Lecture 9: Statistical models and experimental designs Chapters 8 & 9

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1 Lecture 9: Statistical models and experimental designs Chapters 8 & 9 Geir Storvik Matematisk institutt, Universitetet i Oslo 12. Mars 2014

2 Content Why statistical models Statistical inference Experimental design

3 Bioinformatics Concerned with analyzing biological data Very much specified through algorithms Classification: x g(x) {1,..., K }. Hierarchical clustering Statistics: Start by specifying a statistical model, defines an estimation procedure and end up with an algorithm Question: Do we need models and estimation procedures? Important for design of experiments Important for evaluating performance of algorithm

4 Design of experiments Assume two populations Cancer population and control population Two different cod species Interest in which genes that differ Questions: How many individuals from each population? How many of each gender? How many DNA/RNA samples from each individual? Examples of design of experiments (chap 9) Answers require Statistical models Some prior information on parameters

5 Performance Comparing populations: Assume we have estimated that 1030 genes differ How reliable is this estimate? Classical statistics: Look at performance over repeated experiments In practice not possible to repeat experiments Given statistical models: Can simulate experiments on computer Statistical terminology: Resampling/bootstrapping Bayesian approach: Look at properties conditioned on given data Also require statistical models Covered in Chapter 10

6 Statistical inference Three parts Statistical models Statistical estimation Numerical algorithm

7 Statistical models Parametric models ind Y1,..., Y n N(µ, σ) ind Yi = βx i + ε i, ε i N(0, σ 2 ) Non-parametric models ind Y 1,..., Y n F (y) = Pr(Y i y) No assumptions on F. Semi-parametric: Yi = βx i + ε i, ε i F(x) = Pr(ε i x) No assumptions on F.

8 General framework Assume data D Statistical modeling: Specify f (D θ) f : Probability distribution/density for data D θ: Parameters in model ind Linear regression: Y i = βx i + ε i, ε i N(0, σ 2 ) Data D = {(y i, x i ), i = 1,..., n} f : Product of Normal densities θ = (β, σ) Parametric models: θ small Semi-/non-parametric models: θ large

9 Estimation methods Maximum likelihood Define L(θ) = f (D θ) θml = arg max θ L(θ) Bayesian estimation Define a prior g(θ) reflecting our prior knowledge about θ prior to collection of data Calculate (through Bayes theorem) g(θ D) = θbayes = E[θ D] = θ θg(θ D)dθ Both methods rely heavily on f (D θ) g(θ)f (D θ) f (D) With little data, Bayesian methods rely heavily on g(θ)

10 Numerical algorithms Need to calculate θ ML = arg max L(θ); or θ θ Bayes =E[θ D] = θ θg(θ D)dθ Numerical problems, optimization or integration General model classes: Routines available, can be slow! Specific models: Need to implement our self. Possible to use general optimization/integration routines!

11 What kind of model to use? Variance-bias trade off E[(ˆµ µ) 2 ] = Var(ˆµ) + [E(ˆµ) µ] 2 Strong model assumptions can give low variance but potentially high bias Weak model assumptions can give high variance but low bias Bioinformatics Data Y i,j, i = 1,..., n, j = 1,..., p n is number individuals/samples (typically small) p is number of genes (typically large) If repetition is on individuals: Small datasets, need strong model assumptions If repetition is on genes: Large datasets, can use weak model assumptions

12 Examples of models In the book Regression models ANOVA models Survival models

13 Survival models Linking gene expression profiles to survival data Example: Survival of lung cancer, 200 patients, genes, repetition on patients 120 patients still alive when ending study, censored data Data D: {(xi, t i, c i )}, t i is survival time, c i = I(dead). Possible model: λ(t X = x) = lim dt t Pr(t T < t + dt T t, x) dt =λ 0 (t) exp(β T x)

14 Survival models Data D: {(x i, t i, c i )}, t i is survival time, c i = I(dead). λ(t X = x) = lim dt t Pr(t T < t + dt T t, x) dt =λ 0 (t) exp(β T x) Likelihood: L(β D) = c i =1 Pr(T = t i β) c i =0 Pr(T > t i β) Partial likelihood (non-parametric λ 0 (t)) L(β) = i:c i =1 exp(β T x i ) j:t j t i exp(β T x j ), Problems: Non-identifiability due to p >> n

15 p >> n General problem for many regression models Possible approaches Penalizing (partial) likelihood: p log L(θ) γ βj 2 Ridge regression j=1 p log L(θ) γ β j Lasso regression j=1 Reducing the number of covariates Principal component analysis Clustering of genes, using mean from each cluster

16 Experimental design Questions: How many samples How to distribute over populations? How many samples per individual How to distribute wrt gender... General principles Randomization Replication Pooling Blocking

17 Randomization Experimental subjects should be randomly assigned to treatment/conditions be be studied Eliminates unknown factors (bias) Randomization can be at several levels Randomization wrt treatment Randomization wrt equipment/lab/technician Randomization wrt order to analyze samples Not always possible to perform full randomization Mutant-versus-control mice: Not possible to randomize genotype

18 Replication replicates to distinguish replication at different levels (Churchill, 2002; Yang and Speed, 2002). Biological replicates are considered true replicates while technical replicates are replicates at a lower level than the biological ones, often measurements from the same RNA sample (Yang and Speed, 2002; Simon et al., 2003). Because of thecomplexityofmicroarraytechnology, there arepotentiallymany differentlevels of technical replicates. For example, Figure 9.1 shows a microarray experiment that has three levels of replication: the first level is at the cell culture level. Four independent cell cultures were established and each receives one of two treatments. Two RNA samples are obtained from each cell culture and each sample is measured with One population: Y N(µ, σ) two one-color arrays. In this experiment, the independent cell cultures are biological replicates. The replicates at RNA samples and arrays are technical replicates which Need replications to estimate both µ and σ (n > 2) are similar to the repeated measurements. They are less useful for identifying significantly expressed genes between the two treatments. However, technical Replicationsreplicates canarebe essential done in experiments ondesigned different for evaluatinglevels the technology(two and in identifying the sources of variation (Zakharkin et al., 2005). The variability between populations). Figure 9.1 Different levels of replication in a microarray experiment. Regression: Y = β T x + ε, ε N(0, σ). Again need replications to estimate all parameters (n > p + 1) Now: replications on ε, can have different x s

19 Sample size General rule: Need about 5-10 observations for each parameter More specific: Power/performance calculation. Example: X 1,..., X n ind N(µ C, σ), Y 1,..., Y n ind N(µ T, σ) 95% confidence interval for = µc µ T : x ȳ ± 1.96σ 2/n Want width to be δ implying σ 2/n = δ n = σ 2 /δ 2 x ȳ Testing H0 : = 0: Reject H 0 if > 1.96 σ 2/n Want Pr(Reject H 0 = 1) = 0.8, implying Pr( x ȳ > 1.96 = 1) = 0.8 n = 25.98σ2 σ 2/n Need preliminary experiments to get number for σ.