Bayesian estimation of marker dosage in sugarcane and other autopolyploids

Abstract

In sugarcane or other autopolyploids, after generating the data, the first step in constructing molecular marker maps is to determine marker dosage. Improved methods for correctly allocating marker dosage will result in more accurate maps and increased efficiency of QTL linkage detection. When employing dominant markers like AFLPs, single-dose markers represent alleles present as one copy in one parent and null in the other parent, double-dose markers are those present as two copies in one parent and null in the other parent and so on. Observed segregation ratios in the offspring are employed to infer marker dosage in the parent from which the marker was inherited. Commonly, for each marker, a χ ² test is used to assign dosage. Such an approach does not address important practical considerations such as multiple testing and departures from theoretical assumptions. In particular, extra-binomial variation or overdispersion has been observed in sugarcane studies and standard methods may result in fewer correct dosage allocations than the data warrant. To address these shortcomings, a Bayesian mixture model is proposed where all markers are considered simultaneously. Since analytic solutions are not available, Markov chain Monte Carlo methods are employed. Marker dosage allocation for each individual marker employs the estimated posterior probability of each dosage. For a sugarcane study these methods resulted in more markers being allocated a dosage than by standard approaches. Simulation studies demonstrated that, in general, not only are more markers classified but that more markers are also correctly classified, particularly if overdispersion is present.

Appendices

Appendix 1: Informative prior specification for τ _k

Conjugate prior distributions are employed for the means μ _k and precisions ${{\uptau}}_{\text{k}} ,k = 1, \ldots, K.$ A method to determine the hyperparameters as ${\text{logit}}\left( {P_{k} } \right)$ and T _k for the prior distribution of the mean μ _k are outlined in “Priors”. Prior distributions for the means μ _k in autooctoploids are provided in Table 4.

Table 4 Expected segregation ratios for autooctoploids on the logit scale and hyperparameters set for strong priors assuming theoretical segregation ratios

Employing a similar approach for the mean of ${{\uptau}}_{k} \sim {\text{Gamma}}\left( {A_{k} ,B_{k} } \right),$ we can calculate the logit transformed 2.5 and 97.5 percentiles of the theoretical binomial distribution with parameters in Eq. 1 to obtain the expected value ${{\hat{\tau}}}_{\text{k}}$ of ${{\uptau}}_{\text{k}}$. Denoting the percentiles as q _0.025 and q _0.975 on the untransformed scale then for a 95% confidence region on the logit scale

$$ \tilde{s}_{k} \approx {{\left[ {{\text{logit}}\left( {q_{0.0975} } \right) - {\text{logit}}\left( {q_{0.0925} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {{\text{logit}}\left( {q_{0.0975} } \right) - {\text{logit}}\left( {q_{0.0925} } \right)} \right]} 4}} \right. \kern-\nulldelimiterspace} 4} $$

(12)

and so expected value ${{\tilde{\tau}}}_{k} = {1 \mathord{\left/{\vphantom {1 {\tilde{s}_{k}^{2}}}} \right. \kern-\nulldelimiterspace} {\tilde{s}_{k}^{2}}},$, where $\tilde{s}_{k}$ is defined in Eq. 12 and so may be obtained directly from the percentiles q _0.025 and q _0.075 of the binomial distribution with size equal to the number of individuals N _k and probability P _k .

The conjugate prior distribution for the precision τ _k is a Gamma(A _k ,B _k ) which has mean ${{A_{k} } \mathord{\left/ {\vphantom {{A_{k} } {B_{k} }}} \right. \kern-\nulldelimiterspace} {B_{k} }}$ and variance ${{A_{k} } \mathord{\left/ {\vphantom {{A_{k} } {B_{k}^{2} }}} \right. \kern-\nulldelimiterspace} {B_{k}^{2} }}$. Ideally, the mean of the prior distribution would be specified as ${{\hat{\tau}}}_{\text{k}}$ but the variance may be specified by several methods. Those considered here involve setting an interval around either ${{\tilde{\tau}}}_{\text{k}}$ or $\tilde{s}_{k}$.

First, if the prior distribution is specified to have a 95% confidence region around τ _k as ${{\uptau}}_{k} \left( {1 \pm x} \right)$, then the interval has length $2x{{\uptau}}_{k}$ which is 4 SD(τ _k ). A _k and B _k are obtained by equating the mean $A_{k}/B_{k}$ and variance ${{A_{k} } \mathord{\left/ {\vphantom {{A_{k} } {B_{k}^{2} }}} \right. \kern-\nulldelimiterspace} {B_{k}^{2} }}$ to their observed values ${{\hat{\tau}}}_{\text{k}}$ and $2x{{\hat{\tau}}}$, respectively.

In a similar fashion, if the 95% confidence region around s _k is set to $s_{k} \left( {1 \pm x} \right)$ then the SD(τ _k ) is a quarter of the interval on the on the τ _k scale. This is simply

$$ \begin{gathered} {\text{SD}}\left( {{{\uptau}}_{k} } \right) = {\frac{1}{4}}\left( {{\frac{1}{{s_{k}^{2} (1 - x)^{2} }}} - {\frac{1}{{s_{k}^{2} (1 + x)^{2} }}}} \right) \times \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;{\frac{x}{{s_{k}^{2} \left( {1 + x} \right)^{2} \left( {1 - x} \right)^{2} }}} \hfill \\ \end{gathered} $$

and once the observed and expected means and variances are equated then $A_{k} = C.{{\hat{\tau}}}_{k}^{4}$and $B_{k} = C{{\hat{\tau}}}_{k}^{3}$ where $C = x^{ 2} /\left( { 1+ x} \right)\left( { 1- x} \right).$ This produces a narrower prior distribution than specifying limits around τ _k .

Appendix 2: Generating markers with overdispersion

For simulation studies, markers with specified dosage may be generated from a Binomial (n,p) distribution where p is the appropriate segregation proportion in Eq. 1. When overdispersion is present, a beta-binomial distribution may be used with p ~ beta (α,β), where α and β are the first and second shape parameter, respectively (Skellams 1948). If the theoretical segregation ratio P _jk in Eq. 1 is equated to the expected value E(p) = α/(α + β) then simply setting the first shape parameter α fixes the value of $\beta = \alpha \left( {1 - P_{jk} } \right)/P_{jk}$. Note that larger values of α correspond to smaller values of Var $\left( p \right) = ab\left( {a + b} \right)^{2} \left( {a + b + 1} \right)$ which results in less overdispersion.

Appendix 3: Comparison of mixture model options

From Fig. 11 the model with more components performs slightly better in that, on average, the median percentage of correctly allocated markers was higher with more components and the misclassification rate was lower. In general, while results are better when more components are employed at higher ploidy levels, the range may appear to indicate that worse results may actually be obtained for particular data sets. Further investigation reveals that this only occurs for medium to severe overdispersion (see Fig. 12).

Fig. 11

Box plots of the i percentage of markers with dosage correctly allocated and ii misclassified by models with three or four components where equal variances on the logit scale were assumed and strong prior information was incorporated. Three component models may avoid computational problems but could result in more markers being misclassified

Fig. 12

Box plots of the percentage of misclassified markers for non to severely overdispersed data for models with three or four components. The model employed was that of equal variances on the logit scale with strong prior information incorporated. The range of results increases with increasing overdispersion which could result in worse classification for some data sets

While the percentage of correctly allocated markers decreases with increasing threshold (see Fig. 13), the trend becomes more pronounced with increasing overdispersion and ploidy levels. On the other hand, misclassification rates increase with smaller thresholds and increasing ploidy or overdispersion levels. While there is no clear optimal threshold value, it would seem that that a value of around 0.8 is a reasonable compromise and corresponds in some ways to the value of 0.2 which is commonly used in false discovery rate studies and commonly used as a reasonable power when designing experimental studies (Fig. 14).

Fig. 13

Box plots of the percentage of markers with dosage correctly allocated by mixture models for a range of thresholds by four levels of overdispersion (None, Slight, Medium, Severe) and ploidy (4, 6, 8, 10). Dosage is allocated when the posterior probability exceeds the threshold. The models fitted were chosen to be those with the maximum number of components, equal variances on the logit scale were assumed and strong prior information incorporated. The percentage of correctly allocated markers tails off for thresholds larger than 0.8

Fig. 14

Box plots of the percentage of misclassified markers for a range of thresholds with increasing overdispersion (None, Slight, Medium, Severe) varying ploidy levels (4, 6, 8, 10). Dosage is allocated when the posterior probability exceeds the threshold. When there is little or no overdispersion, very few markers are misclassified. However, for moderate to severe overdispersion the misclassification rate decreases as the threshold increases but is greater for higher ploidy levels