Propagating Measurement Uncertainty in Microarray Data Analysis

1. Propagating Measurement Uncertainty in Microarray Data Analysis Magnus Rattray School of Computer Science University of Manchester

2. Talk Outline • Part 1: Affymetrix probe-level analysis • Probabilistic model for oligonucleotide arrays • Estimating credibility intervals • Evaluation on real and spike-in data • Part 2: Propagating uncertainties • A general framework for propagating uncertainties • Example 1: Identifying differentially expressed genes • Example 2: Modified Principal Component Analysis

3. Part 1: Affy probe-level analysis PM – Perfect match DNA probe designed to measure signal MM – Mismatch DNA probe designed to measure background Probes for the same gene differ greatly in their binding affinities, eg. ~10000-50000 probe-sets with 11-20 PM/MM probe-pairs

4. Are mismatch probes useful? • In practice there is specific binding to MM, so some methods ignore MM probes altogether. But… …if fraction is the same for each chip, this term cancels when computing expression ratios.

5. Probabilistic probe-level analysis • Most methods return a single expression level estimate • Probabilistic models provide confidence intervals • Useful for propagating through higher-level analysis • Hopefully, this approach will also improve accuracy A hierarchical Bayesian model (Hein et al. 2005) uses MCMC for Bayesian parameter estimation, but this can be prohibitively slow – a more efficient approach is required.

6. Gamma model for oligo signal: gMOS Models (PM,MM) distribution for each probe-set - PM (background+signal) - MM (background) - signal Mean log-signal where Milo et. al., Biochemical Transactions31, 6 (2003)

7. Modelling probe affinity: mgMOS • PM and MM probes have correlated binding affinities • Use a shared scale parameter for probe-pair • Treat scale parameter as a latent variable • Distribution of PM ( ) and MM ( ) is Improves fit to data

8. Further extensions of the model • Share binding affinity parameter across multiple chips • Include fraction specific binding to MM probe Probe in probe-set on chip Parameter is unidentifiable We estimate an empirical prior from spike-in data Liu et. al., Bioinformatics 21, 3637 (2005).

9. Posterior signal distribution • We estimate the mean signal over a probe-set as • Only the first term is chip & condition specific • Distribution of gives posterior signal distribution • We assume a uniform positive prior on • Approximate posterior of as truncated Gaussian or using a histogram approach (very similar in practice) • Percentiles of provide percentiles of

10. Posterior signal distribution • Posterior becomes more peaked as signal increases • Normal provides good fit for large signals • For low signal there is a long left-hand tail due to the fact that we are measuring • Posterior distribution can be used to put credibility intervals on the estimated expression level

11. Results: Accuracy on real data • 5 time-points, 3 replicates & qr-PCR for 14 genes mgMOS multi-mgMOS RMS error to PCR results Mouse hair-follicle morphogenesis data from Lin et. al. PNAS101, 15955 (2004).

12. Importance of credibility intervals Red boxes show truly differentially expressed genes Left: Log-ratios used to rank genes Right: Credibility intervals used to rank genes 1331 up-regulated genes (1.2 to 4-fold), 12679 invariant Spike-in data from Choe et al Genome Biology 6, R16 (2005).

13. Part 2: Propagating uncertainties • Uncertainties can be propagated as noise where is diagonal covariance matrix for gene • Use your favourite probabilistic model for • Data is not i.i.d. making parameter estimation tricky We consider two popular tasks as examples: (i) Combining replicates and identifying differential expression (ii) Principal Component Analysis (PCA)

14. (i) Combining replicates Simplest model of log-expression is a Gaussian: for replicate in conditions with priors • Parameters are • Hyper-parameters are • We can then calculate the probability of the sign of change in expression level between two conditions:

15. Hyper-parameter estimation Likelihood: Prior: We wish to optimise the log marginal likelihood: The integral is intractable, so we use a variational approximation (popular approach in machine learning). The resulting optimisation resembles an EM-algorithm.

16. Variational approximation E-step: M-step: We use a factorised approximation to the posterior:

17. Results: credibility intervals Data from Lin et. al. PNAS101, 15955 (2004)

18. Identifying differential expression One chip per condition 3 replicates per condition 1331 up-regulated genes (1.2 to 4-fold), 12679 invariant Spike-in data from Choe et al Genome Biology 6, R16 (2005).

19. (ii) Principal Component Analysis • Popular dimensionality reduction technique • Project data onto directions of greatest variation Useful tool for visualising patterns and clusters within the data set Usually requires an ad-hoc method for removing genes with low signal/noise This example from Pomeroy et. al. Nature 415, 436, 2002. Embryonic tumours of the central nervous system.

20. Probabilistic PCA • PCA can be cast as a probabilistic model with -dimensional latent variables • The resulting data distribution is • Maximum likelihood solution is equivalent to PCA Diagonal contains the top sample covariance eigenvalues and contains associated eigenvectors Tipping and Bishop, J. Royal Stat. Soc. 6, 611 (1999).

21. Relationship to Factor Analysis • Probabilistic PCA is equivalent to factor analysis with equal noise for every dimension • In factor analysis for a diagonal covariance matrix • An iterative algorithm (eg. EM) is required to find parameters if precisions are not known in advance In our case we want the precision to be gene and experiment specific – we need a more flexible model

22. PCA with measurement uncertainty • If we let the covariance matrix be gene specific then Probabilistic PCA: Corrupted data model: • The log-likelihood is with • The maximum likelihood solution for the mean is which is no longer the sample mean

23. Likelihood optimisation • The optimal parameters are solutions to a coupled non-linear set of equations (eg. depends on ) • Gradients require inversion of large matrices • An EM-algorithm provides more efficient optimisation • M-step still requires non-linear optimisation • Redundant parameterisation of model gives us a significant speed-up

24. Advantages over standard PCA • Automatically eliminates influence of consistently noisy genes, eg. noisy in all experiments • Automatically chooses no. of principal components because noise “explains away” some of the variation • Down-weights influence of noisy measurements in an experiment specific way • Provides error-bars on the reduced dimension representation of the data • Can be used to “denoise” expression profiles

25. Results: Improved visualisation Under standard PCA 43% of samples are closest to a sample of the same tumour type. For modified PCA this percentage increases to 71%. Data from Pomeroy et. al. Nature 415, 436, 2002.

26. Denoising a data set • We can estimate the uncorrupted data from the noisy measurements as • Denoised profile approaches original as noise is reduced • Denoised data improves performance of clustering Sanguinetti et al. Bioinformatics 21, 3748 (2005).

27. Conclusions • We have developed a computationally efficient probabilistic model for Affymetrix probe-level analysis. • The model provides good accuracy and confidence intervals for gene expression level estimates. • Measurement uncertainties can be propagated through an appropriate probabilistic model. • Example applications to Bayesian t-test and PCA. • Parameter estimation becomes much more difficult, so approximate methods are needed. • Same principal can be applied to other models.

28. Rest of the team: Xuejun Liu, School of Computer Science, University of Manchester. Guido Sanguinetti, Marta Milo & Neil Lawrence, Department of Computer Science, University of Sheffield. Software: www.bioinf.man.ac.uk/resources/puma Papers: Liu et al. “A tractable probabilistic model for Affymetrix probe-level analysis across multiple chips” Bioinformatics 21, 3637 (2005). Sanguinetti et al. “Accounting for probe-level noise in principal component analysis of microarray data” Bioinformatics 21, 3748 (2005). Supported by a BBSRC award “Improved processing of microarray data with probabilistic models” Acknowledgments