Table 1 Main characteristics of the different models
From: Genetic heteroscedastic models for ordinal traits: application to sheep litter size
Models | Number of parameters | Thresholds | Underlying |
|---|---|---|---|
Population | |||
Homoscedastic | |||
TM | \(n\) | \({\mathbf{t}}^{*} = \left\{ {{\mathbf{t}}_{1}^{*} ,{\mathbf{t}}_{2}^{*} , {\mathbf{t}}_{3}^{*} , \ldots ,{\mathbf{t}}_{n - 1}^{*} } \right\}\) | \({\text{Y }}\sim{\text{N}}\left( {0, 1} \right)\) |
Heteroscedastic | |||
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 i | |||
Homoscedastic | |||
TM | \(n\) | \({\text{t}}_{i}^{*} = \left\{ {{\text{t}}_{1i}^{*} , {\text{t}}_{2i}^{*} , {\text{t}}_{3i}^{*} , \ldots , {\text{t}}_{{\left( {n - 1} \right)i}}^{*} } \right\}\) \({\text{t}}_{.i}^{*} = {\text{t}}_{i} + \mu_{i} + u_{i}\) | \({\text{y}}_{i} = {{\upvarepsilon }}_{i}\) |
Heteroscedastic | |||
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 | \(\left( {n + 2} \right)\left( {n - 1} \right)/2\) | \({\text{t}}_{i}^{{{\prime \prime }}} = \left\{ {{\text{t}}_{1i}^{{{\prime \prime }}} , {\text{t}}_{2i}^{{{\prime \prime }}} , {\text{t}}_{3i}^{{{\prime \prime }}} , \ldots , {\text{t}}_{{\left( {n - 1} \right)i}}^{{{\prime \prime }}} } \right\}\) \({\text{t}}_{.i}^{{{\prime \prime }}} = {\text{t}}_{i}^{{{\prime \prime }}} + \mu_{i}^{{{\prime \prime }}} + u_{i}^{{{\prime \prime }}}\) | \({\text{y}}_{i} = {{\upvarepsilon }}_{i}\) |