Controlling mixed directional false discovery rate in multidimensional decisions with applications to microarray studies

Abstract

Time-course microarray experiments harvested samples at several time points. To reveal the dynamic gene expression changes over time, we need to identify the significant genes and detect the patterns of gene expressions, which may bring directional errors. Guo et al. (Biometrics 66(2):485–492, 2010) introduced a mixed directional false discovery rate (mdFDR) controlled procedure, which controls the sum of expected proportions of Type I and Type III errors among all rejections. In this paper, we develop weighted p value procedures for mdFDR control and give out some sufficient conditions to assure the (asymptotic) mdFDR control. Some weights and their estimators are illustrated to satisfy the sufficient conditions. The proposed weighted p value procedures are compared with the existing method by extensive simulations. Based on the proposed weighted p values procedure, we provide multiple CIs which control the false coverage-statement rate (FCR). We use the proposed methods to analyze the time-course microarray data studied in Lobenhofer et al. (Mol Endocrinol 16:1215–1229, 2002). Most of our findings are the same as those obtained by the existing method. In addition, we identify some other important genes, such as CDKN3 and NQO1.

Appendix

Proof of Theorem 1

We show the conclusion to be true following Guo et al. (2010). Without loss of generality, we suppose the non-zero components of ${\varvec{\mu }}_i=(\mu _{i1},\ldots ,\mu _{iq})$ are all positive. Then

$$\begin{aligned} \begin{aligned} {\text{ m }dFDR}&\le \sum _{r=1}^m\sum _{j=1}^q\frac{1}{r}\sum _{i=1}^m\bigg ( I(\mu _{ij}=0)Pr\left\{ P_{ij}\le \frac{w_{ij}r}{mq} \alpha , R=r \right\} \\&\quad +I(\mu _{ij}>0)Pr\left\{ P_{ij}\le \frac{w_{ij}r}{mq} \alpha , R=r, T_{ij}<0 \right\} \bigg )\\&\le \sum _{r=1}^m\sum _{j=1}^q\frac{1}{r}\sum _{i=1}^m\bigg ( I(\mu _{ij}=0) +\frac{1}{2} I(\mu _{ij}>0)\bigg )Pr\left\{ R^{(-i)}=r-1 \right\} \frac{w_{ij}r}{mq} \alpha \\&=\sum _{j=1}^q\sum _{i=1}^m\bigg ( I(\mu _{ij}=0)+\frac{1}{2} I(\mu _{ij}>0)\bigg ) \frac{w_{ij}}{mq} \alpha =\alpha , \end{aligned} \end{aligned}$$

where $I(\cdot )$ is the indicator function and $R^{(-i)}$ is the number of rejections not including the rejection of $H_i$ in the BH step-up procedure with $P_{ij}$ replaced by 0. Note that $P_{ij}$ replaced by 0 leads to $P_{i}=0$. The proof is completed. $\square$

Proof of Theorem 2

Without loss of generality, we suppose the non-zero components of ${\varvec{\mu }}_i$ are all positive. Then

$$\begin{aligned} \begin{aligned} mdFDR&\le \sum _{r=1}^m\sum _{j=1}^q\frac{1}{r}\sum _{i=1}^m\bigg ( I(\mu _{ij}=0)Pr \{P_{ij}\le \frac{ \widehat{w}_{ij}r}{mq} \alpha , R=r \}\\&\quad +I(\mu _{ij}>0)Pr\left\{ P_{ij}\le \frac{ \widehat{w}_{ij}r}{mq} \alpha , R=r, T_{ij}<0 \right\} \bigg )\\&\le \sum _{r=1}^m\sum _{j=1}^q\frac{1}{r}\sum _{i=1}^m\bigg ( I(\mu _{ij}=0)Pr\{P_{ij} \le \frac{ \widehat{w}_{ij,-ij}r}{mq} \alpha , R^{(-i)}=r-1 \}\\&\quad +I(\mu _{ij}>0)Pr\left\{ P_{ij}\le \frac{ \widehat{w}_{ij,-ij}r}{mq} \alpha , R^{(-i)}=r-1, T_{ij}<0 \right\} \bigg ) \\&=\sum _{r=1}^m\frac{1}{mq}\sum _{i=1}^m\sum _{j=1}^q E_{\mathbf{P}^{(-ij)}}\widehat{w}_{ij,-ij} \alpha \big [ I(\mu _{ij}=0)+\frac{1}{2} I(\mu _{ij}>0)\big ]\\&\quad Pr(R^{(-i)}=r-1|\mathbf{P}^{(-ij)}) \\ {}&=\sum _{j=1}^q\sum _{i=1}^m E_{\mathbf{P}^{(-ij)}}\widehat{w}_{ij,-ij}\frac{1}{2mq} \big [ I(\mu _{ij}=0)+ 1\big ]\alpha \le \alpha , \end{aligned} \end{aligned}$$

where $\mathbf{P}^{(-ij)}$ is the collection of $P_{i^{\prime }j^{\prime }}, i^{\prime }=1,\ldots ,m, j^{\prime }=1,\ldots ,q,$ excluding $P_{ij}$.

The proof is completed. $\square$

Proof of Theorem 3

Without loss of generality, we assume $H_{ij}, i=1,\ldots ,m_{0j},$ to be true for each j. Note that, for $i=1,\ldots ,m_{0j},$

$$\begin{aligned} \begin{aligned} \sum _{i^{\prime }=1,i^{\prime }\ne i}^{m_{0j}}I(P_{i^{\prime }j}> \lambda )+1\sim Bin(m_{0j}-1,1-\lambda ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{(m+m_{0j})(1-\lambda )}{m(1-\lambda )+(1-\lambda )\widehat{\pi }_{0j,-ij}}\le \frac{(m+m_{0j})}{m+(1-\lambda )\widehat{\pi }_{0j,-ij}}, \end{aligned} \end{aligned}$$

where Bin(a, b) is the binomial distribution with the parameter (a, b). Suppose $m_{0j}>0$, then we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\sum _{i,j}(1+I(\mu _{ij}=0))E_{\mathbf{P}^{(-ij)}}\widehat{w}^a_{ij,-ij}\le E_{\mathbf{P}^{(-1j)}}\frac{m+m_{0j}}{1+\widehat{\pi }_{0j,-1j}}\\&\quad \le E_{(\mathbf{P}^{(-1j)},DU)}\frac{m+m_{0j}}{1+\widehat{\pi }_{0j,-1j}}\\&\quad \le m E_{(\mathbf{P}^{(-1j)},DU)}\frac{(m+m_{0j})(1-\lambda )}{m(1-\lambda )+(1-\lambda )m\widehat{\pi }_{0j,-1j}}\\&\quad \le m E_{(\mathbf{P}^{(-1j)},DU)}\frac{m+m_{0j}}{m+(1-\lambda )m\widehat{\pi }_{0j,-1j}}\\&\quad \le m E_{(\mathbf{P}^{(-1j)},DU)}\frac{m_{0j}}{(1-\lambda )m\widehat{\pi }_{0j,-1j}}\le m, \end{aligned} \end{aligned}$$

(7.1)

where the subscript DU of the expectation means that it is calculated under the Dirac-uniform configuration, which assumes that the p values corresponding to the false null hypotheses are 0 and the p values corresponding to the true null hypotheses are i.i.d as U(0, 1). The last inequality in Equation (7.1) is true by Sarkar et al. (2012) and the last inequality but one follows from $m_{0j}\ge (1-\lambda )m\widehat{\pi }_{0j,-1j}$ under the Dirac-uniform configuration. Thus, $\widehat{w}^a_{ij}$ satisfy $\sum _{i,j}(1+I(\mu _{ij}=0))E\widehat{w}^a_{ij,-ij}\le 2mq$. Obviously, $\widehat{w}^a_{ij}\le \widehat{w}^a_{ij,-ij}$. By Theorem 2, the data-driven Apro1 can control the mdFDR. The proof is completed. $\square$

Proof of Theorem 5

Without loss of generality, we suppose the non-zero components of ${\varvec{\mu }}_i$ are all positive. Then

$$\begin{aligned} \begin{aligned} mdFDR_{GW}&\le \sum _{r=1}^m\sum _{j=1}^q\frac{1}{r}\sum _{i=1}^m\bigg ( I(\mu _{ij}=0)Pr \{P_{ij}\le \frac{r\widehat{w}_{ij}}{mq} \alpha , R^{(-i)}=r-1 \}\\&\quad +I(\mu _{ij}>0)Pr\{P_{ij}\le \frac{r\widehat{w}_{ij}}{mq} \alpha , R^{(-i)}=r-1, T_{ij}<0 \}\bigg )\\&\le \sum _{r=1}^m\sum _{j=1}^q\frac{1}{r}\sum _{i=1}^m\bigg ( I(\mu _{ij}=0)Pr\{P_{ij} \le \frac{\widehat{w}_{ij,i}r}{mq} \alpha , R^{(-i)}=r-1 \}\\&\quad +I(\mu _{ij}>0)Pr\{P_{ij}\le \frac{\widehat{w}_{ij,i}r}{mq} \alpha , R^{(-i)}=r-1, T_{ij}<0 \}\bigg ), \end{aligned} \end{aligned}$$

where $\widehat{w}_{ij,i}=\sup _{0\le P_{ij}\le 1,j=1,\ldots ,q}\widehat{w}_{ij}$, and $R^{(-i)}$ is the number of rejections not including the rejection of $H_i$ by the first step of the GW procedure with $p_{ij}$ replaced by 0. Then

$$\begin{aligned} \begin{aligned} mdFDR_{GW}&\le \sum _{r=1}^m\sum _{j=1}^q\sum _{i=1}^m E_{\mathbf{P}^{(-ij)}} \frac{\widehat{w}_{ij,i}\alpha }{m}\big [ I(\mu _{ij}=0)\\&\quad +\frac{1}{2} I(\mu _{ij}>0) \big ]Pr(R^{(-i)}=r-1|\mathbf{P}^{(-ij)}) \\ {}&=\sum _{r=1}^m\sum _{j=1}^q\sum _{i=1}^m E_{\mathbf{P}^{(-ij)}}\frac{(w_{ij}+o_p(1)) \alpha }{m}\big [ I(\mu _{ij}=0)+\frac{1}{2} I(\mu _{ij}>0)\big ]\\ {}&\ \ \times Pr(R^{(-i)}=r-1|\mathbf{P}^{(-ij)}) \\ {}&=\frac{1}{2mq}\sum _{i,j}(1+I(\mu _{ij}=0)) w_{ij}\alpha +o(1)\le \alpha +o(1). \end{aligned} \end{aligned}$$

The proof is completed. $\square$