisker plot (Figure 3) to the expert group that consisted of two pharmacokinetists, two molecular biologists and two geneticists. Second, these experts independently gave a decision from six response patterns (Figure 1). Finally, we applied the modified maximum contrast method and the maximum contrast method to this survey dataset, and, by using Kendall’s rank correlation coefficient, evaluated, which method is closer to expert judgment.

##### Figure 3.

Summary statistics and box–whisker plot in questionnaire survey.

Full figure and legend (60K)#### Application to actual gemcitabine pharmacogenomics study

We applied the Kruskal–Wallis test, the maximum contrast method and the modified maximum contrast method to an actual genome-wide pharmacogenomics study on antitumor drugs. The study was performed within the Millennium Genome Project in Japan, and four antitumor drugs were chosen as project targets: irinotecan, fluorouracil, paclitaxel and gemcitabine. One or some combination of the drugs was administered to about 1000 cancer patients with written informed consent at the National Cancer Center, Japan. PK data were also obtained for the drugs except for fluorouracil. In addition, about 1000 DNA samples were extracted from peripheral blood mononuclear cells and 109 365 SNPs were genotyped by Illumina Human-1 BeadChip.

In this application, we emphasize a focus on the gemcitabine pharmacogenomics study. Gemcitabine (2′,2′-difluorodeoxycytidine) is a nucleoside anticancer drug that has a broad spectrum of antitumor activity against various solid tumors, such as nonsmall cell lung cancer and pancreatic cancer. The participants consisted of 256 Japanese gemcitabine-naive cancer patients (mainly pancreatic carcinoma) at the National Cancer Center Hospital and National Cancer Center Hospital East. Details of this analysis group were reported previously.^{27} For gemcitabine PK analysis, 5 ml of heparinized blood was sampled before the first gemcitabine administration, and at 0, 15, 30, 60, 90, 120 and 240 min after the termination of the infusion. The AUC, mean residence time from 0 to infinity, peak concentration (*C*_{max}), clearance (CL m^{−2}), distribution volume based on the terminal phase (Vz m^{−2}) and elimination rate constant (*K*_{el}) were calculated using WinNonlin ver. 4.01 (Pharsight Corporation, Mountain View, CA, USA). For an investigation of the association between these PK parameters and SNPs in Illumina Human-1 BeadChip, we applied the maximum contrast method, the modified maximum contrast method and the Kruskal–Wallis test.

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### Acknowledgements

We thank Professor Isao Yoshimura and Professor Chikuma Hamada at Tokyo University of Science for their valuable advice, and Dr Hiromi Sakamoto and Ms Sumiko Ohnami for the SNP genotyping. Yasunori Sato is a recipient of Official Trainee of the Foreign Clinical Pharmacology Training Program, Japanese Society of Clinical Pharmacology and Therapeutics. This work was initiated while Yasunori Sato was a postdoctoral fellow at the Department of Biostatistics, Harvard School of Public Health, and partially supported in Japan by the program for promotion of Fundamental Studies in Health Sciences of the National Institute of Biomedical Innovation (NiBio).

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