Parameter Estimation for the Exponential-Normal Convolution Model

Transcription

1 Parameter Estimation for the Exponential-Normal Convolution Model Monnie McGee & Zhongxue Chen Department of Statistical Science Southern Methodist University ENAR Spring Meetings, March 26-29, 2006 p.1/19

2 The Affymetrix Chip Human Genome U133 Plus 2.0 Array Courtesy of Affymetrix Some Definitions Probes = 25 bp sequences Probe sets = 11 to 20 probes corresponding to a particular gene or EST Chip contains 54K probe sets ENAR Spring Meetings, March 26-29, 2006 p.2/19

3 Perfect Match vs. Mismatch PM Probe = 25 bp probe perfectly complementary to a specific region of a gene MM Probe = 25 bp probe agreeing with a PM apart from the middle base. The middle base is a transition (A G, C G) of that base ENAR Spring Meetings, March 26-29, 2006 p.3/19

4 Perfect Match vs. Mismatch PM Probe = 25 bp probe perfectly complementary to a specific region of a gene MM Probe = 25 bp probe agreeing with a PM apart from the middle base. The middle base is a transition (A G, C G) of that base Image Courtesy of Affymetrix ENAR Spring Meetings, March 26-29, 2006 p.3/19

5 Central Dogma of MA Analysis Computing Expression Values for each probe set requires three steps: Background correction Normalization Summarization ENAR Spring Meetings, March 26-29, 2006 p.4/19

6 Central Dogma of MA Analysis Computing Expression Values for each probe set requires three steps: Background correction Normalization Summarization Approaches: Microarray Analysis Suite 5.0 (MAS Affymetrix, 2001, 2003) Model Based Expression Index (MBEI - Li and Wong, 2001a,b) Robust Multichip Analysis (RMA - Irizarry et. al., 2003) GeneChip-RMA (Wu, et. al., 2004) Probe Logarithmic Intensity Error Estimation (PLIER - Affymetrix, 2004) ENAR Spring Meetings, March 26-29, 2006 p.4/19

7 The RMA Approach Background Correction under Exponential-Normal Convolution Model. Normalization via Quantile Normalization. Summarization with Median Polish (Tukey, 1977). Biconductor allows user to interchange methods at any step. ENAR Spring Meetings, March 26-29, 2006 p.5/19

8 Exp-Norm Convolution Model The Convolution Model is given by X = S + Y where X = observed probe level intensity S E(α) = true signal Y TN(µ,σ 2 ) = background noise The true signal can be estimated by E(S X = x) = a + b where a = x µ σ 2 α and b = σ. ( φ( a b ) φ( x a b ) ) Φ( a b ) + Φ(x a b ) 1, ENAR Spring Meetings, March 26-29, 2006 p.6/19

9 CM for the Right Brained... ENAR Spring Meetings, March 26-29, 2006 p.7/19

10 Parameter Estimation Background Corrected intensity is E ij = E(S ij X ij ), where i = 1...G, and j = 1,...,J. We need to estimate µ, σ, and α. ENAR Spring Meetings, March 26-29, 2006 p.8/19

11 Parameter Estimation Background Corrected intensity is E ij = E(S ij X ij ), where i = 1...G, and j = 1,...,J. We need to estimate µ, σ, and α. How does BioC estimate the parameters? µ = Mode of observations to the left of the overall mode σ = Sample standard deviation for observations to left of overall mode α = Mode of observations to the right of the overall mode Shown to perform better than most other approaches (Hein, et. al., 2005). ENAR Spring Meetings, March 26-29, 2006 p.8/19

12 Code for Parameter Estimates > bg.parameters function (pm, n.pts = 2ˆ14) { max.density <- function(x, n.pts) { aux <- density(x, kernel = "epanechnikov", n = n.pts, na.rm = TRUE) aux$x[order(-aux$y)[1]] } pmbg <- max.density(pm, n.pts) bg.data <- pm[pm < pmbg] pmbg <- max.density(bg.data, n.pts) bg.data <- pm[pm < pmbg] bg.data <- bg.data - pmbg bgsd <- sqrt(sum(bg.dataˆ2)/(length(bg.data) - 1)) * sqrt(2) sig.data <- pm[pm > pmbg] sig.data <- sig.data - pmbg expmean <- max.density(sig.data, n.pts) alpha <- 1/expmean mubg <- pmbg list(alpha = alpha, mu = mubg, sigma = bgsd) } ENAR Spring Meetings, March 26-29, 2006 p.9/19

13 Simulation Experiment 100 replications for n = 100, 000. True parameter values of µ = 50, 100, σ = 10, 20, and α = 50, 250. Four methods for estimating α: Mean, Median, 75 th percentile, and th percentile of PM values larger than overall mode Five methods of estimating µ Estimate σ using SD of intensities smaller than ˆµ. ENAR Spring Meetings, March 26-29, 2006 p.10/19

14 Estimating µ Estimate µ with 1. Original method programmed in Bioconductor 2. Overall mode (s) of PM intensities 3. Mode of data to the left of 2s 4. Either of 2 or 3 plus a one-step correction, defined by the formula: ( ) [ ( )] s µ s µ φ ασ = ασ Φ ασ σ σ ENAR Spring Meetings, March 26-29, 2006 p.11/19

15 Results MSE for α, when µ = 50, σ = 10, α = 50 Using BioC 1754 ENAR Spring Meetings, March 26-29, 2006 p.12/19

16 Results MSE for α, when µ = 50, σ = 10, α = 50 Using BioC 1754 ˆµ ˆα Given By Mean Median 75% 99.95% s s s s ENAR Spring Meetings, March 26-29, 2006 p.12/19

17 Performance on Cell Line Data PM intensities compared to original curve for ˆµ = 2s + 1 and various estimates of α. Data: SW 480 Colon Cancer cell line with short term freezing of cells. ENAR Spring Meetings, March 26-29, 2006 p.13/19

18 Performance on Spike-In Data ENAR Spring Meetings, March 26-29, 2006 p.14/19

19 Ongoing and Future Work Distribution-Free Convolution Model Find smallest q 1 % of PM intensities Obtain q 2 % of corresponding MM intensities MM intensities are an estimate of background noise ENAR Spring Meetings, March 26-29, 2006 p.15/19

20 Ongoing and Future Work Distribution-Free Convolution Model Find smallest q 1 % of PM intensities Obtain q 2 % of corresponding MM intensities MM intensities are an estimate of background noise Obtain corrected PM intensities (PM ) using a given number k and PM ˆµ if PM > ˆµ + kˆσ PM = min(pm) = l, PM = ˆµ + kˆσ kˆσ otherwise. ENAR Spring Meetings, March 26-29, 2006 p.15/19

21 Some Preliminary Results DFBC with q 1 = 0.30 and q 2 = 0.90 vs. RMA, GCRMA, MAS 5.0, dchip, and PLIER True Positive Rate RMA RMA 75 Nonpar C GCRMA MAS 5.0 RMA nobg Li Wong PLIER False Positive Rate (FPR) ENAR Spring Meetings, March 26-29, 2006 p.16/19

22 Acknowledgments SMU - UTSW Microarray Analysis Group (SMUT-MAG) Faculty Jennifer Cai Jing Cao Tony Ng Richard Scheuermann William Schucany Jyoti Shaw Burke Squires Xinlei Wang Students Zhongxue Chen Kinfe Gedif Drew Hardin Jobayer Hossain Julia Kozlitina Feng Luo ENAR Spring Meetings, March 26-29, 2006 p.17/19

23 References 1. Affymetrix, Inc (2001). "Statistical Algorithms Reference". Data Analysis Fundamentals Technical Manual, Chapter Affymetrix Technical Note: Design and Performance of the GeneChip Human Genome U133 Plus 2.0 and Human Genome U133A Plus 2.0 Arrays (2003) Affymetrix, Inc (2002). Statistical Algorithms Description Document Affymetrix, Inc (2005). Guide to Probe Logarithmic Intensity Error (PLIER) Estimation. 5. Bolstad BM (2004). Low Level Analysis of High-density Oligonucleotide Array Data: Background, Normalization and Summarization. Dissertation. University of California, Berkeley. 6. Hein AK, Richardson S, Causton H, Ambler GK, and Green PJ (2005). BGX: a fully Bayesian integrated approach to the analysis of Affymetrix GeneChip data. Biostatistics, 6, Irizarry RA, Bolstad BM, Collin F, Cope LM, Hobbs B, and Speed TP (2003). Summaries of Affymetrix GeneChip probe level data. Nucleic Acids Research, 31 (4) e15. ENAR Spring Meetings, March 26-29, 2006 p.18/19

24 References Continued 8. Irizarry RA, Hobbs B, Collin F, Beazer-Barclay YD, Antonellis KJ, Scherf U, and Speed TP (2003). Exploration, normalization, and summaries of high density oligonucleotide array probe level data. Biostatistics, 4, Li C and Wong HW (2001). Model-based analysis of oligonucleotide arrays: Expression index computation and outlier detection. Proceedings of the National Academy of Sciences, 98 (1): Li C and Wong HW (2001). Model-based analysis of oligonucleotide arrays: model validation, design issues and standard error application. Genome Biology, 8 (8): research Naef F and Magnasco MO (2003). Solving the riddle of the bright mismatches: Labeling and effective binding in oligonucleotide arrays. Physical Review, Wu Z, Irizarry RA, Gentleman R, Martinez Murillo F, Spencer F (2004) A Model Based Background Adjustement for Oligonucleotide Expression Arrays. Journal of the American Statistical Association, 99, ENAR Spring Meetings, March 26-29, 2006 p.19/19