Incorporation of genetic model parameters for cost-effective designs of genetic association studies using DNA pooling
- Fei Ji^{1},
- Stephen J Finch^{2},
- Chad Haynes^{1},
- Nancy R Mendell^{2} and
- Derek Gordon^{3}Email author
DOI: 10.1186/1471-2164-8-238
© Ji et al; licensee BioMed Central Ltd. 2007
Received: 19 December 2006
Accepted: 16 July 2007
Published: 16 July 2007
Abstract
Background
Studies of association methods using DNA pooling of single nucleotide polymorphisms (SNPs) have focused primarily on the effects of "machine-error", number of replicates, and the size of the pool. We use the non-centrality parameter (NCP) for the analysis of variance test to compute the approximate power for genetic association tests with DNA pooling data on cases and controls. We incorporate genetic model parameters into the computation of the NCP. Parameters involved in the power calculation are disease allele frequency, frequency of the marker SNP allele in coupling with the disease locus, disease prevalence, genotype relative risk, sample size, genetic model, number of pools, number of replicates of each pool, and the proportion of variance of the pooled frequency estimate due to machine variability. We compute power for different settings of number of replicates and total number of genotypings when the genetic model parameters are fixed. Several significance levels are considered, including stringent significance levels (due to the increasing popularity of 100 K and 500 K SNP "chip" data). We use a factorial design with two to four settings of each parameter and multiple regression analysis to assess which parameters most significantly affect power.
Results
The power can increase substantially as the genotyping number increases. For a fixed number of genotypings, the power is a function of the number of replicates of each pool such that there is a setting with maximum power. The four most significant parameters affecting power for association are: (1) genotype relative risk, (2) genetic model, (3) sample size, and (4) the interaction term between disease and SNP marker allele probabilities.
Conclusion
For a fixed number of genotypings, there is an optimal number of replicates of each pool that increases as the number of genotypings increases. Power is not substantially reduced when the number of replicates is close to but not equal to the optimal setting.
Background
Case/control genetic association studies are used as a means of localizing susceptibility genes for a complex disease. With the recent development of technologies that can determine the genotypes for hundreds of thousands of single nucleotide polymorphisms (SNPs) across the human genome, such studies are now being reported in the literature [1–3]. Design issues such as power to detect association using these technologies are also being published [4, 5]. Since a critical requirement for such studies to be sufficiently powered is that the disequilibrium among the disease allele and neighboring marker alleles be large, marker density needs to be high. If the effect size for a complex disease is small (e.g., genotype relative risks [6] on the order of 1.5 to 2), the sample size required to detect association may be thousands of cases and controls [4, 5, 7–9]. Therefore, researchers often consider genotyping technologies such as DNA pooling [10–13] as an initial strategy to identify genomic regions that may harbor susceptibility loci in an effort to reduce cost (time and money) (e.g.,[14, 15]). Advantages of DNA pooling technologies include (a sometimes substantial) reduction in genotyping cost when performing multi-stage association studies to identify disease susceptibility genes. Potential disadvantages include reliance on a number of assumptions related to statistical design and analysis. For example, a key assumption is that the intensity measure has an expected value equal to the allele frequency. Another potential disadvantage is that DNA pooling techniques may not detect disease mode of inheritances that deviate from dominant or recessive modes. For example, DNA pooling techniques will be underpowered to detect disease genes that operate in an over-dominant form.
Sham et al. reviewed currently available technologies for DNA pooling [10]. The statistical analysis of data from pooled DNA studies uses analysis of variance (ANOVA) procedures that have algorithms for calculating power to detect unequal allele probabilities. A major design issue when using DNA pooling technologies is the measurement error as compared with the gold standard method of individual genotyping.
Research has been done regarding specification of study parameter settings to maximize power [10, 16, 17]. The research question addressed in this work is: assuming a certain level of measurement error, what settings of study design parameters maximize the power to detect association? More specifically, we study the sensitivity of power to changes in design parameters (e.g., total sample size, differing numbers of genotypings, number of pools, and genetic model parameters). We present a closed form approximation to the power in terms of the genetic model, pooling measurement error model, and the study parameters (e.g., number of pools, number of replicates per pool, sample size) and we perform a systematic study of the design parameters to identify which have the greatest effect on power to detect association for DNA pooling studies.
Results
The analysis of variance table for a two-stage nested design
ANOVA Table | |||
---|---|---|---|
Source | DF | SS | E(MS) |
Case or control (α) | 1 | $\sum _{i}{\displaystyle \sum _{j}{\displaystyle \sum _{k}({Y}_{i\u2022\u2022}-}}}{Y}_{\u2022\u2022\u2022}{)}^{2}=JK{\displaystyle \sum _{i}({Y}_{i\u2022\u2022}-}{Y}_{\u2022\u2022\u2022}{)}^{2$ | $\frac{JK{\displaystyle \sum {\alpha}_{i}^{2}}}{I-1}+(K{\sigma}_{\overline{P}}^{2}+{\sigma}_{E}^{2})$ |
Pools nested in case or control (P) | 2(J - 1) | $\sum _{i}{\displaystyle \sum _{j}{\displaystyle \sum _{k}({Y}_{ij\u2022}-}}}{Y}_{i\u2022\u2022}{)}^{2}=K{\displaystyle \sum _{i}{\displaystyle \sum _{j}({Y}_{ij\u2022}-}{Y}_{i\u2022\u2022}{)}^{2}$ | $(K{\sigma}_{\overline{P}}^{2}+{\sigma}_{E}^{2})$ |
Replicates (E) | IJ(K - 1) | $\sum _{i}{\displaystyle \sum _{j}{\displaystyle \sum _{k}({Y}_{ijk}-}{Y}_{ij\u2022}{)}^{2}}$ | ${\sigma}_{E}^{2}$ |
The power calculation of the F-test, the standard statistical procedure used when testing allele frequency differences for DNA pooling, requires the non-centrality parameter (NCP) of the test. Its approximate value is given in equation 1 of the Methods and Technical Issues section below. The NCP is a function of the difference between the case and control allele 2 frequencies, the quality of the pooling estimate of these probabilities, the number of cases and controls, the number of replications of DNA measurements of each pool, and the size of each pool.
When the number of replicates K is fixed, the approximate NCP is constant with respect to the number of pools (J). When the number of pools J is larger, the denominator degrees of freedom (df) are larger, so that the power of the F-test is greater. That is, smaller pool sizes T = N/J for larger J, have greater power. The protocol of genotyping each subject has T = 1, which is the most powerful allele frequency testing protocol. That is, if genotype cost is not an issue, it is always most powerful to individually genotype all subjects.
We note that, although results are not presented, we performed analyses similar to those presented in Figures 1 and 2 for a multiplicative MOI. The conclusions were the same, with results being very similar to the dominant MOI results (Figure 2). We omit these results in the interest of brevity.
Maximum power as a function of the number of genotyping(G = J × K), number of replicates giving maximum power (K_{ o }(G)), number of replicates (K) at 95% of the maximum power at specific experimental and genetic parameters and the power at K = 1 when assuming no machine replicability variability (m = 1)
Situation | N | MOI | R _{2} | α | m | MAF | p _{ r } | G = 40 | G = 80 | G = 160 | G = 320 | G = 640 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 10000 | R | 2.2 | 0.0001 | 2.25 | 0.20 | 0.9 | 38%, 2, (2), 82% | 54%, 4, (3–4), 85% | 64%,6, (4–7), 87% | 72%, 10, (5–16), 87% | 77%, 13, (6–27), 88% |
2 | 10000 | R | 2.0 | 0.001 | 2.25 | 0.20 | 0.9 | 43%, 2, (2), 79% | 56%, 4, (3–4), 81% | 65%, 7, (4–7), 82% | 71%, 11, (6–16), 83% | 75%, 16, (7–32), 83% |
3 | 10000 | R | 1.8 | 0.01 | 2.25 | 0.20 | 0.9 | 50%, 2, (2), 79% | 62%, 4, (3–4), 80% | 68%, 7, (5–7), 80% | 72%, 13, (6–16), 80% | 75%, 20, (7–32), 81% |
4 | 10000 | R | 2.2 | 0.0001 | 2.25 | 0.15 | 1 | 69%, 2, (2), 98% | 84%, 4, (3–4), 99% | 90%, 6, (3–7), 99% | 95%, 10, (4–16), 99% | 96%, 13, (4–32), 99% |
5 | 10000 | R | 2.2 | 0.0001 | 2.0 | 0.15 | 1 | 75%, 2, (2), 98% | 87%, 4, (2–4), 99% | 92%, 5, (3–7), 99% | 95%, 8, (3–16), 99% | 97%, 12 (3–32), 99% |
6 | 10000 | R | 2.0 | 0.0001 | 2.25 | 0.15 | 1 | 43%,2, (2), 86% | 59%, 4, (3–4), 89% | 70%, 6, (4–7), 90% | 77%, 10, (5–16), 91% | 82%, 13, (6–28), 91% |
7 | 10000 | R | 2.0 | 0.0001 | 2.0 | 0.15 | 1 | 49%, 2 (2), 86% | 63%, 4, (3–4), 89% | 73%, 5, (4–7), 90% | 79%, 8, (4–16), 91% | 83%, 12, (5–27), 91% |
8 | 2000 | D | 1.5 | 0.0001 | 2.25 | 0.15 | 0.9 | 57%, 3, (2–3), 94% | 73%, 4, (3–6), 96% | 82%, 6, (4–10), 96% | 88%, 10, (4–18), -- | 91%, 13, (5–32), -- |
9 | 2000 | D | 1.5 | 0.0001 | 2.25 | 0.15 | 1 | 65%, 3, (2–3), 97% | 80%, 4, (3–6),98% | 88%, 6, (3–9), 98 | 92%, 10, (4–21), -- | 95%, 13, (4–40), -- |
10 | 2000 | D | 1.5 | 0.0001 | 2.0 | 0.15 | 1 | 70%, 2, (2–3), 97% | 83%, 4, (3–6), 98% | 90%, 5. (3–10), 98% | 94%, 8, (3–20), -- | 96%, 12, (4–37), -- |
Regression modeling results
List of parameters considered in the multiple regression analysis
Parameter | Description | Value |
---|---|---|
N | Number of case (control) subjects | 1000, 2000, 5000, 10000 |
φ | Prevalence of the disease | 0.01, 0.1 |
T | Size of the pool | 25, 50, 100, 250, 500 |
K | Number of replicates of each pool | 1, 2, 4, 8 |
p _{ d } | Disease allele frequency | 0.1, 0.25 |
MOI | Modes of inheritance | dominant, recessive, multiplicative |
R _{2} | Genotype relative risk of homozygote of disease allele | *1.2, 1.5, 2.25, 4 |
q _{1} | Minor SNP marker allele frequency | 0.1, 0.35 |
M | Machine replicability variance factor | 2.05, 2.1, 2.25, 3 |
Among the interaction terms not involving K, p_{ d }× q_{1}, p_{ d }× MOI, R_{2} × T, p_{ d }× R_{2}, and N × T are highly significant (sorted in increasing P-values). The most significant interaction term is p_{ d }× q_{1}. This finding is not surprising as there has been extensive documentation in the statistical genetics literature that power for genetic association is maximized when the difference between the disease allele frequency and the SNP marker allele frequency in coupling with the disease allele is 0, with decreasing power occurring as the difference increases [20–23]. The finding of a significant interaction p_{ d }× MOI between disease allele frequency and disease MOI has also been documented previously, most recently in the work by Skol et al. [4]. The finding underscores the fact that, when all other factors are fixed, the disease allele frequency that gives optimal power differs depending upon the disease MOI.
Discussion
Our results have produced two types of conclusions. The first is that the genetic parameters of the disease being studied are the most important determinants of the power to detect association. This fact is consistent with the association of ApoE with late onset Alzheimer's Disease [24] and recent association results for age-related macular degeneration [1, 3]. In each of these studies, estimated genotype relative risks are approximately 3 for the heterozygote and greater than 9 for the homozygote. In all studies, highly significant associations were observed with less than 500 total cases and controls. Furthermore, for age-related macular degeneration [24], associations were observed for SNP alleles in linkage disequilibrium (LD) with the functional variants. The results from the OLS regression analysis are consistent with this history. The genetic relative risk is the most significant parameter, followed by the sample size. For a fixed genotype relative risk R_{2,} the median power is greatest for dominant MOI, followed by multiplicative and then recessive MOIs. The linear and quadratic terms in the number of replicates K and a number of interactions with K are significant. Since there is an optimal setting of K, this result is expected.
The second type of conclusion is guidance about the choice of the number of genotypings G = J × K and the simultaneous setting of the number of replicates K of the J pools. We have shown that the number of genotypings G = J × K should be as large as possible (holding all other factors constant) to have the greatest power. When G is fixed, we have shown that there is a setting K_{ o }(G) that maximizes the power when all genetic model parameters are specified. The optimal setting increases as G increases. These differences are practically important and suggest that those conducting pooled studies use the program available from the corresponding author to determine optimal settings. In all situations studied, for fixed value of G, power is relatively insensitive to choice of K near K_{ o }(G). Further, when the machine replicability variance factor m is larger than 1, the setting K = 1 has power much less than replicated designs. This suggests that such extensions of these designs as staggered nested designs [18] may have little value in genetic pooling studies.
Our work provides the basis for extending recommendations such as those of Sham et al. [10] to include genetic model parameters. For the very large studies possible with pooling, there is strong evidence that increasing the number of genotypings and increasing the number of replicate measurements of each pool can increase power noticeably. This approach is dependent on the assumption that E(Π_{ i }) = E(Y_{ ijk }), where Π_{ i }is the fraction of the major allele 2 in a randomly selected subject from the i^{th} group; that is, the pooled estimate of the intensity of an allele is in fact an unbiased estimate of the allele 2 frequency. Further work will incorporate designs that formally include validation of this assumption.
Conclusion
Our work extends that of previous researchers who have considered power and sample size calculations for genetic association studies with pooled DNA samples (e.g., [16]). Our extension involves inclusion of genetic model parameters such as disease MOI, disease allele frequency, disease prevalence, marker allele frequency, and genotype relative risks. It is clear from the results of our regression analysis that incorporation of such parameters is important in the design of more powerful genetic association tests. We recommend that researchers incorporate information into their power and sample size calculations for genetic association with pooled DNA, such as choice of numbers of genotypings and the number of replicates that can increase power from such relatively low levels as 40% to 50% to 75% to 80% using the same cases and controls.
Methods
Definitions
N: number of case (control) subjects; we assume equal numbers of cases and controls (balanced design).
J: number of pools; J ≥ 2.
T = N/J: number of individuals in each pool; we assume that case subjects are assigned randomly to case pools and control subjects are assigned randomly to control pools.
K: number of replicates of each pool; we assume that there is no reassignment of subjects in the replications.
G = J × K: number of case (control) genotypings.
Genetic model parameters
We consider a disease associated with a di-allelic gene with allele d associated with increased risk of disease and allele + associated with no increased risk.
p_{ d }: allele frequency of disease locus d allele.
p_{+} = 1 - p_{ d }: allele frequency of disease locus wild-type (+) allele.
φ: prevalence of the disease.
f_{2}: probability of having disease with 2 disease alleles in the genotype = penetrance of dd.
f_{1}: probability of having disease with 1 disease allele in the genotype = penetrance of d+.
f_{0}: probability of having disease with 0 disease alleles in the genotype = penetrance of ++.
Genotype relative risks (GRR)
Modes of Inheritance (MOI)
The three MOIs are characterized by the parameter R.
Multiplicative MOI: The penetrances satisfy the equation $R=\frac{{f}_{1}}{{f}_{0}}=\frac{{f}_{2}}{{f}_{1}}$; that is, ${R}_{2}={R}_{1}^{2}$.
Dominant MOI: R = R_{1} = R_{2}.
Recessive MOI: $R={R}_{2}=\frac{{R}_{2}}{{R}_{1}}$; that is, R_{1} = 1.
SNP marker parameters
q_{1}: allele frequency of minor SNP marker allele 1 (that is, 0 <q_{1}≤ 0.5).
q_{2}: the frequency of the major SNP marker allele 2.
Disequilibrium parameters
D_{max} = min(p_{ d }q_{2}, p_{+}q_{1}).
${R}_{\mathrm{max}\phantom{\rule{0.5em}{0ex}}}^{2}=\frac{{D}_{\mathrm{max}\phantom{\rule{0.5em}{0ex}}}^{2}}{{p}_{d}{p}_{+}{q}_{1}{q}_{2}}$ (see, e.g., [25]).
p_{ r }: measure of linkage disequilibrium between disease gene and SNP marker; here it is a fraction of ${R}_{\mathrm{max}\phantom{\rule{0.5em}{0ex}}}^{2}$ (0 <p_{ r }≤ 1); the examples use p_{ r }= 0.9.
The detailed computation of case and control genotype probabilities which are functions of the disease allele frequency, minor SNP allele frequency, and linkage disequilibrium parameters are documented in the PAWE-3D Helpfile [19].
We use method [26] implemented in the PAWE software [27] to calculate the power of the 2 × 2 test of independence when each subject is individually genotyped and we report these value in Figures 1 and 2.
Case-control frequency of allele 2
Analysis of variance (ANOVA) table for two-stage nested design
Specification of ANOVA model
A_{ ijk }: intensity level of allele 2 in the i^{th} group (i = 0 for cases, 1 for controls), j^{th} pool (j = 1,...,J), k^{th} replicate (k = 1,...,K).
B_{ ijk }: intensity level of allele 1 in the i^{th} group (i = 0 for cases, 1 for controls), j^{th} pool (j = 1,..., J), k^{th} replicate (k = 1,..., K).
${Y}_{ijk}=\frac{{A}_{ijk}}{{A}_{ijk}+{B}_{ijk}}$: fraction of SNP allele 2 estimated in the i^{th} group (i = 0 for cases, 1 for controls), j^{th} pool (j = 1,..., J), k^{th} replicate (k = 1,..., K).
Model:
Y_{ ijk }= μ + α_{ i }+ P_{j(i)}+ σ_{ E }E_{ ijk },
Here, var(Y_{ ijk }) is modeled as the sum of two components of variance. The first, ${\sigma}_{P,i}^{2}=\frac{J{\tau}_{i}^{2}}{N}$, is due to the sampling variation of the frequency of allele 2 in the subjects assigned to each pool. The second, ${\sigma}_{E}^{2}$, is due to the measurement error of the processing of the pooled material. Under an ideal measurement process, ${\sigma}_{E}^{2}$ = 0; we define a parameter m to capture the departure from this ideal. The parameter m (machine replicability variance factor), m ≥ 1, is defined by ${\sigma}_{E}^{2}={\sigma}_{P,i}^{2}(m-1)$, so that m = 1 represents the ideal measurement process and m > 1 models additional variability due to a less than perfect measurement process. The fraction of var(Y_{ ijk }) due to the measurement process is $\frac{m-1}{m}$.
This model is dependent on the assumption that $E({\Pi}_{i})=E(\frac{{A}_{ijk}}{{A}_{ijk}+{B}_{ijk}})=E({Y}_{ijk})$. Also, let $\rho =\frac{\mathrm{max}\phantom{\rule{0.5em}{0ex}}({\sigma}_{P,0}^{2},{\sigma}_{P,1}^{2})}{\mathrm{min}\phantom{\rule{0.5em}{0ex}}({\sigma}_{P,0}^{2},{\sigma}_{P,1}^{2})}$. This value is an indication of the adequacy of the approximation of the NCP in equation (1) below [28].
where ${T}_{ij}={P}_{j(i)}+{\sigma}_{E}{E}_{ij\u2022},{T}_{i\u2022}={\displaystyle \sum _{j=1}^{J}{T}_{ij}}/J,\text{and}{T}_{ij}~N(0,{\sigma}_{P,i}^{2}+\frac{{\sigma}_{E}^{2}}{K})$.
where ${X}_{i}~{\chi}_{J-1}^{2}$. The sum of squares SS_{ P }therefore has the distribution $K({\sigma}_{\overline{P}}^{2}+\frac{{\sigma}_{E}^{2}}{K}){\chi}_{2(J-1)}^{2}$ when the null hypothesis is true, with ${\sigma}_{\overline{P}}^{2}={\sigma}_{P,0}^{2}={\sigma}_{P,1}^{2}$. Further $E(S{S}_{P})=2(J-1)(K{\sigma}_{\overline{P}}^{2}+{\sigma}_{E}^{2})$ under both the null and alternative hypotheses with ${\sigma}_{\overline{P}}^{2}=\frac{J{\overline{\tau}}^{2}}{N}$ and ${\overline{\tau}}^{2}=\frac{{\displaystyle \sum _{i=0}^{1}{\tau}_{i}^{2}}}{2}=\frac{{\displaystyle \sum \mathrm{var}\phantom{\rule{0.5em}{0ex}}({\Pi}_{i})}}{2}$. The distribution of SS_{ P }under the alternative is a weighted sum of independent central chi-square distributions.
which is not dependent upon J, assuming this model. This result is due to the fact that we assumed ${\sigma}_{P,i}^{2}=\frac{{\tau}_{i}^{2}}{T}$, which is an assumption that each individual's variance contributes equally to the variance of the pool. The factor (m - 1) includes the cumulative effect of such sources of additional variability as experimental error, differential variability in processing of individuals, and other sources.
Multiple regression analysis of approximate power
We calculated the approximate power of the experimental design under various values of parameters (Table 3). We then used OLS multiple regression analysis to identify the parameters that had the greatest impact on power, using SAS software [30]. For independent variables, we used all variables listed in Table 3, all two way interactions of these variables, and K^{2}, the square of the number of replicates to incorporate the existence of an optimal number of replicates. We considered type I errors at 0.01, 0.001 and 0.0001 levels. It might be argued that researchers should use 0.0001 or less as a stringent significance level if the design is applied in a genome-wide association study. Since DNA pooling techniques are normally used as 1^{st} stage screening and for 1^{st} stage design, researchers may be more concerned with false negatives than false positives [9, 31].
Declarations
Acknowledgements
This research was supported by the US National Institutes of Health grants NIMH R01 MH071523 (NRM), NIMH 2R01 MH04480114A1 (SJF), and MH44292 (FJ). The authors thank Dr. Yaning Yang for providing comments on our revised version of the manuscript.
Authors’ Affiliations
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