Identification of gene expression patterns using planned linear contrasts
- Hao Li^{1}Email author,
- Constance L Wood^{1},
- Yushu Liu^{1},
- Thomas V Getchell^{2, 4},
- Marilyn L Getchell^{3, 4} and
- Arnold J Stromberg^{1}
https://doi.org/10.1186/1471-2105-7-245
© Li et al; licensee BioMed Central Ltd. 2006
Received: 27 October 2005
Accepted: 05 May 2006
Published: 05 May 2006
Abstract
Background
In gene networks, the timing of significant changes in the expression level of each gene may be the most critical information in time course expression profiles. With the same timing of the initial change, genes which share similar patterns of expression for any number of sampling intervals from the beginning should be considered co-expressed at certain level(s) in the gene networks. In addition, multiple testing problems are complicated in experiments with multi-level treatments when thousands of genes are involved.
Results
To address these issues, we first performed an ANOVA F test to identify significantly regulated genes. The Benjamini and Hochberg (BH) procedure of controlling false discovery rate (FDR) at 5% was applied to the P values of the F test. We then categorized the genes with a significant F test into 4 classes based on the timing of their initial responses by sequentially testing a complete set of orthogonal contrasts, the reverse Helmert series. For genes within each class, specific sequences of contrasts were performed to characterize their general 'fluctuation' shapes of expression along the subsequent sampling time points. To be consistent with the BH procedure, each contrast was examined using a stepwise Studentized Maximum Modulus test to control the gene based maximum family-wise error rate (MFWER) at the level of α_{ new }determined by the BH procedure. We demonstrated our method on the analysis of microarray data from murine olfactory sensory epithelia at five different time points after target ablation.
Conclusion
In this manuscript, we used planned linear contrasts to analyze time-course microarray experiments. This analysis allowed us to characterize gene expression patterns based on the temporal order in the data, the timing of a gene's initial response, and the general shapes of gene expression patterns along the subsequent sampling time points. Our method is particularly suitable for analysis of microarray experiments in which it is often difficult to take sufficiently frequent measurements and/or the sampling intervals are non-uniform.
Background
Recent advances in DNA microarray technologies have made it possible to investigate the transcriptional portion of gene networks in a variety of organisms. When microarray experiments are performed to monitor gene expression over time, researchers can address questions concerning the detection of the cellular processes underlying the observed regulatory effects, inference of regulatory networks and, ultimately, assignment of functions to the genes analyzed in the time courses.
There is a natural connection between gene function and gene expression. Based on our understanding of cellular processes, genes that are contained in a particular pathway, or respond to a common internal or external stimulus, should be co-regulated and consequently, should show similar patterns of expression. Therefore, identifying patterns of gene expression and grouping genes into expression classes may provide much greater insight into their biological functions. A large group of statistical methods, generally referred to as "cluster analysis", have been developed to identify genes that behave similarly across a range of experimental conditions, including time courses. These statistical algorithms can be divided into two classes, depending on whether they are based on 'similarity' measures or not. Methods based on 'similarity' measures rely on defining a distance (or 'dissimilarity') between gene expression vectors; Euclidean distance and/or the Pearson correlation coefficient are the two most commonly used distance measures. Examples of similarity measures-based methods are hierarchical clustering [1], k-means [2], self-organization maps (SOM) [3, 4], and support vector machine (SVM) [5]. These methods do not consider the temporal structure of the data when used to analyze time-course experiments. In addition, some methods could confuse the clusters because the actual expression patterns of the genes themselves become less relevant as clusters grow in size [6].
The clustering methods in the second class are based on statistical models, without defining a 'similarity' measure. Using statistical models to represent clusters changes the question from how close two data points are to how likely a given data point is under the model. Such clustering methods are more commonly used to analyze time-course microarray experiments. Examples of such methods are based on cubic spline [7], ANOVA model [8], autoregressive curves [9], first-order kinetics [10], Hidden Markov Models [11, 12], Bayesian model average [13], order-restricted inference methodology [14], and Gaussian Mixture Models [15–19]. Such approaches may be restricted either by the rigorous assumptions of the stochastic models [9, 11, 12], or by the small number of time points and non-uniform sampling intervals in gene expression data [7, 9, 10].
In gene networks, the level of expression of individual genes changes based on their functional position in the network. Therefore, the most critical information in time course expression profiles is the timing of the changes in expression level for each gene [10], and secondarily is the general shape of its expression pattern [20, 21]. In addition, different genes will be activated or inactivated at each level of a gene network. Therefore it may not be reasonable to expect that the expression levels of those co-expressed genes will go up and down concordantly all the way through the entire sampling period. With the same timing of initial change, genes which share similar pattern of expression for any number of sampling intervals from the beginning should be considered co-expressed at certain level(s) in the gene network. However, statistical methods to analyze these patterns have not yet been reported.
Attention to the multiplicity problem in gene expression analysis has been increasing. Numerous methods are available for controlling the family-wise type I error rate (FWER). Since microarray experiments are frequently exploratory in nature and the sample sizes are usually small, Benjamini and Hochberg [22] suggested a potentially more powerful procedure, the false discovery rate (FDR), to control the expected proportion of errors among the identified differentially expressed genes. A number of studies for controlling FDR have followed [23–29]. In microarray experiments with multi-level treatments, the multiple testing problems are two dimensional. Not only are thousands of genes involved, but for each gene, either pre-selected contrasts or post-hoc comparisons may be needed to characterize its expression pattern. There are very few studies that have investigated how to deal with such multiple-testing problems in the microarray literature [30].
In this manuscript, we propose a different strategy based on planned linear contrasts (pre-selected contrasts) for the analysis of time-course microarray experiments. Specifically, our approach takes into consideration the temporal order in the data, including the timing of a gene's initial response and the general shapes of gene expression patterns along the subsequent sampling time points. Our methods are particularly suitable for analysis of microarray experiments in which it is often difficult to take sufficiently frequent measurements and/or the sampling intervals are non-uniform. We demonstrated our method on the analysis of microarray data from murine olfactory sensory epithelia at five different time points after target ablation.
Results
Example of genes from different classesThree genes from each of the 4 classes were selected to illustrate their expression patterns. P_F: P values for the overall F test; P_t: P values at their initial responses; FC: fold changes at their initial responses, where a negative sign indicates down-regulation.
Class | Gene | P_F | P_t | FC | Gene Function |
---|---|---|---|---|---|
I | Pdcd5 | 5.40E-03 | 6.30E-04 | 1.2 | apoptosis |
Cetn3 | 7.40E-05 | 3.10E-04 | 1.2 | Ca binding | |
Kit | 3.40E-03 | 5.30E-04 | 1.4 | growth factor | |
II | Ccl2 | 3.20E-05 | 3.70E-04 | 4.1 | chemotaxis |
Csf3 | 6.30E-03 | 5.80E-04 | 3.2 | growth factor | |
Bub3 | 2.10E-04 | 1.20E-04 | 1.2 | cell cycle | |
III | Omp | 1.40E-04 | 1.60E-05 | -1.6 | marker protein |
Ptdss2 | 9.00E-05 | 8.00E-04 | -1.4 | enzymatic activity | |
Tfrc | 2.10E-03 | 3.60E-04 | 2.1 | endocytosis | |
IV | Casp6 | 4.50E-03 | 6.60E-04 | -1.4 | Apoptosis |
Cd68 | 4.40E-04 | 2.90E-05 | 2 | macrophage marker | |
Slfn4 | 1.50E-04 | 7.30E-06 | 4 | cell cycle |
Discussion
In this study, we adopted linear models to describe our data and used planned linear contrasts to analyze time-course microarray experiments. We identified 1234 genes with significant changes in expression in a microarray study of murine olfactory epithelium, and 1105 of them were grouped into 4 classes based on the timing of their initial changes. We further categorized these 1105 genes into 41 fluctuation patterns. We also used simple diagrams to illustrate these fluctuation patterns and a series of characters (1, 0, -1) to index these patterns. Although the ANOVA F tests were significant, 129 genes cannot be grouped into any of these 4 classes based on our criteria. A significant ANOVA F test among a group of means indicates that the largest contrast among all possible contrasts is significant. Therefore, a gene with a significant F test does not necessarily have a significant selected contrast. Therefore the expression patterns of these genes should be interpreted carefully.
The critical value $\left|{M}_{{\alpha}_{new},m-2,v}\right|$ used to select significant contrasts is the uniform upper bound for testing a complete set of contrasts regardless of the correlation structure among these contrasts. It is a conservative approach. For planned linear contrasts, the most powerful bound can be found based on the correlation structure of these contrasts [31, 32]. In general, the most powerful bound can't be obtained without knowing the correlation structure among the contrasts [33]. The uniform bound, however, can be obtained from testing a complete set of orthogonal contrasts using the Studentized Maximum Modulus Distribution [34]. In practice, although a little bit conservative, it is straightforward to use this uniform bound to test all contrasts especially when the number of different combinations of contrasts is large.
Our methods emphasized the relative differences between adjacent sampling time points and the direction of the differences. The information about exact magnitudes of gene expressed at each time point was not included in our methods. For example, two genes may have the same pattern index 0 1 -1 0 0, but the magnitude of changes for the two genes may be dramatically different. Therefore, even for genes in the same index groups, their expression patterns should be examined with care.
The temporal order in the data was considered in our methods by the selection sequence but was not parameterized in our model. The information about the differences among sampling intervals were also ignored in our analysis. With small sample sizes and non-uniform sampling intervals, which are very common in biomedical research, our methods may be more straightforward and robust than those commonly in use. With large sample sizes and relative uniform sampling intervals, other methods, such as regression analysis, mixture models, or autoregressive models can be applied.
Conclusion
Linear models were adopted to describe microarray data, and sequences of planned linear contrasts were used to group genes into different expression patterns based on their initial and subsequent changes in expression. Our methods are particularly suitable for analysis of microarray experiments in which it is often difficult to take sufficiently frequent measurements and/or the sampling intervals are non-uniform. Our methods can also be extended to designs with more than one factor.
Methods
Microarray experiments
The goal of this study was to investigate the induction of gene regulation at short time intervals (2, 8, 16, and 48 hrs) following deafferentation of olfactory sensory neurons by target ablation (olfactory bulbectomy, OBX) compared with sham controls [35]. Total RNA was isolated from the olfactory epithelium of 3 male mice per time point (1 GeneChip/mouse). Following hybridization with Affymetrix GeneChips MG U74Av2, 3 chips per time point (a total of 15 GeneChips), the signal intensities were generated by Affymetrix Microarray Suite v5.0.
In our study, all positive control genes and genes that resulted in "absent" calls for all chips across all time points were removed from further analysis. If there was no evidence that these genes were expressed in any of the samples, then these genes can be removed to reduce problems associated with multiple comparisons. Other methods of removing low intensity points were also suggested by Bolstad et al., 2003[36]. All ESTs were also removed from the analysis because the primary aim of these experiments was to identify known genes that were differentially regulated; eliminating ESTs further reduced problems with multiple comparisons. After data filtering steps, 6464 genes remained, and the background-corrected intensities of these genes were subjected to further statistical analyses.
Algorithm and analysis
Statistical model
We use a linear model to describe the experiment. Let Y_{ g }be the vector of observed expression levels for gene g, g = 1, ..., 6464 then
Y _{ g } = Xβ _{ g } + ε _{ g }
where X is the matrix of known constants, β_{ g }= (μ_{g 1}, μ_{g 2}, ..., μ_{ gm }), and m is the number of time points (m = 5 in this study). ε_{ g }is the random error, and we assume ε_{ g }~ MVN(0, σ_{ g }^{ 2 }I ).
Reverse Helmert series
A contrast is a linear combination of parameters for which the coefficients sum to zero. A complete set of orthogonal contrasts is a set of k-1 contrasts in k treatments (or treatment combinations) which provides a complete partitioning of the variability among parameters into mutually exclusive and exhaustive parts. Each contrast in such a set is orthogonal to every other remaining one [37]. One commonly used complete set of orthogonal contrasts is the reverse Helmert series, in which one treatment group is compared with the average of all remaining treatment groups. Subsequent contrasts eliminate the first group and then proceed by comparing one of the remaining groups to the average of the other remaining groups, as show below:
One of the advantages of the Reverse Helmert contrasts is that these contrasts are orthogonal and, hence, contrasts among the sample means are uncorrelated. Basing tests on uncorrelated contrasts avoids the problems inherent in interpreting conditional tests. Adjacent Differences(AD) are sometimes used to identify the point at which initial gene expression occurs. However, these contrasts are not orthogonal and consecutive contrasts have a correlation of 0.5. Consequently, the probability of identifying the correct threshold is lower for AD than for the Reverse Helmert Contrasts.
Clustering genes based on the timing of their initial responses
The reverse Helmert series can test the following m-1 hypotheses sequentially:
Genes will be partitioned into m-1 classes based on the testing results of H_{10} ~ H_{s 0}, where s = m-1. Class 1 contains genes that reject H_{10}; genes that reject H_{20} from the remaining list are grouped into class 2, and so on; Class s includes genes that reject H_{s 0}without rejecting the previous s-1 hypotheses. Therefore Genes in Class 1 are considered to be early responding genes whose expression levels are significantly altered during the first sampling interval, that is, at the 2^{nd} sampling time point. Genes that do not change their expression levels until the 3^{rd} sampling time point are collected in Class 2, and so on. As indicated by the described partition process, genes within a class share the same timing of onset or cessation of expression.
Clustering genes within a class
Genes in each of these above m-1 classes can be further classified based on their 'fluctuation' shapes at the subsequent sampling points. For gene g in class j, where j = 1, 2, ..., s, the following s-j contrasts are needed,
Therefore, a specific sequence of m-1 hypotheses will be performed for each gene to determine its expression pattern. Let ${\widehat{\lambda}}_{g}^{j,k}$ be the unbiased estimate of the contrast corresponding to the hypothesis ${H}_{g}^{j,k}$, in a balanced experiment with sample size n in each treatment group, the statistic
where c_{ i }is the ith coefficient for the contrast, and i = 1, ..., m. MSE_{g} is the usual unbiased estimate of ${\sigma}_{g}^{2}$, and v = N-m is the error degree of freedom (df), where N = mn.
Indexing gene expression patterns
Let the state of the first observation be 0, for gene g, its expression profile can be transformed into a sequence of expression fluctuation as follows:
where k = 1, 2, ..., s is an index, where k = r if it is the rth contrast for gene g. S is the transformed value of the gene expression profiles. Thus an m-time-point expression profile is transformed into an m-1-state sequence of expression fluctuation consisting of a character set (1, 0, -1). Each character in the sequence indicates whether the mean expression level of the gene is significantly up-regulated (1), not altered (0), or significantly down-regulated (-1) at the next time point, while the whole sequence represents the fluctuation pattern of the gene expression. Besides the pattern in which the gene's expression level is unchanged throughout the entire sampling period, there are at most 2 × 3^{m-k-1}fluctuation patterns for genes in Class k. There are no more than 3^{m-1}patterns of expression in total in an m-time-point microarray experiment.
Multiple testing control
An ANOVA F test was performed for each gene to identify the differentially expressed genes. This F test is testing the hypothesis μ_{g 1}= μ_{g 2}, ..., = μ_{ gm }, which is equivalent to test the composite hypothesis Lβ = 0. The BH procedure of controlling FDR at 5% was applied to the P values of the F test. A cutoff point α_{ new }, which is equal to the largest P value considered to be significant, was determined by the above BH procedure. By this procedure, each test for the gene that rejected the F test is at least α_{ new }level test.
For each selected gene, a specific sequence of m-1 contrasts was tested to determine its expression pattern. To be consistent with the BH procedure performed, the family-wise error rates (FWER) for these genes have to be controlled at least at the level of α_{ new }. In this study, we controlled the maximum family-wise error rates (MFWER) by using the Studentized Maximum Modulus distribution [34]. The following theorem in the Appendix outlined the concern of gene-based controlling MFWER at the level of α_{ new }.
Outline of the analysis
- 1.
Linear models were used to describe the data based on the experimental design. For each gene, an ANOVA F test was performed based on the described model, and the corresponding P-value was obtained.
- 2.
To adjust for multiple tests based on the large number of genes, the BH method of controlling FDR [22] at 5% was applied to the P-values obtained above, providing a list of genes (list I) that exhibit significant differences among the means of the 5 sampling points.
- 3.
Using α_{ new }, which equals the largest P-value determined to be significant in step 2 as the cut-off point, we grouped genes in list I into 4 classes based on the timing of their initial responses by testing the reverse Helmert contrasts sequentially. The Studentized Maximum Modulus distribution parameter m-2 = 3 and v = 10 were used in this example study, where α_{ new }= 0.009545 and $\left|{M}_{{\alpha}_{new},m-2,v}\right|$ = |M_{0.009545,3,10}| = 3.8651.
- 4.
Using the same critical value $\left|{M}_{{\alpha}_{new},m-2,v}\right|$, we further clustered genes in each of the above classes by testing appropriate contrasts for the subsequent sampling time points.
- 5.
Based on the results of the m-1 contrasts for each gene, we also can select genes which share similar pattern of expression for any number of sampling intervals from the beginning.
Statistical software
We used the SAS (version 9.0) proc GLM procedure to do model fitting and significance analysis. The SAS program implementing linear models for the olfactory sensory epithelia data is available [38].
Appendix
Theorem For any balanced one-way model with m treatment groups and assuming normality and equal variance σ^{2}, Let λ_{1}, λ_{2}, ..., λ_{ k }be an arbitrary complete set of contrasts such that
Under the null hypothesis, let P be the distribution of the vector λ= [λ_{1}, λ_{2}, ..., λ_{ K }] with mean 0 and covariance matrix ∑, let P_{ K }be the distribution of the vector of a complete set of contrasts $\lambda *=[{\lambda}_{1}^{*},{\lambda}_{2}^{*},\dots ,{\lambda}_{K}^{*}]$ with the covariance matrix ∑_{ K }= Iσ^{ 2 } is the diagonal of ∑ then the gene based maximum family-wised error rate (MFWER) at any level of α of testing a specific sequence of contrasts (list in the Methods) after rejecting the overall F test is achieved by comparing |T_{ i }| with |M_{α,k-1,v}|, where v is the df of error,
and |M_{α,k-1,v}| is the 100(1 - α) percentile from the Studentized Maximum Modulus distribution.
Proof Let λ_{1}, λ_{2}, ..., λ_{ K }be any complete set of contrasts, then let
V_{0} = {0 ≤ i ≤ K: λ_{ i }= 0} and
V_{1} = {0 ≤ j ≤ K: λ_{ j }≠ 0},
let test function
(1) Suppose V_{1} is empty such that λ_{ i }= 0 ∀i, then MFWER is
(2) V_{0} is empty such that λ_{ i }≠ 0 ∀i, then MFWER is 0.
(3) Suppose that neither V_{0} nor V_{1} is empty, then MFWER is
under P_{ K }, based on Sidak's inequality (8) [39], $\underset{i\in {V}_{0},{v}_{0}=K-1}{\mathrm{max}}\left|{T}_{i}\right|$ has a Studentized Maximum Modulus distribution [34] with parameter K-1 and v, let
MFWER = α*(α, m - 1, v) = max{α(α, m - 1, v)} = α,
then q = |M_{α,K-1,v}| is the 100(1-α) percentile from above distribution.
Declarations
Acknowledgements
This work was supported by NIH AG-016824 (TVG) and NIH-P20RR16481 and NSF-EPS-0132295 (AJS). We also wish to thank Donna Wall, Microarray Core Facility, and Radhika Vaishnav, M.S., Department of Physiology, for their expertise.
Authors’ Affiliations
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