Using Sufficient Direction Factor Model to Analyze Latent Activities Associated with Breast Cancer Survival

Summary:

High-dimensional gene expression data often exhibit intricate correlation patterns as the result of coordinated genetic regulation. In practice, however, it is difficult to directly measure these coordinated underlying activities. Analysis of breast cancer survival data with gene expressions motivates us to use a two-stage latent factor approach to estimate these unobserved coordinated biological processes.

Compared to existing approaches, our proposed procedure has several unique characteristics. In the first stage, an important distinction is that our procedure incorporates prior biological knowledge about gene-pathway membership into the analysis and explicitly model the effects of genetic pathways on the latent factors. Secondly, to characterize the molecular heterogeneity of breast cancer, our approach provides estimates specific to each cancer subtype. Finally, our proposed framework incorporates sparsity condition due to the fact that genetic networks are often sparse.

In the second stage, we investigate the relationship between latent factor activity levels and survival time with censoring using a general dimension reduction model in the survival analysis context. Combining the factor model and sufficient direction model provides an efficient way of analyzing high-dimensional data and reveals some interesting relations in the breast cancer gene expression data. The computing codes for our simulation studies and application are publicly available at https://github.com/sbaek306/FACTOR.

1. Introdcution

High-dimensional gene expression data often exhibit complicated correlation patterns. These correlation patterns are frequently the result of coordinated gene regulations related to the underlying data generating process. Groups of genetic pathways or processes can be co-regulated through shared biological signals in a cell. In order for cancer cells to continuously maintain disease progression, multiple sources of coordinated genetic regulations are needed for the cells to escape the mechanisms that normally govern their proliferation, survival, and migration (Feng et al., 2018). In practice, however, it is difficult to measure these underlying coordinated genetic activities directly.

In this paper, we aim to estimate the activity levels of these unobserved underlying biological processes from independent sources via a latent factor approach. In the literature, several approaches have been proposed to model latent factors: such as the supervised principal components approach proposed by Bair et al. (2006), the surrogate variable analysis proposed by Leek and Storey (2007), and the partial least squares methods by Nguyen and Rocke (2002); Li and Gui (2004). In addition, methods based on Bayesian approach were introduced in the literature as well, for example, Bernardo et al. (2003), Carvalho et al. (2008) and Lopes and West (2004), and Lucas et al. (2006).

Our proposed procedure has some unique characteristics for analyzing breast cancer data with gene expressions, which make it different from methodologies presented in the existing literature. An important distinction is that our procedure incorporates prior biological knowledge about gene-pathway membership into the analysis. Motivated by the fact that crosstalks between disrupted pathways occurs frequently in cancer (Wang et al., 2016), our proposed approach explicitly models the effects of gene pathways on the latent processes. As a result, our method offers pathway-based estimates and hence enhances the interpretability of the model.

The proposed framework comprises of two stages. In the first stage, we use gene expression measurements and prior biological knowledge about gene-pathway membership to estimate the latent factor activities. In the second stage, we aim to study how the latent processes jointly affect the survival time of the cancer patients.

In order to consider the molecular heterogeneity of breast cancer in the first stage, we specify breast cancer subtype-specific loading matrices. This allows the linear association between the latent factor activity levels and the exhibited gene expression levels to be different for each breast cancer subtype, and hence, it improves the interpretability and flexibility of the model. Second, we impose a sparsity assumption on the loading matrices. The purpose of this assumption is to incorporate the prior biological knowledge that the expression levels of certain genes can be crucial to the underlying latent processes while other genes are irrelevant and hence the corresponding entries in the loading matrix should be shrunk to zero. The sparsity assumption is taken into account in our factor model analysis procedure through incorporating an L₁ penalty in terms of methodology and is achieved via an Alternating Direction Method of Multipliers (ADMM) optimization procedure in terms of computation.

Even with the successful estimation of the pathway activity levels of each patient, it is still not a straightforward task to link these latent activity levels to breast cancer survival. This is due to the unclear functional relation between patient survival time and his/her individual latent activity levels. We hence restrain from imposing any specific survival model and leave the functional form nonparametric. On the other hand, even when dealing with a seemingly small number of latent processes, our estimation is still subject to the curse of dimensionality due to the nonparametric nature of the relation.

In addition, besides the latent factor activity levels, certain covariates are known to directly link to the survival times of breast cancer patients. For example, the highest breast cancer incidence rates for white women are among the group aged 40 and above (DeSantis et al., 2014). We thus need to include the ages of patients as a clinically relevant variable and incorporate it into the model as well. Thus, as a second stage model, we adopt the flexible sufficient dimension reduction approach. We assume the pathways together with age form several linear combinations, and these combinations jointly affect the survival time in a nonparametric way. We will use the data to determine what is the appropriate number of the linear combinations, what are these combinations, and what is the eventual functional form. To further take into account the censoring nature of the survival times, we incorporate the Martingale technique in combination with semiparametric treatment (Zhao et al., 2017) to provide a consistent estimator that is optimal in terms of achieving the smallest possible estimation variability.

2. Methodological Development

2.1. Data structure and modeling

To be specific, we describe the structure of the data considered in this work and our modeling strategy. Although the model is directly motivated by the breast cancer data, it is sufficiently general and can be applied in similar problems as well.

Let the independent and identically distributed (iid) observations (X_i, Y_i, ∆_i, Z_i, W_i), i = 1, … , n, measured from n patients. Here, ${\text{X}}_i\in {\mathcal{R}}^p$ is the vector of the gene expressions of the ith patient measured on p genes. For the breast cancer data that we consider, n = 978 and p = 19149. Let T_i be the time to event and C_i be the censoring time for the ith patient, then Y_i = min(T_i, C_i) and ∆_i = I(T_i < C_i). It is a common practice to classify breast cancer tumors into one of five subtypes (Parker et al. 2009; Cancer Genome Atlas Network 2012), and we use Z_i to represent the breast cancer subtype of patient i. There are five categorical values that Z_i can be in our data: 1 (Basal), 2 (Her2-enriched), 3 (Luminal B), 4 (Luminal A) and 5 (Normal-like). In a more general setting, we can view Z_i as a stratification index with, say K, different categories, and Z_i can be correlated with X_i. Finally, W_i contains additional covariates that are known to affect the time to event T_i. In the breast cancer study, W_i contains age because of its known causal effect on survival (DeSantis et al., 2014).

We first stratify gene expressions X_i based on cancer subtype Z_i, and then model the relationship between X_i and the latent factor activity levels f_i within each stratum using a factor model:

$${\text{X}}_i={\text{B}}_k{\text{f}}_i+{\text{U}}_i\hspace{0.17em}\hspace{0.17em}\text{when}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Z}_i=k,$$ (1)

where ${\text{f}}_i\in {\mathcal{R}}^q$, ${\text{U}}_i\in {\mathcal{R}}^p$ and ${\text{B}}_k\in {\mathcal{R}}^{p\times q}$ is a subtype-specific loading matrix. For identifiability reason and to to keep the interpretation simple, we assume that cov(f_i) = I_q and ${\text{B}}_k^{\text{T}}{\text{B}}_k$ is diagonal for all k = 1, … , K. The above equation associates the independent latent biological processes (F) with the gene expression measurements (X) through the loading matrix B. The optimal number of latent factors (q) can be determined based on cross-validated mean squared error (MSE). In order to study the effects of pathways on latent activities, we incorporate known prior knowledge about gene-pathway membership matrix (G) and set B_k = GV_k. Hence, the above model can be rewritten as

$${\text{X}}_i=\text{G}{\text{V}}_k{\text{f}}_i+{\text{U}}_i\hspace{0.17em}\hspace{0.17em}\text{when}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Z}_i=k,$$ (2)

Here, G is known and G ∈ {−1, 0, 1}^p×m for p genes and m gene sets (m ⩽ p). G_ij = 1(−1) indicates that gene i is part of the j^th geneset and promote (inhibit) geneset activity; G_ij = 0 indicates that gene i is not part of the j^th geneset. In addition, V_k is a subtype-specific m × q loading matrix for each gene set. The j-th column of V_k is the coefficients of a set of pathways associated with the latent variable j. In other words, the columns of V_k are the estimated effects of pathways on the latent factors.

In the second stage, we model the impact of f_i and W_i on T_i through

$$\text{pr}\left(T_i<t|{\text{f}}_i,{\text{W}}_i\right)=h\left(t,{\mathbf{\beta }}^{\text{T}}{\text{f}}_i+{\mathbf{\alpha }}^{\text{T}}{\text{W}}_i\right),$$ (3)

where $\mathbf{\beta }\in {\mathcal{R}}^{q\times d}$ with d < q, $\mathbf{\alpha }\in {\mathcal{R}}^{\dim \left(\text{W}\right)\times d}$, and h is an unspecified smooth conditional cumulative distribution function. The stage 1 model in (1) (2) and the stage 2 model in (3) jointly fully describe our modeling strategy for the breast cancer data analysis.

2.2. Estimation of pathway activity levels in stage 1 model

In the first stage, we estimate individual latent activity levels based on breast cancer subtypes, which results in estimating subtype-specific loading matrices. To this end, we stratify the samples into five strata according to the Z_i values, with the data in the kth stratum having Z_i = k for all i. In each stratum, say stratum k, we first iteratively estimate the subtype-specific loading matrix B_k and the latent factor f_i for the ith patient in the kth stratum. After obtaining the estimates of B_k and f_i, the estimate for pathway loading matrix V_k can be calculated as: V_k = (G^TG)⁻¹G^TB_k. Next, we describe the iterative procedure to obtain B_k and f in detail.

For simplicity of notation, we omit the stratum index k and write the analysis as if there were only one stratum. Let X = (X₁, … , X_n), F = (f₁, … , f_n), U = (U₁, … ,U_n). Hence the factor model (1) is summarized as X = BF + U. We can interpret the relationship as the following. The expression level of X_ij is linearly related to q independent latent activities, plus some random noise captured in U_ij. If we do not impose any additional knowledge on B, then B and f_i can be estimated using the Singular Value Decomposition (SVD) as in the classical factor model. This is equivalent to minimizing the Frobenious norm of X − BF. In our context, we thus naturally extend the SVD to solving ${\min }_{\text{B},\text{F}}{\Vert \text{X}-\text{BF}\Vert }_F^2$ subject to the constraints that B^TB is diagonal and FF^T = I_q. Note that in the case of several strata, we only require B^TB to be diagonal within each stratum. We do not impose additional constraints on the relation between loading matrices from different strata.

In breast cancer studies, some further prior knowledge is available. Certain breast cancer-related genes are more likely to participate in active biological processes than other genes. To incorporate such knowledge, we impose additional sparsity condition on B through incorporating penalty. For example, using Lasso penalty, we solve the minimization problem ${\min }_{\text{B},\text{F}}{n}^{-1}{\Vert \text{X}-\text{BF}\Vert }_F^2+\lambda {\sum }_{i=1}^q{\Vert {\text{B}}_{i-}\Vert }_1$, where B_i− represents the ith column of B with some components that we do not want to shrink to zero excluded. Since the correlation among genes averaged 0.08 only, we simply used the L₁ penalty. Alternatively, the elastic net penalty can be used if such correlation is a concern. In terms of computation, the detailed algorithm is provided in Supporting Information A.

2.3. Estimation of survival model in stage 2 model

We now establish the relationship between breast cancer survival and latent factor activity levels and other covariates that are linked to breast cancer survival. To this end, we treat ${\text{f}}_i^{\ast }={\left({\text{f}}_i^{\text{T}},{\text{W}}_i\right)}^{\text{T}}$ in the second stage model as covariates, where f_i is individual latent factor activity levels and W_i contains age. This process results in an uncertainty that affects stage 2 model. Jiang et al. (2019) takes this uncertainty into account in the final variance analysis. Letting $\mathbf{\gamma }={\left({\mathbf{\beta }}^{\text{T}},{\mathbf{\alpha }}^{\text{T}}\right)}^{\text{T}}$, then we can write (3) succinctly as

$$\text{pr}\left(T_i<t|{\text{f}}_i^{\ast }\right)=h\left(t,{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right),$$ (4)

where $\mathbf{\gamma }\in {\mathcal{R}}^{\left\{q+\dim \left({\text{W}}_i\right)\right\}\times d}$. In the estimation of γ, all latent factors and ages from all n samples are considered as covariates. Hence, we estimated γ using all individuals together, not separately for each subtype. After estimating γ, we estimated the cumulative hazard function based on each subtype in order to investigate possible different functional dependences. Thus, we are able to obtain a specific survival function based on each cancer subtype as follows:

$$\text{pr}\left(T_i<t|{\text{f}}_i^{\ast }\right)=h_k\left(t,{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{when}\hspace{0.17em}\hspace{0.17em}{Z}_i=k,$$

where h_k(⸳) is the survival function for the kth stratum.

The second stage model (4) merits some remarks. First, we employ d linear summaries by the d columns of γ, which gives the flexibility for the association among components of f_i and W_i. Second, by leaving the link function h(⸳) in (4) unspecified, we can avoid problems that stem from the mis-specification of a model. In order for γ to be identifiable, we parameterize it so that the upper d × d block matrix of γ is the identity matrix I_d. For convenience, we write ${\text{f}}^{\ast }={\left({\text{f}}_u^{\ast \text{T}},{\text{f}}_l^{\ast \text{T}}\right)}^{\text{T}}$, where ${\text{f}}_u^{\ast }\in {\mathcal{R}}^d$ and ${\text{f}}_l^{\ast }\in {\mathcal{R}}^{\left\{q+\dim \left({\text{W}}_i\right)\right\}d}$.

Combining semiparametric treatment and martingale process, Zhao et al. (2017) proposed an estimation procedure for censored survival data that is consistent, asymptotically normal and semiparametric efficient, without imposing any assumption on the conditional distribution of the censoring time given covariates. We use this procedure to obtain the semiparametric efficient estimator by solving the following estimating equations

$$\sum _{i=1}^n{\Delta }_i\frac{{\widehat{\lambda }}_1\left(Y_i,{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right)}{\widehat{\lambda }\left(Y_i,{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right)}\otimes \left[{\text{f}}_{li}^{\ast }-\frac{\widehat{E}\left\{{\text{f}}_{li}^{\ast }{U}_i\left(Y_i\right)|{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right\}}{\widehat{E}\left\{U_i\left(Y_i\right)|{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right\}}\right]=\text{0},$$ (5)

where $\widehat{\lambda }(\cdot )$ is the estimated hazard function, ${\widehat{\lambda }}_1(\cdot )$ is the derivative of $\widehat{\lambda }(\cdot )$ with respect to ${\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }$, and U (t) = I(Y ⩾ t). More specifically,

$$\widehat{\lambda }\left(Y,{\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right)=\sum _{i=1}^n{K}_b\left(Y_i-Y\right)\frac{{\Delta }_i{K}_h\left({\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }-{\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right)}{{\sum }_{j=1}^{n}I\left(Y_j⩾Y_i\right)K_h\left({\mathbf{\gamma }}^T{\text{f}}_j^{\ast }-{\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right)},$$ (6)

$${\widehat{\lambda }}_1\left(Y,{\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right)=\partial \widehat{\lambda }\left(Y,{\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right)/\partial \left({\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right),$$ (7)

where K(⸳) is a kernel function, $K_h(\cdot )=K\left(\cdot /h\right)/h$ and h is a bandwidth. ${\text{K}}_h^{\text{'}}\left(\text{v}\right)$ is the first derivative of K_h(v) with respect to v, and b is also a bandwidth. Similarly, using kernel estimation, we let

$$\widehat{E}\left\{{\text{f}}_{li}^{\ast }{U}_i\left(Y_i\right)|{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right\}=\frac{{\sum }_{j=1}^n{K}_h\left({\mathbf{\gamma }}^{\text{T}}{\text{f}}_j^{\ast }-{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right){\text{f}}_{lj}^{\ast }I\left(Y_j⩾Y_i\right)}{{\sum }_{j=1}^n{K}_h\left({\mathbf{\gamma }}^{\text{T}}{\text{f}}_j^{\ast }-{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right)},$$ (8)

$$\widehat{E}\left\{U_i\left(Y_i\right)|{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right\}=\frac{{\sum }_{j=1}^n{K}_h\left({\mathbf{\gamma }}^{\text{T}}{\text{f}}_j^{\ast }-{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right)I\left(Y_j⩾Y_i\right)}{{\sum }_{j=1}^n{K}_h\left({\mathbf{\gamma }}^{\text{T}}{\text{f}}_j^{\ast }-{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right)},$$ (9)

where $E\left\{U_i\left(Y_i\right)|{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right\}\equiv E\left\{U_i\left(t\right)|{\mathbf{\gamma }}^{\text{T}}{\text{f}}_i^{\ast }\right\}{|}_{t=Y_i}$. Inserting (6), (7), (8) and (9) into (5), we can solve the estimating equation and the solution is the semiparametric efficient estimator of γ. More details about the derivation of the semiparametric efficient estimator can be found in Supporting Information B.

The nonparametric estimators given in (6), (7), (8) and (9) require bandwidth selection. As noted in Assumption A2 in Supporting Information C, a wide range of bandwidths can be used, and it does not affect the asymptotic property, i.e., the final estimator $\widehat{\mathbf{\gamma }}$ is insensitive to the bandwidth selection. For example, we can select the bandwidths by taking the sample size n to suitable power for which Assumption A2 holds, and multiply a constant, such as the variance of the covariates, for proper scaling. We refer to Zhao et al. (2017) for more specific guidelines on bandwidth selection.

Zhao et al. (2017) showed the consistency and asymptotic properties of the efficient estimator obtained by solving the estimating equations (5) under Assumptions A1-A7 in Supporting Information C. In summary, $\widehat{\mathbf{\gamma }}-{\mathbf{\gamma }}_0=o_p\left(1\right)$ and $\sqrt{n}\left(\widehat{\mathbf{\gamma }}-{\mathbf{\gamma }}_0\right)\to \mathcal{N}\left(\text{0},{\left[E\left\{{\text{S}}_{\text{eff}}^{\otimes 2}\left(\mathrm{\Delta },Y,{\text{f}}^{\ast }\right)\right\}\right]}^{-1}\right)$ in distribution, where γ₀ is the true parameter, and S_eff is the efficient score with its specific form given in Supporting Information C, and ${\text{a}}^{\otimes 2}\equiv \text{a}{\text{a}}^{\text{T}}$ for any vector or matrix a. These asymptotic properties are provided through Theorems 1 and 2 in Supporting Information C for convenience. See Zhao et al. (2017) for their proofs.

3. Simulations

3.1. Simulation Settings

Before applying our modeling and estimation methods to analyze the breast cancer data, we first carry out simulations to evaluate the finite sample performance of the methods. We set up three simulation scenarios. The three simulation scenarios vary in the number of strata (for cancer subtypes, K), the number of variables (p), the variance-covariance matrix (Σ) of p random variables, the number of latent variable (q), sample size (n), and the proportion of censoring.

Scenario 1

In scenario 1, we consider two strata (K = 2) for the stage 1 model. We fix Z_i = 1 for i = 1, … , n₁, and Z_i = 2 for i = n₁ + 1, … , n₁ + n₂. The sample size for each stratum is n₁ = n₂ = 250, resulting in n = 500. We let p = 200, q = 4 and d = 1. For each i = 1, … , n, we simulate f_i from the q-dimensional multivariate standard normal distribution and further generate a p × 1 random vector L_i as the following. When Z_i = 1, we simulate L_i from the p-dimensional multivariate standard normal distribution. When Z_i = 2, we simulate L_i from a p-dimensional multivariate normal distribution with mean 0 and variance-covariance matrix Σ_L, where the (i, j) element of Σ_L is 0.5^|i−j| for 1 ⩽ i, j ⩽ p.

To construct the factor loading matrix B_k for each stratum from (1), we perform eigen decomposition on the matrix LL^T, where L = (L₁, … , L_n)^T. We let E₁ be the n × q orthogonal matrix formed by the eigenvectors corresponding to the q largest eigenvalues of LL^T, and let E₂ be the n × q orthogonal matrix formed by the eigenvectors corresponding to the (q + 1)th to the 2qth largest eigenvalues of LL^T. We then form B_k = n^−1/2L^TE_k, for k = 1, 2. Because the eigenvectors corresponding to a symmetric matrix are orthogonal to each other, this construction ensures that ${\text{B}}_k^{\text{T}}{\text{B}}_k$ is diagonal.

We further generate U_is from a p-dimensional multivariate normal distribution with mean 0 and variance-covariance matrix 0.5I_p and then form X_i = B_kf_i + U_i when Z_i = k, for k = 1, 2, i = 1, … , n. To further generate data in the stage 2 model, we set dim(W_i) = 1, and generate W_i from a standard normal distribution. We then form a new covariate ${\text{f}}_i^{\ast }={\left({\text{f}}_i^T,W_i\right)}^{\text{T}}\in {\mathcal{R}}^{q+1}$. We now generate event times from

$$T=\frac{\left|{\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }\right|}{2+{\left({\mathbf{\gamma }}^{\text{T}}{\text{f}}^{\ast }+1.5\right)}^2}+0.5ϵ,$$

where ϵ is uniformly distribution on (0, 1), and γ = (1, 0, 0.3, −0.3, −1)^T. We further generate the covariate dependent censoring times using $C_i=\mathrm{\Phi }\left(2f_{i2}^{\ast }+2f_{i3}^{\ast }\right)+c_1$, where c₁ is used to control the proportion of censoring.

Scenario 2

In scenario 2, we provide the numerical performance under more general setting assuming five strata and eight latent factors, different sizes of (n, p), and dim(W_i) = 4. The detailed setting for scenario 2 are relegated to Supporting Information D and E.

Scenario 3

We design simulation scenario 3 to resemble the real data in Section 4, i.e., K = 5, p = 19, 149, q = 8, dim(W_i) = 1, and d = 1. Unlike Simulation 2, we treat the estimated loading matrices from the real data set in Section 4 as the true loadings. The specific setting and the performance for Simulation 3 are given in Supporting Information D and E.

3.2. Simulation Results

We first evaluate the performance of the stage 1 model and compare our proposed stage 1 model to two other closely related methods: supervise principal component (Superpc) (Bair et al., 2006; Bair and Tibshirani, 2004) and surrogate variable analysis (SVA) (Leek and Storey, 2007) using data generated with simulation scenario 1. Except that the current version of Superpc only allow q to be at most 3, hence we set q = 3 in this comparison. For Superpc and SVA, we used the provided software programs in R/Bioconductor (Superpc, and sva). To account for K cancer subtypes, for Superpc, we stratified the samples into K strata and perform analysis with default parameter values. For SVA, cancer subtypes were treated as covariates adjusted in the model. We compare the estimation of F and calculate the average absolute errors (AAE):${\Vert {\mathcal{P}}_{\widehat{\text{F}}}-{\mathcal{P}}_{\text{F}}\Vert }_{\text{1}}$. Here the projection matrix is calculated as ${\mathcal{P}}_{\text{A}}=\text{A}{\left({\text{A}}^{\text{T}}\text{A}\right)}^{-1}{\text{A}}^{\text{T}}$ for any generic matrix A. The AAEs of our proposed approach, Superpc, and SVA are presented in Table 1. The result suggests that our proposed approach based on iterative ADMM outperforms Superpc and SVA and achieves smallest errors in estimating F.

Table 1:

The average absolute errors (AAE) for the estimation of the latent factors (F) using our proposed approach (Iterative ADMM), Superpc, and SVA using data generated from simulation scenario 1 with q = 3. AAE is calculated as ${\Vert {\mathcal{P}}_{{\widehat{\text{B}}}_k}-{\mathcal{P}}_{{\text{B}}_k}\Vert }_1$. Here the projection matrix is calculated as ${\mathcal{P}}_{\text{A}}=\text{A}{\left({\text{A}}^{\text{T}}\text{A}\right)}^{-1}{\text{A}}^{\text{T}}$ for any generic matrix A.

Censoring Rate	0%	20%	40%
Iterative ADMM	0.0038	0.0042	0.0042
Superpc	0.3199	0.3225	0.3279
SVA	0.0106	0.0115	0.0109

In addition, we report the performance of the estimation of F and B through calculating the corresponding AAEs for our proposed approach in all three simulation scenarios. For stage 2 model, we provided the difference of the true and estimates indices via AAE₃, which is the average of $n^{-1}{\sum }_{i=1}^n{\Vert {\widehat{\mathbf{\beta }}}^{\text{T}}{\widehat{\text{f}}}_i-{\mathbf{\beta }}^{\text{T}}{\text{f}}_i\Vert }_1$. See Table S.5 in Supporting Information E for the results.

Finally, we evaluate the performance of our proposed methods including both stage 1 and stage 2 procedure, and provide the final coefficient estimates, absolute biases, and sample standard deviations in Table S.1 for scenario 1, S.2 for scenario 2, and Table S.3 for scenario 3, respectively. For scenario 1 and 2, we consider three different censoring rates: no censoring, 20%, and 40% censoring. Across all simulations, the estimators have very small biases. It is not surprising that the variability of the estimators under no censoring is in general smaller compared to the cases with censoring. Further as p and n increase in Simulation 2, the estimation performance improves in terms of the Euclidean distances $\Vert \widehat{\mathbf{\gamma }}-\mathbf{\gamma }\Vert$, as shown in Figure S.1. The results in Simulation 3 (Table S.3) continue to have very small bias, while the standard errors are larger due to the small sample sizes in some individual strata.

4. Application

We now apply the two-stage modeling and estimation procedure to the breast cancer survival data. We downloaded breast cancer data from The Cancer Genome Atlas (TCGA) project via the ICGC controlled data portal (https://dcc.icgc.org/releases/release26/Projects/BRCA-US). The data contains 978 female patients with information for gene expression, age, and survival/follow-up time. Following Parker et al. (2009), we categorize the patients in the study into five subtypes: Basal (185, 18.9%), Her2-enriched (107, 10.9%), Luminal B (428, 43.8%), Luminal A (248, 25.4%), and Normal-like (10, 1.0%). For the subtype classification, we employ the prediction analysis of microarray (PAM) algorithm (Parker et al., 2009). Each individual is classified into one among five subtypes using the 50-gene classifier (PAM50). R package ‘genefu’ is available to carry out this algorithm. Among the 978 patients, only 100 events are observed, while the remaining 878 are censored, and hence the censoring rate is about 89.8%, and there is no left- or right-truncation.

For each patient, expression levels at 19,149 genes are measured, i.e., p = 19, 149. In the following analysis, the optimal number of latent factors q was estimated to be 8 based on 10-fold cross-validated MSE per latent factor $\left(\frac{{\Vert \text{X}-\widehat{\text{X}}\Vert }_F^2}{q}\right)$. Inspired by the fact that genetic networks (Gardner et al., 2003) are sparse, we incorporate sparsity condition of B using Lasso penalty, except for 132 breast cancer related-genes reported in the KEGG database. The list of these 132 genes can be found in Table S.4 in Supporting Information E.

We incorporate prior information of gene-pathway membership into the stage 1 analysis. Using the KEGG database, 335 genesets are downloaded, and 91 genesets specific to human diseases are excluded. We use the term geneset (a collection of functionally related genes) and pathway interchangeably throughout the texts. The KEGG pathway database provides graphical diagrams indicating various interactions between genetic molecules. We specify G ∊ {−1, 0, 1}^p×m based on the above information. G_ij = 1(−1) indicates that gene i is part of the j^th geneset and promote (inhibit) geneset activity; G_ij = 0 indicates that gene i is not part of the j^th geneset. For genes that do not belong to any known gene set, a new gene set that contains itself is constructed. Using the ICGC Breast cancer data, m = m′ + m′′=244+12095=12339 (m′=244 known KEGG genesets and m′′=12,095 genesets with genes that do not map to any known KEGG genesets).

We performed the estimation procedure following the description in Section 2. After obtaining the estimate of B_k, we calculate the estimate of ${\text{V}}_k\left(\in {\mathcal{R}}^{m\times q}\right)$ as V_k = (G^TG)⁻¹G^TB_k, and we refer to V_k as the pathway loading matrix. To explore the relationship between known pathways and the latent factors, we examined the subsets of V (named as V′) which describe the loading values of 244 known genesets associated with the eight latent factors. Using the data from Her2-enriched breast cancer, we remove the pathways whose loading values are larger than 5% quantiles and less than 95% quantiles for all latent factors and plot the factor loadings for the remaining 21 pathways in Figure 1.

Figure 1:

Heatmap of top pathways loaded on eight latent factors in patients with Her-2 enriched breast cancer Her-2 enriched.

The top pathways associated with the eight latent factors are presented in Table 2. Several pathways listed in Table 2 are reported to play pivotal roles in breast cancer metastasis in the literature including: fatty acid metabolism, adherens junction, ribosome, antigen processing and presentation, ErbB and relaxin signaling pathway (Monaco, 2017; Burris, 2004; Eroles et al., 2012; Penzo et al., 2019; Zhu et al., 2014; Khoury et al., 2001; Radestock et al., 2008).

Table 2:

Top pathways associated with eight latent factors in patients with Her2-enriched breast cancer.

Latent factor	Factor loading	Top pathways
Factor 1	7.85	Biotin metabolism
	7.24	Proximal tubule bicarbonate reclamation
	−17.86	Relaxin signaling pathway
	−18.58	Ribosome
Factor 2	7.56	Relaxin signaling pathway
	6.20	Protein digestion and absorption
	−3.24	Antigen processing and presentation
	−7.99	Ribosome
Factor 3	2.43	Adherens junction
	1.99	Biotin metabolism
	−1.70	Fatty acid biosynthesis
	−1.79	Relaxin signaling pathway
Factor 4	1.35	Adherens junction
	1.32	ErbB signaling pathway
	−0.83	PPAR signaling pathway
	−3.65	Antigen processing and presentation
Factor 5	1.30	Endocrine and other factor-regulated calcium reabsorption
	1.09	Hematopoietic cell lineage
	−0.96	Neomycin, kanamycin and gentamicin biosynthesis
	−1.44	Relaxin signaling pathway
Factor 6	1.24	PPAR signaling pathway
	1.16	Biosynthesis of unsaturated fatty acids
	−1.34	Relaxin signaling pathway
	−5.06	Ribosome
Factor 7	1.60	Adherens junction
	1.44	ErbB signaling pathway
	−1.55	Biotin metabolism
	−2.21	Hematopoietic cell lineage
Factor 8	1.52	Fatty acid biosynthesis
	0.85	ErbB signaling pathway
	−0.85	Antigen processing and presentation
	−0.88	Biotin metabolism

For example, the results for latent factor 4 in Table 2 and Figure 1 suggest this latent process involves pathway crosstalk between adherens junction, ErbB signaling, PPAR signaling and antigen processing and presentation. Similar collaboration patterns of deregulated pathways are also reported in a microarray gene expression study by Zhu et al. (2014). The study by Khoury et al. (2001) reported that ErbB signal pathway works together with cell-cell junction activity and alters cell motility. PPAR signaling transduction and immuno-inflammatory response (antigen processing and presentation) activities are advantageous for malignant cells to stay alive. In addition, HIF1 signaling pathway indicated in the fourth column in Figure 1 also is found to be a major regulator of ErbB2 that regulates malignant cells from anoikis and metabolic stress caused by decreased matrix adhesion (Whelan et al., 2013). Similarly, the corresponding heatmaps of top pathways in V′ for Basal, Luminal B, Luminal A, and Normal-like subtypes are provided in Figure S.2 to S.5 of Supporting Information E.

In the stage 2 model, we incorporated the latent factor activities levels and age as the covariate W in the second stage model due to its potential association with breast cancer survival. In addition, the dimension of the reduced space d is selected via validated information criterion (VIC), where the candidate dimension at which the VIC value is minimized is chosen as d (Ma and Zhang, 2015). Specifically, for the breast cancer data, the VIC value at d = 1 is 62.05, whereas the VIC values are all greater than 124.27 when d ⩾ 2. Thus we select d = 1. The tuning parameters (λ and ρ) in stage 1 was selected via a tenfold cross-validation, where 90% of observations were randomly selected as training sample and the rest 10% as the testing sample.

We estimated the variability of the estimates through 1,000 bootstrap samples and provide the final results in Table 3, showing that all eight latent activity levels are statistically significant in terms of their relation with survival time, with p-values ranging from 0.0045 to 0.0192. However, having adjusted for the latent activity levels, age is no longer a significant factor, with its p-value more than 0.5. In addition, our results illustrate that the latent factors 2 and 5 would have effects on the time to event T in the opposite direction to the other factors. Since we selected d = 1, we set γ₁ in (4) to be 1 for identifiablility of γ. Relative to the first latent factor, the magnitude of all other factors affecting the time to event varied from 1.7 to 2.5. For example, γ₄ = 1.9, meaning that the fourth latent factor has almost twice impact on the cumulative hazard compared to the first latent factor, and it has the same direction as the first latent factor. Furthermore, for example, since the coefficient of the latent factor 3 from Table 3 is positive; we can expect that the cumulative hazard would decrease with increasing value of the latent factor 3 while holding all other latent factor activity levels fixed.

Table 3:

Analysis of the breast cancer data with censoring. Estimates (est) are obtained from the two-stage model. Sample mean of estimates (mean), sample median of estimates (median), sample standard deviation (std) and sample median absolute deviation (mad) are calculated based on 1000 bootstrap samples. The pathway activity level indices in the second stage survival model are γ_i, i = 2, … , 8, and γ₉ is the index corresponding to age.

	γ2	γ3	γ4	γ5	γ6	γ7	γ8	γ9
est	−2.4591	2.5606	1.9657	−2.1220	2.0584	1.8624	1.7060	0.0878
mean	−2.5368	2.6165	1.9785	−2.1641	2.0389	1.8670	1.7519	0.0568
median	−2.5151	2.5929	1.9345	−2.1199	1.9645	1.8133	1.7360	0.0312
std	0.8940	0.9278	0.7769	0.8572	0.7588	0.7975	0.7378	0.6246
mad	0.8305	0.8353	0.7752	0.8088	0.8069	0.8089	0.7627	0.7770
p-value	0.0045	0.0048	0.0109	0.0116	0.0072	0.0192	0.0176	0.9275

The estimated hazard function $\widehat{\mathrm{\Lambda }}\left(t,{\mathbf{\beta }}^{\text{T}}\text{f}+{\mathbf{\alpha }}^{\text{T}}\text{W}\right)$ is plotted in Figure 2 for four cancer subtypes including Basal, Her2-enriched, Luminal B and Luminal A. Because no terminal events were observed in the Normal-like cancer patients, we do not provide results for this cancer subtype. The upper-left panel in Figure 2 illustrates the plot of $\widehat{\mathrm{\Lambda }}$ as a function of t while the pathway activity levels are fixed at the respective mean value index value ${\widehat{\mathbf{\beta }}}^{\text{T}}{\overline{\text{f}}}_i+{\widehat{\mathbf{\alpha }}}^{\text{T}}{\overline{\text{W}}}_i$ for each cancer subtype. Compared to other cancer subtypes, the cancer subtype Her2-enriched shows much higher estimated cumulative hazard at almost all times, suggesting a much high risk of an event. The other three panels contain plots of $\widehat{\mathrm{\Lambda }}$ as a function of the index ${\mathbf{\beta }}^{\text{T}}\text{f+}{\mathbf{\alpha }}^{\text{T}}\text{W}$ while t is fixed at 600, 1200 and 3000 respectively. These three plots illustrate that the cumulative hazard function is generally a decreasing function of the index ${\mathbf{\beta }}^{\text{T}}\text{f+}{\mathbf{\alpha }}^{\text{T}}\text{W}$. In addition, we provide contour plots of $\widehat{\mathrm{\Lambda }}$ in Figure 3 as a function of both t and the index ${\mathbf{\beta }}^{\text{T}}\text{f+}{\mathbf{\alpha }}^{\text{T}}\text{W}$ for the four cancer subtypes. The numbers in the plots are the values of $\widehat{\mathrm{\Lambda }}$ on the contour. It shows that Her2-enriched cancer subtype has relatively high cumulative hazard.

Figure 2:

Plot of the estimated cumulative hazard functions $\widehat{\mathrm{\Lambda }}$ for four cancer subtypes. Upper-left panel: $\widehat{\mathrm{\Lambda }}$ as a function of t while the index ${\widehat{\mathbf{\beta }}}^{\text{T}}\text{f}+{\widehat{\mathbf{\alpha }}}^{\text{T}}\text{W}$ is fixed at its sample mean. Other panels: $\widehat{\mathrm{\Lambda }}$ as a function of ${\mathbf{\beta }}^{\text{T}}\text{f+}{\mathbf{\alpha }}^{\text{T}}\text{W}$ while t is fixed at 600 (upper-right), 1200 (lower-left) and 3000(lower-right).

Figure 3:

Contour plots of $\widehat{\mathrm{\Lambda }}$ as a function of ${\mathbf{\beta }}^{\text{T}}\text{f+}{\mathbf{\alpha }}^{\text{T}}\text{W}$ and t for each cancer subtype, which are Basal, Her2-enriched, Luminal B, and Luminal A as labeled.

5. Discussion

We have adopted a framework for analyzing high-dimensional breast cancer survival data by combining a factor model and flexible sufficient direction method. There are several features and corresponding benefits. First, for different cancer subtypes, we employ a stratified approach using a latent factor model, i.e., the estimation of factor and loading matrix hinges upon each cancer subtype. Second, we incorporate the prior sparsity knowledge on the factor loading matrix using L₁ penalty, which helps to reduce the number of none zero elements to be estimated in the loading matrix. Third, we incorporated prior gene-pathway membership information so that the pathway-latent factor relationships can be characterized via the pathway loading matrices V_k. Finally, we employed a very flexible general index model in our second stage model which only requires minimal assumptions. This is important because when we form a model using latent factors as covariates, it can be more susceptible to model misspecification.

In the stage one model, V_k is calculated as V_k = (G^TG)⁻¹G^TB_k To ensure invertibility, it is recommended that the pathways defined in G contain genesets that are not linear combinations of other smaller pathways. In some special situations, a generalized inverse can be applied for estimating V_k(Glazko and Emmert-Streib, 2009).

In this study, we used the cross-validated MSE/q to choose the optimal number of latent factors (q). A large body of literature provides many estimation methods for determining the number of factors (Ahn and Horenstein 2013; Bai and Ng 2002; Lam and Yao 2012; Luo and Li 2016), which can be easily adapted into our proposed approach.

Lastly, we would like to point out that research on dimension reduction for p > n is much needed. It will provide alternative ways of analyzing high-dimensional data by applying dimension reduction technique to p > n case more directly than the method introduced in this work.