Report

Achim Tresch UoC / MPIPZ Cologne Statistics treschgroup.de/OmicsModule1415.html [email protected] 1 Multiple Testing Challenge • You test plants/patients/… in two settings (or from different populations). • You want to know which / how many genes are differentially expressed (alternate) • You don’t want to make too many mistakes (declaring a gene to be alternate = differentially expressen when in fact they are null – not differentially expressed). The Multiple Testing Problem • You choose a significance level, say 0.05. • You calculate p-values of the differences in expression. • The p-value of g is the probability that if g is null (not differentially expressed), it would have a test statistic (e.g., t-statistic) at least that large. • You say all genes that differ with p-value ≤ 0.05 are truly different. What’s the problem? The Multiple Testing Problem You are testing many genes at the same time • Suppose that you test 10,000 genes, but no genes are truly differentially expressed. • You will conclude that about 5% of those you called significant are differentially expressed. • You will find 500 “significant” genes. • Bad. The Multiple Testing Problem Bonferroni Correction Bonferroni Correction (FWER control) Pr(at least one gene found diff.expr.) Bonferroni controls the probability by which our list of differentially expressed genes contains at least one mistake = Family-wise error rate (FWER). This is very strict. False Discovery Rate (FDR) estimation A Fundamental Insight • All truly null genes (i.e. not truly differentially expressed) are equally likely to have any p-value. • That is by construction of p-val: under the null hypothesis, 1% of the genes will be in the top 1 percentile, 1% will be in percentile between 89 and 90th and so on. P-val is just a way of saying percentile in null condition. 0 p-value 1 False Discovery Rate (FDR) estimation Idea: The observed p-value distribution is a mixture of • null genes (light blue marbles) and truly different genes (red marbles). • If the chosen test is appropriate, red marbles should be concentrated at the low p-values. Differential gene 0 p-value Non-Differential gene 1 False Discovery Rate (FDR) estimation • We don’t of course know the colors of the marbles/we don’t know which genes are true alternates. • However, we know that null marbles are equally likely to have any p-value. • So, at the p-value where the height of the marbles levels off, we have primarily light blue marbles/null genes. False Discovery Rate (FDR) estimation • Because if all genes/marbles were null, the heights would be about uniform. • Provided the reds are concentrated near the low p-values, the flat regions will be primarily light blues. Absolute frequency • We estimate the baseline of null marbles 0 p-value 1 ≈nondifferential genes False Discovery Rate (FDR) estimation Absolute frequency • Subtracting the “baseline” of true null hypotheses, the remaining balls are primarily red, i.e., they are true alternative hypotheses ≈ differential genes 0 p-value 1 ≈nondifferential genes False Discovery Rate (FDR) estimation • Given a p-value cutoff, we can estimate the rate of false discoveries (FDR) that pass this threshold. Absolute frequency FDR(p-cut) = + ≈ differential genes 0 p-value cutoff p-value 1 ≈nondifferential genes FDR-based p-value cutoff • Given a desired FDR (e.g., 20%), we can find the largest p-value cutoff for which this FDR is achieved. Absolute frequency FDR(p-cut1)= 9% Baseline of nulls 0 p-cut1 = 0.1 p-value 1 FDR-based p-value cutoff • Given a desired FDR (e.g., 20%), we can find the largest p-value cutoff for which this FDR is achieved. FDR(p-cut1)= 9% Absolute frequency FDR(p-cut1)= 20% Baseline of nulls 0 p-cut1 p-cut1 = 0.1 = 0.2 p-value 1 FDR-based p-value cutoff • Given a desired FDR (e.g., 20%), we can find the largest p-value cutoff for which this FDR is achieved. FDR(p-cut1)= 9% Absolute frequency FDR(p-cut1)= 20% FDR(p-cut3)= 52% Baseline of nulls 0 p-cut1 p-cut1 = 0.1 = 0.2 p-cut1 p-value = 0.7 1 FDR-based p-value cutoff • Given a desired FDR (e.g., 20%), we can find the largest p-value cutoff for which this FDR is achieved. FDR(p-cut1)= 9% Absolute frequency FDR(p-cut1)= 20% FDR(p-cut3)= 52% Baseline of nulls 0 p-cut1 p-cut1 = 0.1 = 0.2 p-cut1 p-value = 0.7 1 Example: All null • Consider the all null case (all marbles are blue). • For any p-value cutoff, the estimated FDR will be close to 100%. • For any sensible FDR (substantially below 100%), there will be no suitable p-value cutoff, and the method will not return any gene. • Good. 0 p-value 1 Examples: All alternate • Consider the all alternate case (all marbles are red). • For a large range of p-value cutoffs, the estimated FDR will be close to 0. • For sensible FDR cutoffs (e.g. 20%), the corresponding p-value cutoff will be high. • The method will return many genes • Good. 0 p-value 1 Conclusions • A flat p-value distribution may force us to the far left in order to get a low False Discovery Rate. • This may eliminate genes of interest. • If subsequent validation experiments are not too expensive, we can accept a higher False Discovery Rate (e.g., 20%) • FDR rate and significance level are entirely different things! Gene Set Enrichment Fisher‘s exact test, once more Fisher‘s exact test, once more Gene Ontology Example 559 Gene Ontology Example (immune response) (macromolecule biosynthesis) Kolmogorov-Smirnov Test < 10-10 • Move 1/K up when you see a gene from group a • Move 1/(N-K) down when you see a gene not in group a GO scoring: general problem GO Independence Assumption GO sets light yellow GO Independence Assumption light yellow The elim method The elim method Top 10 significant nodes (boxes) obtained with the elim method Algorithms Summary Evaluation: Top scoring GO term Significant GO terms in the ALL dataset Advantages & Disadvantages for ALL Simulation Study Introduce noise Simulation Study Simulation Study Quality of GO scoring methods 10% noise level 40% noise level Summary