An analytic solution to computation of power and sample size for genetic association studies under a pleiotropic mode of inheritance

Abstract

Our motivation here is to calculate the power of three statistical tests used when there are genetic traits that operate under a pleiotropic mode of inheritance and when qualitative phenotypes are defined by use of thresholds for the multiple quantitative phenotypes. Specifically, we formulate a multivariate function that provides the probability that an individual has a vector of specific quantitative trait values conditional on having a risk locus genotype, and we apply thresholds to define qualitative phenotypes (affected, unaffected), and compute penetrances and conditional genotype frequencies based on the multivariate function. We extend analytic power and minimum-sample-size-necessary (MSSN) formulas for two categorical-based tests (Genotype, Linear Trend Test (LTT)) of genetic association to the pleiotropic model. We further compare MSSN of the Genotype and LTT tests with that of a MANOVA test (Pillai). We approximate MSSN for statistics by linear models using factorial design and ANOVA. With the ANOVA decomposition, we determine what factors most significantly change power/MSSN for all statistics. Finally, we determine what test statistics have smallest MSSN In this work, MSSN calculations are for two traits (bivariate distributions) only (for illustrative purposes). We note that the calculations may be extended to address any number of traits.

Our key findings are that the Genotype test usually has smaller MSSN requirements than the LTT. More inclusive thresholds (top/bottom 25% versus top/bottom 10%) have higher sample size requirements. The Pillai test has much larger MSSN than both the Genotype and Trend tests as a result of sample selection.

With these formulas, researchers can specify how many subjects they must collect to localize genes for pleiotropic phenotypes.

Introduction

In his review article on one hundred years of pleiotropy, Stearns credits the Swiss geneticist Ludwig Plate as being the first to use the term in 1910 [1]. Stearns’ definition was “Pleiotropy refers to the phenomenon in which a single locus affects two or more apparently unrelated phenotypic traits and is often identified as a single mutation that affects two or more wild-type traits.” [1] We translate this definition into a mathematical model in the Methods section.

As of this writing, searching on the term ‘pleiotropy’ under “Topic” in the ISI Web of Science database yields over 11,000 publications. This number suggests that pleiotropy is both a common phenomenon and one that has been well-studied. A significant number of these publications (over 1,300, according to ISI Web of Science) deal with mouse, fly, plants, dogs, chickens, and other animals/organisms. There are a host of statistically powerful techniques available for gene mapping in these model organisms (see, e.g., [2] for mouse).

In humans, there are numerous examples of pleiotropic effects that are correlated with traits and/or diseases. Some examples include: colorectal cancer [3,4], Crohn’s Disease [3,5–10], Alzheimer’s Disease [11–19], and Marfan syndrome [20–22]. Papers by Baumgartner et al. [22] and Solovieff et al. [23] highlight some challenges regarding the study of pleiotropic traits in humans. One challenge is the computation of statistical power and/or minimum sample size necessary (MSSN) for genetic association, a critically important component of any gene mapping work. With these values, researchers may obtain a realistic estimate either of MSSN to establish genetic association, or of the probability of detecting genetic association for a collected sample. Power and MSSN calculations for single-phenotype tests of genetic association have been derived by Mitra [24] for the chi-square test of indepedence on alleles/genotypes and by several authors [25–28] for the linear trend test. From this point forward, we refer to the first and second tests as the Genotype test (since the data collected are genotypes on individuals) and LTT (for Linear Trend Test), respectively.

There have been a number of publications documenting ways to detect and analyze pleiotropic data, most recently for genome-wide association studies [23,29–58], and also reporting methods to determine power and or MSSN for association mapping [43,45,46,59–66]. If one broadens the search to allow for multiple phenotypes that may not be pleiotropic, the list of published methods increases [34,67–85]. Studying these methods, we note that the majority deal with data analysis. We comment that a number of authors who document power for their method do so by simulation (e.g., [45,47]) or for a specific data set (e.g., [6,60,86]).

The purpose of this work is the development of an analytic approach to computing statistical power for a fixed sample size and given significance level, or MSSN (in terms of affected and unaffected individuals) to achieve fixed power at a given significance level for a number of different statistical tests. Our method is threshold-based, in the sense that we transform individuals with quantitative phenotype-vector-values into either affected or unaffected individuals using thresholds. From this point forward, we will use the abbreviations QT for Quantitative Trait/Quantitative Phenotype, and Quantitative Trait Value (QTV) to refer to an individual's quantitative phenotype-vector-values.

Our method is a natural extension of the univariate threshold-selected quantitative trait association power and MSSN calculator (e.g., [87,88]) in that, when the number of phenotypes is one, our method reduces to the univariate method. Some suggested benefits of our method are that: (i) it is based on classical quantitative-genetics mapping methods for selected-sampling; (ii) the mathematics used is well-established and straightforward to implement.

We use a threshold approach because a number of pleiotropic diseases are defined this way. For example, Marfan syndrome and Tourette syndrome are comprised of multiple traits, each of which may be caused by a single gene on the chromosome [2]. The phenotypes caused by these disorders are also quantitative or continuously distributed. That is, individuals may exhibit these traits in varying degrees (e.g., mild to severe). We note that each trait may be defined by thresholds for different QTs. Thresholds are provided below. For all of the syndromes listed below, each of the conditions listed is necessary.

1. Marfan syndrome. According to the Marfan Foundation [89], one definition of Marfan syndrome in the absence of family history [90] is given by:

   1. Aortic Root Dilatation Z score ≥ 2;
   2. Systemic Score ≥ 7 points.
2. For a person to be diagnosed with Tourette syndrome [91], he or she must:

   1. Have ≥ 2 motor tics (for example, blinking or shrugging the shoulders);
   2. Have ≥ 1 vocal tic (for example, humming, clearing the throat, or yelling out a word or phrase), although they might not always happen at the same time;
   3. Have these tics ((a) and (b)) for ≥ 1 year. The tics can occur many times a day (usually in bouts) nearly every day, or off and on;
   4. Have tics that start at ≤18 years of age;
   5. Have symptoms that are not due to taking medicine or other drugs or due to having another medical condition (for example, seizures, Huntington’s disease, or post-viral encephalitis).

Additionally, we make the distinction between pleiotropy and locus heterogeneity. In Tourette’s syndrome, there is a documented evidence of locus heterogeneity [92,93]. Hence, in a particular family, it may be that these traits are “caused by a single gene” with a high penetrance. However, this situation is not what we mean by pleiotropy. For pleiotropy, it must be the same gene causing changes in the multiple phenotypes across families/individuals.

We include a section on derivation of the power/MSSN for Multivariate ANalysis Of VAriance (MANOVA) using the Pillai trace statistic applied to the quantitative measures directly. Our reasons are that: (i) several published methods consider power and/or MSSN for pleiotropic phenotypes using quantitative measures [31,32,36,40,42,45,94–101]; (ii) while there is no uniformly most powerful test for MANOVA using equality of means as the null hypothesis, the Pillai trace statistic has high power under a number of different settings; and (iii) it is robust to several violations of assumptions in the MANOVA model [102,103]. We perform a comparison of the MSSN for the Pillai statistic and our statistics using specified genetic model parameter settings.

Finally, we develop software that performs power and/or MSSN calculations for detecting genetic association with: (i) LTT and Genotype test for threshold-defined phenotypes; and (ii) Pillai's trace statistic for the original phenotypes. We note that this software is an extension of software programs designed to compute power and/or MSSN considering a single locus and a single phenotype. In this work, MSSN calculations are for two traits (bivariate distributions) only. Our calculations may be extended to address any number of traits.

Methods

Note: Notation for much of this section may be found in the Appendix.

Test statistic for one-way Multi-variate ANalysis Of Variance (MANOVA)

Here, we present the test-statistic used to test our multiple null hypotheses when the data are quantitative. Several multivariate mean vectors in a one-way MANOVA may be compared statistically using Wilks’ lambda, Pillai’s trace, Roy’s largest root, or Hotelling-Lawley’s tests [102,103]. Though none of tests is uniformly most powerful, Pillai’s trace statistic is reported to have good power in many scenarios and is robust to deviations from assumptions specified in MANOVA [102]. As an indication of its popularity, Pillai's trace test is the default test in the manova function of the R statistical software package [106]. Wilks’ lambda is equivalent to the likelihood ratio test and it has similar power to Pillai’s statistic in many alternative settings [102,103].

Notation for Pillai statistic

Here, we define the null hypotheses for Pillai’s statistic, and the statistic itself.

g: The number of groups considered for each phenotype; here, g is the number of genotypes at a SNP locus, so that g = 3.

p: The number of phenotypes (response variables).

Definition of Pillai statistic

Here, we present the Pillai trace statistic. It is used to test our multiple null hypotheses when the data are quantitative. As an indication of its popularity, Pillai's trace test is the default test in the manova function of the R statistical software package [106]. Wilks’ lambda is equivalent to the likelihood ratio test and it has similar power to Pillai’s statistic in many alternative settings [102,103].

To begin, let $Y=\left(\begin{array}{c}\hfill Y_1\hfill \\ \hfill Y_2\hfill \\ \hfill ⋮\hfill \\ \hfill Y_g\hfill \end{array}\right)=\text{An }N\times p$ data matrix, where $Y_i=\left(\begin{array}{cccc}\hfill y_{i11}\hfill & \hfill y_{i12}\hfill & \dots \hfill & \hfill y_{i1p}\hfill \\ \hfill y_{i21}\hfill & \hfill y_{i22}\hfill & \dots \hfill & \hfill y_{i2p}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill y_{{\mathit{\text{in}}}_{i}1}\hfill & \hfill y_{{\mathit{\text{in}}}_{i}2}\hfill & \cdots \hfill & \hfill y_{{\mathit{\text{in}}}_{i}p}\hfill \end{array}\right)$ is an n_i × p data matrix, and y_ijk is the j^th observation of the i^th phenotype in the k^th genotype group, the total number of observations being denoted by N = n₁ + ⋯ + n_g. Note that 1 ≤ i ≤ g, 1 ≤ j ≤ n_i for the i^th genotype group, and 1 ≤ k ≤ p. Also, n_i is the number of individuals with the i^th genotype.

Let X denote the N × g design matrix given by $X=\left(\begin{array}{ccc}\hfill 1_{n_1}\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1_{n_g}\hfill \end{array}\right)$, where the matrices 1_ni, 1 ≤ i ≤ g, are of size n_i × 1 and are defined as $1_{n_i}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill ⋮\hfill \\ \hfill 1\hfill \end{array}\right)$. Also, let X′X and $\frac{1}{N}$ X′X be the diagonal g × g matrices given by $\left(\begin{array}{ccc}\hfill n_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \ddots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill n_g\hfill \end{array}\right)$ and $\left(\begin{array}{ccc}\hfill \frac{n_1}{N}\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill \frac{n_g}{N}\hfill \end{array}\right)$, respectively.

The Pillai trace test statistic is defined as: $V={\sum }_{i=1}^s\frac{{\varphi }_i}{1+{\varphi }_i}$, and is based on the s = min(g − 1, p) eigenvalues {ϕ₁ ≥ ⋯ ≥ ϕ_s} of E⁻¹H, where:

$$E=A\prime (Y-X\hat{B})\prime (Y-X\hat{B})A,$$

$$H=N(C\hat{B}A)\prime {(C{(\frac{1}{N}X\prime X)}^{-1}C\prime )}^{-1}(C\hat{B}A).$$

Note that the matrix B̂ is the matrix B with parameters estimated from the data. The matrices C and A are stated below. The estimate of each µ_ij is given by ${\hat{\mu }}_{\mathit{\text{ij}}}=\frac{1}{n_i}{\sum }_{u=1}^{n_i}{y}_{\mathit{\text{iuj}}}$. The Pillai statistic has an F distribution with df₁ = r_Cr_A and df₂ = s(N − r_X + s − r_A) degrees of freedom under the null hypothesis. Note r_C, r_A, and r_X are the ranks of the matrices C, A, and X, respectively.

A. Null hypothesis

We can write a linear hypothesis in a one-way MANOVA as:

$$H_0:\mathit{\text{CBA}}-D_0=0,$$

where $B=\left(\begin{array}{ccc}\hfill {\mu }_{11}\hfill & \hfill \cdots \hfill & \hfill {\mu }_{1p}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\mu }_{g1}\hfill & \hfill \cdots \hfill & \hfill {\mu }_{\mathit{\text{gp}}}\hfill \end{array}\right)$ is a g × p matrix for the p mean vectors. The matrices C and A are determined from a linear null hypothesis.

B. Power and sample size calculations

O’Brien and Shieh [107] summarize the calculation of the power for global effects in one-way MANOVA. The Pillai trace statistic under the alternative hypothesis has a non-central F distribution with df₁ and df₂ degrees of freedom and the non-centrality parameter (NCP) $\lambda =\mathit{\text{Ns}}(\frac{V^{\ast }}{s-V^{\ast }})$, where $V^{\ast }={\sum }_{i=1}^s\frac{{\varphi }_i^{\ast }}{1+{\varphi }_i^{\ast }}$ and ${\varphi }_i^{\ast }$ is the ith largest eigenvalue of:

$${(A\prime \mathit{\sum }A)}^{-1}(\mathit{\text{CBA}}-D_0)\prime {(C{(\mathit{\text{diag}}(p_1,\cdots ,p_g)}^{-1}C\prime )}^{-1}(\mathit{\text{CBA}}-D_0),$$

where $p_j=\frac{n_j}{N}$ or the limit of the ratio as N → ∞. We specify that the phenotype vectors in all groups have the common covariance matrix Σ. This common covariance matrix specification is necessary to derive the NCP. Note that for the threshold-based phenotypes, we need not make such an assumption.

C. Example NCP calculation for two phenotypes

Consider our null hypothesis H₀: µ₀₁ = µ₁₁ = µ₂₁, µ₀₂ = µ₁₂ = µ₂₂ for three genotype groups (i = 0,1,2) with the bivariate phenotypes (j = 1,2), that is, p = 2 and g = 3. Thus, s = min(g − 1, p) = 2. These means are determined using the information in section above (Methods - Notation for quantitative trait).

Let A be the 2 × 2 identity matrix $A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$ and let $C=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right)$. Let $D_0=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$, and let the covariance matrix of the bivariate phenotypes be denoted by $\mathit{\sum }=\left(\begin{array}{cc}\hfill {\sigma }_1^2\hfill & \hfill {\sigma }_1{\sigma }_2\rho \hfill \\ \hfill {\sigma }_1{\sigma }_2\rho \hfill & \hfill {\sigma }_2^2\hfill \end{array}\right)$ with the correlation coefficient ρ. These matrices are specified so that we may test the null hypothesis H₀: µ₀₁ = µ₁₁ = µ₂₁, µ₀₂ = µ₁₂ = µ₂₂ stated above. We can calculate the 2 × 2 matrix Φ* as:

$${\mathrm{\Phi }}^{\ast }={(A\prime \mathit{\sum }A)}^{-1}(\mathit{\text{CBA}}-D_0)\prime {(C{(\mathit{\text{diag}}(p_1,\cdots ,p_g)}^{-1}C\prime )}^{-1}(\mathit{\text{CBA}}-D_0),={\mathit{\sum }}^{-1}(\mathit{\text{CB}})\prime {\left(C\right(\mathit{\text{diag}}(\frac{1}{p_0},\frac{1}{p_1},\frac{1}{p_2}))C\prime )}^{-1}(\mathit{\text{CB}}).$$

The matrix Φ* is used to compute the eigenvalues ${\varphi }_i^{\ast }$, which in turn are used to compute the Pillai statistic and the NCP.

Let us define the terms S_ij as:

$$S_{\mathit{\text{ij}}}=\frac{1}{1-{\rho }^2}\sum _{k=0}^2{p}_k\left(\frac{{\mu }_{\mathit{\text{ki}}}-\overline{{\mu }_i}}{{\sigma }_i}\right)\left(\frac{{\mu }_{\mathit{\text{kj}}}-\overline{{\mu }_j}}{{\sigma }_j}\right),$$

where ${\overline{\mu }}_i={\sum }_{k=0}^2{p}_k{\mu }_{\mathit{\text{ki}}}$. We can simplify the matrix Φ* to be:

$${\Phi }^{\ast }=\left(\begin{array}{cc}\hfill S_{11}-\rho S_{12}\hfill & \hfill \frac{{\sigma }_2}{{\sigma }_1}(S_{12}-\rho S_{22})\hfill \\ \hfill \frac{{\sigma }_1}{{\sigma }_2}(S_{12}-\rho S_{11})\hfill & \hfill S_{22}-\rho S_{11}\hfill \end{array}\right).$$

Note that:

$$V^{\ast }=\sum _{i=1}^2\frac{{\varphi }_i^{\ast }}{1+{\varphi }_i^{\ast }}=\frac{{\varphi }_1^{\ast }+{\varphi }_2^{\ast }+2{\varphi }_1^{\ast }{\varphi }_2^{\ast }}{1+{\varphi }_1^{\ast }+{\varphi }_2^{\ast }+{\varphi }_1^{\ast }{\varphi }_2^{\ast }},$$

$${\varphi }_1^{\ast }+{\varphi }_1^{\ast }=\mathit{\text{trace}}({\Phi }^{\ast })=S_{11}+S_{22}-2\rho S_{12},$$

and

$${\varphi }_1^{\ast }{\varphi }_2^{\ast }=\text{det}({\Phi }^{\ast }),=(S_{11}-\rho S_{12})(S_{22}-\rho S_{11})-\frac{{\sigma }_2}{{\sigma }_1}(S_{12}-\rho S_{22})\frac{{\sigma }_1}{{\sigma }_2}(S_{12}-\rho S_{11}),=(1-{\rho }^2)(S_{11}{S}_{22}-S_{12}^2).$$

Therefore, the NCP λ can be written as:

$$\lambda =N\frac{s{V}^{\ast }}{s-V^{\ast }},=N\frac{s\times \mathit{\text{trace}}({\Phi }^{\ast })+2s\times \text{det}({\Phi }^{\ast })}{s+(s-1)\times \mathit{\text{trace}}({\Phi }^{\ast })+(s-2)\times \text{det}({\Phi }^{\ast })},=N\frac{2(S_{11}+S_{22}-2\rho S_{12})+4(1-{\rho }^2)(S_{11}{S}_{22}-S_{12}^2)}{2+S_{11}+S_{22}-2\rho S_{12}}.$$ (1)

The power of the Pillai’s trace test is obtained by:

$$\text{Pr}(F({\mathit{\text{df}}}_1,{\mathit{\text{df}}}_2,\lambda )\ge f_{\alpha ,{\mathit{\text{df}}}_1,{\mathit{\text{df}}}_2}),$$

where f_α,df1,df2 is the (1 − α) quantile of a central F distribution with df₁ and df₂ degrees of freedom, respectively, and F(df₁, df₂, λ) is a noncentral F random variable with NCP λ, and degrees of freedom df₁ and df₂, respectively. For our example, df₁ = r_Cr_A = 4 and df₂ = s(N − r_X + s − r_A) = 2(N − 3).

Bivariate example

For the remainder of this work (excluding the Discussion), we focus on the bivariate distribution; that is, on pleiotropic diseases with two quantitative traits We do this because results are more easily interpreted, and because we can present graphs of functions such as the cdf.

MSSN calculations using factorial design

We ask the following question: what factors most substantially alter the calculated MSSN when testing for genetic association with a pleiotropic gene affecting two phenotypes?

To answer this question, we use a 2⁴ × 3³ factorial design (see [108]) on a total of seven design variables (factors) to approximate the calculated MSSN with functions of the design variables. These factors are listed in Table 1. Note that, we obtain 2⁴ × 3³ = 432 vectors of factor settings and therefore 432 MSSN calculations. One benefit of the factorial design is that we can look at multiple factors jointly over a broad range of settings and assess the factors that most change the outcome variable. For all MSSN calculations, we specify a fixed power is 0.80 and a significance level is 5 × 10⁻⁸.

Table 1.

Factors (and their settings) used in the MSSN calculations for genetic tests of association.

Factor	Number of settings	Setting-values
pd	2	0.05, 0.330
${\sigma }_1^2$	2	0.05, 0.10
τ1	3	−0.50, 0.00, 0.50
${\sigma }_2^2$	2	0.025, 0.05
τ2	3	−0.50, 0.00, 0.50
ρ	3	0.00, 0.33, 0.67
Percent-Affected and Percent-Unaffected	2	10%, 25%

We use the following abbreviations in the Factor column:

• p_d = Disease allele frequency (DAF);
• ${\sigma }_1^2$ = The variance for the first phenotype’s quantitative trait distribution;
• τ₁ = The dominance-additivity ratio for the first phenotype;
• ${\sigma }_2^2$ = The variance for the second phenotype’s quantitative trait distribution;
• τ₂ = The dominance-additivity ratio for the second phenotype;
• ρ = The correlation among the two phenotypes, or ρ₁₂. While we can consider negative correlations, for bivariate distributions, two phenotypes may always be parameterized so that the correlation is non-negative.

Approximation of calculated MSSN

After we compute all 216 MSSN for the Pillai test, and all 432 MSSN (we compute the number of affected individuals needed, and set the number of unaffected individuals to be equal to the number of affected individuals; that is, r = 1) for the Genotype and LTT tests, we perform a linear model analysis (i.e., ANOVA) on the seven main factors (Table 1) and all two-way interactions. The ANOVA calculations are performed using the methods developed for the R statistical software package [106].

Our rationale for performing the ANOVA with the factorial design is as follows: Equation (1) above and Equations (A8.1) and (A9.1) in the Appendix are closed-form equations that specify the NCPs (from which the MSSN may be calculated). Here, the MSSN is given by n = n(r, w_k, g_ik), where i = affection status, k = genotype. Although they are analytic, it is difficult to identify the variables that are most important. Consequently, we approximate the exact function by a linear model (including all two-way interactions) n̂(r, w_k, g_ik) = µ + µ_r + ⋯. We use 432 settings for our linear model approximation (216 for the Pillai statistic, since it is not dependent upon Percent-Affected and Percent-Unaffected settings) and report the factors that most explain the MSSN.

We note here and in the Results section that we do not attempt to make statistical inferences from our applications of the factorial design and ANOVA. Rather, we are use them as explanatory tools, specifically documenting the factors (main and interaction) that appear to have the most substantial effect on altering MSSN (i.e., those with the largest F-statistics), and then documenting quantitatively whether the results appear to be true. We can do this by computing MSSN considering different settings of the aforementioned factors and checking whether the different settings produce substantially different MSSN estimates.

Results

Factors that most significantly alter genetic association test MSSN

Genotype test

In Table 2, we report the results of our ANOVA for the Genotype test. Overall, this statistic had the smallest MSSN requirements on average for any setting of factor-settings in Table 1. This result is notable, since the Genotype test has 2 degrees of freedom (df), so one might expect the LTT to have lower MSSN values. Also, the Genotype test is applied to categorical data, and it is generally true that for quantitative data, quantitative-based tests such as Pillai's will require smaller MSSN than tests on categorical data. We discuss this point further in the Discussion.

Table 2.

Results of Analysis of Variance (ANOVA) for main effects and all two-way interactions - Genotype test.

Factor	Df	SSQFactor	F Statistic	φ2
Percent-Affected	1	1560721	33815.121	0.314
ρ	2	1723434	18670.263	0.347
${\sigma }_1^2$	1	612234	13264.884	0.123
${\sigma }_2^2$	1	308685	6688.076	0.062
pd	1	303127	6567.645	0.061
ρ × Percent-Affected	2	103543	1121.697	0.021
${\sigma }_1^2\times \text{Percent-Affected}$	1	46967	1017.613	0.009
pd × Percent-Affected	1	40969	887.657	0.008
${\sigma }_1^2\times \rho$	2	63336	686.134	0.013
${\sigma }_1^2\times {\sigma }_2^2$	1	25551	553.597	0.005
τ1	2	46357	502.194	0.009
${\sigma }_2^2\times \text{Percent-Affected}$	1	22923	496.648	0.005
${\sigma }_2^2\times \rho$	2	31059	336.47	0.006
pd × ρ	2	23991	259.896	0.005
τ2	2	16522	178.984	0.003
τ1 × Percent-Affected	2	5191	56.235	0.001
$p_d\times {\sigma }_2^2$	1	2162	46.851	0
τ2 × Percent-Affected	2	2892	31.331	0.001
τ1 × ρ	4	4434	24.018	0.001
${\tau }_1\times {\sigma }_2^2$	2	1723	18.67	0
τ1 × τ2	4	3101	16.796	0.001
${\sigma }_1^2\times {\tau }_2$	2	1041	11.28	0
τ2 × ρ	4	1606	8.7	0
$p_d\times {\sigma }_1^2$	1	282	6.11	0
${\sigma }_2^2\times {\tau }_2$	2	97	1.051	0
${\sigma }_1^2\times {\tau }_1$	2	74	0.799	0
pd × τ1	2	70	0.753	0
pd × τ2	2	2	0.02	0
Residuals	379	17493
Total		4964426

The values in the column labeled "Factor" are defined in Table 1. The column SSQ_Factor is the sum of squares for the given factor. The column labeled "φ²" lists each factor's proportion of the overall Sum of Squares. That is, ${\phi }^2=\frac{{\mathit{\text{SSQ}}}_{\text{Factor}}}{{\mathit{\text{SSQ}}}_{\text{Total}}}$. All values with the exception of the last column are computed using methods developed for the R statistical software package [106].

In Table 2, the factors are sorted from largest F-statistic to least. Also, we report the value φ², the respective factor's proportion of the overall Sum of Squares. Specifically, ${\phi }^2=\frac{{\mathit{\text{SSQ}}}_{\text{Factor}}}{{\mathit{\text{SSQ}}}_{\text{Total}}}$ (values provided in Table 2). Based on F-statistics and the φ² values, we may infer that there are five main factors that most substantially affect the number of affected individuals needed to detect association. These are, in order of F-statistic (rounded to nearest integer from Table 2), Percent-Affected (F-Statistic = 33,815), ρ (Correlation) (F-Statistic = 18,670), ${\sigma }_1^2(\text{F-statistic}=\mathrm{13,265})$, and ${\sigma }_2^2(\text{F-statistic}=\mathrm{6,688})$, and p_d (F-statistic = 6,568). Along with their two-way interaction-terms (a total of ten), these five factors account for 98% of the proportion of the total Sum of Squares (SSQ_Total) (Table 2). The dominance-additivity ratios, τ₁ and τ₂ had relatively small impact on the calculated MSSN. This result suggests that the Genotype test is equally powerful when the QTL operate under either an additive or non-additive mode of inheritance.

That is, researchers need not focus on whether their traits of interest deviate from an additive mode of inheritance when performing MSSN calculations.

Given these results, we performed a regression analysis in which we used the five main-effect terms, and their two-way interaction. Results of the regression analysis are provided in Table 3. As main be seen in Table 3 and Equation (2) below, there are actually six "main" effects terms, since there are three settings for the correlation factor ρ, and hence we need two separate variables. Our goal is to compute the coefficients of the fitted sample size equation:

$$\widehat{n_A}={\beta }_0+{\sum }_{d=1}^D{\sum }_{i=1}^{{\psi }_d}{\beta }_i{x}_i+{\sum }_{d=1}^D{\sum }_{f=2}^D{\sum }_{i=1}^{{\psi }_d}{\sum }_{j=1}^{{\psi }_f}{\beta }_i{\beta }_j{x}_i{x}_j+e,\text{ where }e~N(0,{\epsilon }^2).$$ (2)

Table 3.

Coefficients for linear regression model using five most significant main factors - Genotype test.

Factor and Setting	Coefficient Estimate	Standard Error	t-Statistic
(Intercept)	154.718	3.449	44.853
Percent-Affected = 25	139.272	3.688	37.767
ρ = 0.33	81.701	4.123	19.816
ρ = 0.67	185.045	4.123	44.882
${\sigma }_1^2=0.10$	−43.27	3.688	−11.734
${\sigma }_2^2=0.05$	−38.942	3.688	−10.56
pd = 0.33	−21.689	3.688	−5.882
Percent-Affected = 25, ρ = 0.33	31.973	3.688	8.67
Percent-Affected = 25, ρ = 0.67	75.548	3.688	20.487
Percent-Affected = 25, ${\sigma }_1^2=0.10$	−41.708	3.011	−13.852
Percent-Affected = 25, ${\sigma }_2^2=0.05$	−29.137	3.011	−9.677
Percent-Affected = 25, pd = 0.33	−38.954	3.011	−12.937
ρ = 0.33, ${\sigma }_1^2=0.10$	−25.374	3.688	−6.881
ρ = 0.33, ${\sigma }_2^2=0.05$	−18.002	3.688	−4.882
ρ = 0.33, pd = 0.33	−17.221	3.688	−4.67
ρ = 0.67, ${\sigma }_1^2=0.10$	−59.121	3.688	−16.032
ρ = 0.67, ${\sigma }_2^2=0.05$	−41.421	3.688	−11.233
ρ = 0.67, pd = 0.33	−36.489	3.688	−9.895
${\sigma }_1^2=0.10,{\sigma }_2^2=0.05$	30.763	3.011	10.217
${\sigma }_1^2=0.10$, pd = 0.33	3.232	3.011	1.073
${\sigma }_2^2=0.05$, pd = 0.33	8.949	3.011	2.972

Here, we present results of a linear regression using the five most significant factors from Table 2. We include all two-way interactions of these factors. An example description of the factors is as follows: 'ρ = 0.33' means, if the settings of correlation is 0.33, use the coefficient 81.701 (second column) when computing the fitted value. Otherwise, use 0. We compute coefficients for the other main factors in the same manner. For two-way interactions, consider the example 'Percent-Affected = 25, p_d = 0.33'. Here, if the disease allele frequency setting is 0.33 and the Percent-Affected setting is 25, then the coefficient used for the fitted values is −38.954, 0 otherwise. All values are computed using methods (specifically, the lm and summary commands) developed for the R statistical software package [106]. All values in the last three columns are rounded to three decimal places.

Here, D is the number of factors (five in this case), and ψ_z is the number of df for the z^th factor, 1 ≤ z ≤ D. Also, 1 ≤ d < f ≤ D, and β_iβ_j = 0 if i, j are settings for the same factor. This form of the fitted equation is used for all test statistics (Genotype, LTT, Pillai).

From Table 3, we compute the fitted function as:

$$\hat{n}=154.718+139.272x_1+81.701x_2+185.045x_3-43.27x_4-38.942x_5-21.689x_6+31.973x_1{x}_2+75.548x_1{x}_3-41.708x_1{x}_4-29.137x_1{x}_5-38.954x_1{x}_6+0.00x_2{x}_3-25.374x_2{x}_4-18.002x_2{x}_5-17.221x_2{x}_6-59.121x_3{x}_4-41.421x_3{x}_5-36.489x_3{x}_6+30.763x_4{x}_5+3.232x{x}_6+8.949x_5{x}_6,$$ (3)

where:

$$x_1=\{\begin{array}{c}1,\text{if Percent}-\text{Affected}=25\%\hfill \\ 0,\text{if Percent}-\text{Affected}=10\%\hfill \end{array},$$

$$x_2=\{\begin{array}{c}1,\rho =0.33\hfill \\ 0,\rho =\text{otherwise}\hfill \end{array},$$

$$x_3=\{\begin{array}{c}\hfill 1,\rho =0.67\hfill \\ \hfill 0,\rho =\text{otherwise}\hfill \end{array},$$

$$x_4=\{\begin{array}{c}1,{\sigma }_1^2=0.10\hfill \\ 0,{\sigma }_1^2=0.05\hfill \end{array},$$

$$x_5=\{\begin{array}{c}1,{\sigma }_2^2=0.050\hfill \\ 0,{\sigma }_2^2=0.025\hfill \end{array},$$

$$x_6=\{\begin{array}{c}1,p_d=0.33\hfill \\ 0,p_d=0.05\hfill \end{array}.$$

Reviewing the coefficients in Equation (2), we observe that increasing the Percent-Affected from 10 to 25 percent produces a substantial increase in MSSN (approximately 139 individuals; coefficient for variable x₁). The next largest coefficient is for the correlation term ρ in the variance-covariance matrix Σ. Increasing the correlation from 0 (uncorrelated phenotypes) to 0.33 produces an MSSN increase of approximately 82 individuals (coefficient for variable x₂) and increasing the correlation from 0 to 0.67 produces an MSSN increase of 185. This coefficient is the single largest coefficient in the fitted equation (1). Coefficients for the other main-effects are smaller, but significantly non-zero.

For the interaction terms, the larger coefficient in Equation (3) in absolute value is for the pair (Percent − Affected, ρ). When Percent − Affected equals 25 and ρ equals 0.67, the increase in MSSN is approximately 76. With the exception of the pair ( ${\sigma }_1^2$, p_d) and the pair ( ${\sigma }_2^2$, p_d), coefficients for all other interaction terms are greater than 15 in absolute value (Equation (2), Table 3). These results are consistent with the F-statistic values in Table 2.

Finally, a review of the results in Table 3 suggests that MSSN is decreased the most when ${\sigma }_1^2=0.10$, since every coefficient that contains ${\sigma }_1^2=0.10$ (with the exception of coefficients for the third to last- and second to last-rows of Table 3) is negative. This result is consistent with the fact that increasing the QTL variance increases the separation among the component MVN distributions, thereby making it easier to determine genotype from QT values.

In Figure 1, we present a plot of the fitted values (using Equation (3)) versus the analytic MSSN (n = n_A + n_U) determined using the NCP (Appendix; Equations (A8.1) and (A8.2)). The coefficients of the trend line, computed using the method in Excel, are consistent with the finding that the analytic MSSN are accurately approximated by a linear combination of the six variables x₁, …, x₆ (Table 3) and their two-way interactions. We base this conclusion on the fact that the trend line intercept is 0.0005 (close to 0) and the slope is exactly 1. From this we may conclude that, for the parameter settings considered in Table 1, only five of the seven factors are needed to approximate the analytic MSSN, and that among them, Percent-Affected/Unaffected and the correlation ρ make the greatest change. Since Percent-Affected/Unaffected is the only variable that researchers can control, to decrease MSSN requirements, one should decrease the Percent-Affected value to a 10% threshold (set x₁ to 0 in Equation (3)). Doing so will decrease the fitted MSSN by approximately 139 individuals (coefficient of x₁ in Equation (3)). In the Appendix, we compute analytic MSSN over a range of Percent-Affected/Unaffected values for the Genotype test and the LTT, and document that, as the Percent-Affected/Unaffected setting approaches 0%, so does the MSSN (Appendix; Figure A4).

Figure 1.

LTT

The results of the LTT test are very similar to those of the Genotype test, although the MSSN requirements are generally higher. We place results of our analyses in the Appendix (Table A2). Also, see the Discussion section.

Pillai test

We provide results of our ANOVA for the Pillai test in Table 4. Overall, this statistic had the largest MSSN requirements for any set of factor-settings in Table 1. Note that the factor "Percent-Affected/Unaffected" is not used when computing MSSN requirements for the Pillai statistic, because we use QTVs on all individuals, not just those whose values are above/below a threshold. Hence, we compute the ANOVA for a total of 432/2 = 216 vectors of settings from Table 2.

Table 4.

Results of Analysis of Variance (ANOVA) for main effects and all two-way interactions - Pillai test.

Factor	Df	SSQFactor	F Statistic	φ2
${\sigma }_1^2$	1	4626159	5804.04	0.679
${\sigma }_2^2$	1	502017	629.836	0.074
ρ	2	891480	559.231	0.131
${\sigma }_1^2\times {\sigma }_2^2$	1	237057	297.415	0.035
${\sigma }_1^2\times \rho$	2	247063	154.984	0.036
τ1 × τ2	4	83801	26.284	0.012
pd	1	16801	21.078	0.002
${\sigma }_2^2\times \rho$	2	22746	14.269	0.003
pd × ρ	2	12947	8.122	0.002
τ1	2	9030	5.665	0.001
τ2	2	9030	5.665	0.001
$p_d\times {\sigma }_1^2$	1	2308	2.896	0
τ1 × ρ	4	6988	2.192	0.001
τ2 × ρ	4	6988	2.192	0.001
pd × τ1	2	1403	0.88	0
pd × τ2	2	1403	0.88	0
${\sigma }_1^2\times {\tau }_1$	2	1243	0.78	0
${\sigma }_1^2\times {\tau }_2$	2	1243	0.78	0
$p_d\times {\sigma }_2^2$	1	28	0.035	0
${\sigma }_2^2\times {\tau }_1$	2	16	0.01	0
${\sigma }_2^2\times {\tau }_2$	2	16	0.01	0
Residuals	173	137891
Total		6817658

The legend for this table is virtually identical to the legend for Table 2, with the exception that the "Percent-Affected" factor is not considered, because the Pillai statistic is computed on all individuals. All values with the exception of the last column are computed using methods developed for the R statistical software package [106].

As in Table 2, the factors considered in our ANOVA are sorted from largest F-statistic to least, and we report the φ² values (listed in Table 4). Considering F-statistics and the φ² values, we infer that there are three main terms that most substantially affect the MSSN to detect association. These are, in order of F-statistic (rounded to nearest integer), ${\sigma }_1^2(\text{F-statistic}=\mathrm{5,804}),{\sigma }_2^2(\text{F-statistic}=630)$, and ρ (F-statistic = 559). The three two-order interactions of these terms are: ${\sigma }_1^2\times {\sigma }_2^2(\text{F-statistic}=297),{\sigma }_1^2\times \rho (\text{F-statistic}=155)$ and ${\sigma }_2^2\times \rho (\text{F-statistic}=14)$. These six main and interaction factors account for approximately 96% of the proportion of the total Sum of Squares (SSQ_Total) (Table 4 -last column). These results suggest that a linear function of the top five factors (like Equation (3) for the Genotype test) provide a very close approximation to the actual MSSN for all 216 vectors of settings from Table 1.

Using the results in Table 4, we perform a regression analysis in which we select the three main-effect terms (a total of four variables, given the two settings of correlation) and their two-way interactions. We present results in Table 5.

Table 5.

Coefficients for linear regression model using three most significant main factors and all interactions - Pillai test.

Factor	Coefficient Estimate	Standard Error	t-Statistic
(Intercept)	651.081	8.089	80.491
${\sigma }_1^2=0.10$	−277.541	10.232	−27.126
${\sigma }_2^2=0.05$	−173.81	10.232	−16.987
ρ = 0.33	150.831	10.852	13.898
ρ = 0.67	215.882	10.852	19.893
${\sigma }_1^2=0.10,{\sigma }_2^2=0.05$	132.513	10.232	12.951
${\sigma }_1^2=0.10$, ρ = 0.33	−78.614	12.531	−6.273
${\sigma }_1^2=0.10$, ρ = 0.67	−165.614	12.531	−13.216
${\sigma }_2^2=0.05$, ρ = 0.33	−6.512	12.531	−0.52
${\sigma }_2^2=0.05$, ρ = 0.67	39.915	12.531	3.185

In this table, we report linear regression analysis coefficients for the three most significant factors from Table 4. Also, we include all two-way interaction terms. Similar to Table 3, we have the following factor descriptions: ' ${\sigma }_1^2=0.10$' means, if the setting of the first phenotype's QTL variance is 0.10, use the coefficient −277.541 (second column) when computing the fitted value. Otherwise, use 0. We compute coefficients for the other main factors in the same manner. Computation for the interaction factors is described in the legend for Table 3. All values are computed using methods (specifically, the lm and summary commands) developed for the R statistical software package [106].

From Table 5, we compute the fitted function as:

$$\hat{n}=651.081-277.541x_1-173.81x_2+150.831x_3+215.882x_4+132.513x_1{x}_2-78.614x_1{x}_3-165.614x_1{x}_4-6.512x_2{x}_3+39.915x_2{x}_4$$ (4)

where:

$$x_1=\{\begin{array}{c}1,{\sigma }_1^2=0.10\hfill \\ 0,{\sigma }_1^2=0.05\hfill \end{array},$$

$$x_2=\{\begin{array}{c}1,{\sigma }_2^2=0.050\hfill \\ 0,{\sigma }_2^2=0.025\hfill \end{array},$$

$$x_3=\{\begin{array}{c}\hfill 1,\rho =0.33\hfill \\ \hfill 0,\rho =\text{otherwise}\hfill \end{array},$$

$$x_4=\{\begin{array}{c}\hfill 1,\rho =0.67\hfill \\ \hfill 0,\rho =\text{otherwise}\hfill \end{array}.$$

Studying Equation (4), we note that changes in main factors result in changes of at least 174 individuals. For example, increasing ${\sigma }_1^2$ from 0.05 to 0.10 reduces the MSSN by 278 individuals in Equation (3). Similarly, increasing the correlation ρ from 0 to 0.33 increases the MSSN by 151. For the interaction terms, the largest change is −166, occurring when ${\sigma }_1^2$ is 0.10 and ρ is 0.67. The smallest change in MSSN occurs when ${\sigma }_2^2$ is 0.05 and ρ is 0.33.

In Figure 2, we plot the fitted values (using Equation (4)) versus the analytic MSSN (n = n_A + n_U) determined using the Pillai NCP (Appendix). As with Figure 1, the coefficients of the trend line, computed using the method in Excel, are consistent with the finding that the analytic MSSN are accurately represented by a linear combination of all terms in Equation (4) (Trend line intercept is 0.0004, slope is 1.0). In contrast to the Genotype test results, for the Pillai test, we require only three of the six factors to approximate the analytic MSSN (Table 6 and Figure 3). Also, MSSN requirements are decreased most substantially by increasing the QTL variances ${\sigma }_1^2$ and ${\sigma }_2^2$ and decreasing the correlation ρ.

Figure 2.

Table 6.

Percentiles for MSSN ratios with different test statistics.

Percentile	Ratio of MSSNs
	LTT/Genotype	Pillai/Genotype(10%)	Pillai/Genotype(25%)	Pillai/LTT(10%)	Pillai/LTT(25%)
Minimum	0.95	1.59	0.94	1.20	0.74
Median	1.35	3.41	1.95	2.62	1.65
Mean	1.26	3.37	1.98	2.66	1.64
Maximum	1.64	5.28	3.14	4.18	2.45

In this table, we use the abbreviations "LTT(x%)" and "Genotype(x%)" to mean the MSSNs for the LTT and Genotype tests, respectively, when the Percent-affected/unaffected settings are x (x = 10% or 25%). Also, each column's pair of tests corresponds to the same numbered column in Figure 3. For example, the first pair of tests is LTT and the Genotype test. The same pair is considered in the first column of Figure 3.

Figure 3.

What method produces the smallest MSSN requirements?

So far, we have answered the questions of what factors most substantially alter MSSN requirements and by how much for the Genotype test, Pillai test, and LTT test (Appendix) for the factor settings in Table 1. An equally important question is, what statistic produces the smallest analytic MSSN requirements for any vector of factor settings in Table 1? To answer this question, we compute the five sets of differences:

1. LTT(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff) − Genotype(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff);

2. Genotype(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 10) − Pillai(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 10);

3. Genotype(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 25) − Pillai(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 25);

4. LTT(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 10) − Pillai(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 10);

5. LTT(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 25) − Pillai(p_d, ${\sigma }_1^2$, τ₁, …, ρ, Percent − Aff = 25);

Each of the differences in MSSN is computed as a function of the parameter settings. For example, if p_d = 0.33, ${\sigma }_1^2=0.10$, τ₁= 0.0, ${\sigma }_2^2=0.5$, τ₂= 0.5, ρ = 0.0, Percent − Aff = 25, then difference (i) is:

• Analytic MSSN for LTT for vector (0.33, 0.10, 0.0, 0.05, 0.5, 0.0, 25) − Analytic MSSN for Genotype test for vector (0.33, 0.10, 0.0, 0.05, 0.5, 0.0, 25).

The differences (ii) – (v) are computed with a fixed value for the last parameter (Percent − Aff). The reason is that the Pillai test is a function of only six parameters in Table 1; as noted previously, it is not a function of the Percent-Affected parameter. For each of the differences (i) – (v), we present the empirical distributions of the results in the form of box plots. These box plots may be found in Figure 3.

Note that the difference (i) is computed over 432 vectors, while differences (ii) – (v) are computed over 216 vectors.

Some of the key findings resulting from a study of this figure are that the Genotype test usually has the smallest sample size (previously mentioned), and that the Genotype test and the LTT test almost always require smaller analytic MSSN than the Pillai test. In fact, viewing the four right-most box-plots, the greatest difference among Pillai and any of the other test statistics, where Pillai requires a smaller sample size, is for the (LTT(25%) - Pillai test) box plot (the right-most one in Figure 3). The difference is 124 (Outlier for LTT(25%) - Pillai, Figure 3). In results not shown, this difference occurs for the vector of settings p_d = 0.33, ${\sigma }_1^2=0.10$, τ₁= −0.50, ${\sigma }_2^2=0.025$, τ₂= 0.50, ρ = 0.67, Percent − Aff = 25. For this vector, the LTT analytic MSSN is 477, and the Pillai test analytic MSSN is 353.

In Table 6, we present differences in Figure 3 as ratios. Lehmann and Rosano [109], among others, defines these ratios as asymptotic relative efficiencies. We report the minimum, median, mean, and maximum ratios for all pairs of test statistics. In this way, we can compare results across columns. The smallest median and mean values, 1.35 and 1.26 respectively, are for the LTT/Genotype MSSN ratio. This results suggests that the MSSNs for these two test statistics are most similar. The largest median and mean values of 3.41 and 3.37 are for the Pillai/Genotype(10%) MSSN ratio. This result is consistent with the fact that the Genotype(10%) - Pillai MSSN box plot has the lowest Range of Differences (vertical axis) in Figure 3.

For all ratios below the median ratio of 1.35 for the LTT/Genotype MSSN ratio, every vector has the DAF setting p_d = 0.05. This result suggests that LTT and Genotype test MSSNs are most similar for smaller disease allele frequencies.

Finally, we note that we have developed software to perform these calculations. This software will be made available online within the near future. Researchers who want stand-alone copies of the software may contact the first author.

Discussion

In this work, we present the method, the Genotype test, for computing asymptotic power and MSSN calculations for genetic association with pleiotropic traits. In our design, affection status is defined through thresholds. We include computations for power and MSSN for MANOVA by applying Pillai’s statistic.

The first observation we make is that we can specify a multivariate function to compute probabilities for pleiotropic phenotypes (Formulas (A1) and (A2) in the Appendix). Also, we derive categorical data from the QTVs and apply the Genotype test and LTT to the categorical data (Equation A4 in the Appendix). Furthermore, we compute analytic power and MSSN formulas for the Genotype test and LTT (Formulas (A8.1) and (A9.1)), as well as analytic power and MSSN formulas for the Pillai MANOVA test applied to all QTVs.

Our ANOVA results for the factorial designs indicate that, for the Genotype test, the factors that most substantially alter MSSN are correlation among the two QTs (ρ) and the Percent-Affected/Unaffected settings. From the results in Table 3 and Equation (3), we see that MSSN decreases with a decrease in the correlation and a change of the Percent-Affected/Unaffected setting from 25% to 10%. Changes in these two factors reduce MSSN for the LTT as well (results not shown). We comment that we use the ANOVA to provide an numerical approximation (with linear and two-way interaction terms) to the analytic formulas for MSSN. The factors we consider in the approximation are those with the largest F-statistic values.

For the Pillai test, Analytic MSSN is described accurately by settings in three factors and their interactions: ${\sigma }_1^2,{\sigma }_2^2$, and ρ (Table 5 and Equation (4)). Increases in QTL variances ${\sigma }_1^2,{\sigma }_2^2$ reduce the MSSN, while a decrease in the correlation ρ produces a decrease in the MSSN.

When comparing all of the MSSNs for all tests, we see that the Genotype test usually requires the smallest MSSN to achieve 80% power at the 5E-08 significance level for the vector of settings in Table 1. We draw this conclusion by studying the box plots of MSSN differences for all pairs of test statistics. The only test statistic that has a smaller MSSN than the Genotype test for any significant portion of vector-settings is the LTT. In fact, for 110/432 (25%) of the vectors, the LTT has as small or smaller MSSN than the Genotype test. However, the maximum difference is 14 individuals, and the relative efficiency is never less than 95% (Table 6).

While this work focuses on sample size calculations, through use of the NCP, we can just as easily perform power calculations for a fixed sample size. The conclusions we draw about the three statistics are the same (e.g., Genotype test has largest power on average for different vectors of Factor settings, followed by LTT, etc) (data not shown).

What if the SNP we are studying is in linkage disequilibrium (LD) with the disease gene but not the gene itself [23]? In such circumstances, we use the method implemented by others (e.g., [87,88]) to perform power and MSSN calculations of threshold-selected quantitative trait loci that are in LD with a disease locus.

A final and very important issue to address is that fact that the Pillai test, applied to quantitative data for all individuals, has larger MSSN values than either the Genotype test or the LTT. Our explanation for this result is that our design focuses on MSSN calculations before any data are collected. Also, our focus is on gene mapping, not on tests of linearity. If one were conducting a population-based study, where phenotype and genotype values were collected on all individuals, and all three test statistics were applied to all individuals, then the Pillai statistic would typically have the smallest sample size requirement. Consider the following example of vector settings: p_d = 0.05, ${\sigma }_1^2=0.10$, τ₁ = 0.0, ${\sigma }_2^2=0.05$, τ₂ = 0.50, ρ = 0.0, Percent-Affected-Phenotype 01 = (Top) 100%, Percent-Affected-Phenotype 02 = (Top) 50%, Percent-Unaffected-Phenotype 01 = (Lower) 100%, Percent-Unaffected-Phenotype 02 = (Lower) 50%. The parameter settings (with the exception of the Percent-Affected and Percent-Unaffected) are taken from Table 1.

Regarding the affection thresholds, imagine a square. If we draw a horizontal line through the square, cutting it in half, affected individuals are those subjects whose pair of QTVs are in the upper half of the square, and unaffected individuals are those subjects whose pair of QTVs are in the lower half. With these thresholds, we use all of the individuals for the Genotype test and LTT, as well as the Pillai test.

Applying our formulas, we compute that MSSNs are: 1471 for the Genotype test, 1387 for the LTT, and 326 for the Pillai test for 5E-08 significance level. The Pillai MSSN is much lower than for either of the categorical data-based tests.

Similarly, if we define affection by using a vertical line rather than a horizontal line, our MSSNs are: 836 for the Genotype test, 785 for the LTT, and 326 for the Pillai test (the Pillai statistic is not dependent upon threshold settings). That is, the Pillai MSSN is less than half of either of the categorical data-based tests

Another practical issue regarding the lower values for ‘Percent-Affected’ (like 10%) is that, for small or moderate MSSN, one may not observe individuals with phenotypes in these region. For small and moderate MSSNs, the thresholds may be theoretically desirable, but impractical. In such circumstances, one might have no choice but to increase the Percent-Affected threshold.

Appendix

Notation for quantitative trait model

y: (y₁, y₂, …, y_p) = a set of p random quantitative-trait (QT) phenotype values; note that this means there are p phenotypes. From this point forward, we shall use the term phenotype to mean a continuous random variable, represented by the notation y_i.

n_A: Number of affected individuals.

n_U: Number of unaffected individuals.

Note: We use the term affected throughout this work. We could also use the term case. We make the same statement for unaffected and control.

r: Ratio of to = n_U/n_A;

Indices

1 ≤ i ≤ p: Index for phenotype (see above);

0 ≤ k ≤ 2: Index for genotype at SNP locus; this value is the number of disease or increaser alleles in the SNP genotype.

Genetic model parameters

${\sigma }_i^2$, 1 ≤ i ≤ p: Quantitative Trait Locus (QTL) Variance of the phenotype y_i; that is, it is contribution to the variance of the population's i^th quantitative trait from the QTL. Note that this quantity is the genetic component of the population phenotype variance (specified in this work as N(0,1)).

${\sigma }_{R_i}^2$, 1 ≤ i ≤ p: Error variance of the phenotype y_i; using Fisher’s partitioning [104], we have ${\sigma }_{R_i}^2=1-{\sigma }_i^2$. Note that the error variance is the common (phenotype-specific) variance for each of the normal components that make up the i^th mixture distribution.

δ_i, 1 ≤ i ≤ p: Dominance of the disease allele for the phenotype y_i;

In this work, we restrict δ_i to the range −1 ≤ δ_i ≤ 1, although in theory the dominance may range between −∞ and ∞ [105].

p_d: Frequency of the disease (“increaser”) allele at the SNP locus of interest;

p₊: Frequency of the wild-type (“null”) allele at the SNP locus of interest; note that p_d + p₊ = 1.

NOTE: The parameters p_d and p₊ should not be confused with the number of phenotypes p.

a_i, 1 ≤ i ≤ p: Additive term for the phenotype y_i;

${\tau }_i=\frac{{\delta }_i}{a_i}$, 1 ≤ i ≤ p: Dominant-additive ratio for the phenotype y_i;

m_i, 1 ≤ i ≤ p: Mean term for the phenotype y_i;

ρ_ij: Correlation between the variables y_i and y_j.

w_k, 0 ≤ k ≤ 2: Weight of k^th (coded) genotype in the LTT;

From Fisher’s work [104,105], we can compute the means µ_ik from the dominance δ_i and the disease allele frequency p_d. Fisher shows:

1. $a_i=\sqrt{\frac{{\sigma }_i^2}{2p_d{p}_{+}{(1+{\tau }_i(p_{+}-p_d))}^2+4{(p_d{p}_{+}{\tau }_i)}^2}},$

2. δ_i = τ_ia_i,

3. m_i = (−1)[(p_d)²a_i + 2p_dp₊δ_i − (p₊)²a_i],

4. µ_i0 = m_i − a_i

   µ_i1 = m_i + δ_i.

   µ_i2 = m_i + a_i

5. π_k, 0 ≤ k ≤ 2: Mixing proportion for the component-distribution $N({\mu }_{\mathit{\text{ik}}},{\sigma }_{R_i}^2)$, determined by the genotype frequencies at the trait locus; because we are studying pleiotropy, the mixing proportions are independent of the phenotype index i. Note $N({\mu }_{\mathit{\text{ik}}},{\sigma }_{R_i}^2)$ is a univariate normal distribution with mean µ_ik and variance ${\sigma }_{R_i}^2$.

Furthermore, as documented by Lynch and Walsh [105] (among others), the genetic variance ${\sigma }_i^2$ may be decomposed into the sum of an additive variance component ( ${\sigma }_{a_i}^2$), and a dominance variance component ( ${\sigma }_{{\delta }_i}^2$). As Lynch and Walsh report:

1. ${\sigma }_{a_i}^2=2p_d{p}_{+}{\alpha }^2$, where α = [a_i + δ_i(p₊ − p_d)];

2. ${\sigma }_{{\delta }_i}^2={(2p_d{p}_{+}{\delta }_i)}^2$.

From these equations, it is straightforward to see that the genetic variance for the i^th phenotype is a function of the a_i, the additive term for the phenotype y_i, the disease allele frequency p_d, and the dominance δ_i.