Assessing Population Level Genetic Instability via Moving Average

Abstract

Tumoral tissues tend to generally exhibit aberrations in DNA copy number that are associated with the development and progression of cancer. Genotyping methods such as array-based comparative genomic hybridization (aCGH) provide means to identify copy number variation across the entire genome. To address some of the shortfalls of existing methods of DNA copy number data analysis, including strong model assumptions, lack of accounting for sampling variability of estimators, and the assumption that clones are independent, we propose a simple graphical approach to assess population-level genetic alterations over the entire genome based on moving average. Furthermore, existing methods primarily focus on segmentation and do not examine the association of covariates with genetic instability. In our methods, covariates are incorporated through a possibly mis-specified working model and sampling variabilities of estimators are approximated using a resampling method that is based on perturbing observed processes. Our proposal, which is applicable to partial, entire or multiple chromosomes, is illustrated through application to aCGH studies of two brain tumor types, meningioma and glioma.

1 Introduction

Genetic analyses of a variety of cancers have suggested that losses and gains of DNA sequences (copy number variations) are associated with cancer development and progression [22]. Chromosomal loss in a tumor cell is typically associated with underexpression of tumor suppressor genes whose activity checks cellular growth, while copy number gains tend to be associated with overexpression of oncogenes that promote tumor growth [10]. Array-based comparative genomic hybridization (aCGH) and other genotyping methods offer a high-throughput approach to compare copy number between tumor DNA and reference samples. Test (tumoral) and reference (normal) samples are, respectively, labeled with red and green fluorescent tags and then mixed. The combined sample is subsequently hybridized to microarrays with the chip probes corresponding to location-specific clones across the entire genome [19]. At each location, the copy number alterations are measured by the log₂ ratio of the fluorescence intensities of the two colors. Reference sample is typically diploid and so, in the ideal setting, in the absence of contaminated cells, its DNA would have a normal copy number of 2. On the other hand, regions of copy number loss in the tumoral sample would have a copy number of 1 (corresponding to a loss of one chromosomal copy) or 0 (corresponding to homozygous deletion), while genomic regions of copy number gain would have copy numbers of 3 (trisomy) or more (polysomy or amplification). This translates into log₂ ratios of −1, 0 and ≥0.58, corresponding to copy number loss, no-change, and gain, respectively. It is important to note that, due to both biological and experimental reasons, the observed log₂ ratios may deviate from these theoretical values [12].

To analyze DNA copy number data, various statistical methods have been developed to identify aberrant genomic regions that are associated with different types of tumor. A common approach is to determine threshold values for defining gains and losses accounting for data variability. For example, Pollack et al. [23, 24], Weiss et al. [33] and Aguirre et al. [1] estimated the variability of the log₂ ratios that correspond to no genetic alteration, under the assumption that log₂ ratios corresponding to no genetic alteration in the tumor are normally distributed with mean zero. A three-component mixture model approach was considered by Hodgson et al. [11] where the components correspond to copy number loss, gain and no-change. These methods require the normality assumption and independence between clones. Recently, Purdom and Holmes [26] proposed procedures that can be used to analyze non-normally distributed genetic markers. Another commonly used approach considers segments of common log₂ ratio means and aims to identify change-points of the means. For example, Olshen et al. [19] proposed a circular binary segmentation (CBS) algorithm that identifies the change points through successive comparison of segments of the chromosome. Local significance was evaluated via permutation tests and a pruning algorithm was used to control the number of change points. Picard et al. [21] identified change points for the sequence of log₂ ratios using a penalized likelihood-based approach. These methods also require the assumption of independence between clones. More recently, various methods have been developed to incorporate the possible dependence between clones. Fridlyand et al. [8] proposed a discrete-state Hidden Markov Model (HMM) approach assuming that given the genetic states at all nearbylocations on one side of a given location, the genetic state at that location depends only on the true state at the immediately adjacent location. The “states” correspond to segments of common mean and a change in state corresponds to a change-point. The number of mean levels on each chromosome was determined by an AIC criterion. Engler et al. [6] accounted for the dependence of the data by using a pseudolikelihood function to fit a Gaussian mixture model with a Hidden Markov structure. Rueda and Diaz-Uriarte [27] employed a rather computationally intensive method based on a Hidden Markov Model without pre-specifying the number of states. Other Hidden Markov models have been examined by, for example, Shah et al. [29] and Guha et al. [9]. Lai et al. [13] propose a Bayesian stochastic change-point model for segmentation and to obtain inference on aberrations, though their model does not allow for covariates.

Most existing methods are derived under relatively strong model assumptions. Violation of these assumptions may lead to incorrect conclusions about the association between the genetic instability and clinical outcomes of interest. Also, most existing literature focuses primarily on point estimation of the parameters of interest without accounting for the sampling variability in such estimators. Furthermore, most of these methods do not account for possibly inflated type I error rate while making simultaneous inference along the entire genome. With the advancement of technology, the number of locations available is becoming increasingly large. Traditional methods such as the Bonferroni adjustment or control of the false discovery rate [2] may be too conservative to detect regions of interest when there is high correlation between adjacent regions. To overcome such difficulties, we propose a simple graphical approach for assessing population-level genetic alterations over the entire genome based on a moving average technique. To summarize the population level genetic instability for the population of samples, we consider average instability levels across various regions of interest and propose non-parametric estimates that do not require the commonly employed normality assumptions. When there are covariates that may affect the genetic instability, we propose the use of possibly mis-specified working models to approximate the association between the log₂ ratios and the covariates and develop inferential procedures for the average covariate effect that do not require the models to hold. In Sect. 2, we provide a general framework for inference about the population level genetic instability as well as the average covariate effect on the genetic instability. We illustrate the proposed procedures using aCGH data from a meningioma study and from a glioma study in Sect. 3. Some concluding remarks are given in Sect. 4.

2 Inference Procedures for the Covariate Effect on Genetic Instability

Let X_ik(t) denote the log₂ DNA copy number ratio of the ith subject at location t of the kth measurement and let Z_ik denote the p × 1 baseline covariate of the ith subject corresponding to the kth measurement, for k = 1, …, K, i = 1, …, n and $t\in \mathcal{T}$. Here, we allow each subject to have multiple repeated measurements of the copy number at each location and assume that K, the total number of measurements, is a fixed constant. Furthermore, the n sets of clustered observations for each location $t\{X_{i1}\left(t\right),\dots ,X_{iK}\left(t\right),t\in \mathcal{T},{\mathbf{Z}}_{i1},\dots ,{\mathbf{Z}}_{iK}\}$ are independent and identically distributed. Without loss of generality, let $\mathcal{T}=[0,\tau ]$.

We are interested in assessing level genetic instability within various patient populations indexed by the covariate information Z and comparing the genetic instability between different patient populations. To this end, we propose to approximate the association between the covariate Z and X(t) through the marginal regression working model

$$X_{ik}\left(t\right)=g\left\{{\gamma }_{\mu }{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right\}+h\left\{{\gamma }_{\sigma }{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right\}{∊}_{ik}$$ (1)

where ε_ik is a mean zero random variable, $\overrightarrow{\mathbf{Z}}={(1,{\mathbf{Z}}^{\prime })}^{\prime }$, g(·) and h(·) are pre-specified strictly increasing functions with h(·) > 0, ${\gamma }_{\mu }\left(t\right)={({\alpha }_{\mu }\left(t\right),{\beta }_{\mu }{\left(t\right)}^{\prime })}^{\prime }$ and ${\gamma }_{\sigma }\left(t\right)={({\alpha }_{\sigma }\left(t\right),{\beta }_{\sigma }{\left(t\right)}^{\prime })}^{\prime }$. For instance, often g(·) is a simple linear function such as g(x) =x and h(·) is an exponential function such as h(x) exp(x), so we have $X_{ik}\left(t\right)={\alpha }_{\mu }\left(t\right)+{\beta }_{\mu }{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}+\exp ({\alpha }_{\sigma }\left(t\right)+{\beta }_{\sigma }{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik})\in ik$. Thus, β_μ(t) quantifies the average effect of Z on the population mean level of genetic instability at location t. The covariate vector β_σ(t) quantifies the average effect of Z on the variation of genetic instability at location t. Thus, this model allows tumors to have larger or smaller variability depending on their characteristics captured by Z. Note that we do not impose a Normality assumption on ε_ik. In the simple setting where interest lies in assessing the population average genetic instability, we may employ the null model with no covariates and thus summarize it by μ(t) = g{α_μ(t)}. If Z is an index of cancer subtype, then one may summarize the difference between the cancer subtypes based on β_μ(t) and β_σ(t).

To obtain an estimate for the regression parameter γ_μ(t), we consider the following simple estimating equation:

$$\sum _{i=1}^n\sum _{k=1}^{K_i}{\overrightarrow{\mathbf{Z}}}_{ik}\{X_{ik}\left(t\right)-g\left({\gamma }^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right)\}=0$$

Let ${\widehat{\gamma }}_{\mu }$, denote the solution to the above estimating equation. An estimate of ${\gamma }_{\sigma }\left(t\right)$ may be obtained as ${\widehat{\gamma }}_{\sigma }\left(t\right)$, the solution to

$$\sum _{i=1}^n\sum _{k=1}^{K_i}{\overrightarrow{\mathbf{Z}}}_{ik}[{\{X_{ik}\left(t\right)-g\left({\widehat{\gamma }}_{\mu }{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right)\}}^2-h{\left({\gamma }^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right)}^2]=0$$

It follows from the same argument as given in Tian et al. [30] that $\widehat{\gamma }\left(t\right)={({\widehat{\gamma }}_{\mu }{\left(t\right)}^{\prime },{\widehat{\gamma }}_{\sigma }{\left(t\right)}^{\prime })}^{\prime }$ is always convergent to ${\gamma }_0\left(t\right)={({\gamma }_{\mu 0}{\left(t\right)}^{\prime },{\gamma }_{\sigma 0}{\left(t\right)}^{\prime })}^{\prime }$, where ${\gamma }_0\left(t\right)$ is the unique solution to

$$E\left(\begin{array}{c}\hfill \sum _{k=1}^{K_i}{\overrightarrow{\mathbf{Z}}}_{1k}\{X_{1k}\left(t\right)-g\left({\gamma }_{\mu }^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right)\}\hfill \\ \hfill \sum _{k=1}^{K_i}{\overrightarrow{\mathbf{Z}}}_{1k}[{\{X_{1k}\left(t\right)-g\left({\gamma }_{\mu }^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right)\}}^2-h{\left({\gamma }_{\sigma }^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right)}^2]\hfill \end{array}\right)=0$$

regardless of the adequacy of the working model (1). ${\gamma }_0\left(t\right)$ and ${\gamma }_{\sigma 0}\left(t\right)$ are the respective true values of ${\gamma }_{\mu }\left(t\right)$ and ${\gamma }_{\sigma }\left(t\right)$ in (1) when it holds; and is a valid summary of the average effect of Z on X(t) even when the working model (1) fails to hold. We next develop inference procedures about the average effects without requiring model (1) to hold.

To make inference about ${\gamma }_0\left(t\right)$ at a given location t, we note that $n^{\frac{1}{2}}\{\widehat{\gamma }\left(t\right)-{\gamma }_0\left(t\right)\}$ is asymptotically equivalent to $n^{-\frac{1}{2}}{\sum }_{i=1}^n{\mathbf{U}}_i\left(t\right)$ and converges in distribution to a multivariate normal with mean 0 and covariance matrix ${\Sigma }_{\gamma }\left(t\right)=E\left\{{\mathbf{U}}_i{\left(t\right)}^{\otimes 2}\right\}$, where ${\mathbf{U}}_i\left(t\right)={[{\mathbf{U}}_{\mu i}{\left(t\right)}^{\prime },{\mathbf{U}}_{\sigma i}{\left(t\right)}^{\prime }]}^{\prime }$

$${\mathbf{U}}_{\mu i}\left(t\right)=\sum _{k=1}^{K_i}{\mathbb{A}}_{\mu }^{-1}\left(t\right){\overrightarrow{\mathbf{Z}}}_{ik}\{X_{ik}\left(t\right)-g\left({\gamma }_{\mu 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right)\}$$

$${\mathbf{U}}_{\sigma i}\left(t\right)=\sum _{k=1}^{K_i}{\mathbb{A}}_{\sigma }^{-1}({\mathbb{A}}_{\mu \sigma }\left(t\right){\mathbf{U}}_{\mu i}\left(t\right)+{\overrightarrow{\mathbf{Z}}}_{ik}[{\{X_{ik}\left(t\right)-g\left({\gamma }_{\mu 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right)\}}^2-h{\left({\gamma }_{\sigma 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{ik}\right)}^2])$$

${\mathbb{A}}_{\mu }\left(t\right)={\sum }_{k=1}^{K_i}E\left\{\stackrel{.}{g}\left({\gamma }_{\mu 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right){\overrightarrow{\mathbf{Z}}}_{1k}^{\otimes 2}\right\}$, ${\mathbb{A}}_{\sigma }\left(t\right)=2{\sum }_{k=1}^{K_i}E\{\stackrel{.}{h}\left({\gamma }_{\sigma 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right)\times h\left({\gamma }_{\sigma 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right){\overrightarrow{\mathbf{Z}}}_{1k}^{\otimes 2}\}$, ${\mathbb{A}}_{\mu \sigma }\left(t\right)=2{\sum }_{k=1}^{K_i}E[\{X_{1k}\left(t\right)-g\left({\gamma }_{\mu 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right)\}\stackrel{.}{g}\left({\gamma }_{\mu 0}{\left(t\right)}^{\prime }{\overrightarrow{\mathbf{Z}}}_{1k}\right)\times {\overrightarrow{\mathbf{Z}}}_{ik}^{\otimes 2}]$, and for any vector a, ${\mathbf{a}}^{\otimes 2}={\mathbf{aa}}^{\prime }$. For any specific location t, one may construct confidence intervals for ${\gamma }_{\mu 0}\left(t\right)$ and ${\gamma }_{\sigma 0}\left(t\right)$ based on this large sample normal approximation.

In practice, it is often of interest to simultaneously assess the population level instability across a region of t. Unfortunately, it is unclear whether $n^{\frac{1}{2}}\{\widehat{\gamma }\left(t\right)-{\gamma }_0\left(t\right)\}$ converges a process in t due to the unknown correlation structure between as X(t) and X(s). As a result, ${\text{sup}}_{t\in \tau }\mid n^{\frac{1}{2}}\{\widehat{\gamma }\left(t\right)-{\gamma }_0\left(t\right)\}\mid$ may not converge in distribution and is difficult to construct confidence bands for ${\gamma }_0\left(t\right)$. To overcome such a difficulty, we propose to take the moving average approach by assessing the average genetic stability across various small regions of interest. Specifically, we consider to make joint inference about

$${\Gamma }_0\left(t\right)=\frac{1}{2b_0}{\int }_{t-b_0}^{t+b_0}{\gamma }_0\left(s\right)ds,\text{for}t\in [b_0,\tau -b_0]$$

where b₀ is a pre-specified positive constant half window width. Γ₀(t) is essentially a smoothed version of γ₀(t) and summarizes the average covariate effect on the genetic instability in the region s ∈ [t – b₀, t + b₀]. Based on $\widehat{\gamma }\left(t\right)$, one may obtain a simple plug-in estimate for Γ₀(t),

$$\widehat{\Gamma }\left(t\right)=\frac{1}{2b_0}{\int }_{t-b_0}^{t+b_0}\widehat{\gamma }\left(s\right)ds$$ (2)

We show in the appendix that under mild regularity conditions, the process $\widehat{\mathcal{W}}\left(t\right)=n^{\frac{1}{2}}(\widehat{\Gamma }\left(t\right)-{\Gamma }_0\left(t\right))$ converges weakly in to a zero-mean Gaussian process, $\mathcal{W}\left(t\right)$.

To approximate the distribution of $\widehat{\mathcal{W}}\left(t\right)$, we propose to use the simple perturbation resampling method similar to what has been considered in Cai et al. [3] and Park and Wei [20]. Let ${\mathcal{N}}_{1,\dots ,n}=\{{\mathcal{N}}_i,i=1,\dots ,n\}$ denote n i.i.d copies of standard nor-random variables generated independent of the data. For any given set of ${\mathcal{N}}_{1,\dots ,n}$, let

$${\widehat{\mathcal{W}}}^{\ast }\left(t\right)=\frac{1}{n^{\frac{1}{2}}2b_0}\sum _{i=1}^n{\int }_{t-b_0}^{t+b_0}{\widehat{\mathbf{U}}}_i\left(t\right){\mathcal{N}}_i$$

where ${\widehat{\mathbf{U}}}_i\left(t\right)$ is obtained by replacing all the theoretical quantities in U_i(t) by their empirical counterparts. It is not difficult to show that conditional on the data, ${\widehat{\mathcal{W}}}^{\ast }\left(t\right)$ converges weakly to the Gaussian process $\mathcal{W}\left(t\right)$. Therefore, one may approximate the distribution of $\widehat{\mathcal{W}}\left(t\right)$ by the conditional distribution ${\widehat{\mathcal{W}}}^{\ast }\left(t\right)$.

In practice we can approximate the distribution of $\widehat{\mathcal{W}}\left(t\right)$ by generating a large number, M, independent samples of $\{{\mathcal{N}}_i,i=1,\dots ,n\}$. For the mth sample, we obtain a realization ${\widehat{\mathcal{W}}}_{\left(m\right)}^{\ast }\left(t\right)$, ${\widehat{\mathcal{W}}}^{\ast }\left(t\right)$, m = 1, …, M. At any given location t and for any constant vector c we may construct 100(1 – α)% pointwise confidence interval for ${\mathbf{c}}^{\prime }{\Gamma }_0\left(t\right)$ as

$${\mathbf{c}}^{\prime }\widehat{\Gamma }\left(t\right)\pm z_{\alpha }\widehat{\sigma }\left(t\right),\text{where}{\widehat{\sigma }}^2\left(t\right)=\frac{1}{M}\sum _{m=1}^M{\left\{{\mathbf{c}}^{\prime }{\widehat{\mathcal{W}}}_{\left(m\right)}^{\ast }\left(t\right)\right\}}^2$$

and z_α is the 100(1 – α/2)th percentile of the standard normal. Furthermore, a 100(1 – α)% simultaneous confidence interval for $\{{\mathbf{c}}^{\prime }{\Gamma }_0\left(t\right),b_0\le t\le \tau -b_0\}$ can be obtained as

$${\mathbf{c}}^{\prime }\widehat{\Gamma }\left(t\right)\pm s_{\alpha }\widehat{\sigma }\left(t\right)$$

where s_α is the 100(1 – α)th percentile of

$$\left\{\underset{t\in [b_0,\tau -b_0]}{\text{sup}}\mid {\mathbf{c}}^{\prime }{\widehat{\mathcal{W}}}_{\left(1\right)}^{\ast }\left(t\right)∕\widehat{\sigma }\left(t\right)\mid ,\dots ,\underset{t\in [b_0,\tau -b_0]}{\text{sup}}\mid {\mathbf{c}}^{\prime }{\widehat{\mathcal{W}}}_{\left(M\right)}^{\ast }\left(t\right)∕\widehat{\sigma }\left(t\right)\mid \right\}$$

3 Examples

3.1 Meningioma Study

We now evaluate the population level genetic instability based on an aCGH dataset from a meningioma study, conducted at the Massachusetts General Hospital. Meningiomas are tumors of the coverings of the brain and account for approximately 15% of all primary brain tumors. Most of these tumors are benign but they may also be atypical (an intermediate grade) or malignant. The dataset consists of DNA copy number profiles of 72 sporadic solitary meningiomas represented as log₂ ratio along chromosomes 1–22 and X. The pathological subclassification lists 34 subjects as being benign, 25 atypical and 13 malignant cases. Goals of this study include the development of a useful graphical method of displaying the data, the identification of regions of loss or gain that are shared by patients with this type of tumor differences between the pathological subtypes.

First we evaluate the population level genetic instability, based on the mean log₂ aCGH ratio level, on chromosome 1 for the benign tumor subtype. The log₂ aCGH ratio at a total of 1339 locations were recorded for chromosome 1. The population mean of the can be estimated log₂ aCGH ratio, ${\gamma }_{\mu }\left(t\right)=E\left\{X_i\left(t\right)\right\}$, can be estimated empirically by ${\widehat{\gamma }}_{\mu }\left(t\right)=n^{-1}{\sum }_{i=1}^n{X}_i\left(t\right)$. Here, each subject has one observation with K = 1 and for simplicity, we drop the subscript k. The raw mean process is shown in Fig. 1(a). We propose to quantify the population level genetic instability based on ${\Gamma }_{\mu }\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\gamma }_{\mu }\left(s\right)ds$, which can be estimated by ${\widehat{\Gamma }}_{\mu }\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\widehat{\gamma }}_{\mu }\left(s\right)ds$. In Fig. 1(b), we show the estimated process $\{{\widehat{\Gamma }}_{\mu }\left(t\right);b_0\le t\le \tau -b_0\}$ along with their 95% simultaneous this confidence bands. In this and other figures in this example, the shaded regions on the upper level and the lower level represent the regions that the population version of the displayed process is significantly greater or less than 0 (respectively). In this and subsequent analyses, for the moving averages, we choose a priori, overlapping fixed windows of size b₀ = 10 and 95% confidence bands are based on perturbing the original process M 500 = times. Here b₀ = 10 is chosen to ensure that a minimal amount of information is lost due to averaging across the window. Since the simultaneous confidence intervals cover 0 for most of the regions, we conclude that there appears to be little genetic instability at chromosome 1 for patients with benign tumor.

Fig. 1.

The Raw and moving average mean log₂ aCGH ratio levels (solid curve) and 95% simultaneous confidence bands (dashed lines) on Chromosome 1 for the benign tumor patients

Next we compare the population average genetic instability at chromosome 1 among the three tumor subtypes. To this end, we consider the simple regression model where g(·) is the identity function and h(·) = 1:

$$E\{X_i\left(t\right)\mid Z_i\left(1\right),Z_{i\left(2\right)}\}={\beta }_0\left(t\right)+{\beta }_1\left(t\right)Z_{i\left(1\right)}+{\beta }_2\left(t\right)Z_{i\left(2\right)}$$

where Z_i(1) = 1 if subject i has an atypical tumor and 0 otherwise, and Z_i(2) = 1 if subject i has a malignant tumor and 0 otherwise. The reference group, corresponding to Z_i(1) = Z_i(2) = 0, consists of the patients classified as ‘benign’. Figure 2 presents the point and simultaneous interval estimates of ${\Gamma }_{{\beta }_1}\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\beta }_1\left(s\right)ds$ and ${\Gamma }_{{\beta }_2}\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\beta }_2\left(s\right)ds$.

Fig. 2.

The moving average mean difference in log₂ aCGH ratio levels (solid curve) with 95% simultaneous confidence bands (dashed lines) at Chromosome 1 (a) between ‘atypical’ and benign; and (b) between ‘malignant’ and benign

The results suggest that on average, the level of genetic instability at chromosome 1 is higher for atypical tumors, and is even more extreme for malignant tumors, when compared to benign tumors. For both atypical and malignant tumors, there appears to be regions of deletion at locations 0 ~ 700 and regions of amplification at locations 700 ~ 1200.

To assess the genetic instability across the entire genome, we obtained point estimates of Γ_β1(·) and Γ_β2(·) for all 23 chromosomes and obtained simultaneous confidence intervals adjusting for all the locations considered. Note that the moving average is only applied within chromosomes. The left panel in Fig. 3 shows the moving average process related to Γ_β1(·) and the corresponding graph related to Γ_β2(·) is displayed in the right panel. In general, there appear to be differential levels of genetic instability for the three tumor types. Differences in genetic instability between the benign and atypical tumors are most significant at chromosomes 1, 14 and 23. Differences are also apparent in non-trivial regions at chromosomes 2, 5, 6, 7, 11, 15, 16, 17, 19, 20 and 22. Differences between the benign and malignant tumors have a similar pattern, but are more extreme except for chromosome 22. Additional differences are apparent at chromosomes 8, 9, 10, 12 and 18.

Fig. 3.

The moving average mean difference in log₂ aCGH ratio levels (solid curve) and 95% simultaneous confidence bands (dashed lines) across all chromosomes (a) between atypical and benign; and (b) between malignant and benign

3.2 Glioma Study

The second example is from a glioma study [16] that consists of 47 patients diagnosed with two subtypes of gliomas: 22 were diagnosed with oligodendroglioma (OLIGO) and 25 were diagnosed with glioblastoma (GBM). Gliomas are the most commonly diagnosed primary brain tumors, accounting for approximately 45-50% of all primary brain tumors. Among high-grade gliomas, OLIGOs have a more favorable prognosis than GBMs [14]. GBMs are resistant to most available therapies, while OLIGOs are often chemosensitive. Currently, gliomas are classified according to de-fined histological features characteristic of the presumed normal cell of origin. Such methods, however, may lack diagnostic accuracy and reproducibility in many cases (Okada et al. [18]). To develop more objective approaches to glioma classification, investigations have focused on molecular genetic analyses [14]. We are interested in comparing genetic profiles between these two subtypes of gliomas based on the aCGH data, available on chromosomes 1, 7 and 19, for subjects in this study. Until recently, aCGH could only be reproducibly performed on frozen tissue samples and with significant tissue amounts. For brain tumors, however, paraffin-embedded tissue blocks from small stereotactic biopsies may be the only tissue routinely available. The development of methods to analyze formalin-fixed, paraffin-embedded material therefore has the potential to impact molecular diagnosis in a significant way [16]. For each subject in this study, DNA from both frozen tissues and FFPE tissues were analyzed. To perform such analyses, we consider the model where g() is the identity function and h(x) exp = (x):

$$X_{ik}\left(t\right)={\beta }_0\left(t\right)+{\beta }_1\left(t\right)Z_{ik\left(1\right)}+{\beta }_2\left(t\right)Z_{ik\left(2\right)}+\text{exp}\{{\alpha }_0\left(t\right)+{\alpha }_1\left(t\right)Z_{ik\left(1\right)}+{\alpha }_2\left(t\right)Z_{ik\left(2\right)}\}{\epsilon }_{ik}$$ (3)

for k = 1, 2 and n = 1, …, 47. Here, X_ik(t) is the log₂ ratio of the copy numbers at location t obtained using the kth tissue type of the ith subject i, Z_ik(1) is the tissue type indicator with Z_ik(1) = 1 for FFPE tissues and 0 otherwise, Z_ik(2) is the indicator of tumor subtype with Z_ik(2) = 1 for GBM tumors and 0 otherwise. Estimates of β(t) and α(t) in model (3) are obtained using data from each of the three chromosomes. These are used to generate point estimates and simultaneous interval estimates for the processes ${\Gamma }_{{\beta }_{\mathcal{l}}}\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\beta }_{\mathcal{l}}\left(s\right)ds$, ${\Gamma }_{\text{exp}\left({\alpha }_l\right)}\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}\text{exp}\left({\alpha }_l\left(s\right)\right)\left(s\right)ds$. For example, the processes for l = 2 capture the local average difference between the tumor subtypes in the mean and variance of genetic instability processes. The results shown in Fig. 4 suggest that GBM has a greater effect on the population mean genetic instability over the three chromosomes, achieving statistical significance in regions of chromosomes 1 and 19 and in a spiked region on chromosome 7. This confirms known genetic alterations of these tumors. The effect of tumor subtype on the variability seem to vacillate across the chromosomes.

Fig. 4.

The moving average processes (solid curve) and 95% simultaneous confidence bands (dashed lines) showing local average effects of tumor subtype on the mean and variance of log₂ aCGH ratio levels. The shaded regions on the upper level and the lower level represent the regions that the processes are significantly different from the null

4 Simulations

We assess the performance of our methods with simulation studies. We specified the parameters of the simulation models based on a subset of the aCGH dataset from the meningioma study analyzed in the example. We generated 1500 datasets with n = 90 and 300 subjects, where an equal number of subjects each had benign, atypical, and malignant brain tumors. The simulated datasets consist of log₂ ratios of DNA copy number profiles of 500 locations along one chromosome. Each subject has K = 1 observation. To obtain the simulated log₂ ratios, we first generated the true copy numbers of the samples from a one-step random Markov process with transition probabilities dependent on tumor type and estimated empirically from chromosome 1 of the meningioma dataset. We then adapted the models of Engler et al. [6] and assumed a normal distribution dependent on the true copy number so that for a patient i with a true copy number of C_i(t) at location t has an observed log₂ copy number ratio of x_i(t) with probability:

$$P(x_i\left(t\right)\mid C_i\left(t\right),\mu ,\sigma )~\{\begin{array}{cc}N({\mu }_L,{\sigma }_L^2),\hfill & \text{if}{C}_i\left(t\right)<2\hfill \\ N({\mu }_0,{\sigma }_0^2),\hfill & \text{if}{C}_i\left(t\right)=2\hfill \\ N({\mu }_G,{\sigma }_G^2),\hfill & \text{if}{C}_i\left(t\right)>2\hfill \end{array}\phantom{\}}$$

where subscripts L, 0, and G signify copy number loss, no change, and gain, respectively. In this way, we may investigate how the means and variances of the log₂ ratios affect the performance of our methods. Additionally, although subjects may have a common lesion with the same transition probabilities, they may have different change-points.

For the analysis of the simulated datasets, we assume a simple regression working model

$$E\{X_i\left(t\right)\mid Z_i\left(1\right),Z_{i\left(2\right)}\}={\beta }_0\left(t\right)+{\beta }_1\left(t\right)Z_{i\left(1\right)}+{\beta }_2\left(t\right)Z_{i\left(2\right)}$$

where Z_i(1) = 1 if subject i has an atypical tumor and 0 otherwise, and Z_i(2) = 1 if subject i has a malignant tumor and 0 otherwise. We are interested in obtaining inference about ${\Gamma }_{{\beta }_j}\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\beta }_j\left(s\right)ds$ for j = 1, 2 through point and interval estimates. We chose b₀ = 10. Table 1 shows the average bias, standard error, and coverage probabilities from two locations for the setting when μ_L = −0.431, μ₀ = −0.001, μ_G = 0.408, σ_L = 0.202, σ₀ = 0.109, σ = G = 0.219. We compare our estimates to the true parameter vector Γ_μ obtained via Monte Carlo methods with n = 30000. In general, our procedure works well in finite sample with respect to both point and interval estimations. The perturbation procedure provides standard error estimates close to the true sampling standard errors and the resulting pointwise confidence intervals have coverage levels close to their nominal value. However, we did find that when the sample size is small (n = 90, 30 subjects per tumor group), the simultaneous confidence intervals tend to have empirical coverages lower than the desired levels. When the sample size is increased to n = 300 (100 subjects per group), both the pointwise and simultaneous confidence intervals have coverages close to the nominal level. For example, the coverage level of the simultaneous interval for Γ_β1 was about 86% when n = 90 and about 93% when n = 300. This is likely due to the fact that the simultaneous convergence of the moving averages as a process requires a reasonably large sample size. Thus, for small samples, we recommend the use of more exact procedures such as the permutation based tests [17] to control for the overall type I error controlled.

Table 1.

Average bias, empirical standard errors (Emp. SE), average standard error estimates (Est. SE), and coverage probabilities (CovP) of 95% pointwise confidence intervals for Γ_μ at two locations. All values are multiplied by 100

		Bias	Emp. SE	Est. SE	CovP
n = 90	Γβ0 (1)	0.01	1.37	1.33	93.60
	Γβ0 (253)	−0.02	1.24	1.33	96.54
	Γβ1 (1)	0.06	2.21	2.27	94.99
	Γβ1 (253)	0.23	2.23	2.27	94.30
	Γβ2 (1)	0.09	2.31	2.28	93.95
	Γβ2 (253)	−0.07	2.23	2.28	95.25
n = 300	Γ0(1)	−0.10	0.73	0.74	95.20
	Γ0(253)	0.00	0.72	0.74	94.80
	Γβ1 (1)	0.11	1.25	1.25	95.40
	Γβ1 (253)	−0.05	1.26	1.25	95.00
	Γβ2 (1)	0.14	1.23	1.26	95.80
	Γβ2 (253)	0.01	1.24	1.25	95.20

5 Discussion

We developed a new procedure for assessing the population level genetic instability and the effect of covariates on the genetic instability with array CGH data. The use of the moving average technique allows for simultaneous assessment across the entire genome, without requiring specification of the unknown complex correlation structure between regions. It is important to note that data normalization (e.g. Feuk et al. [7], Scherer et al. [28]) and removal of local trends such as genomic waves [5, 15] are important initial steps that should be applied prior to performing population level regression type of procedures to infer about covariate effects on the genetic variability.

We proposed the use of working models to approximate the association between covariates and the population level instability. In general, the validity of the proposed location scale type models for a given dataset can be examined using model checking procedures developed in Cai and Zheng [4]. However, our procedures for making inference about the average covariate effect are valid even if the fitted working models fail to hold. The moving average technique induces correlation between neighboring windows, which allows us to simultaneously assess the covariate effects across the entire genome. Our proposed simultaneous inference provides a useful tool to assess various aspects of the genetic instability profile across the entire genome while controlling for an overall type I error. The procedures proposed in the paper focus primarily on the assessment of genetic instability profile at the population average level. When interest lies in the classification of individual tumors, one may collect subject level information to characterize the tumor. If such information is included as covariates, then classification procedures can be formed by predicting the probability of each tumor having aberrations. Such classification methods warrant further research.

To ensure the validity of the simultaneous inference, the half window width parameter b₀ is a pre-determined constant. For our analyses we have chosen b₀ to be set at 10 and for array CGH data we find that this is rather informative and allows the moving average process to resemble the original (raw) process with sufficient smoothness to ensure the validity of the simultaneous inference about the moving average process. However, it is important to note that with the next generation sequencing technology, the window width may need to be altered to accommodate the finer measurements.

Appendix

Let $t\in \tau =[0,\tau ]$ be such that $n^{1∕2}\{\widehat{\gamma }\left(t\right)-{\gamma }_0\left(t\right)\}$ is asymptotically equivalent to a sum of independent and identically distributed terms, $n^{-\frac{1}{2}}{\sum }_{i=1}^n{\mathbf{U}}_i\left(t\right)$. Here, we assume that U_i(t) has total variation bounded by a constant. It is not difficult to see that under the location-scale working model and with the assumption that X_ik(t) has total variation bounded by a constant, this condition is satisfied. In this section, we show that for b₀ > 0, and $t\in {\tau }^{\prime }=[b_0,\tau -b_0]$, the process

$$\widehat{\mathcal{W}}\left(t\right)=n^{1∕2}\{\widehat{\Gamma }\left(t\right)-{\Gamma }_0\left(t\right)\}$$

where ${\Gamma }_0\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}{\gamma }_0\left(s\right)ds$, $\widehat{\Gamma }\left(t\right)={\left(2b_0\right)}^{-1}{\int }_{t-b_0}^{t+b_0}\widehat{\gamma }\left(s\right)ds$ converges weakly to a zero mean Gaussian process $\mathcal{W}\left(t\right)$ with continuous sample paths and covariance matrix function ${\rm Y}(s,t)=E\left\{{\mathbf{L}}_1\left(t\right){\mathbf{L}}_1{\left(s\right)}^{\prime }\right\}$, where ${\mathbf{L}}_1\left(r\right)={\left(2b_0\right)}^{-1}\times {\int }_{r-b_0}^{r+b_0}{\mathbf{U}}_1\left(u\right)d\left(u\right)\text{and}s,t\in {\tau }^{\prime }$.

To this end, We first note that

$$\widehat{\mathcal{W}}\left(t\right)=n^{1∕2}\{\widehat{\Gamma }\left(t\right)-{\Gamma }_0\left(t\right)\}=\frac{n^{1∕2}}{2b_0}{\int }_{t-b_0}^{t+b_0}\{\widehat{\gamma }\left(s\right)-{\gamma }_0\left(s\right)\}d\left(s\right)$$

which is asymptotically equivalent to $n^{-1∕2}{\left(2b_0\right)}^{-1}{\sum }_{i=1}^n{\int }_{t-b_0}^{t+b_0}{\mathbf{U}}_i\left(s\right)d\left(s\right)$. For brevity, let ${\mathbf{L}}_i\left(t\right)={\int }_{t-b_0}^{t+b_0}{\mathbf{B}}_i\left(s\right)d\left(s\right)$ where ${\mathbf{B}}_i=2b_0^{-1}{\mathbf{U}}_i$ so that $\widehat{\mathcal{W}}\left(t\right)$ is asymptotically equivalent to $\frac{1}{\sqrt{n}}{\sum }_{i=1}^n{\mathbf{L}}_i\left(t\right)$ for $t\in {\tau }^{\prime }$. To show the desired convergence, we need to verify the conditions (i)–(v) of the Functional Central Limit Theorem of Pollard [25, Chap. 10]. Since there exists M such that $\parallel {\mathbf{B}}_i\left(t\right)\parallel \le M$ for all i = 1, …, n and t ∈ [0, τ], we can rewrite B_i(t) as ${\mathbf{B}}_i^{+}\left(t\right)-{\mathbf{B}}_i^{-}\left(t\right)$ where both ${\mathbf{B}}_i^{+}\left(t\right)$ and ${\mathbf{B}}_i^{-}\left(t\right)$ are positive functions bounded by M. Writing ${\mathbf{L}}_i\left(t\right)={\int }_0^{t+b_0}{\mathbf{B}}_i^{+}\left(u\right)du-{\int }_0^{t-b_0}{\mathbf{B}}_i^{+}\left(u\right)du-{\int }_0^{t+b_0}{\mathbf{B}}_i^{-}\left(u\right)du+{\int }_0^{t-b_0}{\mathbf{B}}_i^{-}\left(u\right)du$, each term has pseudo-dimension at most 1 with the same envelope Mτ. Therefore {L_i(t), i = 1, …, n} is Euclidean and hence manageable. Condition (ii), the existence of limiting variance covariance matrix follows from law of large numbers. It follows from law of large numbers that the limiting covariance matrix of $\frac{1}{\sqrt{n}}{\sum }_{i=1}^n{\mathbf{L}}_i\left(t\right)$ converges to $E\left\{{\mathbf{L}}_1\left(t\right){\mathbf{L}}_1{\left(s\right)}^{\prime }\right\}$. Since $\parallel {\mathbf{B}}_i\left(t\right)\parallel \le M$, we can choose the envelope $F_{ni}=2M{b}_0$ for L_i(t). It is then obvious that conditions (iii) and (iv)

$$\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}E\left(F_{ni}^2\right)<\infty ,\sum _{i=1}^{n}E(F_{ni}^2;F_{ni}>∊)\to 0$$

hold. Condition (v) is trivial since ρ_n(s, t) ≡ ρ(s, t). Therefore $\widehat{\mathcal{W}}\left(t\right)=n^{1∕2}\{\widehat{\Gamma }\left(t\right)-{\Gamma }_0\left(t\right)\}$ converges in distribution to the same mean zero Gaussian process with covariance matrix function ${\rm Y}(s,t)=E\left\{{\left(2b_0\right)}^{-2}{\int }_{t-b_0}^{t+b_0}{\mathbf{U}}_1\left(u\right)du{\int }_{s-b_0}^{s+b_0}{\mathbf{U}}_1{\left(u\right)}^{\prime }du\right\}$.