Data analysis and binary regression for predictive discrimination. using DNA microarray data. (Breast cancer) discrimination. Expression array data

Transcription

1 West Mike of Statistics & Decision Sciences Institute Duke University wwwstatdukeedu IPAM Functional Genomics Workshop November Two group problems: Binary outcomes ffl eg, ER+ versus ER ffl eg, lymph node + versus lymph node DNA microarray data: expression levels of ß genes (sequences) ffl RNA from tumour, tumour location, time point, in ffl ER+, ER ffl Discriminatory patterns of expression? ffl Predictive classification of tumours 44, 4,? ffl Which genes are implicated? Surprises? ffl Which tumours depart from general patterns? How? ffl etc Nevins Genetics Joseph Dressman Genetics Holly Marks Surgery & Cancer Center Jeff Blanchette Surgery & Cancer Center Carrie Spang ISDS & Genetics Rainer Zuzan ISDS & Genetics Harry Mike West ISDS for Bioinformatics & Computational Biology Center for Genome Technology Center Microarray data: Affymetrix arrays ffl ß genes (sequences) ffl Data issues: imaging, probe cell specific expression data summaries in commercial software ffl Estimates of expression level by gene: Absolute difference ffl Here: log (max(; AbsDiff)) 4 Data analysis and binary regression for predictive discrimination using DNA microarray data (Breast cancer) discrimination Collaborators Expression array data

2 ffl Binary regression: many predictor variables ffl Possibly many interacting genes relate to status ffl Singular factor projection of expression data reduces dimension with no loss of information summarises important structure" in expression data ffl Principal components decomposition Variances and correlations in expression fully explained" by small ffl of factors number ffl Expression of (many) genes driven" by (few) factors X = ADF Factor loadings matrix A = [a;:::;a n ] ffl patterns/relationships among genes Latent factors are rows of F ffl patterns/relationships among arrays: n << p factors Supergenes=Factors: linear combinations of expression Factors drive" expression levels: gene i on array j : xi;j = a i; f;j + a i; f;j + :::+ a i;n f n;j Notation: ffl x i;j is expression level of gene i on microarray j ffl p genes, n arrays: n << p ffl p = ± genes, n = arrays X = [x; x;:::;x n ] = x; x; x;n x; x; x;n x p; x p;n xp; C A Projecting large-scale expression data Singular value (factor) decomposition 4 4 F F F F Arrays on supergene factors: Coloured for ER+/ER- Summary expression data

3 Weight vectors a; a;:::; Regression on (many) genes reduces to regression on (few) supergenes ffl n parameters, sample size n ffl Ignore stable" factors X fi = F = DA fi ffl Use of stochastic regularisation: priors on elements j independent (orthogonality) j ο N(;fi j ) with prior on fi j neutral: implied priors for classification probability p(x j ) ffl Efficient analysis to estimate ffl Markov chain Monte Carlo model fitting ffl Microarray j; expression profile x j ffl Binary classification: (ER+) or (ER) ffl array j is ER+ is p(x j ) ffl Standard probit model: p(x j ) = Φ(x j fi) Linear regression on gene expression, mapped to probability scale ffl x j fi = P p i= fi i x i;j fi i is regression coefficient on gene i ffl Statistical analysis: estimate coefficients, uncertainty ffl depends on design data X ffl New arrays: new parameter, new priors ffl Out-of-sample prediction: New tumours ffl SVD analysis of all arrays ffl Underlying latent factor model genesis ffl SVD regression as a limiting case Consistent priors for and underlying gene coefficients fi as new ffl arises data ffl Generalised g-prior" Factor weight vectors genes 4 Supergenes in binary regression modelling Binary regression modelling Theoretical context and issues

4 Latent factor model for gene expression: tumour i ffl i ο N(; I) and ffl i ο N(; Ψ) xi = B i + ffl i ffl patterns explained by (a few) latent factors: k =dim( i ) ffl residual/idiosyncratic terms ffl i Outcomes: ο N( i ; ) yi ffl outcomes regress on latent factors in x i indirect regression on x i ffl different outcomes relate to different latent factors ffl Efficient analysis of regression on n << p supergenes ffl Posterior (samples) for supergene vector ffl Compute posterior (samples) fi = AD ffl Bayesian/model justification of generalised inverse to = DA fi ffl Latent factor model defines p(y i ; x i ; i ) ffl Implied p(y i jx i ): regression of y i on x i ffl Linear regression coefficient fi = H ffl H depends on B; Ψ Some implications: ffl Prior on implies generalised gprior on fi ffl Limiting case: Ψ! leads to SVD regression 4 Critical predictive assessment of discriminatory performance ffl One-at-a-time analysis ffl Take out microarray j Fit model Predict status of tumour j ffl Repeat for all arrays j Underlying latent factor models Regression on genes via supergenes Underlying latent factor models: SVD regression case Cross-validation (honest) prediction

5 Marginalising over fi implies with kernel covariance matrix with ffl correlations between arrays y ο N(; K) K = F TF + I T = diag(fi ;:::;fi n) ffl effective dependence structure with respect to classification ffl key role of T ffl effective non-linear classifier First run of tumours/arrays only Kernel regression structure Fitted classification Classification probabilities and outcomes array kernel correlations ER arrays array kernel correlations ER+ arrays Estimated kernel correlation structure Gene 4 beta 4 x Supergene gamma 4 Expression weights: Genes & SuperGenes ER status: Estimated regression coefficients

6 One-at-a-time analysis ffl Heterogeneity in data: noise" from many irrelevant" genes? ffl Screen to smaller subsets - eg, raw correlations with ER+= status ffl Select top k" and fit model on k genes ffl Oestrogen receptor status example: k = Multiple genes refine classification: minor effects Collective effects in addition to primary gene Interesting cases (ER),, 4 (ER+) ffl Tumour : Classified ER+ (non-duke diagnosis) ffl Reclassify as ER+ and refit model: Perfect" classification Overall top 4 Cross-validatory predictions Validation classification probabilities and outcomes Cross-validation predictions: Top Validation classification probabilities and outcomes Validation classification probabilities and outcomes 4 4 Cross-validation predictions Gene screening

7 ffl ps protein gene Some top genes: up favours ER+ ffl mrna for oestrogen receptor ffl cytochrome p4 iib (hiib) mrna ffl intestinal trefoil factor mrna ffl hepatoma mrna for serine protease hepsin ffl insulin like growth factor binding protein [ placenta ] ffl pnb mrna ffl c-myb gene ffl ccaat displacement protein ffl clone 4 mrna sequence ffl nat gene for arylamine n-acetyltransferase ffl ffl breast cancer, oestrogen regulated liv- protein mrna ffl Similar patterns: ER+ or ER? ffl High uncertainty about Pr(ER+) Oestrogen gene marginally down" - Ps and Liv- higher ffl regulated by oestrogen receptor Both ffl Other up for ER+" genes high on arrays, 4 ffl Mixed story in data on arrays, 4 ffl High classification uncertainty results Other regulators of Ps, Liv-? ER status determination? Evolving from to +? F 4 4 hhkb cd4 4 4 cytochrome p4 iib intestinal trefoil factor 4 4 ps oestrogen Expression levels of some top genes 4 F F F Arrays on pairs of factors: Top Tumours, 4

8 Classification probability for tumour Choice of point estimates" - Mean values conservative" initial arrays, later arrays ffl Same breast cancer arrays, classified by axilliary lymph nodal status: lymph node-negative breast cancer primary, versus primary, lymph node-positive ffl Expression: expected to be highly heterogeneous ffl Data confirms this: Analysis of all + genes no clear discrimination expected none found ffl Clearer picture based on Top " - Similar story to ER ffl Data issues: Consistency of samples : +/: most extreme" Case 4: +/ Case F F F F Arrays on supergene factors 4 mean value % interval Classification and uncertainty Validation classification probabilities and outcomes Cross-validation predictions: -at-a-time/top Breast cancer nodal status

9 ffl alk-4 mrna Overall top Some top genes: up favours Node ffl bloom syndrome protein (blm) mrna ffl wilm tumour-related protein ffl mrna for actin-related protein ffl retinoid x receptor beta (rxr-beta) ffl fkbp-rapamycin associated protein (frap) ffl ribosomal protein s4 ffl histone h ffl receptor tyrosine kinase ligand lerk- precursor (eplg) mrna ffl mrna for kiaa gene ffl mrna for glycerol kinase ffl Whitehead Institute, Lander group Golub et al Science, initial arrays, later arrays ffl leukemias: ALL () and AML () ffl easily" identified on non-genetic bases ffl samples (/) on training arrays ffl 4 samples (/4) on validation arrays ffl MIT (Whitehead) study: 4 some difficulty in predictive classification of validation cases Cross-validation predictions: Top 4 4 Validation classification probabilities and outcomes F MIT ALL/AML leukemia study 4 4 F F F Arrays on supergene factors: Top

10 ffl Hybridisation problems: RNA quality ffl Fluorescent image scanning (registration, resolution) ffl Global normalisation of expression, array to array global scaling non-linearities induced by varying hybridisation quality ffl Local issues: scratches, patches, All distort expression summaries ffl Pixel-level image model for background ffl Bayesian image analysis: (non-negative) expression level parameters 4 F Leukemias: factors F Top genes on four leukemia arrays x 4 x 4 Array 4 Array 4 Top genes 4 Top genes x 4 x 4 Array Array 4 Top genes 4 Top genes Data issues with Affymetrix arrays Classification probabilities and outcomes Validation predictions on top

11 ffl probe sequences per gene averaging" of pixel values within probe cells averaging" of probe cell averages empirically based: global reliability? ffl Marked variability across probes for some genes ffl mer specific hybridisation intensity Use all data: measures per gene 4 Applications/extensions ffl Other outcomes: eg, genomic predictor of treatment outcome ffl Multiple outcomes: eg, cancer stages/states ffl Measured outcomes: eg, time to remission ffl Exploration of relationships among genes ffl Combining expression profiles with other clinical data Statistical models ffl Refine empirical" singular factor method ffl Latent supergene factors - to de-noise" singular factor method ffl Accounting for measurement errors in expression summaries ffl Non-linear regressions 4 More data issues Alternatives: ffl Model mer-specific hybridisation intensities (Li & Wong ) Futures Probe effects 4 x 4 Oestrogen levels across probes 4 4 Oestrogen probe effects 4 4 4