Theoretical Basis of the Beavis Effect
- Shizhong Xu1
- 1 Author e-mail: xu{at}genetics.ucr.edu
Abstract
The core of statistical inference is based on both hypothesis testing and estimation. The use of inferential statistics for QTL identification thus includes estimation of genetic effects and statistical tests. Typically, QTL are reported only when the test statistics reach a predetermined critical value. Therefore, the estimated effects of detected QTL are actually sampled from a truncated distribution. As a result, the expectations of detected QTL effects are biased upward. In a simulation study, William D. Beavis showed that the average estimates of phenotypic variances associated with correctly identified QTL were greatly overestimated if only 100 progeny were evaluated, slightly overestimated if 500 progeny were evaluated, and fairly close to the actual magnitude when 1000 progeny were evaluated. This phenomenon has subsequently been called the Beavis effect. Understanding the theoretical basis of the Beavis effect will help interpret QTL mapping results and improve success of marker-assisted selection. This study provides a statistical explanation for the Beavis effect. The theoretical prediction agrees well with the observations reported in Beavis's original simulation study. Application of the theory to meta-analysis of QTL mapping is discussed.
THE primary goal of genetic mapping experiments is to identify the locations of genes that affect variable expression of a trait among individuals. Most researchers also use the data from a sampled population to estimate the genetic effects of quantitative trait loci (QTL). Information about the magnitude of the genetic effects of significant QTL is useful in prioritizing subsequent uses of the loci as candidate genes for consideration in genetic engineering and marker-assisted selection (Lande and Thompson 1990). Many statistical methods produce unbiased or asymptotically unbiased estimates of parameters. However, this property does not apply to QTL effects estimated from genome scans. This phenomenon was first discussed by Lande and Thompson (1990).
Almost all QTL mapping procedures can detect QTL with large effects, but not all can detect QTL with intermediate and small effects. Quantitative traits are defined as traits controlled primarily by intermediate and small QTL. Beavis (1994, 1998) designed a large-scale simulation experiment to evaluate the efficiency of interval mapping for detecting and estimating polygenes. Here-after, this simulation experiment is called the Beavis experiment. Beavis (1998) simulated either 10 or 40 independent QTL, examining situations where each QTL explained a proportion of the phenotypic variance ranging from 0.75 to 9.5% with sample sizes ranging from 100 to 1000. The Beavis experiment showed that the average estimates of phenotypic variances associated with correctly identified QTL were greatly overestimated if only 100 progeny were evaluated, slightly overestimated if 500 progeny were evaluated, and fairly close to the actual magnitude when 1000 progeny were evaluated. When the sample size was small, say 100, the statistical power of detecting a small QTL was as low as 3% and the estimated effects were typically inflated 10-fold. This phenomenon has since been referred to as the Beavis effect and has formed the basis of a number of subsequent analyses (e.g., Otto and Jones 2000; Hayes and Goddard 2001).
Estimating the number of quantitative trait loci is another goal of QTL mapping experiments (Sillanpaa and Arjas 1998). Otto and Jones (2000) acknowledged that the estimated number of QTL using molecular markers (interval mapping) can be more informative than that obtained from analyses using only phenotype (Lande 1981; Zeng 1992). However, a QTL is reported from the analysis only if it is detected. Therefore, the distribution of the detected QTL is actually inferred from censored data due to missing small-effect QTL. Otto and Jones (2000) incorporated the Beavis effect into their maximum-likelihood analysis to recover the potential number of QTL using the number of detected QTL, so called “detecting the undetected.” It should be emphasized that the purpose of Otto and Jones (2000) was to infer the number of QTL for a quantitative trait within a single experiment. Some QTL, however, may have large effects and some may have small effects, and estimating the total number depends on the distribution of QTL effects. If the overall distribution of the effects is described by an exponential distribution, the distribution of detected QTL effects becomes a truncated exponential distribution after incorporating the Beavis effect. This should not be confused with the original Beavis experiment where all simulated QTL had an equal genetic effect.
Hayes and Goddard (2001) conducted a meta-analysis for QTL mapping in livestock to infer the distribution of the effects of genes. They fit the estimated gene effects surveyed from all QTL mapping experiments in pigs and cattle to a gamma distribution. The difference between this meta-analysis and the usual analysis is that estimated genetic effects from multiple experiments were fit in a model rather than estimated from observed measurements. As a consequence, they had to take into account two additional sources of error for the estimated gene effects: the experimental error due to limited sample sizes and bias caused by data censoring. They found that the estimated genetic effects fit a gamma distribution very well. Again, one should not confuse this meta-analysis with the Beavis experiment because Hayes and Goddard (2001) intended to infer the distribution of genetic effects throughout the genome from multiple experiments. Even if two QTL were identified roughly at the same location by two investigators, they were still treated as different QTL because the estimated positions were not precise enough to be assigned to exactly the same position within the genome. The original Beavis experiment demonstrated that a bias exists in estimating QTL effects. Herein, I present a theoretical basis for the Beavis effect. The theory predicts the amount of bias as a function of the type of progeny used in the experiment, the estimation procedure, the marker density, and the sample size.
STATISTICAL THEORY
Marker analysis: The theory is developed using a backcross (BC) mating design, which provides the simplest genetic model in QTL mapping. The result is then extended to other progeny derived from common mating designs. Let yj be the phenotypic value of a quantitative trait measured from individual j sampled from a BC population. The linear model describing yj is
Let â be the estimated genetic effect and
The variance of x depends on the type of progeny. In a BC population without segregation distortion, half of the individuals will be homozygous and half heterozygous, leading to
If the positions of all QTL are known a priori (uncensored), the estimate of each QTL effect is approximately unbiased (depending on the method used) so that the distribution of â can be assumed to be approximately normal, i.e., N(a,
In the practice of QTL mapping, estimated genetic effects are reported only for significant QTL. Thus, the reported QTL consists of a censored sample so that the distribution of the estimated QTL effects conditional on statistical significance is a truncated normal distribution with a mean and variance different from those of the original normal distribution.
I now proceed to calculate this truncated distribution. Let us assume that a z-test statistic is used for the significance test so that the critical value for the z-test statistic is defined as Z1–α/2, where α is a controlled type I error rate. A QTL is reported only if |z| > Z1–α/2, where z = â/σâ is the z-test statistic. In other words, all the QTL reported satisfy |â| > σâZ1–α/2, i.e., â <–σâZ1–α/2 or â > σâZ1–α/2. The two-tailed test leads to the possibility that even if a is positive, it may be detected as a significantly negative effect due to sampling. Denote the truncated â by âT so that
Often, when using likelihood methods, the LOD score criterion may be used for QTL detection. The critical value of the LOD score can be converted into the critical value of the z-test statistic, using the following approximate relationship,
According to the standard statistical machinery of truncated normal distributions
The phenotypic variance may be partitioned into the true QTL variance (
The one-tailed test is a special case of the two-tailed test in that we simply set ξ1 = – ∞ and ψ1 = 0. For the two-tailed test, we can further simplify B into
The proportion of phenotypic variance explained by the QTL is defined as
We now extend the results to other types of progeny. For simplicity, we assume that dominance is absent. In an F2 population, there are three possible genotypes whose genotypic values are defined as a for the homozygote with the “high allele,” 0 for the heterozygote, and –a for the homozygote with the “low allele.” The linear model given in Equation 1 applies here in the F2 population except that the x variable is now defined as xj = 1, 0, –1 for the three genotypes, respectively. Without segregation distortion, the variance of x in an F2 population is
The method also can be applied to double-haploid (DH) and recombinant inbred line (RIL) populations. In both DH and RIL, the heterozygous genotype is absent. The x variable is defined as xj = 1 for one homozygote and xj =–1 for the other homozygote. The two types of homozygote have an equal frequency, and thus
Note that
Interval mapping: In interval mapping, a QTL may be identified at an intermediate position between markers by inferring the genotype of the QTL from flanking marker information. This will affect both the parameter estimates and the statistical tests of inference. We need to substitute the variance of x by the variance of the estimated x, denoted by
Predicted genotypic indicator variable and its probability distribution for the four flanking marker genotypes in backcross progeny
The method used to derive
For a DH population, we define x̂j in a slightly different way (see Table 2) but still use (19) to calculate
Predicted genotypic indicator variable and its probability distribution for the four flanking marker genotypes in double-haploid progeny
Predicted genotypic indicator variable and its probability distribution for the nine flanking marker genotypes in F2 progeny
In an F2 population, there are nine flanking marker genotypic classes. The definitions of x̂j's and their probabilities for the nine classes are given in Table 3. The same formula (Equation 19) is used to calculate
—Relationship between and the position of a QTL flanked by two fully informative markers in an interval of 20 cM in an F2 population in the limit of large n.
NUMERICAL EVALUATION
Bias was evaluated numerically by considering the following factors: the sample size, the genetic effect measured by
The numerical evaluation was conducted only in BC populations because the general trends are similar for all types of progeny (data not shown). In addition, only a one-tailed test was evaluated. The one-tailed test is a special case of the two-tailed test with ξ1 = – ∞ and ψ1 = 0. The functional relationships between the size of the detected QTL and the true sizes are shown in Figure 2. The diagonal lines in the first column of Figure 2 represent the case where aT = a and those in the second column represent the case where
—Functional relationships of the detected QTL parameters to the true parametric values under various sample sizes and LOD score critical values in BC mapping. The diagonal lines represent the relationship between the parameter of detected QTL and the true parametric value when the sample size is infinitely large (unbiased relationship). Note that as the LOD criterion increases, the bias associated with detecting a very small QTL (a → 0) becomes worse but the probability of detecting such a QTL becomes much smaller (not shown). Note that in the Beavis experiment, LOD = 2.5 and n = 100, 500, 1000.
The functional relationships between
If the average estimated
We now use the parameter values of the Beavis experiment to compare the biases with those observed by Beavis (1998). Table 4 shows the original data reported by Beavis as well as the predicted biases for both the QTL effects and their variances. The average QTL position is in the middle of a 20-cM interval. However, for an interval this short, the position of a detected QTL rarely coincided with the true position. We observed that the detected position within the interval varied almost uniformly across the interval. Therefore, we choose an average
DISCUSSION
The Beavis effect describes a phenomenon that occurred in the Beavis experiment where all QTL were simulated to have the same effect and distributed independently throughout the genome. The average effect of the detected QTL was biased upward due to censoring. It is more likely that QTL effects vary across the genome and the distribution of the QTL effects may be described by a negative exponential distribution (Xu 2003). In addition, some QTL may be linked within the same chromosomes and thus they do not segregate independently. On the one hand, the Beavis effect will cause the estimated number of QTL to be biased downward (Beavis 1994, 1998; Otto and Jones 2000), because the undetected QTL are not reported. On the other hand, the average effect of the detected QTL will be biased upward.
Using the Beavis effect to interpret results of a meta-analysis of QTL mapping is more straightforward. If a QTL mapping experiment can be repeated many times, the average effect of a chromosome location calculated only from the significant replicates will definitely be biased unless this QTL is detected in all replicates. If one considers incorporating a particular marker into a marker-assisted selection program for an economically important quantitative trait, the Beavis effect will affect the decision. The investigator may decide to search the literature to see how much genetic variance is accounted for by this marker from all published experiments.
The theory developed here helps predict the potential bias in the estimated effect of QTL. The theory may also be used to correct the bias but should be used with caution. Let
Comparisons of predicted and observed (estimated) biases in estimated QTL effects and variances from Beavis F2 simulation experiments
—Changes in effect and variance of detected QTL as a function of their true parametric values as sample size increases for various sizes of QTL and LOD score critical values in BC mapping.
The theory developed herein applies to segregating populations with two alternative genotypes. For populations with more than two alternative genotypes, e.g., F2, the model is restricted to either the additive or the dominance model but not both. This is because the test statistic utilized is the z-test statistic, which is a 1-d.f. test. Further investigation is necessary to predict the biases in both the additive and dominance effects using a 2-d.f. test. Similar extensions can be made for a test with >2 d.f., e.g., four-way crosses (Xu 1996) and diallel crosses (Rebai and Goffinet 1993).
Comparisons of predicted and estimated biases in estimated QTL effects and variances from our own F2 simulation experiments at sample size 100
The variance of the genotype indicator (x) determines the estimation error of the QTL effect and thus plays an important role in the Beavis effect. When the QTL under investigation is tightly linked to a fully informative marker,
The theory of the Beavis effect is derived on the basis of a single QTL model. However, it applies approximately to multiple QTL experiments, as shown in the Beavis experiment where multiple QTL were actually simulated (Tables 4 and 5). When we predict the bias, the residual variance includes the environmental variance plus the sum of the variances of the remaining QTL excluding the one in question. For example, when we predicted the bias for the QTL explaining 3% of the phenotypic variance, the residual variance was chosen as 100 – 3 = 97, where 100 is the total phenotypic variance. If the data were analyzed by a multiple-QTL model, e.g., composite interval mapping (Jansen 1993; Zeng 1994), the residual variance should include only the environmental variance given that the remaining QTL have been included in the model. When QTL are linked, the situation will be complicated and further investigation is necessary.
Acknowledgments
The author thanks Dr. Chenwu Xu for his help in the simulating experiment. The author also appreciates editor S. Otto and two anonymous reviewers for their constructive comments and suggestions on the manuscript. This work was supported by the National Institutes of Health grant R01-GM55321 and the U.S. Department of Agriculture National Research Initiative Competitive Grants Program 00-35300-9245.
Footnotes
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Communicating editor: S. Otto
- Received November 11, 2002.
- Accepted August 26, 2003.
- Copyright © 2003 by the Genetics Society of America