Eshelman School of Pharmacy, University of North Carolina at Chapel Hill, NC, USA

Renaissance Computing Institute, University of North Carolina at Chapel Hill, NC, USA

NHEERL Environmental Carcinogenesis Division, United States Environmental Protection Agency, Research Triangle Park, NC, USA

Department of Psychiatry, University of North Carolina at Chapel Hill, NC, USA

Department of Biostatistics, University of North Carolina at Chapel Hill, NC, USA

Abstract

Background

Improvements in high-throughput technology and its increasing use have led to the generation of many highly complex datasets that often address similar biological questions. Combining information from these studies can increase the reliability and generalizability of results and also yield new insights that guide future research.

Results

This paper describes a novel algorithm called BLANKET for symmetric analysis of two experiments that assess informativeness of descriptors. The experiments are required to be related only in that their descriptor sets intersect substantially and their definitions of case and control are consistent. From resulting lists of n descriptors ranked by informativeness, BLANKET determines **shortlists **of descriptors from each experiment, generally of different lengths p and q. For any pair of shortlists, four numbers are evident: the number of descriptors appearing in both shortlists, in exactly one shortlist, or in neither shortlist. From the associated contingency table, BLANKET computes Right Fisher Exact Test (RFET) values used as scores over a plane of possible pairs of shortlist lengths

Conclusions

Researchers seek within a universe of descriptors some minimal subset that collectively and efficiently predicts experimental outcomes. Ideally, any smaller subset should be insufficient for reliable prediction and any larger subset should have little additional accuracy. As a method, BLANKET is easy to conceptualize and presents only moderate computational complexity. Many existing databases could be mined using BLANKET to suggest optimal sets of predictive descriptors.

Background

In contemporary high-throughput experiments, very many descriptor values can be measured, leading to the issue of correction for multiple testing to minimize false positives at the cost of a high number of false negatives. Reconciliation entails compromises that are to some extent arbitrary. A deterministic method is needed for selecting a minimal, distinguished set of descriptors that collectively provide effective, efficient prediction. Researchers can subsequently investigate members of such a subset to determine exactly how they are related (e.g. are they genetically or chemically related?) and perhaps why they should be inherently associated with predictions (e.g. are some members of the shortlists components of a certain biochemical pathway?).

Meta-analysis is the general body of knowledge that addresses integrating results from multiple experimental programs on one topic; the purpose of this paper is to suggest inclusion of BLANKET as an additional technique

Lists of descriptors ranked by informativeness are often encountered in the general pursuit of relationships among diseases, physiological processes, and the action of small molecule therapeutics. Notable examples include the Connectivity Map by Lamb et al. and the generation of quantitative structure-activity relationships (QSAR)

Regarding genes as descriptors (that is, expression of mRNAs or proteins), a vast, public repository of data that should support discovery of distinguished descriptor lists is supported by the Gene Expression Omnibus (GEO) project. GEO predominantly stores gene expression data generated by microarray technology

Methods

Presented first are two synthetic examples. Suppose the number of distilled descriptors n = 500 and the ranked list for Experiment A is simply labeled 1, 2, ..., 500. Suppose in the ranked list for Experiment B, the first ten are a random permutation of 1, 2, ..., 10, and the other 490 are a random permutation of 11, 12,...,500. We would expect BLANKET to suggest an optimal subset of the first ten just as is shown in Figure

BLANKET applied to comparison of two illustrative lists of 500 descriptors

**BLANKET applied to comparison of two illustrative lists of 500 descriptors**. The first ten from Experiment A appear in scrambled order within the first ten of Experiment B. BLANKET suggests that four combinations of shortlists are sufficiently coincidental to meet a p-value of .05. Note that three of the four selected shortlist pairs (p, q) have unequal numbers of selected descriptors.

As a second synthetic example, suppose Experiment A descriptors again have canonical ordering 1,2,...,500 while Experiment B has the same with local permutations from weighted noise. Figure

BLANKET applied to a second set illustrative lists of n = 500 descriptors

**BLANKET applied to a second set illustrative lists of n = 500 descriptors**. Descriptors in Experiment A are ranked in canonical order 1, 2, ..., 500. To the same ranking, weighted noise is added to arrive at an Experiment B ranking of 7, 26, 17, 32, 21, 34, 12, 46, 49, 14, 57, 54, 67, 19, 61, 28, 1, 15, 82,... BLANKET finds no shortlists of length at most 10 that meet the criteria for significance (p-value ≤ .05, corresponding to RFET < .00105). However, BLANKET finds three shortlist pairs as shown of length < 20 that do meet the same (RFET < .00800). Thus BLANKET, not knowing the effects of noise, would recommend to the researcher these descriptors for further investigation. Note the characteristic sharp decline in RFET values near the chosen shortlists. This example is relevant to the case of one experiment performed with great accuracy and the other with substantial noise. Note that each selected shortlist pair (p, q) has unequal numbers of descriptors selected from both experiments (p ≠ q).

Next the BLANKET method will be used to evaluate data from a classic microRNA (miRNA) microarray paper by He et al.

He et al. measured in quadruplicate for cases and for controls 190 mature miRNA levels for normal B-cells and several cell lines including OCI-Ly4 and OCI-Ly7. Thus Experiment A includes the 190-by-8 output matrix of Normal (control) B-cell miRNA values versus OCI-Ly4 (case) miRNA values, and as Experiment B the same from OCI-Ly7 values. This yields p-values and hence rankings of the two lists of 190 miRNAs.

We tested the hypothesis that the same miRNAs can differentiate control B-cells from both of the two cases OCI-Ly4 and OCI-Ly7 by attempting to find informative subsets of the 190 probes.

The BLANKET for these data is shown as the surface in Figure

BLANKET applied to ranked list date of 190 descriptors from He et al. (He et al. 2005)

**BLANKET applied to ranked list date of 190 descriptors from He et al. (He et al. 2005)**. BLANKET suggests that three combinations of shortlists are sufficiently coincidental to meet a p-value of .05. Note that each selected shortlist pair (p, q) has unequal numbers of descriptors selected from both experiments (p ≠ q).

BLANKET multiple comparison-corrected significance threshold values for p-value 0.05.

**n**

**p, q ≤ 20**

**p, q ≤ 10**

100

0.00424

0.00192

200

0.00549

0.00150

300

0.00666

0.00119

400

0.00657

0.00104

500

0.00800

0.00105

It is already obvious from the heatmap in Figure

This alignment invites investigation because the bases near the 5' terminus (the "seed region") are generally thought by miRNA researchers to be most important in terms of targeting and gene regulation

BLANKET is next applied to suggest shortlists of genes from experiments with lung adenocarcinoma measurements versus control tissue measurements in microarray studies by Stearman et al. and Bhattacharjee et al.

Ranked by lowest p-values, the top 1000 genes in each experiment can be selected. The intersection of the lists can be reranked to a list of 289 genes that are possibly informative in both experiments. BLANKET yields the surface in Figure

BLANKET applied to comparison of 289 genes within lung cancer microarray studies of Stearman and Bhattacharjee

**BLANKET applied to comparison of 289 genes within lung cancer microarray studies of Stearman and Bhattacharjee**. One pair of shortlists with 14 descriptors from the first and eight from the second yields a RFET score = .0044; this is less than .0066, the level that insures a p-value significance level (.05) for any shortlists with twenty or fewer members from a universe of 300 members.

From the Stearman data BLANKET chooses 14 genes: GPC3, SOX4, GRK5, ADH1B, CLEC3B, MFAP4, TEK, FH1, AOC3, TBX2, COX7A1, TGFBR3, MYLK, VWF. BLANKET chooses 8 genes from the Bhattacharjee data: HYAL2, GRK5, SPOCK2, ENO1, SEMA5A, CDH5, VWF, COX7A1. Thus the intersection is {GRK5, COX7A1, VWF} with RFET = 4.42E-03.

Interestingly, several additional papers connect some of the shortlisted genes with lung cancer. Regarding GPC3, Powell et al. used microarrays to identify GPC3 as one of several genes the expression of which was lower in the healthy lung tissue of smokers than in nonsmokers and was lower in tumor tissue than in healthy tissue

In summary, there are potential lung cancer connections with genes in the Stearman-Bhattacharjee BLANKET shortlists. This illustrates the main output of BLANKET, namely, suggestions to researchers of small subsets of genes especially worthy of further investigation.

The next analysis pertains to an instance in which BLANKET does not suggest informative shortlists; this example compares results of Stearman with another lung cancer study by Beer et al.

BLANKET applied to comparison of lung cancer microarray studies of Beer and Stearman

**BLANKET applied to comparison of lung cancer microarray studies of Beer and Stearman**. The method fails to find a threshold pair with low RFET score < .00800, which would be sufficient for shortlists with up to 20 members to have statistical significance in a universe of 489 descriptors. This surface is more organized than random BLANKETs, since there is a sharp decrease from 1 to low values, but it is less organized than those in Figures 3 and 4.

Lastly, BLANKET applied to the third possible combination of tests (Beer-Bhattacharjee) is not interesting. It finds a very small subset pair (2,5). Such small shortlists are evident by inspection because the top two for Beer are ADH1B and CLEC3B while the top five for Bhattacharjee are GPC3, SOX4, GRK5, ADH1B, CLEC3B.

Implementation

Following is a pipeline (for R code see Additional File

**R program for BLANKET**. R program for BLANKET. This program yields a value that can be tested in Table

Click here for file

Considerable preprocessing might be needed to derive the ranked lists. This is because the raw data (e.g. mRNA microarrays) typically have many thousands of descriptors, from which one distills hundreds that are significantly up or down in case versus control; the researcher might wish to treat up- and downregulated genes separately. The intersection of the two lists must be found and then a selection made of the topmost descriptors (such as the top 500) of the two lists. Some of the top 500 in one list might not be in the top 500 of the other, so a second intersection is needed to yield a list of genes with different rankings in the two experiments, that is, somewhat fewer, shared, ranked descriptors suitable for BLANKET. Real data tested in preparation of this paper yielded an intersection n = ~250 to ~450 descriptors.

Using the list of shared descriptors and selecting the top p descriptors from A and the top q descriptors from B yields a contingency table. Over the discrete plane of all possible pairs, RFET values can be represented as a blanket-like surface.

Table

After finding shortlist pairs that provide RFET values lower than the values in Table

All such values and corresponding descriptors should be noted by the researcher. All genes that achieve a level of informativeness discovered in a BLANKET selection might be considered. That is, the union of all the descriptors in the shortlists might be informative, as well as, of course, the intersection. If several pairs of shortlists fulfill this condition, then minimizing the RFET values or minimizing p+q might yield especially interesting shortlist pairs.

Results

If two experiments of case versus control have substantially overlapping descriptor sets and a consistent, binary categorization of outcome, then standard statistical analyses can provide two lists, ranked by informativeness, of the shared descriptors. The ranked lists suggest two questions:

Question 1: From the results of the two experiments, is there a minimal subset of descriptors that predicts experimental outcome much better than smaller subsets and about as well as any larger subsets?

Question 2: If existence of such a minimal subset is indicated, then what are its members?

The focus here is on one method that answers these questions. We call our method BLANKET; this is not an acronym, but merely a term suggestive of a blanket-like surface suspended above a plane of shortlist length pairs.

Suppose two experiments such as microarray analyses investigate informativeness of descriptors relative to a property of samples. Here a descriptor (predictor) is any tested type of measurement, such as detection of messenger RNA of a certain gene in a microarray experiment (a continuous variable) or presence or absence of a certain chemical substructure in a compound evaluated for toxicity (a binary variable). A property of the samples could be case versus control, survival time, or another characteristic or outcome of interest. BLANKET treats the set of shared descriptors as two ranked lists.

The informativeness of each descriptor, considered in isolation, can be determined by a by t-test, z-test, or other method; our only requirement is that informativeness analysis for each experiment yields a ranked list. The two experiments might use the same or different definitions of informativeness.

The basic idea of BLANKET is consideration of **shortlists **of descriptors from each experiment, say the top p of n descriptors of Experiment A and the top q of the same n descriptors of Experiment B. For any such pair of shortlists, four numbers are evident: the number of descriptors appearing in both shortlists, in exactly one shortlist, or in neither shortlist. The sum of all four is n; the sum of the first two is p; and the sum of the first and third is q. BLANKET computes Right Fisher Exact Test (RFET) values used as scores over a discrete plane of all possible pairs of shortlist lengths (so all (p, q) with 0 ≤ p, q ≤ n). BLANKET then chooses one pair or a few pairs with RFET score less than a threshold; the threshold depends upon n and upper bounds of shortlist lengths. The threshold has been determined by simulations and represents a quality of RFET value achieved by only 5% of random lists. A further property of a pair (p, q) selected by BLANKET is parsimony, that is, that no other pair (p', q') exists with p' ≤ p, q' ≤ q, p'+q' < p+q, and an RFET score that also survives the threshold. Multiple shortlists could be scored by smallness of p+q.

Furthermore, we seek to represent the information to the researcher in a visual form such as an Excel spreadsheet surface graph that invites assessment based upon a researcher's experience with data of a given type, much in the manner of the commonplace heatmap.

Theoretical basis

Our approach is to consider the RFET value for all combinations of shortlist lengths 10 or 20 within ranked descriptor lists of length n = 100, 200, 300, 400, or 500. In the grid of lengths, this can be thought of as the examination of all p-by-q rectangles of RFET values within a given nxn square, subject to p ≤ n and q ≤ n. The RFET attaining the minimum nominal p-value is then compared to the null distribution of such minimum p-values, obtained via permutation, which assumes that the orderings of the two lists of descriptors are random. The corresponding 0.05 quantile values are used as rejection thresholds for controlling the overall Type I error at 0.05.

Formally, the approach is the single-step Westfall-Young permutation p-value for potentially correlated tests, which controls the family-wise error and avoids the excessive conservativeness of Bonferroni bounds

Discussion

The term BLANKET reflects the shapes of the surfaces in Figures

To our knowledge, BLANKET is a novel means for nominating distinguished subsets of descriptors from data from two experiments. BLANKET suggests shortlists (subsets) of genes from each list, where the shortlists achieve a certain level of informativeness individually. The subsets then collectively differentiate case from control. While the preprocessing considers the full ranked lists, BLANKET does not make global declarations. That is, BLANKET ignores very uninformative descriptors but can tolerate descriptors with marginal p-values provided they consistently appear among the best found of ranked lists.

Other related scores that might be substituted for the RFET score are Pearson's chi-square test and the G-test

Another meta-analysis paper is that of Blangiardo and Richardson

First is their use of a given number (101) of bins so that a bin could contain all of a subset of descriptors with close p-values. Second, the hypergeometric distribution is the score of the paired bins as shortlists. (By definition, hypergeometric distribution is the chance probability of exactly a given intersection size of subsets of p and q elements from a universe of n elements; RFET is the probability of that number of intersection elements or more, limited by max {min {p, q}}. Thus RFET is a decreasing sum of a finite number of hypergeometric terms, the first of which was the score used by Blangiardo and Richardson.)

Third, and perhaps most importantly, they only scored shortlists of equal length, hence the diagonal of the discrete space of all combinations of bin sizes. By contrast we consider all "rectangular" combinations of shortlists lengths that lie within a larger "square" (such as 20-by-20) of combinations.

The significance of the restriction of consideration to shortlists of equal length can be illustrated as follows. Suppose that two experiments test 100 descriptors for case and control informativeness, providing two ranked lists. Suppose the first experiment is very accurate but the second is not; perhaps the second employs a noisier technology but still might provide a degree of confirmation. Suppose the top four descriptors in the first experiment appear in rank positions 5, 10, 15, 20 of the second, and that no other descriptors in the top twenties are shared. BLANKET correctly selects the top 4 of the first list and the top 20 of the second, with RFET = .00124 < .00424 in Table

Regarding other related papers, we note that Hess and Iyer reported that Fisher's combined p method applied to microarray data from spike-in experiments with RT-qPCR validation usually compared favorably to other methods

Breitling et al., devised the "rank product" method which in simplest form uses multiplication across N experiments of the reciprocal of rank positions of N descriptors, leading to a kind of global ranking

An enhancement of BLANKET in gene expression analysis of microarrays might include consistency of fold change. That is, the researcher might require that the genes in the intersection of shortlists all have fold change > 1 or all have fold change < 1 for case versus control. Doing this for randomly generated ranked lists and random fold changes would result in a table like Table

Conclusions

The BLANKET method provides a visual representation of optimal selections of subsets of informative descriptors. A key observation in our real data is that there can be an abruptly lower (better) RFET score value, going from a plateau of almost 1 to a valley floor of almost 0 values as shortlist lengths are slightly incremented. Furthermore, if upper limits on the shortlist lengths are specified as 10 or 20, then our simulations provide values for RFET scores that allow rejection of the null hypothesis with 95% certainty. In such circumstances, BLANKET can suggest a sharp distinction between slightly too few and slightly too many descriptors, that is, a classifier based upon optimal collective informativeness.

Authors' contributions

All three wrote sections of the paper. CDJ conceived an initial version of the blanket algorithm; WOW executed early applications and R code and contributed much of the text; DOP contributed refinements regarding applications; and FAW contributed the random permutation design and simulations, R code, and theoretical foundations and analysis.

Acknowledgements

This paper reflects several enhancements prompted by critical review of an earlier version; the authors thank the reviewers. Support for our research has included grants from an Anonymous Donor, Stanley Medical Research Foundation grant 08R-1978 "Herpesviruses in Schizophrenia Risk," NIH grant 5P01ES014635-02 "The System of Response to DNA Damage Suppresses Environmental Melanomagenesis," and NIH grant 2R01GM066940-05A1 "Predictive QSAR Modeling." The content is solely the responsibility of the authors, not the funding institutions. This article was reviewed by the National Health and Environmental Effects Research Laboratory, US Environmental Protection Agency, and approved for publication. Approval does not signify that the contents necessarily reflect the views and policies of the Agency nor does the mention of trade names or commercial products constitute endorsement or recommendation for use.