# Genetic heteroscedastic models for ordinal traits: application to sheep litter size

- Samira Fathallah
^{1}, - Loys Bodin
^{1}Email author and - Ingrid David
^{1}

**48**:30

https://doi.org/10.1186/s12711-016-0202-4

© Fathallah et al. 2016

**Received: **6 February 2015

**Accepted: **14 March 2016

**Published: **1 April 2016

## Abstract

### Background

Classical genetic canalization models, which accommodate the mean and variance of a trait separately, provide a flexible approach to take heteroscedasticity for continuous traits into account. However, this model is not appropriate for discrete traits. The aim of this work was to propose heteroscedastic threshold models suitable for the genetic analysis of ordinal data.

### Methods

In order to first fit the mean and variance of ordinal traits separately, we extended the classical threshold model (TM) for discrete data by introducing non-genetic and genetic factors of heterogeneity on the variance of its underlying variable, which leads to a homothetic threshold model HTM and its alternative parameterization HTM’ in which the thresholds of different individuals are linked by a homothetic-translation. Relaxing the constraint between the thresholds led us to propose an independent threshold model ITM that was more flexible than HTM’ but required the estimation of more parameters. TM, HTM and ITM models were applied to study 19,671 records on litter size in Romane sheep.

### Results

Both HTM and ITM were able to disentangle the link between the mean and variance that holds in the classical homoscedastic threshold model. The results obtained for the litter size of Romane ewes showed that the data was best fitted with HTM compared to ITM and TM. The correlations between the observed and predicted variances were equal to 0.6 and 0.2 for HTM and TM, respectively. These analyses showed the existence of a genetic component for the heterogeneity of litter size in sheep that was taken into account in HTM.

### Conclusions

HTM is the most suitable model to study the variability of litter size in sheep. It accommodates both the mean and variance separately while requiring the estimation of only a few parameters.

## Background

Robustness can be viewed as the ability to maintain a stable phenotype regardless of the environmental conditions. In line with this approach, Scheiner and Lyman [1] proposed a model which considers that the expression of a trait is controlled by two sets of genes. One set controls the level of the performance and the other set controls the environmental variability of the trait. Various authors developed heteroscedastic models (i.e. canalization models) to jointly estimate the influence of various factors on both the mean and the variability of a trait [2–5].

These models were implemented with frequentist [6, 7] or Bayesian methods [8, 9], and applied to a large variety of continuous traits (birth weight of piglets, birth weight of rabbits, weight of chickens, etc.).

The methods described above mainly target continuous traits. Theoretically, these linear models do not analyze discrete traits adequately because of their non-normal distributions. However, some previous empirical studies showed that the performance of linear and non-linear models was similar in terms of goodness-of-fit and predictive ability [10, 11] when analyzing the level of discrete livestock data. However, it is not known whether these properties are still true when dealing with the variability of discrete data.

The main objective of this research was to propose new models that take the heterogeneous individual variability of ordinal traits into account using derived threshold models. The models are described consecutively in the “Methods” section. “Heteroscedastic extensions of the threshold model” are presented in the first subsection, and a “Numerical application” of these models to litter size in Romane sheep is presented in the second subsection.

## Methods

### Heteroscedastic models for ordinal traits

#### Homothetic threshold models for ordinal traits

#### Extension of the homothetic threshold model into the independent threshold model

The total number of parameters in this model is \(\left( {n + 2} \right)\left( {n - 1} \right)/2\). It should be noted that ITM cannot be parameterized like HTM using a fixed set of thresholds and a model on the liability \(l\). In this model, fixing the thresholds to modify the parameterization would imply that the underlying variable would no longer be Gaussian.

Due to the lack of constraints between the thresholds (except for the obvious relationship \(- \infty < \tau_{1.} < \tau_{2.} < \cdots < \tau_{n - 1.} < \infty\)), ITM fits a much larger range of situations than a homothetic threshold model, which assumes a strong constraint either on the homothetic translation relationship between thresholds (HTM’) or on the normality of the liability (HTM).

*k*

^{t}

^{h}indicator function is equal to 1 is:

As for ITM, there are \(\left( {n + 2} \right)\left( {n - 1} \right)/2\) parameters to estimate i.e. \(n - 1\) thresholds and \(n\left( {n - 1} \right)/2\) variance covariance parameters.

Main characteristics of the different models

Models | Number of parameters | Thresholds | Underlying |
---|---|---|---|

Population | |||

| |||

TM | \(n\) | \({\mathbf{t}}^{*} = \left\{ {{\mathbf{t}}_{1}^{*} ,{\mathbf{t}}_{2}^{*} , {\mathbf{t}}_{3}^{*} , \ldots ,{\mathbf{t}}_{n - 1}^{*} } \right\}\) | \({\text{Y }}\sim{\text{N}}\left( {0, 1} \right)\) |

| |||

HTM | \(n + 2\) | \({\mathbf{t}} = \left\{ {0, 1, {\mathbf{t}}_{3} , \ldots , {\mathbf{t}}_{n - 1} } \right\}\) | \({\text{Y }}\sim{\text{N}}\left( {\mu + u,f^{2} (\eta ,{\mathbf{v}}} )\right)\) |

HTM’ | \(n + 2\) | \({\mathbf{t^{\prime}}} = \left\{ {{\mathbf{t}}_{1}^{'} ,{\mathbf{t}}_{2}^{'} , {\mathbf{t}}_{3}^{'} , \ldots ,{\mathbf{t}}_{n - 1}^{'} } \right\}\) | \({\text{Y }}\sim{\text{N}}\left( {0, 1} \right)\) |

ITM | \(\left( {n + 2} \right)\left( {n - 1} \right)/2\) | \({\mathbf{t}}^{{{\prime \prime }}} = \left\{ {{\mathbf{t}}_{1}^{{{\prime \prime }}} ,{\mathbf{t}}_{2}^{{{\prime \prime }}} , {\mathbf{t}}_{3}^{{{\prime \prime }}} , \ldots ,{\mathbf{t}}_{n - 1}^{{{\prime \prime }}} } \right\}\) | \({\text{Y }}\sim{\text{N}}\left( {0, 1} \right)\) |

Animal | |||

| |||

TM | \(n\) | \({\text{t}}_{i}^{*} = \left\{ {{\text{t}}_{1i}^{*} , {\text{t}}_{2i}^{*} , {\text{t}}_{3i}^{*} , \ldots , {\text{t}}_{{\left( {n - 1} \right)i}}^{*} } \right\}\) \({\text{t}}_{.i}^{*} = {\text{t}}_{i} + \mu_{i} + u_{i}\) | \({\text{y}}_{i} = {{\upvarepsilon }}_{i}\) |

| |||

HTM | \(n + 2\) | \({\text{t}}_{i} = \left\{ {0, 1 {\text{t}}_{3i} , \ldots , {\text{t}}_{{\left( {n - 1} \right)i}} } \right\}\) | \({\text{y}}_{i} = {{\mu }}_{i} + {u_{i}} + {{\upvarepsilon }}_{i}\) |

HTM’ | \(n + 2\) | \({\text{t}}_{i}^{'} = \left\{ {{\text{t}}_{1i}^{'} , {\text{t}}_{2i}^{'} , {\text{t}}_{3i}^{'} , \ldots , {\text{t}}_{{\left( {n - 1} \right)i}}^{'} } \right\}\) \({\text{t}}_{.i}^{'} = {\text{t}}_{i}^{'} + \mu_{i}^{'} + u_{i}^{'} + {\text{v}}_{i}^{'}\) | \({\text{y}}_{\text{i}} = {{\upvarepsilon }}_{i}\) |

ITM | \(\left( {n + 2} \right)\left( {n - 1} \right)/2\) | \({\text{t}}_{i}^{{{\prime \prime }}} = \left\{ {{\text{t}}_{1i}^{{{\prime \prime }}} , {\text{t}}_{2i}^{{{\prime \prime }}} , {\text{t}}_{3i}^{{{\prime \prime }}} , \ldots , {\text{t}}_{{\left( {n - 1} \right)i}}^{{{\prime \prime }}} } \right\}\) \({\text{t}}_{.i}^{{{\prime \prime }}} = {\text{t}}_{i}^{{{\prime \prime }}} + \mu_{i}^{{{\prime \prime }}} + u_{i}^{{{\prime \prime }}}\) | \({\text{y}}_{i} = {{\upvarepsilon }}_{i}\) |

### Numerical application

#### Data

Features of Romane sheep

| |

Number of records | 19,671 |

Number of animals | 11,073 |

Number of sires | 1096 |

Number of dams | 6550 |

Mean | 2.13 |

| |

Single | 17.70 |

Twin | 55.01 |

Triplet | 24.18 |

Quadruplet and + | 3.11 |

The fixed effects included in the analysis were selected to be as similar as possible to those found in others studies [15, 16]. This was done by comparing the likelihood-ratio of all the possible linear mixed models which included the following fixed effects: age of ewe (nine levels), time interval between lambing in months (seven levels), parity (seven levels), age of the ewe at first lambing in months (nine levels), season of lambing (two levels), as well as the random effects: flock and ewe effect. For this step of the analysis, the ewe effect was considered as random but the genetic relationships between individuals were ignored.

#### Estimation of parameters and comparison of methods

The three threshold models TM, HTM’ and ITM described above were applied to this data. The classical TM, used as a basis for comparison, included the previously selected fixed effects and the ewe as both a random genetic effect (\(u^{*}\)) and a random permanent environmental effect \(\left( {p^{*} } \right)\) (i.e. \({\mathbf{t}}_{i}^{*} = \varvec{x}_{i} {\varvec{\beta}} + u_{i}^{*} \mathbf1^{{\mathbf{T}}} + p_{i}^{*} \mathbf1^{{\mathbf{T}}}\)). The homothetic threshold model HTM’ considered that fixed, genetic and permanent effects were common for all thresholds and threshold-specific random genetic effects with constrained genetic (co)variance to ensure homothetic translation (i.e. \({\mathbf{t}}_{i}^{'} = \varvec{x}_{i} {\varvec{\beta}} + u_{i}^{'} \mathbf1^{{\mathbf{T}}} + p_{i}^{'} \mathbf1^{{\mathbf{T}}} + {\mathbf{v}}_{\varvec{i}}^{\varvec{'}}\)). Finally, an additional extension resulting in the independent threshold model ITM was fitted and it considered that the fixed and permanent effects were common for all thresholds and included a genetic effect that was specific to each threshold with no constrained genetic (co)variance (i.e. \({\mathbf{t}}_{i}^{{{\prime \prime }}} = \varvec{x}_{i} {\varvec{\beta}} + p_{i}^{{{\prime \prime }}} \mathbf1^{{\mathbf{T}}} + {\mathbf{v}}_{\varvec{i}}^{{{\prime \prime }}}\)).

ASReml software [17] was used to implement all the models.

*y*in each subgroup \(\hat{\sigma }_{m}^{2}\) was calculated in order to calculate:

The subgroup number was defined in order to get a sufficient amount of data in each subgroup to obtain a good estimation of the subgroup variance. The intensity of the dispersion of each model was also assessed by calculating the correlation, either for the mean or the variance, between the observed values and the predicted values on the observable scale.

## Results

### Comparison of models

Values obtained from likelihood-ratio, BIC tests, B criteria, and mean squared error (MSE) between observed and fitted values in the different models

Models/tests | Likelihood-ratio (LRT) | BIC test | B | MSE of mean | MSE of variance |
---|---|---|---|---|---|

TM | TM versus HTM': ≫11 (Reject H0 TM versus ITM: ≫16 (Reject H0 | 54,005 | 0.013 | 0.07 | 0.13 |

HTM’ | 49,517 | 0.51 | 0.07 | 0.10 | |

ITM | 50,663 | 0.42 | 0.07 | 0.10 |

Values obtained from mean squared error of prediction (MSEP) and B criteria on the validation subset in the different models

Models/tests | MSEP | B criteria |
---|---|---|

Threshold model (TM) | 0.473 | 0.005 |

Homothetic threshold model (HTM’) | 0.471 | 0.363 |

Independent threshold model (ITM) | 0.481 | 0.371 |

The results showed that the homoscedastic model should be rejected and that the models that account for genetic heterogeneity of litter size distribution should be favored. Among the heteroscedastic models, BIC values were lower with HTM’ than with ITM. MSE estimates of the mean were similar for the three models, whereas MSE estimates of the variance differed and were higher with TM than with HTM’ (0.13 with TM vs. 0.10 with HTM’). The differences between MSEP values are small, which is explained by the fact that this parameter can estimate prediction error for the level of the trait only and not for its variability. To evaluate the ability of the models to predict variance, we calculated the B criteria on the validation subset by discretizing the whole data into 200 adjacent subgroups. The results are in Table 4. Based on the B criteria, HTM’ and ITM were more efficient than TM (0.36 for HTM’, 0.37 for ITM vs. 0.005 for TM).

### Homoscedastic threshold model used as reference (TM)

Correlations between the predicted and observed means as well as the predicted observed variance with the different models

Model | Predicted mean versus observed mean | Predicted variance versus observed variance |
---|---|---|

TM | 0.80 | 0.21 |

HTM’ | 0.79 | 0.61 |

ITM | 0.78 | 0.59 |

### Homothetic threshold model (HTM’)

Genetic variance and correlation between thresholds for Romane sheep using the HTM’ model

Thresholds/genetic variance | \(\sigma_{{u^{\prime}.1}}^{2}\) | \(\sigma_{{\nu^{{\prime }} .2}}^{2}\) | \(\sigma_{{\nu^{{\prime }} .3}}^{2}\) |
---|---|---|---|

0.04 | 0.13 | 0.13 | |

0.05 | 1 | ||

0.20 |

The correlation between observed and predicted means was high (\(r = 0.79\)) and not different from the value obtained with TM. This contrasts with the correlation between predicted and observed variances (\(r = 0.61\)). Although the prediction of the variances was still lower than that of the means, it was much higher than with TM. HTM’ could therefore break the link between the predicted means and variances and the B parameter was equal to 0.51 (Table 3) even if the correlation between these variables was still high (\(r = 0.88\)). The scatterplot in Fig. 1b shows this variability of the predicted variance for a given predicted mean.

Observed values (prolificacy, variance) and estimated values with HTM’ [thresholds, different gap between the first thresholds (\({{\Delta }}_{1 - 2}\)), ratio (R), prolificacy, variance, and percentage of the simple (% LS 1), double (% LS 2), triplet (% LS 3) and quadruplet (% LS 4)] for two individuals

Observed values | Estimated values with HTM’ model | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ind | Prolificacy | Variance | \(\tau_{1}^{'}\) | \(\tau_{2}^{'}\) | \(\tau_{3}^{'}\) | \({{\Delta }}_{1 - 2}\) | \(R\) | Prolificacy | Variance | % LS1 | % LS2 | % LS3 | % LS4+ |

1 | 2.00 | 0.50 | −0.63 | 0.92 | 2.16 | 1.55 | 1.8 | 2.09 | 0.54 | 16 | 53 | 25 | 6 |

2 | 2.00 | 0.33 | −0.89 | 0.86 | 2.28 | 1.75 | 1.8 | 2.02 | 0.38 | 21 | 64 | 15 | 0 |

### Independent threshold model (ITM)

Genetic variance and correlation between thresholds for Romane sheep using the ITM model

Thresholds/genetic variance | \(\sigma_{{\nu^{\prime\prime}.1}}^{2}\) | \(\sigma_{{\nu^{\prime\prime}.2}}^{2}\) | \(\sigma_{{\nu^{\prime\prime}.3}}^{2}\) |
---|---|---|---|

0.035 | 0.631 | 0.430 | |

0.138 | 0.921 | ||

0.176 |

Spearman rank correlations between the HTM’ and ITM models

First threshold | 0.988 |

Second threshold | 0.989 |

Third threshold | 0.988 |

## Discussion

A genetic heteroscedastic model was previously proposed [6] to analyze categorical variables and break down the link between mean and variance that holds with the classical homoscedastic threshold model. This model originated from the combination of two base models: the homoscedastic threshold model for categorical data [19] and the structural heteroscedastic model for continuous variables [2]. In this paper, we studied further extensions of this model by analyzing the relationship that exists between categories and in particular by releasing the main constraint.

In this paper, we propose two kinds of heteroscedastic models, both with different parameterizations. The first type (HTM/HTM’) was fully derived from the canalization model proposed for continuous traits [2] with which it was easy to draw comparisons. The second type consisted in relaxing the relationship that exists between thresholds (translation in TM or homothetic translation in HTM) and led to a model that considers each threshold independently i.e. ITM. We demonstrated that this model was equivalent to a constrained multiple-trait model MTM on each class of the ordinal trait. This second type of model is much more flexible than HTM and its application in selection would be of particular interest because it allows for each class to be controlled through a multiple selection index [20] with specific environmental effects and economic weights for each class. Modeling each class of the trait to control its distribution or modeling only the most important class are trivial issues and have already been considered [21], however the correct multiple trait model must include a constraint on the sum of the probabilities of the \(n - 1\) binary variables. In spite of their potential interest, the use of this kind of model (ITM or MTM) for genetic evaluation and selection would be limited since they require a larger number of parameters to be estimated than HTM or HTM’ and would probably show convergence problems when the number of classes exceeds three. Furthermore, their biological interpretation is difficult; for instance we could not link the parameters of ITM or MTM to the biological model of Schneider and Lyman [22], which assumes the existence of different genes that control the mean and the variance of a trait.

We compared TM, HTM’ and ITM through the analysis of data on litter size from both Romane and Rouge de l’Ouest sheep (the latter results are not shown). The low heritability of the liability to litter size obtained with an homoscedastic threshold model (i.e. TM) agreed with previous studies that used TM [23–25]. However, LRT, BIC, and MSE parameters showed that this homoscedastic model was less successful than both heteroscedastic models. Consequently, the distribution of sheep litter size shows heteroscedasticity of genetic origin as revealed by both the homothetic HTM’ and independent threshold ITM models.

Taking the genetic heteroscedasticity in HTM’ and the subsequent transformation of the parameters to those of HTM into account did not substantially change the heritability of the liability to litter size compared to TM. However, with HTM, the additive genetic variance of the variability of the trait was smaller than the corresponding variance of its mean. This is in line with results in the literature that showed that when additive genetic variances of the mean and the variability of a trait were jointly estimated, their values were smaller for the variability than for the mean [6, 9, 26–31]. This seems to be a common rule regardless of the methodology used for such estimations, the trait or the species. In contrast, the genetic correlation between the mean of a trait and its variability is highly variable and covers the whole range of its variation interval. For instance, in a mouse population, genetic correlations of −0.93 and 0.97 were reported for litter size and individual birth weight, respectively [29]. In our study, we found a negative genetic correlation between the mean and the variability of sheep litter size with HTM’ (\(\rho = - 0.40\)). As highlighted by Yang et al. [32], this parameter might be affected by an artefact due to the scale of measurement or skewness of the data, although with the threshold model no such skewness affected this correlation. However, a study on another French sheep breed (Rouge de l’Ouest) for which the mean prolificacy (prolificacy = 1.80; LS3+ = 12.5 %) is lower than in Romane sheep (results not shown) reported a similar value i.e. −0.25.

Relaxing the homothetic constraint between the thresholds of HTM’ did not improve the fit of the model or the predictive ability of the data as might be expected, but instead increased the BIC values due to the larger number of parameters to estimate in ITM. Moreover, although the constraint on the genetic correlation between thresholds was relaxed in ITM, the correlation between the second and third thresholds remained close to 1 and the correlation between the first and second thresholds was high; in addition, the Spearman rank correlations between the thresholds of HTM’ and ITM were close to 1 (Table 9). This suggests that the true nature of the link between the thresholds is almost a homothetic translation in sheep and that the distributions of litter size classes are not independent but linked by a strong common law. It also means that the concept of sensitivity thresholds, which transform individual Gaussian variables that differ in mean and variance into the observed categories, might be relevant.

Nonetheless in our study, only genetic effects were considered as threshold-specific in ITM in order to compare the different models using LRT or BIC criteria. Of course, threshold-specific fixed effects and permanent effects could be included in ITM. In the same way, threshold-specific fixed effects and permanent effects could be included in HTM’. Unfortunately, HTM with a threshold-specific permanent effect did not converge and threshold-specific fixed factors with proportional effects cannot be implemented in ASReml. It may be possible to estimate fixed effects in the residual variance using an iterative process that tests the influence of each effect on the variance of the model. We did not test such algorithms on our data and it would probably be a slow and difficult process. Another solution would be to modify existing software for canalizing selection of a continuous trait in order to take into account the discrete characteristics of the data.

## Conclusions

This study shows that we can model the variabilities (mean and variance) of a discrete trait using a suitable model, namely the HTM’ model. Thus, the canalization of discrete traits is possible and the gap between the first two thresholds estimated for each individual in this model could be used to estimate the breeding value for the variability of litter size independently from that of the mean.

## Declarations

### Authors’ contributions

SF, ID and LB jointly designed the study and discussed the results. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the CORAM for the funding and Helene Munduteguy for linguistic revision and referees for their helpful contributions.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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