# Minimal sizes of cases with a susceptible genotype and minimal odds ratios among susceptible individuals in case-control studies.

**ABSTRACT** Disease risk elevation due to an environmental factor only for individuals with a susceptible genotype is a typical example of gene-environment interaction. In order to identify risk factors interacting with susceptible genotypes in case-control studies, presumptions on minimal size of cases with the susceptible genotype (S (min)) and odds ratio (OR) among the susceptible individuals (OR(susceptible)) are useful.

Proportion of exposed cases (P(1)) and OR for whole cases (OR(whole)) statistically detectable in a case-control study can be calculated in a conventional method. P(1) was assumed to be a weighted sum of the exposed among cases with the genotype (P(x)) and cases without the genotype (equal to proportion of the exposed among controls, P(0)), i.e., S P(x) + (1 - S) P0, where S is the size (proportion) of cases with the genotype. For each calculated P(1), S became the minimum (S(min)) in case of P(x) = 1. OR(susceptible) was calculated by {P(x) (1 - P(0))} / {(1 - P(x)) P(0)}.

S(min) and OR(susceptible) were listed for the combinations of the above components. For example, a detectable P(1) was 0.638 for P(0)=0.5 in a case-control study with 200 cases (N(1)) and 200 controls (N(0)), when a error of a two-sided test was 0.05 with an 80% of power. In case of P(1)=0.638, OR(whole) was 1.77, producing S(min) = 0.277 for infinite OR(susceptible). It indicates that an environmental factor cannot be detected in case that a high-risk genotype frequency is less than 0.277.

If the size of cases with a susceptible genotype is expected to be less than S(min), case-control studies are unlikely to detect a significant OR of the environmental factor.

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Page 1

Asian Pacific Journal of Cancer Prevention, Vol 6, 2005

165

Minimal Sizes of Classes and Odds Ratios for Case-Control Studies

Asian Pacific J Cancer Prev, 6, 165-169

RESEARCH COMMUNICATION

Introduction

Recent development of genotyping methods allows us

to examine the hypothesis that environmental factors cause

a disease for individuals with a susceptible genotype.

Although not perfect, it was exemplified by the finding that

smoking causes lung cancer more frequently in those with

low enzyme activity genotypes of carcinogen detoxification

enzyme genes (Kiyohara et al., 2002; Mohr et al., 2003).

Epidemiologically, such phenomena are termed as a gene-

environment interaction, which is defined with a relative

risk ratio of environmental exposure for those with a

Minimal Sizes of Cases with a Susceptible Genotype and Minimal

Odds Ratios among Susceptible Individuals in Case-control

Studies

Nobuyuki Hamajima1, Hironori Mutoh2, Hidetaka Eguchi3, Hiroyuki Honda2

1 Department of Preventive Medicine / Biostatistics and Medical Decision Making, Nagoya University Graduate School of Medicine,

Nagoya, Japan 2 Department of Biotechnology, School of Engineering, Nagoya University, Nagoya, Japan. 3Department of Radiobiology/

Molecular Epidemiology, Radiation Effects Research Foundation, Hiroshima, Japan.

Corresponding to: Nobuyuki Hamajima, M.D., Ph.D., M.P.H., Department of Preventive Medicine / Biostatistics and Medical Decision

Making, Nagoya University Graduate School of Medicine, 65 Tsurumai-cho, Showa-ku, Nagoya 466-8550 Japan, TEL:+81-52-744-

2133, FAX:+81-52-744-2971, e-mail: nhamajim@med.nagoya-u.ac.jp

Abstract

Objective: Disease risk elevation due to an environmental factor only for individuals with a susceptible genotype

is a typical example of gene-environment interaction. In order to identify risk factors interacting with susceptible

genotypes in case-control studies, presumptions on minimal size of cases with the susceptible genotype (Smin) and

odds ratio (OR) among the susceptible individuals (ORsusceptible) are useful.

Model: Proportion of exposed cases (P1) and OR for whole cases (ORwhole) statistically detectable in a case-control

study can be calculated in a conventional method. P1 was assumed to be a weighted sum of the exposed among cases

with the genotype (Px) and cases without the genotype (equal to proportion of the exposed among controls, P0), i.e.,

S Px + (1 - S) P0, where S is the size (proportion) of cases with the genotype. For each calculated P1, S became the

minimum (Smin) in case of Px = 1. ORsusceptible was calculated by {Px (1 - P0)} / {(1 - Px) P0}.

Results: Smin and ORsusceptible were listed for the combinations of the above components. For example, a detectable

P1 was 0.638 for P0=0.5 in a case-control study with 200 cases (N1) and 200 controls (N0), when α α α α α error of a two-sided

test was 0.05 with an 80% of power. In case of P1=0.638, ORwhole was 1.77, producing Smin=0.277 for infinite ORsusceptible.

It indicates that an environmental factor cannot be detected in case that a high-risk genotype frequency is less than

0.277.

Interpretation: If the size of cases with a susceptible genotype is expected to be less than Smin, case-control studies

are unlikely to detect a significant OR of the environmental factor.

Key Words: gene-environment interaction – genetic polymorphism – sample size – case-control studies

genotype relative to those without it, or a relative risk ratio

of genotype for the exposed relative to the unexposed

(Khoury and Flanders, 1996; Hamajima et al., 1999;

Brennan, 2002). Since the elucidation of the interactions is

useful for individualized disease prevention, researches on

the interactions have been becoming popular in the field of

epidemiology (Mucci et al., 2001; Kang, 2003). The targeted

genotypes are selected from commonly observable ones,

which are called “polymorphism” genotypes.

When the genotype interacting with an environmental

factor is known, a sample size to detect the odds ratio (OR)

of the factor in a case-control study can be calculated based

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Nobuyuki Hamajima et al

Asian Pacific Journal of Cancer Prevention, Vol 6, 2005

166

on the genotype frequency with a conventional method

(Hwang et al., 1994; Garcia-Closas and Lubin, 1999). On

the contrary, the sample size cannot be calculated in case

that the genotype frequency is unknown. In order to detect

environmental factors in case-control studies including both

subjects with and without the susceptible genotype, we had

better have presumptions on the size (proportion) of

individuals with the genotype and the OR among them. This

paper aims to demonstrate minimal size of cases with the

susceptible genotype to detect a significant environmental

factor in case-control studies, as well as minimal required

OR for individuals with the susceptible genotype.

Statistical Models

We recognized that there was a subgroup of cases with a

genotype susceptible to an environmental factor. In order to

calculate minimal detectable odds ratios of the environmental

factor among those with the genotype (ORsusceptible), the

following steps were made, as shown in Chart.

2.1. A proportion of exposed cases (P1) producing a

significant result in a case-control study with N0 controls

and N1 cases was calculated based on a significance level

(α), statistical power (1-β), and proportion of exposed

controls (P0), using the below conventional formula for a

sample size calculation (Donner, 1984).

Results

Since a large number of combinations exist, those with

α=0.05 in a two-sided test (Zα=1.96), 1-β=0.80 (Zβ=0.842),

and N0=N1 (M=1) were calculated as examples. Table 1

shows the calculated P1, ORwhole, and Smin, when N0 is fixed

to be 200, 500, 1,000, or 2,000, and P0 to be 0.05, 0.1, 0.3,

0.5 or 0.8. For example, a detectable P1 was 0.638 for P0=0.5

in a case-control study with 200 cases (N1) and 200 controls

(N0), when α error of a two-sided test was 0.05 with an 80%

of power. In case of P1=0.638, ORwhole was 1.77, producing

Smin=0.277 for infinite ORsusceptible. It indicates that an

environmental factor cannot be detected in case that a high-

risk genotype frequency is less than 0.277. Figure 2 depicts

the relationship between Smin and N0 for given P0. The

minimal size of cases with the genotype (Smin) increased with

the proportion of the exposed in controls (P0) and decreased

with the number of controls (N0).

0

0.1

0.2

0.3

0.4

0.5

0.6

0500100 0

N0

150 02000

Smin

Smin

P0=0.05

P0=0.1

P0=0.3

P0=0.5

P0=0.8

P0=0.8

Number of Controls (N0)

P0=0.05

P0=0.1

P0=0.3

P0=0.5

Figure 1. Proportions of the Exposed among Controls

(P0) and Cases (P1). P1 is the Average Proportion for Cases with

a Susceptible Genotype (Px) and Cases with no Susceptible

genotype (P0). The Area Surrounded by a Dotted Line is the Same

as the Shadowed Areas. S is the Size in Proportion of Cases with a

Susceptible Genotype.

Figure 2. Minimal Size of Susceptible Cases Enabling to

Detect a Significant Odds Ratio (Smin) According to

Sample Sizes (N0, in Case of N0=N1) and Proportion of

the Exposed among Controls (P0)

P0

0

Px

ControlsCases

P1

01S

Proportion of the exposed

01

With a

susceptible

genotype

With no

susceptible

genotype

With a susceptible genotype +

With no susceptible genotype

where P is defined with (P0 + M P1) / (1 + M ), M with the

ratio of N1 / N0, and Zα and Zβ with the values derived from

a normal distribution with mean=0 and variance=1 for a

given significance level (α) and statistical power (1-β),

respectively.

2.2. Odds ratio for whole subjects (ORwhole) was obtained

by P1 (1-P0) / P0 (1-P1).

2.3. P1 was also defined with a weighted average

calculated by S Px + (1 - S) P0, as shown in Fig 1. In this

formula, Px and P0 were the proportions for the exposed in

cases with and without the susceptible genotype,

respectively. S was the size in proportion for cases with the

genotype. It was assumed that the environmental exposure

does not elevate the risk of disease for cases without the

genotype. Accordingly, the proportion of the exposed among

them was set to be the same as that among the controls, i.e.,

P0.

2.4. Smin was defined as the S in case of Px=1. It was the

minimum of S, because Px was the maximum at 1.

2.5. ORsusceptible was calculated with {Px (1-P0)}/{P0 (1-

Px)}.

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Minimal Sizes of Classes and Odds Ratios for Case-Control Studies

Table 1. Detectable Proportion of the Exposed among

Cases (P1), Odds Ratio for Whole Subjects (ORwhole),

Minimal Size of Cases with a Susceptible Genotype (Smin)

according to Number of Controls (N0) and Proportion

of Exposed Controls (P0), under a Significance Level (α α α α α)

= 0.05 for a Two-sided Test with Statistical Power (1-β β β β β)

=0.8

N0

P0=0.05P0=0.1P0=0.3P0=0.5P0=0.8

P1

0.435

0.384

0.359

0.341

200

500

1,000

2,000

0.130

0.096

0.081

0.071

0.200

0.160

0.141

0.128

0.638

0.588

0.563

0.544

0.900

0.866

0.848

0.834

ORwhole

1.79

1.45

1.31

1.21

200

500

1,000

2,000

2.84

2.02

1.67

1.46

2.25

1.71

1.47

1.32

1.77

1.43

1.29

1.19

2.25

1.62

1.39

1.26

Smin

0.192

0.120

0.084

0.059

200

500

1,000

2,000

0.084

0.048

0.033

0.022

0.111

0.066

0.045

0.031

0.277

0.176

0.125

0.088

0.499

0.330

0.239

0.171

Figire 3. Detectable Minimal ORsubgroup in a Case-control

Study with 200 Cases and 200 Controls According to Size

of Cases with a Susceptible Genotype (S) and Proportion

of the Exposed among Controls (P0)

Figure 4. Detectable Minimal ORsubgroup in a Case-control

Study with Half of the Controls Exposed (P0=0.5),

According to Size of Cases with a Susceptible Genotype

(S) and Number of Controls (N0)

Figure 3 shows ORsusceptible in a case-control study with

200 cases and 200 controls according to size of cases with

the genotype (S) and proportion of the exposed controls (P0).

Since all the cases with the genotype were to be the exposed

at Smin, the ORsusceptible was infinite at Smin. In case of S > Smin,

the ORsusceptible decreased with S, and was equal to ORwhole at

S=1. Figure 4 shows ORsusceptible in case of P0= 0.5 according

to N0 (=N1). As N0 was larger, ORsusceptible was smaller in a

given S. Table 2 lists the detectable ORsusceptible according to

S for different P0 and N0.

The above results can be used for the following examples.

1) When a case-control study has only 200 cases (N1) and

200 controls (N0), smoking can not be evaluable as a risk

factor of male colon cancer in the following condition. Those

with the susceptible genotype (S) are assumed to be 20%

among the cases, and smokers are 50% among the controls

(P0). Table 1 provides Smin = 0.277 for N0=N1=200 and

P0=0.5, which is larger than the assumed S (0.2). 2) When a

30% of male colon cancer cases (S) have a genotype

susceptible to smoking, ORsusceptible more than 3.85 would be

detected in a case-control study with 500 male cases (N1)

and 500 male controls (N0), in an area where smokers are

50% among the male population (P0) as indicated in Table

2.

Discussion

We know intuitively that risk factors affecting a small

proportion of individuals may not be detected in a study,

because of the effect dilution. Accordingly, even with a high

penetrance, rare genotypes are not examined in association

studies. As Shpilberg et al stated, “A twofold risk for 1000

exposed versus nonexposed people could be an average

twofold risk for all 1000 exposed or a 20-fold risk for 100

exposed individuals“ (Shpilberg et al., 1997). In case-control

studies, however, there were no reference tables on the

proportion of susceptible individuals. To date, several papers

have been reporting required sample sizes for unmatched

case-control studies to detect a gene-environment or gene-

gene interaction (Hwang et al., 1994; Garcia-Closas and

Lubin, 1999; Gauderman, 2002a; Gauderman, 2002b,

Selinger-Leneman et al., 2003). But, their view is different

from the present report. Tables and Figures presented in this

paper provide useful information to avoid studies impossible

to detect the significant results. The newly introduced

concept, Smin, is an important measure when case-control

studies are planned taking account of a susceptible subgroup

in the study subjects.

In the present paper, the size of susceptible cases was

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0.00. 20. 4 0.6 0.81.0

S

ORsubgroup

N0=200

N0=500

N0=1000

N0=2000

Size of Cases with a Susceptible Genotype (S)

Detectable ORsusceptible

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0.00.2 0.40. 60.81. 0

S

ORsubgroup

P0=0. 05

P0=0. 1

P0=0. 3

P0=0. 5

P0=0. 8

Size of Cases with a Susceptible Genotype

Detectable ORsusceptible

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Asian Pacific Journal of Cancer Prevention, Vol 6, 2005

168

References

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used, not of susceptible controls which represent the

population without disease under study. Generally, the size

of susceptible cases is larger than the size of susceptible

controls (Scontrol). Although Tables and Figures could

similarly be made using Scontrol, the size of susceptible cases

(S) was adopted here. The S seems easier to be understood

and estimated by clinicians, who are faced with patients.

In conclusion, this paper provided the useful figures

when case-control studies on environmental factors

interacting with genotypes are designed. These figures are

applicable for OR of a genotype interacting with

environmental factors, and also for gene-gene interactions

to be derived from case-control studies based on high-

throughput SNP analysis (Marnellos, 2003; McLeod and

Yu, 2003).

Acknowledgements

This work was supported in part by a Grant-in-Aid for

Scientific Research on Special Priority Areas of Cancer from

the Ministry of Education, Culture, Sports, Science and

Technology of Japan.

Chart for the Calculation Steps

1.Calculation of P1 to obtain a significant result from given

P0, N0, N1, significance level, and statistical power.

Calculation of ORwhole from P0 and P1.

Calculation of Px from P0, P1, and given S.

Calculation of Smin in case of Px = 1.

Calculation of ORsusceptible from P0, Px, and S.

2.

3.

4.

5.

N0: Number of controls

N1: Number of cases

P0: Proportion of the exposed among controls

Px: Proportion of the exposed among cases with a susceptible

genotype

P1: Proportion of the exposed among cases, which is defined

with S Px + (1 – S) P0

S : Size (proportion) of cases with the susceptible genotype

Smin: The minimal S, i.e., S in case of P1=1

ORwhole: Odds ratio for whole cases

ORsusceptible: Odds ratio for individuals with the susceptible

genotype.

Table 2. Detectable OR for Individuals with a Genotype

Susceptible to Environmental Factor (ORsusceptible)

according to Size of Cases with the Susceptible Genotype

(S), Proportion of Exposed Controls (P0), and Number

of Controls (N0), under a Significance Level (α α α α α) = 0.05

for a Two-sided Test with Statistical Power (1-β β β β β) =0.8

N0

S=0.1S=0.2S=0.3 S=0.5S=0.7 S=1

P0=0.05

200

500

1,000

2,000

107

19.8

10.7

6.72

15.5

7.40

4.90

3.50

8.80

4.86

3.44

2.60

5.05

3.15

2.40

1.93

3.73

2.49

1.98

1.66

2.84

2.02

1.67

1.46

P0=0.1

200

500

1,000

2,000

N.E.

20.5

9.28

5.55

13.4

5.94

3.93

2.86

6.86

3.83

2.78

2.16

3.85

2.52

2.00

1.67

2.88

2.04

1.69

1.47

2.25

1.71

1.47

1.32

P0=0.3

200

500

1,000

2,000

N.E.

N.E.

18.6

5.81

85.4

6.00

3.42

2.40

6.96

3.22

2.30

1.82

3.09

2.05

1.67

1.45

2.26

1.69

1.46

1.31

1.79

1.45

1.31

1.21

P0=0.5

200

500

1,000

2,000

N.E.

N.E.

N.E.

16.4

N.E.

15.9

4.33

2.59

24.7

3.85

2.43

1.84

3.48

2.09

1.67

1.43

2.31

1.67

1.43

1.29

1.77

1.43

1.29

1.19

P0=0.8

200

500

1,000

2,000

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

N.E.

8.43

N.E.

N.E.

5.85

2.66

584

3.43

2.14

1.65

4.10

2.12

1.65

1.41

2.25

1.62

1.39

1.26

N.E.: ORsusceptible does not exist.

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Minimal Sizes of Classes and Odds Ratios for Case-Control Studies

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