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Smyth (2004): limma, a tool for differential expression analysis. It fits gene wise linear models with moderated t statistics. The t statistics is based on empirical Bayes approach which borrows information across genes, resulting in far more stable inference when the number of arrays is small.
Basic model assumptions: \[ y_g = (y_{g1}, \cdots, y_{gn})^\intercal \]
\[ E(y_g) = X \alpha_g \quad var(y_g) = W_g \sigma_g^2 \]
The parameters of interest is \(\beta_g = C^\intercal \alpha_g\).
We assume the linear model is fitted to the response for each gene to obtain estimators, \[ var(\hat{\alpha}_g) = V_g s_g^2 \quad var(\hat{\beta}_g) = C^\intercal V_g C s_g^2 \]
Let \(v_{gj}\) be the \(j\)th diagonal element of \(C^\intercal V_g C\). \[ \hat{\beta}_{gj}|\beta_{gj}, \sigma_g^2 \sim N(\beta_{gj}, v_{gj} \sigma_g^2) \quad s_g^2 | \sigma_g^2 \sim \frac{\sigma_{g}^2}{d_g} \chi_{d_g}^2 \] where \(d_g\) is the residual degrees of freedom of the linear model for gene g. Under these assumptions, \[ t_{gj} = \frac{\hat{\beta}_{gj}}{s_g \sqrt{v_{gj}}} \sim T_{d_g} \]
Prior on \(\sigma_g^2\): \[ \frac{1}{\sigma_g^2} \sim \frac{1}{d_0 s_0^2} \chi_{d_0}^2. \] For any \(j\), we assume \[ P(\beta_{gj} \neq 0) = p_j \] \(p_j\) is the expected proportion of truly differentially expressed genes. For those non-zero coefficients, we assume \[ \beta_{gj}|\sigma_g^2, \beta_{gj} \neq 0 \sim N(0, v_{oj} \sigma_{g}^2) \]
Under above priors, the posterior mean of \(1/\sigma_g^2 | s_g^2\) is \[ \tilde{s}_g^2 = \frac{d_0 s_0^2 + d_g s_g^2}{d_0 + d_g} \]
The moderated t statistics is \[ \tilde{t}_{gj} = \frac{\hat{\beta}_{gj}}{\tilde{s}_g \sqrt{v_{gj}}} \sim T_{d_g + d_0} \text{ under the null } \beta_{gj} = 0 \]
The odds that the \(g\)th gene has non-zero \(\beta_{gj}\) is \[ O_{gj} = \frac{p(\beta_{gj} \neq 0 | \tilde{t}_{gj}, s_g^2)}{p(\beta_{gj} = 0 | \tilde{t}_{gj}, s_g^2)} = \frac{p_j}{1-p_j} \frac{p(\tilde{t}_{gj}| \beta_{gj} \neq 0)}{p(\tilde{t}_{gj}| \beta_{gj} = 0)} = \frac{p_j}{1-p_j} (\frac{v_{gj}}{v_{gj}+v_{0j}})^{1/2} \exp (\frac{\tilde{t}_{gj}^2}{2}\frac{v_{0j}}{v_{gj}+v_{0j}}) \]
The hyperparameters can be estimated using empirical Bayes approach.
Smyth, Gordon K. 2004. “Linear Models and Empirical Bayes Methods for Assessing Differential Expression in Microarray Experiments.” Statistical Applications in Genetics and Molecular Biology 3 (1). De Gruyter.