Robust test method for time-course microarray experiments
- Insuk Sohn^{1, 3}Email author,
- Kouros Owzar^{2, 3},
- Stephen L George^{2, 3},
- Sujong Kim^{4} and
- Sin-Ho Jung^{2, 3}
DOI: 10.1186/1471-2105-11-391
© Sohn et al; licensee BioMed Central Ltd. 2010
Received: 8 March 2010
Accepted: 22 July 2010
Published: 22 July 2010
Abstract
Background
In a time-course microarray experiment, the expression level for each gene is observed across a number of time-points in order to characterize the temporal trajectories of the gene-expression profiles. For many of these experiments, the scientific aim is the identification of genes for which the trajectories depend on an experimental or phenotypic factor. There is an extensive recent body of literature on statistical methodology for addressing this analytical problem. Most of the existing methods are based on estimating the time-course trajectories using parametric or non-parametric mean regression methods. The sensitivity of these regression methods to outliers, an issue that is well documented in the statistical literature, should be of concern when analyzing microarray data.
Results
In this paper, we propose a robust testing method for identifying genes whose expression time profiles depend on a factor. Furthermore, we propose a multiple testing procedure to adjust for multiplicity.
Conclusions
Through an extensive simulation study, we will illustrate the performance of our method. Finally, we will report the results from applying our method to a case study and discussing potential extensions.
Background
The objective of a time-course microarray experiment is to study the temporal dynamics of the expression profile for each gene. For many of these experiments, the primary objective is to identify genes for which these temporal profiles depend on a phenotypic, experimental or environmental factor. We mention three examples next. Wang and Kim [1] identify genes in Caenorhabditis elegans for which the expression level depends on the dauer state. Graham et al. [2] obtain RNA expressions from kidney tissue from patients ranging between 27 and 92 years old, to identify genes whose expression profiles are age dependent while adjusting for other phenotypic factors. Sekiguchi et al. [3] study the mRNA expression profiles of peripheral blood cells in patients with rheumatoid arthritis receiving a TNF-α inhibitor drug. Henceforth, for notational brevity we refer to these temporal expression profiles as gene time-trajectories or simply time-trajectories whenever understood from the context.
There is an extensive recent body of literature on statistical methodology for identifying genes whose time-trajectories depend on a factor. We provide a brief summary of representative works. Park et al. [4] propose a permutation-based two-way ANOVA model. Luna and Li [5] propose a statistical framework based on a shape-invariant additive error model utilizing periodically expressed guide genes. Storey et al. [6] estimate gene expression time-trajectories using splines and then approximate the null sampling distribution of the goodness of fit statistic using a bootstrap method. Sohn et al. [7] extend this method to carry out the inference using permutation resampling. Hong and Li [8] discuss a functional hierarchical model for detecting temporally differentially expressed genes between two experimental conditions for cross sectional designs, where the gene expression profiles are treated as functional data and modelled by basis function expansions. Finally, Angelini et al. [9] use a Bayesian hierarchical model along with Bayes factors for the inference.
The remainder of this paper is organized as follows. In the next section, we provide a technical overview of the proposed method. Thereafter, we evaluate the procedure empirically using an extensive simulation study and by applying it to a published case study. We finalize the paper by a discussion of the empirical results and by considering extensions.
Methods
- 1.
Compute the the F-test statistics ) from the observed (non-permuted) data.
- 2.
From the b-th (b = 1, ..., B) permutation sample compute a permutation replicate .
- 3.
Single-step procedure to control the FWER:
- (a)
From the b-th permutation data, calculate .
- (b)
For gene j, calculate the adjusted p-value as .
- (c)
For a specified FWER level α, consider gene j significant if .
Results
We investigate the performance of our method by conducting an extensive simulation study. This will be followed by a discussion of the application of our method to the Caenorhabditis elegans dauer developmental data discussed in [1]. These discussions will be limited to the two-sample case (i.e., K = 2). For notation brevity, we will refer to genes whose time-trajectory depends on the factor as prognostic and non-prognostic otherwise. Similarly, we will refer to the genes whose corresponding FWER P-value is less than the given nominal level as significant or non-significant otherwise. For all of these illustrations, as in [6], we will employ a B-spline basis with knots placed at the 0, 1/(p - 1),2/(p - 1), ..., (p - 2)/(p - 1), 1 quantiles of the observed time points pooled across both samples. For these illustration we set p = 4. Our method is developed within the framework of regression for estimation, and permutation resampling for the inference. For additional notational brevity we will adopt the acronym -PERM to refer to our method. To put the discussions in comparative perspective, alongside our results, we will provide those obtained by the permutation method of [7] and the bootstrap method by [6]. Given that these are regression methods, we will denote them by -PERM and -BOOT respectively. All of these analyses are carried out using the R statistical environment [14]. The function rq from the quantreg package [11] is used to estimate the quantile regression functions.
Simulation Study
The first term, μ_{ kj }(t_{ l }), denotes the time-trajectory function at time t_{ l }for group k. For non-prognostic genes, we will set μ_{1j}(t) = μ_{2j}(t) = 0 while for prognostic genes, we will specify μ_{1j}(t) = 0 and μ_{2j}(t) = 1.5e^{ -t }respectively. The error terms are mutually independent and identically distributed according to a standard normal law. The second term in the model, a_{ klij }, represents the random outlier which will assume a value of 4, 0, or -4 with probabilities π/2, 1 - π and π/2 respectively. In the case of a normal law, the mean and median coincide. As such, in the absence of the outlier effect (i.e., a_{ klij }= 0 almost surely, or equivalent π = 0) the quantile function g_{ kj }(t) = μ_{ kj }(t) for all t > 0.
For these simulations, we will adopt a design similar to the Caenorhabditis elegans dauer developmental data, by choosing 11 time points t = 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, and 12. We will generate four replicates at each time-point from each group (i.e., n_{1l}= n_{2l}= 4 for all l).
Illustration of the empirical FWER based on a nominal two-sided 0.05 level, with m= 200 genes, N= 200 simulation samples and B= 200 resampling replications per sample. The outlier law realizes values -4,0 or 4 with probabilities π/2, 1 - π and π/2 respectively. Under simulation model 1, the outlier effect is added to all time-points. Under simulation model 2, the outlier effect is only added to the first and last time-points.
Simulation 1 | Simulation 2 | ||||||
---|---|---|---|---|---|---|---|
π | ρ | -PERM | -PERM | -BOOT | -PERM | -PERM | -BOOT |
0 | 0 | 0.035 | 0.060 | 0.060 | |||
0.3 | 0.040 | 0.035 | 0.035 | ||||
0.6 | 0.040 | 0.040 | 0.040 | ||||
0.05 | 0 | 0.040 | 0.040 | 0.040 | 0.065 | 0.075 | 0.155 |
0.3 | 0.040 | 0.055 | 0.055 | 0.050 | 0.065 | 0.160 | |
0.6 | 0.060 | 0.045 | 0.055 | 0.075 | 0.055 | 0.130 | |
0.1 | 0 | 0.035 | 0.040 | 0.050 | 0.035 | 0.040 | 0.355 |
0.3 | 0.040 | 0.045 | 0.055 | 0.050 | 0.075 | 0.355 | |
0.6 | 0.060 | 0.045 | 0.045 | 0.050 | 0.055 | 0.295 |
As shown in Table 1, under the simulation model 1, where the outlier effects are identically distributed over the time-points, all three methods control the type I error rate. Under simulation model 2, where only the first and last time-points are contaminated by the outlier effect, the -BOOT method however fails to control the type I error rate. In this case, the type I error rate based on the -BOOT method is seemingly inflated by a factor of three when π = 0.05 or by a factor of six when π = 0.1. This can be explained by the fact that the parametric regression bootstrap is based on the assumption of homoscedasticity of the error terms.
Under simulation model 1, the error terms as the sum of the outlier and measurement error components, although no longer normally distributed, are identically distributed within and among all time-points and groups. Under simulation model 2, the error terms are no longer homoscedastic. As such, it is not surprising that the -BOOT method may not be adequate in this situation.
To investigate the global power (i.e., the probability of rejecting at least one null hypothesis) of this procedure, we generate 10 prognostic genes and 190 non-prognostic genes. A correlation structure similar to that of the FWER case is specified. The corresponding results are reported in Table 2.
Illustration of the empirical Power based on a nominal two-sided 0.05 FWER level, with m= 200 genes, N= 200 simulation samples and B= 200 resampling replications per sample. The outlier law realizes values -4,0 or 4 with probabilities π/2, 1 - π and π/2 respectively. Under simulation model 1, the outlier effect is added to all time-points. Under simulation model 2, the outlier effect is only added to the first and last time-points.
Simulation 1 | Simulation 2 | ||||||
---|---|---|---|---|---|---|---|
π | ρ | -PERM | -PERM | -BOOT | -PERM | -PERM | -BOOT |
0 | 0 | 0.960 | 0.995 | 0.995 | |||
0.3 | 0.880 | 0.940 | 0.925 | ||||
0.6 | 0.750 | 0.855 | 0.845 | ||||
0.05 | 0 | 0.910 | 0.800 | 0.790 | 0.890 | 0.880 | 0.970 |
0.3 | 0.835 | 0.755 | 0.755 | 0.845 | 0.825 | 0.900 | |
0.6 | 0.680 | 0.630 | 0.605 | 0.835 | 0.805 | 0.935 | |
0.1 | 0 | 0.760 | 0.485 | 0.480 | 0.895 | 0.740 | 0.930 |
0.3 | 0.675 | 0.415 | 0.425 | 0.660 | 0.585 | 0.885 | |
0.6 | 0.650 | 0.410 | 0.370 | 0.730 | 0.530 | 0.895 |
we evaluated the empirical power and FWER under the simulation model 1 and 2 for π = 2. The result of π = 2 have a similar trend the results of π = 0.05 and π = 0.1 under simulation model 1 and 2. We also evaluated the different proportions for 20 prognostic genes (10%) and 180 non-prognostic genes (90%) and 5 prognostic genes (2.5%) and 195 non-prognostic genes (97.5%). These results have a similar trend the result of 10 prognostic genes (5%) and 190 non-prognostic genes (95%).
Case Study
In this section, we will summarize the results from applying our proposed method to the Caenorhabditis elegans dauer data. Wang and Kim [1] use cDNA microarrays to profile gene expression time-trajectories during the transition from the dauer state to the non-dauer state (experimental group) and after feeding of starved first laval stage worms (control group). The cDNA microarray expressions were measured on m = 18, 556 genes. For the experimental group, the worms are harvested at 0, 1.5, 2, 3, 4, 5, 6, 7, 8, 10, and 12 hours after feeding with three to four replicates at each time-point. For the control group, are harvested 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, and 12 hours after feeding withe four replicates at each time-point. For this illustration, we have set the t = 1.5 time-point in the experimental group to t = 1. This data set is available for download from http://cmgm.stanford.edu/~kimlab/dauer/.
Discussion
where ∫(g''(t))^{2}dt is a so called penalty term. The amount of smoothing is determined by the parameter λ ∈ (0, ∞). The estimation procedure used in this paper is based on a similar regularization approach where the terms (y_{ i }- g(t_{ i }))^{2} are replaced by ρ(y_{ i }- g(t_{ i })) and the penalty term ∫(g''(t))^{2}dt is replaced ∫|g''(t)|dt.
Conclusion
We proposed a robust method for identifying genes whose time trajectories depend on a phenotypic or experimental factor. Furthermore, we proposed a multiple testing procedure to adjust for multiplicity. Our method is based on regression type estimator. Through an extensive simulation study, we observed our method accurately control the FWER and the mean model by regression is not robust to the outliers.
Declarations
Acknowledgements
The authors thank reviewers and the associated editor for insightful comments, which substantially improved the paper. Partial support for this research was provided by grant from the National Cancer Institute, CA142538 [KO, SHJ, SLG].
Authors’ Affiliations
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