scDesign3 marginal distribution for genes (original) (raw)
Introduction
In this tutorial, we explain different forms of the function that can be used when fitting the marginal distribution for each gene.
Notation
The following notations are used:
- \({\mathbf{Y}} = [Y_{ij}] \in \mathbb{R}^{n \times m}\): the cell-by-feature matrix with \(n\) cells as rows, \(m\) features as columns, and \(Y_{ij}\) as the measurement of feature \(j\) in cell \(i\); for single-cell sequencing data, \({\mathbf{Y}}\) is often a count matrix.
- \(\mathbf{X} = [\mathbf{x}_1, \cdots, \mathbf{x}_n]^T \in \mathbb{R}^{n\times p}\): the cell-by-state-covariate matrix with \(n\) cells as rows and \(p\) cell-state covariates as columns; example covariates are cell type, cell pseudotime, and cell spatial locations.
- \(\mathbf{Z} = [\mathbf{b}, \mathbf{c}]\): \(\mathbf{b} = (b_1, \ldots, b_n)^T\) has \(b_i \in \{1, \ldots, B \}\) representing cell \(i\)’s batch, and \(\mathbf{c} = (c_1, \ldots, c_n)^T\) has \(c_i \in \{1, \ldots, C \}\) representing cell \(i\)’s condition.
For each feature \(j=1,\ldots,m\) in every cell \(i=1,\ldots,n\), the measurement \(Y_{ij}\)—conditional on cell \(i\)’s state covariates \(\mathbf{x_i}\) and design covariates \(\mathbf{z}_i = (b_i, c_i)^T\)—is assumed to follow a distribution \(F_{j}( \cdot~|~\mathbf{x}_i, \mathbf{z}_i~;~\mu_{ij}, \sigma_{ij}, p_{ij})\), which is specified as the generalized additive model for location, scale and shape (GAMLSS). The various specifications of \(f_{jc_i}(\cdot)\), \(g_{jc_i}(\cdot)\), and \(h_{jc_i}(\cdot)\) are summarized in the next section. \[\begin{equation} \begin{cases} Y_{ij}~|~\mathbf{x}_i, \mathbf{z}_i &\overset{\mathrm{ind}}{\sim} F_{j}( \cdot~|~\mathbf{x}_i, \mathbf{z}_i~;~\mu_{ij}, \sigma_{ij}, p_{ij})\\ \theta_{j}(\mu_{ij}) &= \alpha_{j0} + \alpha_{jb_i} + \alpha_{jc_i} + f_{jc_i}(\mathbf{x}_i) \\ \log(\sigma_{ij}) &= \beta_{j0}+ \beta_{jb_i} + \beta_{jc_i} + g_{jc_i}(\mathbf{x}_i) \\ \operatorname{logit}(p_{ij}) &= \gamma_{j0} + \gamma_{jb_i}+ \gamma_{jc_i}+ h_{jc_i}(\mathbf{x}_i) \\ \end{cases} \, \end{equation}\]
Summary
| Covariate type | Covariate form | Function form | Explaination | Geometric meaning | Code Example |
|---|---|---|---|---|---|
| Discrete cell type | \(x_i \in \left\{1, \ldots, K_C\right\}\) | \(f_{jc_i}(x_i) = \alpha_{jc_ix_i}\) | Cell type \(x_i\) has the effect \(\alpha_{jc_ix_i}\); for identifiability, \(\alpha_{jc_ix_i} = 0\) if \(x_i = 1\) | One intercept for each cell type | mu_formula = “cell_type” |
| Continuous pseudotime in one lineage | \(x_i \in [0,\infty)\) | \(f_{jc_i}({x}_i) = \sum_{k = 1}^Kb_{jc_ik}(x_{i})\beta_{jc_ik}\) | \(b_{jc_ik}(\cdot)\) is a basis function of cubic spline; \(K\) is the dimension of the basis | A curve along the pseudotime | mu_formula = “s(pseudotime)” |
| Continuous pseudotimes in \(p\) lineages | \(\mathbf{x}_i = (x_{i1}, \ldots, x_{ip})^T \in [0,\infty)^{p}\) | \(f_{jc_i}(\mathbf{x}_i) = \sum_{l = 1}^p \sum_{k = 1}^Kb_{jc_ilk}(x_{il})\beta_{jc_ilk}\) | \(b_{jc_ilk}(\cdot)\) is a basis function of cubic spline; \(K\) is the the dimension of the basis (default \(K=10\)) | One curve along each lineage | mu_formula = “s(pseudotime1, k = 10, by = l1, bs = ‘cr’) + s(pseudotime2, k = 10, by = l2, bs = ‘cr’)”, \(p = 2\) in this case |
| Spatial location | \(\mathbf{x}_i = (x_{i1}, x_{i2})^T \in \mathbb{R}^{2}\) | \(f_{jc_i}(\mathbf{x}_i) = f_{jc_i}^{\operatorname{GP}}(x_{i1}, x_{i2}, K)\) | \(f_{jc_i}^{\operatorname{GP}}(\cdot, \cdot, K)\) is a Gaussian process smoother; \(K\) is the dimension of the basis (default \(K=400\)) | A smooth surface | mu_formula = “s(spatial1, spatial2, bs = ‘gp’, k = 400)” |
| Note: For simplicity, we only show the form of \(f_{jc_i}(\cdot)\) because \(g_{jc_i}(\cdot)\) and \(h_{jc_i}(\cdot)\) have the same form. |