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Cytochrome P450 1A1 Polymorphism and Childhood Leukemia: An Analysis of Matched Pairs Case-Control Genotype Data

School of Public Health, University of California at Berkeley, Berkeley, California

Requests for reprints: Steve Selvin, School of Public Health, University of California at Berkeley, Berkeley, CA 94720. Phone: 510-642-3241; Fax: 510-643-5163. E-mail: selvin{at}stat.berkeley.edu

	Abstract

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

 
The association between the genotypic frequencies of the cytochrome P450 1A1 polymorphism and the risk of childhood leukemia is explored with the data from a matched case-control study. The data are displayed in a 3 x 3 case-control array, and the discordant pair counts are assessed for quasi-independence, homogeneity, and symmetry. This statistical approach is contrasted to the more typical analysis of matched data based on a conditional logistic model and estimated odds ratios. The statistical analysis of 175 matched pairs (part of a large study of potential environmental/genetic influences on the risk of childhood leukemia) showed no evidence of an association between cytochrome P450 1A1 genotype frequencies and case-control status.

	Introduction

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The possible association between cytochrome P450 1A1 (CYP1A1) genetic polymorphism and disease has been explored by several investigators. Some examples are the investigations of Ladonna et al. (1) on Spanish toxic oil syndrome, Ishibe et al. (2) on breast cancer, and Kim et al. (3) on cervical cancer and several reports on the association with childhood leukemia (e.g., refs. 4, 5). These analyses, as well as most analyses of matched case-control genetic data, are summarized in terms of ratios of discordant pairs of observations (odds ratios) estimated from conditional logistic regression models. The following statistical approach to the analysis of the CYP1A1/leukemia matched data is based on a 3 x 3 case-control array that allows an assessment of three fundamental statistical issues (i.e., independence, homogeneity, and symmetry of the discordant pairs). In addition, this approach is contrasted to the more typical use of odds ratios estimated from a conditional logistic model.

	Data

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To describe the CYP1A1 polymorphism and the risk of acute lymphoblastic leukemia, data collected as part of the Northern California Childhood Leukemia Study are used. These matched case-control observations were abstracted from a large number of genetic/environmental variables that potentially influence the risk of childhood leukemia. The cases are children ages 0 to 14 years old with newly diagnosed leukemia (1995 to 1999) obtained from major hospitals in the San Francisco Bay Area. Comparison with California State Cancer Registry data shows that >90% of the eligible children were ascertained. The control children were randomly selected from birth certificate records and matched to cases with respect to sex, age, race, and county of birth. A more extensive description of this far-ranging study is found elsewhere (6). The CYP1A1/leukemia data consisting of 175 matched pairs of acute lymphoblastic leukemia cases and their controls (117 concordant and 58 discordant pairs) are given in Table 1.

	Quasi-Independence

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

. The values p_i represents the case genotypic frequencies, and q_j represents the control genotypic frequencies. The quantity N represents the "total number of pairs" that would have occurred if the genotypic frequencies were independent and the data were randomly sampled. Specifically, the value N = n / p_iq_j for i j = 1, 2, and 3 where n represents the total observed number of discordant pairs.

The maximum likelihood estimates of the p_i and q_j frequencies are found by iterative techniques (7) or an application of a specialized log-linear model (8). Both procedures are designed to estimate the genotypic frequencies excluding the concordant pairs from consideration (truncated). For the Northern California Childhood Leukemia Study data (Table 1), these estimated genotypic frequencies are

A natural estimate of the expected number of case-control discordant pairs based on the quasi-independence model is _ij = _ij where = n / _ij for i j = 1, 2, and 3. These expected counts estimated from the data in Table 1 are displayed in Table 2.

In the context of the analysis of matched case-control data, another form of this same ² test is called a "test for the consistency of the odds ratios" (9). The noninformative concordant pairs are excluded because the increased correlation within pairs due to the matching process tends to increase the number of concordant pairs relative to the number predicted by a model postulating independence.

	Marginal Homogeneity

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

 
Quasi-independence does not imply marginal homogeneity (identical case and control genotypic frequency distributions). The issue of marginal homogeneity in general is the subject of several statistical articles (e.g., refs. 10-12). The expected row/column totals, under the conjecture of marginal homogeneity, are estimated by

where n_i. = _jf_ij, n._j = _if_ij and f_ij represents the number of pairs in the (i,j)^th cell.For the CYP1A1/leukemia data, the estimated homogeneous marginal totals are

	Symmetry

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

 
When the case-control discordant pairs are independent and the marginal frequencies are homogeneous, the expected counts of case-control pairs create a symmetrical array; that is, when case-control status is unrelated to the genotypic frequencies, the counts within the three kinds of discordant pairs (f_ij versus f_ji) differ by chance alone. Under these conditions (independence + homogeneity = symmetry), the maximum likelihood estimates of the genotypic frequencies (denoted _i) are

where f_{ij} = f_ij + f_ji. Because the number of independent parameters equals the number of independent observations (2), the maximum likelihood estimates _i can be derived by equating expected and observed frequencies (13). The specific estimated genotype frequencies from the acute lymphoblastic leukemia data are

These estimated genotypic frequencies produce an estimate of the expected counts of matched case-control discordant pairs (denoted F_ij') where

The estimated "sample size" is ' = n / _ij for i j = 1, 2, and 3. The Northern California Childhood Leukemia Study matched pairs data (Table 1) produce the estimates displayed in Table 3. These expected discordant pairs are quasi-independent, and the marginal frequencies are homogeneous.

The ² test for symmetry requires three independent estimated parameters giving a test statistic with 3 df (6 – 3 = 3). In addition, this test statistic partitions into three independent components each with 1 df.

In addition, the three estimated genotypic frequencies _i lead to a compact form of a ² test statistic to evaluate marginal homogeneity in a 3 x 3 case-control array. The test statistic is

The test statistic X_H² summarizes deviations from marginal homogeneity and has an approximate ² distribution with 2 df when the expected marginal frequencies are homogeneous. The ² expression for testing homogeneity requires four independent estimated parameters yielding 2 df (6 – 4 = 2). For the Northern California Childhood Leukemia Study data, the ² value is X_H² = 0.479 (P = 0.787).

	Conditional Logistic Model

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

 
The additive conditional logistic model applied to genotype frequency data collected in a matched design yields estimates of the logarithms of the three ratios of discordant pairs (denoted b_i). When the genotypic frequencies are the same for both cases and controls, the model estimated ratio within all three kinds of discordant pairs is 1 (b₁ = b₂ = b₃ = 0). Furthermore, the additive model requires that b₁ + b₃ = b₂. For the CYP1A1/leukemia data, these estimated log-ratios are

₁ = –0.170 for AA/AG discordant pairs,

₂ = 0.110 for AA/GG discordant pairs, and

₃ = 0.280 for AG/GG discordant pairs.

1

The estimated log-ratios are directly related to the expected cell counts generated by the quasi-independence model. In symbols, the estimates from the quasi-independence model give

In other words, both models generate identical expected counts contained in a 3 x 3 case-control array (Table 2).

From another prospective, both models necessarily conform to the relationship that

or

The requirement that D = 1 is essentially the definition of quasi-independence or model additivity (no interaction) within a 3 x 3 array.

The distribution of the estimated value

has estimated variance given by

These two estimates provide a computationally easy and additional assessment of the correspondence between the observed and the expected counts generated by the quasi-independence or the additive conditional logistic models.

The estimate of log() and its estimated variance from the CYP1A1/leukemia data are log() = –1.264 and = 2.332 yielding the test statistic

The value X_Q'² has an approximate ² distribution with 1 df when the data (Table 1) randomly differ from the expected counts generated by the quasi-independence or the conditional logistic models (Table 2). In general, this ² statistic (X_Q'²) and the ² statistic (X_Q²) will be similar, particularly for case-control arrays with many discordant pairs.

It should also be noted that the likelihood ratio test based on the additive conditional logistic model addresses only the hypothesis that b₁ = b₂ = b₃ = 0 or the ratios of the corresponding discordant pairs are 1.0. The score likelihood ratio test statistic is identical to the previously described test for marginal homogeneity _H².

	Discussion

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

 
A symmetrical case-control array of matched pairs data indicates that no association likely exists between case-control status and genotypic frequencies. When statistical evidence emerges of nonsymmetry, two issues become important (i.e., independence and marginal homogeneity). That is, two reasons exist for a significant lack of symmetry: the failure of the genotypic frequencies to be independent (failure of the additive conditional logistic model to reflect the data) and the failure of the matched data to have the same case and control genotypic frequencies or both. These two sources of deviation for a symmetrical array are easily identified and indicate different dimensions of the association between genotypic frequencies and disease risk.

In general, the likelihood ratio statistic estimated from an additive conditional logistic model addresses only the issue of marginal homogeneity of the case-control array because an additive model explicitly requires the discordant matched pairs to be independent. That is, substantial differences in the ratios of discordant pairs can exist in a 3 x 3 case-control array with perfectly homogeneous marginal frequencies X_H² = 0 when the genotypic frequencies that determine the numbers of discordant pairs are not independent. Without an assessment of the quasi-independence model X_Q² or equivalently the consistency of the odds ratios (b₁ + b₃ = b₂), inferences from matched case-control data are potentially biased and even potentially misleading. As with statistical models in general, goodness-of-fit is a critical issue.

The interpretation of quasi-independence of two variables is not different in principle from the interpretation in most contingency tables. Quasi-independence becomes an issue when specific cell frequencies are truncated from consideration. In a matched pairs design, the frequencies on the diagonal cells of the case-control array (the concordant pairs) are not included in the analysis. Nevertheless, two variables are not independent (or not quasi-independent) when the occurrence of one changes the probability of the occurrence of the other. For example, case-control status and genotypic frequencies are not quasi-independent when P(AA | AG is a control) is not equal to P(AA case). A phenotype frequency among the matched pair cases will not be quasi-independent when, for example, cases and controls have differing racial compositions and the phenotypic frequencies under investigation differ among races. In fact, nonindependence potentially arise whenever the controls fail to be a random sample of the population from which the cases were selected.

The ² test of symmetry is a simultaneous evaluation of both independence and homogeneity. Applied to the CYP1A1/leukemia data, this test produces no evidence of an association between genotypic frequencies and case-control status (X_S² = 1.184 with P = 0.757). It then becomes a foregone conclusion that ² tests of quasi-independence and marginal homogeneity (X_Q² = 0.712 and X_H² = 0.479) consists of two nonsignificant pieces.

	Footnotes

 
Grant support: U.S. Environmental Health Sciences research grants R01 ES09137 and PS42 ES04705.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 9/24/03; revised 12/22/03; accepted 2/ 9/04.

	References

Top Abstract Introduction Data Quasi-Independence Marginal Homogeneity Symmetry Conditional Logistic Model Discussion References

 

1. Ladonna MG, Izuierdo-Martinez M, Posada de la Paz M, et al. Pharmacogeneic profile of xenobiotic enzyme metabolism in survivors of the Spanish toxic oil syndrome. Environ Health Perspect 2001;109:369-75.[Medline]
2. Ishibe N, Hankinson SE, Coditz GA, et al. Cigarette smoking, cytochrome P450 1A1 polymorphism and breast cancer in the Nurses' Health Study. Cancer Res 1998;58:667-71.[Abstract/Free Full Text]
3. Kim Jk, Lee CG, Park YG, et al. Combined analysis of germline polymorphisms of p53, GSTMI, GSTTI, CYP1A1, and CYP2E1: relation to the incidence rate of cervical carcinoma. Cancer 2000;88:2082-91.[CrossRef][Medline]
4. Krajiinovic M, Labuda D, Richer C, Karimi S, Simrett D. Susceptibility of childhood acute lymphoblastic leukemia: influence of CYP1A1, CYP2D6, GSTMI and GSTTI genetic polymorphism. Blood 1999;93:1496-501.[Abstract/Free Full Text]
5. Infante-Rivard C, Krajinovic M, Labuda D, Sinnet D, Parental smoking, CYP1A1 genetic polymorphism and childhood leukemia. Cancer Causes & Control 2000;11:547-53.[CrossRef][Medline]
6. Ma X, Buffler PA, Selvin S, Matthay KK, Wiencke JL, Reynolds P. Daycare attendance and risk of childhood acute lymphoblastic leukemia. Br J Cancer 2002;86:1419-24.[CrossRef][Medline]
7. Bishop YMM, Fienberg SE, Holland PW. Discrete multivariate analysis: theory and practice. Cambridge (MA): MIT Press; 1975.
8. Freeman DH. Applied categorical data analysis. New York (NY): Marcel Dekker, Inc.; 1987.
9. Breslow NE, Day NE. Statistical methods in cancer research. Vol 1a. Lyon (France): IARC Scientific Publication; 1908. No. 32.
10. Mandansky A. Test of homogeneity for correlated samples. J Am Stat Assoc 1963;58:97-119.
11. Bhapkar VP. On tests of marginal symmetry and quasi-symmetry in two and three-dimensional contingency tables. Biometrics 1979;35:417-26.
12. Stuart A. A test for homogeneity of the marginal distribution in a two-way classification. Biometrika 1955;42:412-6.[Free Full Text]
13. Bailey NTJ. Testing the solubility of maximum likelihood equations in the routine application of scoring methods Biometrics 1951;7:268-74.
14. Bowker AH. A test for symmetry in contingency tables. J Am Stat Assoc 1948;42:572-4.

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