Improving the power for detecting overlapping genes from multiple DNA microarray-derived gene lists
- Xutao Deng^{1, 2},
- Jun Xu^{1} and
- Charles Wang^{1, 2}Email author
https://doi.org/10.1186/1471-2105-9-S6-S14
© Deng et al; licensee BioMed Central Ltd. 2008
Published: 28 May 2008
Abstract
Background
In DNA microarray gene expression profiling studies, a fundamental task is to extract statistically significant genes that meet certain research hypothesis. Currently, Venn diagram is a frequently used method for identifying overlapping genes that meet the investigator's research hypotheses. However this simple operation of intersecting multiple gene lists, known as the Intersection-Union Tests (IUTs), is performed without knowing the incurred changes in Type 1 error rate and can lead to loss of discovery power.
Results
We developed an IUT adjustment procedure, called Relaxed IUT (RIUT), which is proved to be less conservative and more powerful for intersecting independent tests than the traditional Venn diagram approach. The advantage of the RIUT procedure over traditional IUT is demonstrated by empirical Monte-Carlo simulation and two real toxicogenomic gene expression case studies. Notably, the enhanced power of RIUT enables it to identify overlapping gene sets leading to identification of certain known related pathways which were not detected using the traditional IUT method.
Conclusion
We showed that traditional IUT via a Venn diagram is generally conservative, which may lead to loss discovery power in DNA microarray studies. RIUT is proved to be a more powerful alternative for performing IUTs in identifying overlapping genes from multiple gene lists derived from microarray gene expression profiling.
Background
Nowadays many microarray-based studies adopt complex experimental design involving multiple treatments, cell lines/tissues, multiple dosages, time points, phenotypes and so on [1–4]. These studies are often involved with complex research hypotheses. For instance in one of our previous studies [1], we were interested in identifying differentially expressed genes (DEGs) in responding to common bile duct ligation in bone marrow stem cells (BMSCs) compared against primary hepatocytes. Two DEG sets from BMSCs and hepatocytes were identified respectively, and the overlapping genes across the two cell types were obtained. The overlapping genes produced across the two cell types allowed the identification of common biological pathways, ontological classes, and biological mechanisms across the two cell types in responding to the treatment.
The intersection operations on multiple gene lists are equivalent to performing multiple tests for the combined hypotheses on every single gene. Although there are many statistical tests proposed for gene expression studies [5–8], the problems of obtaining overlapping gene sets based on multiple tests were overlooked in microarray-based studies. To obtain genes that satisfy the specific hypotheses, researchers simply overlap the gene sets from multiple gene sets and visualized them in Venn diagrams. However, because of lacking multiplicity adjustment, this procedure overlooks the changes of statistical properties, i.e., power, type 1 error rate, p-values, during the intersection operations. This type of multiple testing for finding overlapping genes is known as the Intersection-Union Test (IUT). Despite some early efforts [9–11], the statistical properties and adjustment algorithms of IUT are not well established. Berger has proved that IUT without multiplicity adjustment is a level-α test [10], when the individual tests were controlled at type 1 error rate α. However, the family wise error rate (FWER) for IUT α' is generally much smaller than α. Therefore, performing IUT without multiplicity adjustment would be very conservative and result in too many false negatives.
In this paper, we show that current overlapping operation, applying no p-value adjustment for IUT, is overly conservative in general. As a result, current microarray studies suffer from low power in detecting overlapping genes and therefore limit its use in biological data mining. We developed an analytical solution, named as Relaxed IUT (RIUT) for the multiplicity adjustment of IUTs under certain conditions. We theoretically proved that our proposed method is a less conservative and more powerful than current approaches. We demonstrated the superiority of RIUT for detecting overlapping genes in simulated data sets and complex microarray-based toxicogenomic studies.
Results
Monte-Carlo simulation results of RIUT
As an example to showcase the power of RIUT, the mRNA expression of a given gene is tested whether it is significantly altered by a drug treatment in multiple tissues. Suppose gene expressions were measured in m different tissues and one is interested in the overlapping DEGs. For each tissue, a two-sample t-test is performed between a treatment group and a control group, each containing n replicates to obtain a list of significant genes for that tissue. Then we have an IUT that is constructed by m individual tests, each for a different tissue:
H_{0i}: the drug has no effect in the i th tissue, i.e., μ_{ ti }= μ_{ ci },
H_{ Ai }: the drug has effect in the i th tissue, i.e., μ_{ ti }≠ μ_{ ci }, 1 ≤ i ≤ m,
where μ_{ ti }and μ_{ ci }denote the expression mean of the treatment and the control groups respectively of the i th tissue. The hypotheses for IUT are H_{0}: the drug shows no effect on at least one tissue vs. H_{ A }: the drug shows effect on all tissues.
Monte-Carlo estimates of type 1 error rate α^{'}(%)
μ _{t 2} | RIUT | BIUT | Fisher | Stouffer |
---|---|---|---|---|
0.0 | 0.047 | 0.003 | 0.050 | 0.050 |
0.5 | 0.044 | 0.006 | 0.090 | 0.085 |
1.0 | 0.035 | 0.015 | 0.210 | 0.188 |
1.5 | 0.035 | 0.027 | 0.405 | 0.327 |
2.0 | 0.040 | 0.039 | 0.632 | 0.476 |
2.5 | 0.049 | 0.048 | 0.821 | 0.621 |
3.0 | 0.048 | 0.048 | 0.931 | 0.729 |
3.5 | 0.047 | 0.047 | 0.982 | 0.808 |
4.0 | 0.051 | 0.051 | 0.995 | 0.875 |
4.5 | 0.048 | 0.048 | 0.999 | 0.907 |
5.0 | 0.046 | 0.046 | 1.000 | 0.930 |
Monte-Carlo estimates of power (%)
μ _{t 2} | α = 0.05 | α = 0.01 | ||
---|---|---|---|---|
RIUT | BIUT | RIUT | BIUT | |
0.5 | 0.057 | 0.014 | 0.007 | 0.001 |
1.0 | 0.059 | 0.029 | 0.006 | 0.003 |
1.5 | 0.077 | 0.064 | 0.008 | 0.006 |
2.0 | 0.085 | 0.081 | 0.014 | 0.012 |
2.5 | 0.101 | 0.101 | 0.020 | 0.020 |
3.0 | 0.103 | 0.103 | 0.023 | 0.023 |
3.5 | 0.109 | 0.109 | 0.030 | 0.030 |
4.0 | 0.105 | 0.105 | 0.024 | 0.024 |
4.5 | 0.112 | 0.112 | 0.027 | 0.027 |
5.0 | 0.104 | 0.104 | 0.024 | 0.024 |
Simulation results of pooled instances
γ | K _{0} | K _{1} | RIUT | BIUT | ||
---|---|---|---|---|---|---|
V (type 1) | S(power) | V (type 1) | S(power) | |||
0.1 | 9899 | 101 | 482(0.049) | 12(0.119) | 24(0.002) | 1(0.010) |
0.2 | 9607 | 393 | 447(0.047) | 39(0.099) | 32(0.003) | 6(0.015) |
0.3 | 9087 | 913 | 394(0.043) | 98(0.107) | 29(0.003) | 8(0.009) |
0.4 | 8427 | 1573 | 363(0.043) | 126(0.080) | 33(0.004) | 21(0.013) |
0.5 | 7560 | 2440 | 310(0.041) | 202(0.083) | 30(0.004) | 28(0.011) |
0.6 | 6470 | 3530 | 270(0.042) | 264(0.075) | 22(0.003) | 35(0.010) |
0.7 | 5023 | 4977 | 209(0.042) | 317(0.064) | 23(0.005) | 59(0.012) |
0.8 | 3534 | 6466 | 119(0.034) | 410(0.063) | 10(0.003) | 83(0.013) |
0.9 | 1893 | 8107 | 75(0.040) | 483(0.060) | 10(0.005) | 84(0.010) |
Identifying overlapping genes that respond to multiple drug treatment
This example illustrates how RIUT algorithm can be used in real DNA microarray-based multiple-testing problems. Originally generated from the MAQC project [2], this data set consists of rat RNA samples that came from six treatment/tissue groups. The treatment/tissue groups were aristolochic acid/liver, aristolochic acid/kidney, riddelliine/liver, comfrey/liver, control/liver and control/kidney. There were six biological replicates in each treatment/tissue group. mRNA expression profiles were obtained using four commercial platforms including Affymetrix (Rat Genome 230 2.0), Agilent (Whole Rat Genome Oligo Microarray, G4131A), Applied Biosystems (Rat Genome Survey Microarray) and GE Healthcare (RatWhole Genome Bioarray, 300031) in five different labs with two labs using the Affymetrix microarray platform. Totally, 180 chips were obtained and the cross-platform probe-mapping gave rise to 4609 genes commonly detected across four platforms.
Number of differentially expressed genes in all drug treatment groups
Platforms | IUT : A | IUT : B | IUT : C | |||
---|---|---|---|---|---|---|
RIUT | BIUT | RIUT | BIUT | RIUT | BIUT | |
Applied Biosystems | 1160 | 1011 | 1143 13% | 1008 | 940 | 763 |
Agilent | 362 | 262 | 525 | 413 | 452 | 159 |
GE Healthcare | 697 | 528 | 862 | 723 | 639 | 415 |
Affymetrix (Site 1) | 359 | 251 | 502 | 382 | 422 184% | 175 |
Affymetrix (Site 2) | 524 | 375 | 724 | 556 | 521 | 289 |
Detecting genes with time-course and dose-response effect to chemical treatment
To further demonstrate the applicability of RIUT in microarray studies, we focus on the rat cadmium toxicogenomic data set [3, 18]. This study employed a more complex study design, in which both gene expression and cytotoxicity changes were profiled in a multi-dose multi-time-point setting. Briefly, primary rat hepatocytes were isolated and were exposed to three different doses of cadmium acetate (0, 1.25 and 2.0 μM) for 2 h. Cells were collected at 0, 3, 6, 12 and 24 h in all three groups (0, 1.25 and 2.0 μM Cd) for cytotoxicity evaluation by lactase dehydrogenase (LDH) leakage as well as for mRNA expression profiling by DNA microarray. Affymetrix GeneChip^{®} oligonucleotide arrays (RatTox U34) were used for mRNA expression profiling. There are 972 probe sets representing ~800 important toxicology-related genes in the RT U34 array. The microarray experiment was repeated using primary hepatocytes from 3 animals, each with 2 replicates (independent cultures) for each dosage (3 dosage levels) at each time point (5 time points), resulting in a total of 90 chips (3 animals • 2 replicates • 3 doses • 5 time points). The 2 replicates were averaged in our analysis.
Firstly, to identify differentially expressed genes in responding to cadmium treatment at each time point, two-sample t-tests were performed between the treatment (1.25 and 2.0 μM Cd) and control at each time point. Secondly, to identify the genes with persistent differentially expression due to cadmium exposure across different time points, the overlapping of the DEGs at both short term (3 h) and long term (12 h) were identified using our proposed method. The second research question was then formulated an IUT problem which could be solved using our proposed method.
KEGG pathways Identified and the enrichment p-values
KEGG Pathways | Fisher's Exact Test P-Value |
---|---|
Metabolism of xenobiotics by cytochrome P450 | 0.004 |
MAPK signaling pathway | 0.028 |
Porphyrin and chlorophyll metabolism | 0.030 |
Discussion
RIUT can be improved in three ways to be applicable in more scenarios: (1) estimating unknown nuisance parameters; (2) dealing with more than 2 individual tests; and (3) combining non-independent tests. For the scenario (1), the current approach for estimating π with an arbitrary λ was originally proposed by Storey [14] for estimating false discovery rate. The authors proposed a bootstrap method for finding optimal λ which can also be used here. It should be noted that Theorem 2 is valid no matter how the λ is chosen and how good π is estimated. For IUTs consisting of more than 2 individual tests, it is difficult to obtain the analytical solution as Theorem 1. However, we can apply a step-up procedure which agglomeratively applies RIUT on its least significant individual test. It is more difficult to extend our procedure for non-independent tests. It may need a resampling-based algorithm to incorporating correlation structure of multiple tests and dealing with non-normality issues. Resampling needs intensive computation which can be largely offset by today's powerful and inexpensive computing facility. Resampling of IUT is based on resampling of individual tests which can be conveniently performed by either bootstrap or permutation. Bootstrap and permutation were discussed in many literatures [12, 27, 28].
Conclusion
Our study demonstrated that the current unadjusted IUT approaches were overly conservative, which resulted in loss of power in finding overlapping genes in microarray-based gene expression studies. Our proposed RIUT was analytically proved to be a more powerful and less conservative approach than the current unadjusted IUT. The power improvement is more apparent in tests with weak and moderate effect sizes. This is also demonstrated in Monte-Carlo simulations and real case studies. In addition, certain known biologically relevant pathways were identified using the RIUT-derived overlapping genes which were not detected by using the traditional BIUT.
Appendix
Let X denote the random vector of data values. Suppose the probability distribution of X depends on an unknown parameter θ. The set of possible values for θ will be denoted by Θ. Suppose we have m individual tests and let R_{ i }denote a rejection region for a level-α test of H_{0i}: θ ∈ Θ_{ i }versus H_{ Ai }: θ ∈ ${\Theta}_{i}^{c}$, 1 ≤ i ≤ m, where Θ_{ i }is a specified subset of Θ and ${\Theta}_{i}^{c}$ is its complement. Then IUT tests the union of sets against an intersection of sets. ${H}_{0}:\theta \in {\Theta}_{0}={\displaystyle {\cup}_{i=1}^{m}{\Theta}_{i}}$ versus ${H}_{A}:\theta \in {\Theta}_{0}^{c}={\displaystyle {\cap}_{i=1}^{m}{\Theta}_{i}^{c}}$, with the rejection region $R={\displaystyle {\cap}_{i=1}^{m}{R}_{i}}$. In other words, the IUT rejects only if all of the tests reject.
Berger's Theorem: IUT with rejection region R is a level-α test of H_{0} versus H_{ A }.
Proof. For any θ ∈ Θ_{0} and for any 1 ≤ l ≤ m, we have ${P}_{\theta}(R)={P}_{\theta}({\displaystyle {\cap}_{i=1}^{m}{R}_{i}})\le {P}_{\theta}({R}_{l})\le \alpha $. Therefore, is IUT is a level-α test. ▪
Methods
IUT and UIT
Suppose a gene on which a number of α-level hypothesis tests were performed, represented as t_{1}, t_{2},..., t_{ m }, where m is the number of individual tests. Each test t_{ i }tests the null hypothesis H_{0i}versus alternative hypothesis H_{ Ai }. We can combine all tests into a Union-Intersection Test (UIT) which rejects if any of the t_{ i }rejects. We can also combine all tests into an Intersection-Union Test (IUT) which rejects if all the t_{ i }reject. The UIT tests the hypothesis H_{0} = {all H_{0i}are true} against H_{ A }= {at least one H_{0i}is false} and the IUT tests the hypothesis H_{0} = {at least one H_{0i}is true} against H_{ A }= {all H_{0i}are false}. IUT and UIT were named from the fact that their null and alternative hypothesis can be described by set intersections and unions. (see Appendix for details).
The FWER of UIT is defined as α' = Pr(Reject at least one H_{0i}| all H_{0i}are true). It is well known that in general α' ≠ α. For example, if α = 0.05 and m = 5, α' would be about 0.23 when all individual tests are independent. Therefore there is a need to adjust α' for IUT and there exist many procedures to do so, ranging from simple Bonferroni correction to computer-intensive resampling-based correction [12, 13]. The FWER IUT is α' = Pr(Reject all H_{0i}| at least one H_{0i}is true). It is also obvious that α' ≠ α for IUT in general. However unlike the well studied UIT, there is no known procedure for adjusting the FWER α' for IUT. The unadjusted IUT, also known as the Berger's approach, denoted as BIUT, suggests that the overall unadjusted p-value for IUT is
p = max p_{ i }, 1 ≤ i ≤ m, (1)
where p_{ i }is the p-value for individual tests t_{ i }. Berger proved [10] that the unadjusted IUT is a level-α test if all t_{ i }are level-α tests. Berger's also showed that the above IUT is a size-α test under certain trivial case such as the case when exactly one H_{0i}is true while all the other H_{0i}are false. However, the unadjusted approach is not a size-α test in general. For example, when considering two independent individual tests t_{1} and t_{2}, the chance of rejecting both hypotheses is α^{2} rather than α if both H_{01} and H_{02} are true. Nonetheless, due to its simplicity, the unadjusted IUT approach was implicitly adopted by current microarray studies when overlapping genes were taken from several significant gene lists. This BIUT is equivalent to the Venn diagram in obtaining overlapping genes from multiple significant gene lists.
Exact solution for IUT consisting of independent tests
where P_{j} denote the distribution for the p-value of the j th hypothesis under null hypothesis for IUT H_{0} = {at least one H_{0i}is true}. The least significant p-value p is an observed statistic and the random variable max P_{ j }is the test statistic under H_{0}. This definition is intuitive as p-value measures the probability of false positive under null hypothesis, where a false positive of IUT means that all the p-values under H_{0} are less than the observed p. Similar to the above equation, we have the following relationship between the IUT FWER α', and individual test type 1 error rate α.
α' = Pr(all P_{ j }≤ α |H_{0}) 1 ≤ i, j ≤ m (3)
Unlike UIT, the null hypothesis H_{0} of IUT is a composite hypothesis and contains nuisance parameters. However under certain conditions, it is possible to derive the analytical solution for α'. We obtain the following theorems:
Theorem 1
where the true probabilities of alternative hypotheses are Pr(H_{A 1}) = π1 and Pr(H_{A 2}) = π2; the type 2 error rate for t_{1} and t_{2} are β1 and β2.
▪
where p = max(p_{1}, p_{2}).
Theorem 2
The RIUT procedure is universally at least as powerful as the unadjusted IUT, such that p' ≤ p.
▪
where n is the total number of genes, λ is a chosen fixed value at 0.25 and p_{ i }(j) represents the observed p-value for the i th test on the j th gene.
Declarations
Acknowledgements
This article has been published as part of BMC Bioinformatics Volume 9 Supplement 6, 2008: Symposium of Computations in Bioinformatics and Bioscience (SCBB07). The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/9?issue=S6.
Authors’ Affiliations
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