Use of Maximum Likelihood-Mixed Models to select stable reference genes: a case of heat stress response in sheep

BMC Molecular Biology

Table 3 ML-Mixed models description

MID	Model	Variance structure	n° (co)variance parameters
1	y_ijk = μ + t_i + g_j + a_k + tg_ij + ta_ik + ga_jk + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ta~N(0,σ²_ta); ga~N(0,σ²_ga); e~N(0,σ²e)	5
2	y_ijk = μ + t_i + g_j + a_k + tg_ij + ta_ik + ga_jk + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ta~N(0,σ²_ta); ga~N(0,σ²_ga); e~N(0,σ²e_j)	20
3	y_ijk = μ + t_i + g_j + a_k + tg_ij + ta_ik + ga_jk + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ta~N(0,σ²_ta); ga~N(0,σ²_ga); e~N(0,σ²e_ij)	36
4	y_ijk = μ + t_i + g_j + a_k + tg_ij + ta_ik + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ta~N(0,σ²_ta); e~N(0,σ²e_j)	19
5	y_ijk = μ + t_i + g_j + a_k + tg_ij + ta_ik + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ta~N(0,σ²_ta); e~N(0,σ²e_ij)	35
6	y_ijk = μ + t_i + g_j + a_k + tg_ij + ga_ik + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ga~N(0,σ²_ga); e~N(0,σ²e_j)	19
7	y_ijk = μ + t_i + g_j + a_k + tg_ij + ga_ik + e_ijk	a~N(0,σ²_a); tg~N(0,σ²_tg); ga~N(0,σ²_ga); e~N(0,σ²e_ij)	35
8	y_ijk = μ + tg_ij + ta_ik + e_ijk	e~N(0,σ²e_ij)	32

In the Model column the fixed effects were t = treatment (2 levels) and g = gene (16 levels); the random effects were a = sample (15 levels), tg = interaction treatment × gene (32 levels), ta = interaction treatment × sample (29 levels), ga = interaction gene × sample (240 levels), and e = residual. Residual variances: σ²e = homoskedastic residuals and σ²e_j = heteroskedastic residual for the gene effect (16 residual variances); σ²e_ij = heteroskedastic residual for the treatment × gene effect (32 residual variances). Model 8 was proposed by Andersen et al [5], where txg y txa interactions are treated as fixed effects.
MID - model identification

ISSN: 1471-2199