Model setting

Consider \(n\) independent subjects with measurements \(\{ y_i, \boldsymbol{x}_i, \boldsymbol{z}_i \}_{i=1}^{n}\), where \(\boldsymbol{x}_i = (x_{i1}, \cdots, x_{iq})^{\top}\) and \(\boldsymbol{z}_i = (z_{i1}, \cdots, z_{ip})^{\top}\). Here for subject \(i\), \(y_i\) is the response variable, and \(\boldsymbol{x}_i\) and \(\boldsymbol{z}_i\) are the first and second type of imaging features, respectively. Consider the heterogeneity model: \[\begin{equation}\nonumber y_i = \boldsymbol{x}_i^{\top} \boldsymbol{\beta}_i + \boldsymbol{z}_i^{\top} \boldsymbol{\gamma}_i + \epsilon_i, \ i = 1, \cdots, n, \end{equation}\] where \(\boldsymbol{\beta}_i\) and \(\boldsymbol{\gamma}_i\) are the \(q\)- and \(p\)-dimensional vectors of unknown regression coefficients, respectively, and \(\epsilon_i\) is the random error with \(\text{E}(\epsilon_i) = 0\) and \(\text{Var}(\epsilon_i) = \sigma^2\). Intercept is omitted for the simplicity of notation. We consider linear regression for a continuous response, which matches the data analysis in Section 4. Note that the proposed approach is potentially applicable to other types of response/model. Each subject is flexibly modeled to have its own regression coefficients, and two subjects belong to the same (sub)subgroup if and only if they have the same regression model/coefficients.

Significantly advancing from the existing literature, we consider a more sophisticated heterogeneity structure as sketched in the lower panel of Figure \(\ref{scheme}\) (a), where \(\boldsymbol{\beta}_i\)’s define a ``rough’’ heterogeneity structure with \(K_1\) subgroups, and \(\boldsymbol{\gamma}_i\)’s define a more refined heterogeneity structure with \(K_2\) sub-subgroups. Denote \(\{ \mathcal{G}_{1}^{*}, \cdots, \mathcal{G}_{K_1}^{*} \}\) as the collection of subject index sets of the \(K_1\) subgroups, and \(\{ \mathcal{T}_{1}^{*}, \cdots, \mathcal{T}_{K_2}^{*} \}\) as the collection of subject index sets of the \(K_2\) sub-subgroups. The hierarchy of heterogeneity amounts to a nested structure. That is, there exists a mutually exclusive partition of \(\{1, \cdots, K_2\}\): \(\{\mathcal{H}_{1}, \cdots, \mathcal{H}_{K_1} \}\) satisfying \(\mathcal{G}_{k_1}^{*} = \bigcup_{k_2 \in \mathcal{H}_{k_1}} \mathcal{T}_{k_2}^{*}\), $1 k_1 K_1 $, $1 k_2 K_2 $.

Reguarlized estimation

For simultaneous estimation and determination of the heterogeneity structure, we propose the penalized objective function: \[\begin{equation}\label{obj} \begin{aligned} &Q(\boldsymbol{\beta},\boldsymbol{\gamma}) = \frac{1}{2} \sum_{i=1}^{n} (y_i - \boldsymbol{x}_i^{\top} \boldsymbol{\beta}_i - \boldsymbol{z}_i^{\top} \boldsymbol{\gamma}_i)^2 \\ &~~~~~ + \sum_{1 \leqslant j < m \leqslant n}p \left(\sqrt{\| \boldsymbol{\beta}_j - \boldsymbol{\beta}_m \|_2^2 + \| \boldsymbol{\gamma}_j - \boldsymbol{\gamma}_m \|_2^2 }, \lambda_1 \right) + \sum_{1 \leqslant j < m \leqslant n} p\left(\| \boldsymbol{\beta}_j - \boldsymbol{\beta}_m \|_2, \lambda_2 \right), \end{aligned} \end{equation}\] where \(\boldsymbol{\beta} = (\boldsymbol{\beta}_1^{\top}, \cdots, \boldsymbol{\beta}_{n}^{\top} )^{\top}\), \(\boldsymbol{\gamma} = (\boldsymbol{\gamma}_1^{\top}, \cdots, \boldsymbol{\gamma}_{n}^{\top} )^{\top}\), and \(p(\cdot, \lambda)\) is a concave penalty function with tuning parameter \(\lambda > 0\). In our numerical study, we adopt MCP (Minimax Concave Penalty, ), and note that SCAD (Smoothly Clipped Absolute Deviation Penalty) and some other penalties are also applicable. Consider \((\widehat{\boldsymbol{\beta}}, \widehat{\boldsymbol{\gamma}} )=\underset{ \boldsymbol{\beta},\boldsymbol{\gamma} }{\mathrm{argmin}} \ Q(\boldsymbol{\beta},\boldsymbol{\gamma})\). Denote \(\{\widehat{\boldsymbol{\alpha}}_1 , \cdots, \widehat{\boldsymbol{\alpha}}_{\widehat{K}_1} \}\) and \(\{\widehat{\boldsymbol{\delta}}_1 , \cdots, \widehat{\boldsymbol{\delta}}_{\widehat{K}_2} \}\) as the distinct values of \(\widehat{\boldsymbol{\beta}}\) and \(\widehat{\boldsymbol{\gamma}}\), respectively. Then \(\{ \widehat{\mathcal{G}}_{1}, \cdots, \widehat{\mathcal{G}}_{\widehat{K}_1} \}\) and \(\{ \widehat{\mathcal{T}}_{1}, \cdots, \widehat{\mathcal{T}}_{\widehat{K}_2} \}\) constitute mutually exclusive partitions of \(\{1, \cdots, n\}\), where \(\widehat{\mathcal{G}}_{k_1} = \{i: \widehat{\boldsymbol{\beta}}_i = \widehat{\boldsymbol{\alpha}}_{k_1}, i=1, \cdots, n \}\), \(k_1=1, \cdots, \widehat{K}_1\) and \(\widehat{\mathcal{T}}_{k_2} = \{i: \widehat{\boldsymbol{\gamma}}_i = \widehat{\boldsymbol{\delta}}_{k_2}, i=1, \cdots, n \}\), \(k_2=1, \cdots, \widehat{K}_2\). Collectively, they fully determine the heterogeneity structure.

References:

• Ren, M., Zhang, Q., Zhang, S., Zhong, T., Huang, J. & Ma, S. (2022+). Hierarchical cancer heterogeneity analysis based on histopathological imaging features. Biometrics. doi:10.1111/biom.13426