Classification. Methods Course: Gene Expression Data Analysis -Day Five. Rainer Spang

Transcription

1 Classification Methods Course: Gene Expression Data Analysis -Day Five Rainer Spang

2 Ms. Smith DNA Chip of Ms. Smith Expression profile of Ms. Smith Ms. Smith

3 properties of Ms. Smith The expression profile a list of 30,000 numbers... that are all properties of Ms. Smith... some of them reflect her health problem (a tumor)... the profile is a digital image of Ms. Smith s tumor How can these numbers tell us (predict) whether Ms. Smith has tumor type A or tumor type B?

4 Looking for similarities? Ms. Smith Compare her profile to profiles of people with tumor type A and to patients with tumor type B

5 Training and Application There are patients of known class - the trainings samples - There are patients of unknown class - the new samples - Ms. Smith

6 Statistical Learning Use the trainings samples to learn how to predict new samples Ms. Smith

7 Prediction with 1 gene Color coded expression levels of trainings samples A B Ms. Smith Ms. Smith Ms. Smith type A type B borderline Which color shade is a good decision boundary?

8 Data optimal decision rule Use the cutoff with the fewest misclassifications on the trainings samples Smallest training error Decision boundary Distribution of expression values in type B Distribution of expression values in type A Training error

9 Overfitting The decision boundary was chosen to minimize the training error The two distributions of expression values for type A and B will be similar but not identical in a set of new cases We can not adjust the decision boundary because we do not know the class of the new samples Test errors are in average bigger then training errors Test error This phenomenon is called overfitting

10 Accumulating information across genes The top gene The average of the top 10 genes ALL vs. AML Golub et al.

11 Using a weighted average With good weights you get an improved separation

12 The geometry of weighted averages Calculating a weighted average is identical to projecting (orthogonally) the expression profiles onto the line defined by the weights vector

13 Hyperplanes A B 2 genes 3 genes Together with an offset 0 the weight vector defines an orthogonal hyperplane that cuts the data in two groups

14 Linear Signatures A B

15 Nearest Centroids

16 Diagonal Linear Discriminant Analysis (DLDA) Rescale axis according to the variances of genes

17 Discriminant Analysis The data often shows evidence of non identical covariances of genes in the two groups Hence using LDA, DLDA or NC introduces a bias but a good bias

18 Gene Filtering Rank genes according to a score Choose top n genes Build a signature with these genes only Still weights, but most of them are zero Note that the data decides which are zero and which are not Limitation: You have no chance to find these two genes

19 How many genes Is this a biological or a statistical question? Biology: How many genes carry diagnostic information? Statistics: How many genes should we use for classification? The microarray offers genes or more

20 Finding the needle in the haystack A common myth: Classification information is restricted to a small number of genes, the challenge is to find them

21 The Avalanche Aggressive lymphomas with and without a MYC-breakpoint MYC-neg MYC-pos Verbundprojekt maligne Lymphome

22 Genes, Bias & Overfitting The gap between training error and test error becomes wider There is a good statistical reason for not including hundreds of genes in a model even if they are biologically effected

23 Soft Thresholding The shrunken centroid method and the PAM package Tibshirani et al 2002 genes genes genes genes genes genes genes genes genes genes genes genes genes genes genes

24 Centroid Shrinkage

25 How much shrinkage is good in PAM? Train Train Train Select Train Train Train Select Train Train cross validation Compute the CV-Performance for several values of D Pick the D that gives you the smallest number of CV- Misclassifications Adaptive Model Selection PAM does this routinely

26 Model Selection Output of PAM Small D, many genes poor performance due to overfitting High D, few genes, poor performance due to lack of information underfitting - The optimal D is somewhere in the middle

27 Signatures you miss Assume protein A binds to protein B and inhibits it The clinical phenotype is caused by active protein A Predictive information is in expression of A minus expression of B Naïve Idea: Don t calculate weights based on single gene scores but optimize over all possible hyperplanes

28 Two different signatures based on the same genes Calling signature genes markers for a certain disease is misleading!

29 Only one of these problems exists Problem 1: No separating line Problem 2: Many separating lines Why is this a problem?

30 What about Ms. Smith? This problem is also related to overfitting... more soon

31 Prediction with 30,000 genes With the microarray we have more genes than patients Think about this in three dimensions There are three genes, two patients with known diagnosis (red and yellow) and Ms. Smith (green) There is always one plane separating red and yellow with Ms. Smith on the yellow side and a second separating plane with Ms. Smith on the red side OK! If all points fall onto one line it does not always work. However, for measured values this is very unlikely and never happens in praxis.

32 The overfitting disaster From the data alone we can not decide which genes are important for the diagnosis, nor can we give a reliable diagnosis for a new patient This has little to do medicine. It is a geometrical problem.

33 Consequences If you find a separating signature, it does not mean (yet) that you have a top publication in most cases it means nothing.

34 Meaningless vs. meaningful signatures There always exist separating signatures caused by overfitting - meaningless signatures - Hopefully there is also a separating signature caused by a disease mechanism - meaningful signatures We need to learn how to find and validate meaningful signatures

35 How to distinguish a meaningful signature from a meaningless signature? The meaningless signature might be separating small training error -... but it will not be predictive large error in applications The goal is not a separating signature but a predictive signature: Good performance in clinical practice!!!

36 Back to the problem of many separating hyperplanes Which hyperplane is the best?

37 Support Vector Machines Fat planes: With an infinitely thin plane the data can always be separated correctly, but not necessarily with a fat one. Again if a large margin separation exists, chances are good that we found something relevant. Large Margin Classifiers

38 Maximal Margin Hyperplane There are theoretical results that the size of the margin correlates with the test (!) error (V. Vapnik) SVMs are not only optimized to fit to the training data but for predictive performance directly

39 No separable training set Penalty of error: distance to hyperplane multiplied by a parameter c Balance over- and underfitting

40 Independent Validation The accuracy of a signature on the data it was learned from is biased because of the overfitting phenomenon Validation of a signature requires independent test data Test error

41 Test Sets Split data into test and training data ok ok mistake

42 Selection Bias The test data must not be used for gene selection or adaptive model selection, otherwise the observed accuracy is biased Selection bias

43 Cross Validation Train Train Eval Train Train Train Train Train Eval Train You can not evaluate a fitted classification model (= signature) using cross validation Cross validation only evaluates the algorithm with which the signature was build Gene selection must be repeated for every relearning step in the cross validation In the loop gene selection

44 Leave one out Cross-Validation 1 Train Train Eval Train Train Train Essentially the same Train Train Eval But you only leave one sample out at a time and predict it using the others Good for small training sets 1 Train

45 Performance Estimation Estimators of performance have a variance which can be high. The chances of a meaningless signature to produce 100% accuracy on test data is high if the test data includes only few patients Nested 10-fold- CV Variance from 100 random partitions Reuse of the same data no sample variance

46 External Validation and Documentation Documenting a signature is conceptually different from giving a list of genes, although is is what most publications give you In order to validate a signature on external data or apply it in practice: - All model parameters need to be specified - The scale of the normalized data to which the model refers needs to be specified Add on normalization

47 Establishing a signature Split Data into Training and Test Data Test data only: Internal validation Full quantitative specification External Validations Training data only: Machine Learning - select genes - find the optimal number of genes - learn model parameters

48 DOs AND DONTs : 1. Decide on your diagnosis model (PAM,SVM,etc...) and don t change your mind later on 2. Split your profiles randomly into a training set and a test set 3. Put the data in the test set away... far away 4. Train your model only using the data in the training set (select genes, define centroids, calculate normal vectors for large margin separators, perform adaptive model selection...) don t even think of touching the test data at this time 5. Apply the model to the test data... don t even think of changing the model at this time 6. Do steps 1-5 only once and accept the result... don t even think of optimizing this procedure

49 Questions?

50 The bias variance trade off

51 Which model is best? Experience: Linear models work fine Sparse data: Expression data in high dimensions is sparse. The data does not contain information to identify non linear structures adequately, even if they exist.