The mash fit (EZ V1 model) favor three covariance components. One is null (no effect, 84.6%). The other two components show the standardized effects are positively correlated (14.5%). The covariance component \(11'\) (5.4%) shows that the standardized effects are similar, which does not mean the raw effects are similar. The other covariance component has the formate \(D11'D\) (9.1%), D is a diagnoal matrix, which means the standardized effects are different in size, but they are strongly correlated.

Similarity of standardized effects does not mean the similarity of raw effects

  • The covariance is for the standardized effect, \(S_{j}^{-1}\beta_{j}\).

    For example, \[ S_{j}^2 = \left(\begin{array}{c c} 0.5^2 & 0 \\ 0 & 1 \end{array}\right) \] Using EZ model

    \[\begin{align*} \left(\begin{array}{c} \hat{\beta}_{j1}/0.5 \\ \hat{\beta}_{j2} \end{array}\right) | \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) &\sim N\left(\left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right), S_{j}^2 \right) \\ \left(\begin{array}{c} \beta_{j1}/0.5 \\ \beta_{j2} \end{array}\right) | \hat{\pi} &\sim \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N(0,\left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right)) \end{align*}\] The posterior becomes \[\begin{align*} \left(\begin{array}{c} \beta_{j1}/0.5\\ \beta_{j2} \end{array}\right) | \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( \left(\begin{array}{c} \mu_{j1} \\ \mu_{j2} \end{array}\right),c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \right) \\ &= \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \left(\begin{array}{c} 4\hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) , c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \right) \\ &= \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( c \left(\begin{array}{c} 4\hat{\beta}_{j1} + \hat{\beta}_{j2} \\ 4\hat{\beta}_{j1} + \hat{\beta}_{j2} \end{array}\right) , c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \right) \\ \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) | \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( \left(\begin{array}{c} 0.5 \mu_{j1} \\ \mu_{j2} \end{array}\right), c \left(\begin{array}{c c} 0.5^2 & 0.5 \\ 0.5 & 1 \end{array}\right) \right) \\ &= \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( c \left(\begin{array}{c} 2\hat{\beta}_{j1} + 0.5 \hat{\beta}_{j2} \\ 4\hat{\beta}_{j1} + \hat{\beta}_{j2} \end{array}\right) , c \left(\begin{array}{c c} 0.5^2 & 0.5 \\ 0.5 & 1 \end{array}\right) \right) \\ \beta_{j1} - \beta_{j2}|\left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left(-2\hat{\beta}_{j1}-0.5\hat{\beta}_{j2}, 0.25\right) \end{align*}\]

    \[ P(\beta_{j1}-\beta_{j2} = 0|\hat{\beta}_{j}, \hat{\pi}) = \hat{\pi}_{0} \] \[ lfsr = min\left[P(\beta_{j1}-\beta_{j2} \leq 0|\hat{\beta}_{j}, \hat{\pi}), P(\beta_{j1}-\beta_{j2} \geq 0|\hat{\beta}_{j}, \hat{\pi})\right] \]

    Since the \(S_{j}\)’s diagonal elements are not equal, the \(\beta_{j1}\) and \(\beta_{j2}\) would not have similar magnitude.

  • If the diagonal of \(S_{j}\) are equal, like the case in the mash paper (in the paper, \(s_{jr}\) are all around 0.1) \[ S_{j} = \left(\begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array}\right) \]

    \[\begin{align*} \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) | \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) &\sim N\left(\left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right), S_{j}\right) \\ \left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right) | \hat{\pi} &\sim \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N(0,\left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right)) \\ \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) | \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( \left(\begin{array}{c} \mu_{j1} \\ \mu_{j2} \end{array}\right), \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \left(\begin{array}{c c} 2 & 1 \\ 1 & 2 \end{array}\right)^{-1} \right) \\ &= \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( c \left(\begin{array}{c c} \hat{\beta}_{j1} + \hat{\beta}_{j2} \\ \hat{\beta}_{j1} + \hat{\beta}_{j2} \end{array}\right), c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \right) \\ \beta_{j1} - \beta_{j2}|\left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left(0, 0\right) \end{align*}\]

Standardized effects are strongly positive correlated.

\[ S_{j} = \left(\begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array}\right) \] \[\begin{align*} \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) | \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) &\sim N\left(\left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right), S_{j}\right) \\ \left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right) | \hat{\pi} &\sim \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N(0,D\left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right)D) \end{align*}\] \(\Rightarrow\) \[\begin{align*} D^{-1} \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) | \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) &\sim N\left(D^{-1}\left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right), D^{-1}S_{j}D^{-1}\right) \\ D^{-1} \left(\begin{array}{c} \beta_{j1} \\ \beta_{j2} \end{array}\right) | \hat{\pi} &\sim \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N(0,\left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right)) \end{align*}\] \[\begin{align*} D^{-1} \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) | \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( \left(\begin{array}{c} \mu_{j1} \\ \mu_{j2} \end{array}\right), c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \right) \\ &= \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( c \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) D \left( \begin{array}{c} \hat{\beta}_{1} \\ \hat{\beta}_{2} \end{array} \right), c \left( \begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) \right) \\ \left(\begin{array}{c} \beta_{j1}\\ \beta_{j2} \end{array}\right) | \left(\begin{array}{c} \hat{\beta}_{j1} \\ \hat{\beta}_{j2} \end{array}\right) &\propto \hat{\pi}_{0} \delta_{0} + \hat{\pi}_{1} N\left( c D \left(\begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right) D \left( \begin{array}{c} \hat{\beta}_{1} \\ \hat{\beta}_{2} \end{array} \right), c D\left( \begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array}\right)D \right) \end{align*}\]

\(D^{-1}\left( \begin{array}{c} \beta_{1} \\ \beta_{2} \end{array} \right)\) have similar effect size, but \(\left( \begin{array}{c} \beta_{1} \\ \beta_{2} \end{array} \right)\) have different effect size.

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