# Computing power of quantitative trait locus association mapping for haploid loci

- Derek Gordon
^{1}Email author and - Andrew R Zinn
^{2}

**10**:261

https://doi.org/10.1186/1471-2105-10-261

© Gordon and Zinn; licensee BioMed Central Ltd. 2009

**Received: **13 February 2009

**Accepted: **23 August 2009

**Published: **23 August 2009

## Abstract

### Background

Statistical power calculations are a critical part of any study design for gene mapping. Most calculations assume that the locus of interest is biallelic. However, there are common situations in human genetics such as X-linked loci in males where the locus is haploid. The purpose of this work is to mathematically derive the biometric model for haploid loci, and to compute power for QTL mapping when the loci are haploid.

### Results

We have derived the biometric model for power calculations for haploid loci and have developed software to perform these calculations. We have verified our calculations with independent mathematical methods.

### Conclusion

Our results fill a need in power calculations for QTL mapping studies. Furthermore, failure to appropriately model haploid loci may cause underestimation of power.

## Background

Statistical power calculations are a critical part of any study design for gene mapping. With regards to quantitative trait locus (QTL) mapping, it is typically assumed that there are two allele at the QTL, and hence three genotypes (e.g., see Lynch and Walsh [1]). For each genotype, there is a corresponding mean quantitative value that is determined by the biometric model developed by Falconer and others [2]. To determine the mean of the heterozygote, we need a parameter known as the dominance parameter. Software to compute power for the biallelic situation has been made available in the Genetic Power Calculator, developed by Purcell et al. [3] However, for haploid organisms or loci, there is no such parameter because there are no heterozygotes. Thus, there is a different set of calculations necessary to compute power for association. There has been an extensive amount of literature written on the analysis of haploid data (see, e.g., Wu et al. [4, 5], Tulsieram et al. [6]) To our knowledge, there has been no power calculator developed for QTL mapping with haploid loci. This is an oversight, since there are several common situations in humans involving loci that are haploid or hemizygous [7, 8].

The purpose of this work is to mathematically derive the biometric model for haploid loci, and to compute power for QTL mapping when the loci are haploid.

## Implementation

The program for computing power is written in C++ and may be created on any computer that has a C++ compiler, for example those computers with a Unix or Linux operating system.

*t*-test is computed and reported to the screen. Following this, the user is asked about the locus-specific quantitative trait heritability

*Q*. When this value is entered, the program determines the means of the allele 0 group and the allele 1 group, as well as the variance in each group. Finally, the user is asked to specify the number of replicates for which power is computed. We recommend at least 100,000 replicates. See Figure 1 for an example of how data are entered, and see additional file 1. Below, we place an example run.

## Results and discussion

### Calculations of simulation power as function of Locus specific QTL heritability and increaser allele frequency

*Q*and

*p*that we specified in the Methods. There are several conclusions we may draw from studying this graph. The first is that, as

*Q*increases, the power increases for any value of

*p*. The second conclusion is that, if

*Q*is held constant, the power differs minimally over the range of

*p*. In fact, if we consider the set of consecutive differences ((power for

*Q*= 0.05 and

*p*= 0.25) – (power for

*Q*= 0.05 and

*p*= 0.1)), ((power for

*Q*= 0.05 and

*p*= 0.5) – (power for

*Q*= 0.05 and

*p*= 0.25)),..., ((power for

*Q*= 0.1 and

*p*= 0.9) – (power for

*Q*= 0.1 and

*p*= 0.75)), the maximum difference is 0.012, and the average difference is -0.00025. The maximum difference occurs for the difference ((power for

*Q*= 0.07 and

*p*= 0.25) – (power for

*Q*= 0.07 and

*p*= 0.1)). We note that the power was calculated using the executable file that is compiled from the material in Additional File 2.

List of Abbreviations

| Description |
---|---|

| Sample size |

| Significance level |

| Threshold for |

| Variance of quantitative trait locus (QTL) |

| Frequency of QTL increaser allele |

| Mean QTL value in group "0" (those with non-increaser allele) |

| Mean QTL value in group "1" (those with increaser allele) |

| Number of replicates |

By contrast, if we consider the set of consecutive differences ((power for *Q* = 0.06 and *p* = 0.1) – (power for *Q* = 0.05 and *p* = 0.1)), ((power for *Q* = 0.07 and *p* = 0.1) – (power for *Q* = 0.06 and *p* = 0.1)),..., ((power for *Q* = 0.1 and *p* = 0.9) – (power for *Q* = 0.09 and *p* = 0.9)), the maximum difference is 0.182 and the average difference is 0.085. The maximum difference occurs for ((power for *Q* = 0.06 and *p* = 0.75) – (power for *Q* = 0.05 and *p* = 0.75)).

Another result from this graph is that power of at least 80% may be achieved when the Locus specific QTL heritability *Q* is at least 0.07, suggesting that genes for QTLs with reasonable variance may be mapped even when the significance level is stringent.

### Real data example – EFHC2 gene SNP rs7055196 typed in 45, X Turner Syndrome subjects

Zinn et al. genotyped 97 45, X Turner Syndrome subjects at the SNP rs7055196. They also phenotyped the subjects for Facial Affect Fear Recognition score and performed a *t*-test to determine whether there was a significant difference in the scores based on the allele at the rs7055196 locus [9]. They found no significant difference among the two groups (49.8 +/- 26.5 for those with A allele versus 46.2 +/-26.1 for those with G allele; p-value = 0.67; see Figure 1 of Zinn et al.).

## Conclusion

To our knowledge, this is the first work to determine the power of QTL mapping for haploid loci or organisms. We compute power by simulation, and our results with exact analytic power suggest that the power by simulation is highly accurate. The most common application of these results is SNP association studies of loci on the X or Y chromosomes in male subjects. Other situations where haploidy arises in humans include chromosome disorders such as 45,X Turner syndrome [7] and segmental aneuploidies such as Cri-du-Chat (5p-) syndrome [8]. Last, the power to map QTLs in haplo/diploid organisms such as *Saccharomyces cerevisiae* can be increased by studying organisms in the haploid state [10].

Natural extensions of this work are situations such as gene-gene interactions (epistasis of different kinds), gene-environment interactions, and multi-trait mapping. This is the subject of future research. The purpose of this work is to provide researchers with the simplest non-trivial example of power calculations for haploid data.

## Availability and requirements

**Project name**: *t*-test power calculator for haploid data

**Project home page**: http://compgen.rutgers.edu/ttesthap/

**Operating system(s)**: Unix Solaris, Linux

**Programming language**: C++

**Other requirements**: None.

**License**: None.

**Any restrictions to use by non-academics**: None.

## Declarations

### Acknowledgements

The authors gratefully acknowledge grant NIH NS35554 to ARZ. Also, the authors gratefully acknowledge the help of Dr. Tara Matise, who created the home page for their software.

## Authors’ Affiliations

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## Copyright

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