Gene set enrichment analysis for nonmonotone association and multiple experimental categories
 Rongheng Lin^{1}Email author,
 Shuangshuang Dai^{2},
 Richard D Irwin^{3},
 Alexandra N Heinloth^{4},
 Gary A Boorman^{5} and
 Leping Li^{1}Email author
DOI: 10.1186/147121059481
© Lin et al; licensee BioMed Central Ltd. 2008
Received: 25 July 2007
Accepted: 14 November 2008
Published: 14 November 2008
Abstract
Background
Recently, microarray data analyses using functional pathway information, e.g., gene set enrichment analysis (GSEA) and significance analysis of function and expression (SAFE), have gained recognition as a way to identify biological pathways/processes associated with a phenotypic endpoint. In these analyses, a local statistic is used to assess the association between the expression level of a gene and the value of a phenotypic endpoint. Then these genespecific local statistics are combined to evaluate association for preselected sets of genes. Commonly used local statistics include tstatistics for binary phenotypes and correlation coefficients that assume a linear or monotone relationship between a continuous phenotype and gene expression level. Methods applicable to continuous nonmonotone relationships are needed. Furthermore, for multiple experimental categories, methods that combine multiple GSEA/SAFE analyses are needed.
Results
For continuous or ordinal phenotypic outcome, we propose to use as the local statistic the coefficient of multiple determination (i.e., the square of multiple correlation coefficient) R^{2} from fitting natural cubic spline models to the phenotypeexpression relationship. Next, we incorporate this association measure into the GSEA/SAFE framework to identify significant gene sets. Unsigned local statistics, signed global statistics and onesided pvalues are used to reflect our inferential interest. Furthermore, we describe a procedure for inference across multiple GSEA/SAFE analyses. We illustrate our approach using gene expression and liver injury data from liver and blood samples from rats treated with eight hepatotoxicants under multiple time and dose combinations. We set out to identify biological pathways/processes associated with liver injury as manifested by increased blood levels of alanine transaminase in common for most of the eight compounds. Potential statistical dependency resulting from the experimental design is addressed in permutation based hypothesis testing.
Conclusion
The proposed framework captures both linear and nonlinear association between gene expression level and a phenotypic endpoint and thus can be viewed as extending the current GSEA/SAFE methodology. The framework for combining results from multiple GSEA/SAFE analyses is flexible to address practical inference interests. Our methods can be applied to microarray data with continuous phenotypes with multilevel design or the metaanalysis of multiple microarray data sets.
Background
Microarray technology profiles the expression levels of thousands of genes simultaneously, providing a snapshot of transcript levels in the cells/tissues being studied. Analysis of microarray data typically involves examining thousands of genes with relatively small sample sizes, and thus is challenging for statisticians [1, 2]. Moreover, deriving useful biological knowledge from these gene sets is often a long and arduous task. Recently, however, methods have been developed that incorporate existing knowledge of biological pathways/processes into the analysis. For example, gene set enrichment analysis (GSEA) tests whether a known set of genes is associated with a phenotypic difference [3]. In GSEA, an enrichment score (ES) for a predefined gene set, usually a known biological pathway/process, is calculated and statistical significance is evaluated. GSEA was refined by [4] and a more general framework, significance analysis of function and expression (SAFE) was proposed by [5]. Furthermore, ES vectors for gene sets can be defined for individual samples and used for classification [6]. In these procedures, an association measure between a phenotypic endpoint and expression levels is calculated and subsequently used as the basis for further evaluation of the association between gene sets and the same phenotypic endpoint. Additional work on GSEA type methods can be found in [7–11].
Numerous statistics have been proposed to identify differentially expressed genes when the phenotype is binary [12–17]. All these statistics can be used to measure the association between gene expression levels and a binary phenotype. When a phenotypic endpoint is continuous, correlation analysis between gene expression and the phenotypic variable emerges naturally. A correlation measurement needs to be selected based on the goal of the investigation, biological interpretation, statistical properties and computational feasibility. The commonly used Pearson productmoment correlation coefficient measures linear association, whereas the nonparametric Spearman correlation measures monotone trend. A brief summary of several commonly used correlation measurements and their limitations can be found in [18]. Alternatively, the standardized Ztype statistics resulting from univariate Cox proportional hazard model has been proposed as the local statistics for the situation where continuous survival time with censoring is the phenotypic endpoint [5].
In biological research using microarray technology, gene expression data are often collected under multiple experimental conditions, along with traditional pathological endpoints [19]. One practical inferential goal is to identify biological pathways/processes that are associated with the endpoint and thus gain insight into biological mechanisms of tissue response to the experimental conditions. Under the framework of GSEA/SAFE, inference is based on assessing the association between expression levels of predefined sets of genes and the phenotypic endpoint. When associations between the phenotypic endpoint and the gene expressions are likely to be nonlinear or nonmonotone, the local statistics, which is the measurement of gene specific association to the endpoint, should be able to capture these, while a linear or monotone situation should still be accommodated. We propose to use the coefficient of multiple determination R^{2} from a natural cubic spline model [20]. The spline model allows us to capture potential nonlinear and nonmonotone trends. The quantity R^{2} is then used as the correlation in GSEA/SAFE to identify gene sets that are associated with the phenotypic endpoints.
When experimental conditions include multiple categories, different biological mechanisms might be expected and separate association analyses for each category are desired before aggregating to arrive at the final conclusion. We carry out separate enrichment analysis in each category and propose an ad hoc procedure to combine the results.
In this article, the words "association" and "correlation" are used interchangeably in most situations.
Results and discussion
We describe the methods for separate analysis in each category in Section 2.1 and 2.2, then inference across categories in Section 2.3. The methods are then illustrated using National Center of Toxicogenomics (NCT) compendium data in Section 2.4.
The relevant R code is available at http://people.umass.edu/rlin/niehs/r2/R2GSEA.R.
2.1 Nonlinear association measurement
In all methods for the evaluation of gene sets discussed in the introduction, the association measure between a gene set and an endpoint is built from the association measures between individual genes and the endpoint. We would like a gene specific association measure to have several properties. First, it should be scale free, i.e., it would not change when data are linearly transformed. This property guarantees that in data preprocessing, different bases in log transformation (e.g., log_{2} or log_{10}) give the same inferential conclusion. Second, it should accommodate nonlinear or nonmonotone associations between variables. Third, it should allow limited data points. Fourth, it should be easy to compute and interpret. These considerations exclude the usage of many nonlinear correlation functions and measures, e.g., [21]. We propose to use the coefficient of multiple determination R^{2} in natural cubic spline regression models to measure the gene specific association. This quantity can be interpreted as the fraction of the variation in endpoints that can be explained by the gene expression data. When the association is linear, this quantity is approximately equal to the square of Pearson correlation.
to assess the association between X_{ j }and Y, where $\widehat{y}({x}_{ij})={\widehat{f}}_{c,{g}_{j}}({x}_{ij})$ can be estimated by regression method accommodating possible nonlinear trends. Here we use natural cubic splines, which are cubic splines with linear extension out of the boundary knots [20, 22]. This flexible approach guarantees that both nonlinear and linear trends are well captured. When a linear trend is the only desired type of association, we can simply change the splines into linear terms and the proposed methods can still be used.
The calculation of the quantity ${R}_{c,j}^{2}$ has two main steps:

Specify inside knots at b evenly spread quantiles of X_{ j }. Generate Bspline basis matrix for natural cubic spline B(X_{ j })_{n × (b+2)}from X_{ j }(b + 1 spline terms and one intercept term). We used ns() function in the splines package of R[23].

Regress Y on B(X_{ j }) and obtain the ${R}_{c,j}^{2}$ value. We use QR decomposition routine qr() in R directly to avoid extraneous computation for other statistics such as coefficient estimates, standard errors, etc., that would occur if using the linear model regression routine lm().
The piecewise cubic splines approach, unlike the regular b + 1 degree polynomials, automatically provides the roughness penalty [24], which control the degree of ripples in fitted curve. Gene specific quantiles are easy to compute and guarantee enough sample points between knots. Some alternative ways, e.g., using global quantiles of all X_{ j }, do not have this property and would make poor fitting for some genes. For the number of knots to be used in splines model, "a minimum of 5 points should be between knot locations" has been suggested [25]. The selection of b depends on the available sample size and desired flexibility in capturing nonlinear trend.
2.2 Adapted GSEA procedure
Given a gene set in an array, there have been two types of set specific global statistics [8]: "selfcontained" one using only genes in the set and "competitive" one using all genes in the array. Approaches proposed in [26, 27] belong to the first type and GSEA/SAFE approach belongs to the second type. Computationally, when there are only a small number of gene sets to test, "selfcontained" approaches are much faster by avoiding the genes out of the sets.
GSEA method is designed to test whether a prebuilt gene set is associated with the phenotype. Its kernel KolmogorovSmirnov statistic detects the difference of gene positions in the ranked gene list between one gene set and its complementary set. The method can thus highlight a gene set that has similar associations (and ranks) for each gene in the set. When the identified set of genes cluster close to the bottom of the ranked list ordered by association strength, the set is likely to be of limited interest to biologists [27–29]. Also, the method is biased toward those wellstudied gene sets [8].
Belonging to the "selfcontained" type, Goeman et al. (2004)'s approach [26] proposed to fit a random effect model for each gene set and has the potential to accommodate nonlinear terms conveniently. It assumed the coefficients of genes are independent and otherwise a proper covariance matrix needs to be set up. When used with nonlinear terms, the independence assumption thus need to be justified or more exploration on the proper covariance matrix specification will be expected. Once independence assumption holds or proper covariance matrix can be specified, the approach will be very attractive in computation.
GSEA/SAFE (and SAMGS [27]) calculate one measurement for each gene and then for each gene set, accumulate the genes' signals into one for the whole gene set. For each gene, there is only one association measurement. Goeman et al. (2004) fits models directly for each gene set. The coefficients of the same gene in different gene sets could then be different and do not have direct interpretation of association strength, while there exist many other advantages for "self contained" approaches [8].
We construct the global statistics measuring the association between the phenotype and the prebuilt sets of genes based on ${R}_{c,j}^{2}$ as local statistics. We adapt the modified version of GSEA [4] where genes with stronger association are assigned higher weight. The method does not require the independence assumption between genes. In each category c we define enrichment score (ES) for gene set s:

Rank all J genes to form list L = {g_{1}, ..., g_{ J }} decreasing in ${R}_{c,j}^{2}$.

Evaluate the fraction of genes in s ("hits") weighted by ${R}_{c,j}^{2}$ and the fraction of genes not in s ("misses") present up to a given position k in L.$\begin{array}{ccc}{P}_{hit}(s,c,k)& =& {\displaystyle \sum _{{g}_{j}\in s,j\le k}\frac{{R}_{c,j}^{2}}{{N}_{R}},\text{where}{N}_{R}={\displaystyle \sum _{{g}_{j}\in s}{R}_{c,j}^{2}}},\hfill \\ {P}_{miss}(s,c,k)& =& {\displaystyle \sum _{{g}_{k}\notin s,j\le k}\frac{1}{J{N}_{s}},\text{where}{N}_{s}\text{isthenumberofgenesin}s.}\end{array}$

ES(s, c) is the signed maximum deviation from zero of P_{ hit }(s, c, k)  P_{ miss }(s, c, k) over k.
For a ranked gene list based on ${R}_{c,j}^{2}$, our inferential interest is on the top of the list, rather than both extremes as when ranking is based on tstatistics or Pearson correlations. This feature makes those sets with both positively and negatively correlated genes easier to identify than if using signed correlation measurements. Gene sets clustering at the top of the list will have positive maximum deviation and those clustering at the bottom of the list will have negative maximum deviation. Consequently, we use one sided pvalues rather than twosided pvalues to indicate the significance of gene sets. A gene set cluster at the bottom of the list, which is off our interest, will have a pvalue close to 1 and will not be highlighted. A signed local statistics, unsigned ES and twoside pvalues were used in [4].
We calculate these pvalues with permutations. We permute the phenotypic endpoint variable Y 1000 times to generate 1000 permuted data sets and calculate the ES values for each permutation. These ES values form the null distribution of ES to estimate the pvalue of the observed ES value. This permutation method retains the correlation structure among the genes. More discussion on the permutation details is deferred to Section 2.4.2.
2.3 Inference across experimental categories
In the standard framework of GSEA/SAFE, analysis is carried out only for one category. However, with multiple categories of experimental settings, "identifying biological pathways" that are associated with phenotypic changes in most categories" is a practically important inferential goal. When association patterns differ under different categories, pooling data is not a good option for either biological interpretation or statistical inference.
For a gene set s, separate analyses in C categories generate a pvalue vector of length C. At first glance, Fisher's method of combining pvalues [30] could be used along with false discovery rate (FDR) to select a threshold [31, 32]. However, the goal of our method is to identify gene sets that are related to phenotypic changes for most of the categories, while Fisher's method could lead to very high significance even if strong association is observed in only one category. Therefore, we propose a subjective but stringent criterion to evaluate gene sets across categories.
where P_{s,(M)}is the M^{ th }smallest of the C pvalues P_{s,1}, P_{s,2}, ⋯, P_{s, C}. By adjusting the value of p_{0}, we can control the number of sets that meet the criteria.
An example given in Section 2.4 calculates α(p_{0}) in similar way for more complicated practical inferential goal. One benefit of this approach is that it does not require pvalues in very high precision, avoiding computational burden in permutation based hypothesis testing. See Section 2.4.2 for further discussion.
where S is the number of gene sets and A is the cardinality of A. The bound is inflated because it assumes that none of S gene sets are associated with the phenotypic changes. When A = 0, i.e., none of the gene sets is identified, FDR is 0 by definition. Since in most of cases, α(p_{0}) is so small that this bound is small enough for practical usage. When more advanced FDR control methods are desired, extra assumptions, e.g., the independence between gene sets might be needed.
We note that under the global null hypothesis, it is straightforward to show that random variable P_{s,(M)}follows the Beta(M  1, C  M) distribution. We can use pvalues of observed P_{s,(M)}to select gene sets and evaluate the type I error rate, which provides an equivalent alternative to equation (3). By this approach, the setspecific FDR can be calculated based on the empirical distribution of setspecific pvalues [31, 32].
One alternative approach for inference across categories is to control the category specific FDR and then combine the results. Depending on the goal, the unions or intersections of C lists of gene sets identified by proper FDR thresholds can be used. However, approaches to estimate the FDR usually require very precise estimation of pvalues [33] and 1000 permutations can produce pvalues with the granularity up to 0.001. Several methods have been proposed on this issue: assuming the normalized null of ES for different sets are the same and pooling them to obtain a global null distribution with much finer granularity [3, 4]; using a linear combination of tstatistics as the global statistic, which asymptotically has normal distribution [7]; using more permutations (eg., 10,000 times) to generate a setspecific null distribution [5]. However, asymptotic results for R^{2} (and thus ES) may not be available as discussed further in additional file 1, section 1. For a large data set with many categories, 10,000 permutations would constitute a substantial computational burden. More importantly, it is unclear how the categorywise FDRs should be combined into one interpretable summary statistic. To the best of our knowledge, no literature is available on this issue.
2.4 Analysis of NCT compendium data
NCT compendium microarray data. In time rows, the numbers from 0 to 3 indicate vehicle, low, medium and high doses.
Compound  1.2.Di  1.4.Di  Brom  Diqu  Gala  Mono  NNitr  Thio 

6 Hrs  0–3  0–3  0–3  0–4  0–3  0–3  0–3  0–3 
24 Hrs  0–3  0–3  0–3  0–4  0–3  0–3  0–3  0–3 
48 Hrs  0–3*  0–3  0–3  0–4  0–3  0–2^{†}  0–3  0–3 
Replicates  4  4  4  6  4  4  4  4 
Array totals  34  36  36  72  36  32  36  36 
We assume that the ALT depends on time and dose in a way that is functionally related to gene expression levels, i.e., e_{ i }in equation 1 contains time and dose information. We let b, the number of inside knots as described in Section 2.1, equal 4. It is close to the maximum number we can afford with 32–72 samples per compound, following the suggestion "a minimum of 5 points should be between knot locations" [25]. With the degrees of freedom equal 5 (4 inside knots), the model is flexible enough to capture major biologically plausible departures from linear or monotone trend. The same spline space has been used in [13] to calculate a gene's posterior "probability" of differential expression through logistic regression.
To build enrichment scores (ES) indicating the associations between ALT and gene sets from R^{2}, prebuilt gene sets are needed. A total of 466 prebuilt gene sets are prepared with methods detailed in additional file 1, section 2. In our implementation, each compound is analyzed separately in each tissue. As described in Section 2.2, we obtain 466 pvalues (one per gene set) for each of 8 compounds per tissue under the null hypothesis that the gene expression levels in the tissue are not associated with ALT for this compound.
Let ${P}_{s,c}^{A}$ be the pvalue of set s for compound c in liver and ${P}_{s,c}^{B}$ in blood. To call one set significant, we require at least 6 of 8 (a subjective definition of "most") pvalues ${P}_{s,1}^{L},{P}_{s,2}^{L},\cdots ,{P}_{s,8}^{L}$ be less than p_{0}, i.e., M = 6, C = 8. In this article, we use p_{0} = 0.05 and 0:1 such that the number of identified gene sets are manageable.
Using formula (3),
α_{ L }(0.05) = α_{ B }(0.05) = 4.0 × 10^{7}, α_{ L }(0.1) = α_{ B }(0.1) = 2.3 × 10^{5}
By definition, we have A_{ LB }(p_{0}) ⊆ A_{ L }(p_{0}), A_{ LB }(p_{0}) ⊆ A_{ B }(p_{0}) and if p_{0} ≤ ${{p}^{\prime}}_{0}$, A_{ l }(p_{0}) ⊆ A_{ l }(${{p}^{\prime}}_{0}$), l = L, B, LB. Generally, we have
α_{ LB }(0.05) ≤ α_{ L }(0.05) = 4.0 × 10^{7}, α_{ LB }(0.1) ≤ α_{ L }(0.1) = 2.3 × 10^{5}
2.4.1 Analysis results
In general, sets are more significant in liver than in blood. Using p_{0} = 0.05, we identified 38 gene sets (A_{ L }(0.05)) in liver and 2 gene sets (A_{ B }(0.05)) in blood. Five sets are identified for A_{ LB }(0.1) while none of them attain the criteria of A_{ LB }(0.05). The FDR bound (Equation 4) for all reported A_{ l }(p_{0}), l = L, B, LB, p_{0} = 0.05, 0.1 are smaller than 4 × 10^{4}. Biological interpretation of these sets are provided in additional file 1, section 3.
We note that by setting p_{0} = 0.05 and 0:1, we have used a relatively conservative approach to select a small number of the most significant sets that represent good candidates for further study. For completeness, we provide all intermediate results from step 2 in the additional file 2 (and available upon request).
We also carried out a twoway (gene expression and ALT) clustering analysis using software Cluster [35] for each of the 5 sets from A_{ LB }(0.1). Heat maps are provided in the additional file 3. One can see that liver and blood have different patterns of gene expression, supporting our procedure of analyzing liver and blood separately. In addition, this difference suggests that it would be difficult to use blood gene expression to predict liver gene expression or vice versa in this data set. Some identified sets contain both genes positively or negatively correlated to ALT, indicating that in calculating ES values for general GSEA/SAFE procedure, genes should be ranked according the absolute value of their associations to the phenotypic endpoint.
2.4.2 Permutation size and method
Permutations have been widely used in microarray data analysis to establish the null distribution of the test statistics [4, 5, 36]. The number of permutations chosen depends on statistical assumptions and the inferential goal and procedure. The aggregation method assuming the same null for all gene sets requires much fewer (1,000) permutations [4] than would a gene set specific null (10,000) [5]. However, the same null assumption does not always hold and when computationally affordable, we prefer to use gene set specific null [5]. In this study, we use gene set specific null approach with only 1000 permutations due to computational restrains. Using the example shown in the manual of R package SAFE [37] in Bioconductor [38] released with [5], we observe that the YekutieliBenjamini's resampling based FDR estimate [39] for a specific set [32] can range from 0.09 to 0.38 with different random seeds and a permutation size 1000. This illustrates the need to use a large number of permutations when setting the threshold based on a quantity requiring a good estimate of the pvalue and motivated us to check whether permutation number 1000 is large enough in our study.
The number of identified sets and overlaps of two runs.
Random seed 1  Random seed 2  Overlap  

A_{ L }(0.05)  38  37  36 
A_{ B }(0.05)  2  5  2 
A_{ LB }(0.05)  0  0  0 
A_{ L }(0.1)  74  70  68 
A_{ B }(0.1)  28  30  27 
A_{ LB }(0.1)  5  5  5 
We randomly permuted the outcome variable ALT to generate the null distribution of ES. However, for a particular compound, samples taken at the same time point shared one control group. This raises the possibility that the observed association between ALT and gene expression might be influenced by the resulting statistical dependency. To evaluate this possibility, we redid the permutation while keeping the dependency structure. For example, in case of compound 1,4dichlorobenzene, there are 3 dose levels and 3 time points, 4 replicates for each dosetime combination. The 12 samples for each time point sharing the same controls comprise a group. We randomly matched groups of ALT levels with groups of gene expression levels. There are a total of 3! = 6 possible group matches. In each matched group of 12 samples, we then randomly permuted ALT levels. Among the eight compounds, two did not have balanced subgroups for this type of permutation. We thus used only the other 6 compounds and ran this twostage permutation 1000 times to generate 1000 "null" data sets per compound per tissue.
Since we only have 6 compounds in the new permutations, we cannot apply the "6 of 8" criterion to identify associated sets in the new permutation results. However, the Pearson correlation and Spearman rank correlation of P (and Z) values between the two permutations in each compound and tissue are all higher than 0.96, indicating the strong consistency of results between the two different permutation strategies. Thus our identified sets based on pvalues would be almost the same with the two permutations methods.
Conclusion
We present an inferential framework for selecting gene sets that are associated with a continuous phenotypic endpoint for microarray data with multiple categories of experimental conditions. We proceed in three steps: 1) compute gene specific R^{2} between gene expression and endpoints for each individual category; 2) score the association of sets of genes with endpoints based on the gene specific association measurements; 3) combine category specific inference results to identify sets of genes that are associated with endpoint in most of the categories. When the phenotype is binary, many statistics used for identifying differentially expressed genes can be used in step 1. However, these statistics might not work well for continuous and nonlinear/nonmonotonic relationships between endpoint and gene expression. Based on natural cubic spline regression, our proposed R^{2} not only captures nonlinear associations between endpoint and gene expression but also accommodates any existing linear association between the two variables. In step 2, we adapt the framework of GSEA/SAFE using our R^{2} as the association measurement. The combination of steps 1 and 2 can be regarded as a special case of the GSEA/SAFE procedure in a generalized sense. In step 3, we combine the results from multiple categories for each gene set and give a conservative FDR upper bound. This step depends on a subjective choice of threshold that can be tailored to the inferential goal. Different thresholds would result in different numbers of significant sets.
Assuming that the experimental information is fully represented by gene expression levels (Equation 1), we calculate the gene specific R^{2} separately for each category. For some genes, this assumption might be violated if the curves relating gene expression level and endpoints are different for the same category at different experimental conditions. In this case, a pooled approach inside each category might yield reduced R^{2} values. However, limited sample size of the category might not allow splitting the data into subcategories. Also, the curve (X_{ ij }, Y_{ i }) itself is driven by the experimental conditions, e.g., time and dose. If we fix all these factors, then pairs (X_{ ij }, Y_{ i }) may distribute around an average point and we would not be able to capture the main association trend.
We use the quantity R^{2} to screen strong associations between endpoint and gene expression levels rather than to select the best model for predicting the endpoint from the gene expression level. Since the spline's degrees of freedom is fixed, it is not necessary to use a penalty term to control overfitting. Finer model adjustment for purposes beyond screening can be considered after associated gene sets are identified.
We illustrate our method using the NCT compendium data in which the expression values from 20,500 genes in both rat liver and blood were analyzed at 3–4 dose levels and 3 time points for 8 hepatotoxicants. We are interested in identifying predefined gene sets that are associated with liver injury indicated by the ALT activity level in blood for most of the 8 hepatotoxicants. For this compendium data, we did observe nonlinear association between ALT and gene expression. This might be due to the fact that ALT level increases as the degree of liver injury increases and, on the other side, the gene expression levels can be either upregulated or downregulated.
In the data example, we fixed 4 inner knots at 20%, 40%, 60%, and 80% quantiles of the gene expression. This model space was selected by manually examining scatter plots between gene expression levels and ALT levels. We believe that this model space is rich enough to account for biologically plausible nonlinear/nonmonotonic trend. If extra degrees of freedom are desired, a graph similar to Figure 2 can be used to evaluate the benefits. Although our proposed R^{2} can capture nonlinear associations between ALT and gene expression and lead to biologically meaningful inferences, computational feasibility restricts us from considering more complicated models, e.g., with two or more genes jointly in the model. While a high R^{2} indicates strong association, a low R^{2} does not always imply no association. Also, weak association may not be well captured by R^{2}. However, this concern is alleviated in GSEA/SAFE procedure, because the method is designed to accumulate the weak local signals from individual genes into a stronger global signal at gene set level.
We compare the R^{2} from a nonlinear model and a linear model in Figure 2 to illustrate the need for using a nonlinear model. Further performance comparison based on identified sets is difficult without knowing the true state of nature of whether a specific gene set is associated with the phenotypic endpoint. Also, we believe that in our case, it will not provide a very helpful reference to pursue a computational simulation, which has to include arguable assumptions that expressions of genes from one pathway follow some convenient multivariate distribution (e.g., multivariate normal distribution) and the outcome variable follows a skew distribution. As shown in the additional file 1, section 1, the null distributions of R^{2} will be very sensitive to these assumptions.
In summary, we describe a framework of GSEA/SAFE for microarray data in which the gene expression data were generated under multiple categories of experimental conditions and the phenotypic endpoint was continuous. The proposed association measure R^{2} successfully captured nonlinear trends between the gene expression levels and endpoint. The R^{2} was then incorporated into the GSEA/SAFE procedure in per category analysis. The usage of R^{2} has the advantage over the tstatistics or Pearson correlation in identifying the gene sets with both genes positively and negatively correlated to endpoints. Inference across categories serves to identify gene sets, and the corresponding functional pathways whose alteration plays a role in related biological mechanism. Our method is general and can be applied to GSEA/SAFE analysis of microarray data with other continuous phenotype or multiple GSEA/SAFE analyses.
Finally, it is important to note that, in GSEA/SAFE analyses, global statistics is usually designed to test whether the distribution of local statistics within a gene set is different from that of genes outside the gene set. Proper local statistics and global statistics should be selected carefully to avoid the situation where a gene set is statistically significant but actually clusters in a region of weak association in the gene list and is off biological interest. In this article, we have used unsigned local statistics R^{2}, signed global statistics and oneside pvalue to eliminate this possibility.
Declarations
Acknowledgements
This research was supported by the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences. We thank the NCT for the compendium data used in this analysis. We thank Clarice Weinberg, David Umbach and Grace Kissling for insightful discussion and for critically reading the manuscript. We thank Shyamal Peddada, Dmitri Zaykin and Bhanu Singh for helpful discussion. We thank Robert Bass for creating the web site for R code downloading. We thank the Computational Biology Facility at NIEHS for computing time. We also thank anonymous referees for helpful comments which greatly improve our work.
Authors’ Affiliations
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