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Table 1 Models considered in simulation studies using penalized regression (with penalty \(p_{\lambda }\)) for a binary confounder \(L_i \in \{0,1\}\)

From: Feature selection and causal analysis for microbiome studies in the presence of confounding using standardization

Model

Objective function

Conditional Std

\(\sum _l\frac{1}{2n_l} \sum _i I(L_i=l) \left( y_i - \beta _0^l - \sum _j{\tilde{A}}_{ij} \beta _j^l\right) ^2 + \sum _{l,j} p_{\lambda _l}(\beta _j^l)\)

Select L

\(\frac{1}{2n} \sum _i \left( y_i - \beta _0 - L_i\beta _\ell - \sum _j {\dot{A}}_{ij} \beta _j\right) ^2 + \sum _j p_{\lambda }(\beta _j) + p_{\lambda }(\beta _\ell )\)

Select L EffMod

\(\frac{1}{2n} \sum _i\left( y_i - \beta _0 - L_i\beta _\ell - \sum _{l,j} I(L_i=l) {\tilde{A}}_{ij}\beta _j^l\right) ^2 + \sum _l\sum _{j} p_{\lambda }(\beta _j^l) + p_{\lambda }(\beta _\ell )\)

Require L

\(\frac{1}{2n} \sum _i \left( y_i - \beta _0 - L_i\beta _\ell - \sum _j {\dot{A}}_{ij} \beta _j\right) ^2 + \sum _j p_{\lambda }(\beta _j)\)

Require L EffMod

\(\frac{1}{2n} \sum _i\left( y_i - \beta _0 - L_i\beta _\ell - \sum _{l,j} I(L_i=l) {\tilde{A}}_{ij}\beta _j^l\right) ^2 + \sum _l\sum _j p_{\lambda }(\beta _j^l)\)

Ignore L

\(\frac{1}{2n} \sum _i \left( y_i - \beta _0 - \sum _j {\dot{A}}_{ij} \beta _j\right) ^2 + \sum _j p_{\lambda }(\beta _j)\)

Ignore L EffMod

\(\frac{1}{2n} \sum _i \left( y_i - \beta _0 - \sum _{l,j} I(L_i=l) {\tilde{A}}_{ij} \beta _j^l\right) ^2 + \sum _l\sum _j p_{\lambda }(\beta _j^l)\)

  1. \({\tilde{A}}_{ij}\) denotes microbiome feature j centered and scaled within each stratum; \({\dot{A}}_{ij}\) denotes microbiome feature j centered and scaled across all observations, regardless of stratum