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Design of Experiments

Panu Somervuo, March 20, 2007. Design of Experiments. Problem formulation Setting up the experiment Analysis of data. Problem formulation. what is the biological question? how to answer that? what is already known? what information is missing?

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Design of Experiments

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  1. Panu Somervuo, March 20, 2007 Design of Experiments • Problem formulation • Setting up the experiment • Analysis of data

  2. Problem formulation • what is the biological question? • how to answer that? • what is already known? • what information is missing? • problem formulation  model of the biological system

  3. Control Test Setting up an experiment • what kind of data is needed to answer the question? • how to collect the data? • how much data is needed? • biological and technical replicates • pooling • how to carry out the experiment (sample preparation, measurements)?

  4. Analysis of data • preprocessing • filtering & outlier removal • normalization • statistical model fitting • hypothesis testing • reporting the results, documentation

  5. Everything depends on everything problem formulation model of the system analysis of data statistical tests setting up the experiment number of samples

  6. Practical guidelines • blocking unwanted effects (e.g. dye effect) • randomization (avoid systematic bias by randomizing e.g. the order of sample preparations) • replication (replicate measurements can be averaged to reduce the effect of random errors) group2 group1 group2 group1 cy3 cy3 cy3 cy5 cy5 cy5

  7. Control Test log transform, normalization y = µ+F1+F2+...+error

  8. Control Test Pairwise sample comparison vs modeling • pairwise sample comparison is easy and straightforward • instead of comparing samples as such, we can construct a model for the measurements and then perform comparisons

  9. Mathematical model of data • try to capture the essence of a (biological) phenomenon in mathematical terms • here we concentrate on linear models: observation consists of effects of one or more factors and random error • factor may have several levels (e.g. factor sex has two levels, male and female)

  10. Examples of models • single factor: y = µ + gene + error • two factors: y = µ + treatment + gene + error • two factors including interaction term: y = µ + treatment + gene + treatment.gene + error • four factors: y = µ + treatment + gene + dye + array + error normalization, log transform

  11. From model to experimental design y = µ + drug + sex + drug.sex + error factor 1, drug: 3 levels factor 2, sex: 2 levels 3x2 factorial design:

  12. Analysis of variance • ANOVA can be used to analyse factorial designs y = µ + drug + sex + drug.sex + error • summary(aov(y~drug*sex,data=data)) • Df Sum Sq Mean Sq F value Pr(>F) • drug 2 2.86750 1.43375 51.3582 3.644e-08 *** • sex 1 1.26042 1.26042 45.1493 2.673e-06 *** • drug:sex 2 0.06583 0.03292 1.1791 0.3302 • Residuals 18 0.50250 0.02792 • --- • Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

  13. Multiple pairwise comparisons • ANOVA tells that at least one drug treatment has effect, but in order to find which one we perform all pairwise comparisons: • TukeyHSD(aov(y~drug*sex,data=data,"drug") • Tukey multiple comparisons of means • 95% family-wise confidence level • factor levels have been ordered • Fit: aov(formula = y ~ drug * sex, data = data) • $drug • diff lwr upr • A-0 0.0625 -0.1507113 0.2757113 • B-0 0.7625 0.5492887 0.9757113 • B-A 0.7000 0.4867887 0.9132113

  14. Control Test Benefits of (good) models • after fitting the model with data, model can be used to answer the questions e.g.: • is there dye effect? • is the difference of gene expression levels in two conditions statistically significant? • is there interaction between gene and another factor? • simple pairwise sample comparisons cannot give answers to all of these questions simultaneously y=µ+F1+F2+...+error

  15. What is a good model? • good model allows us to get more detailed results • best model and parametrization is application specific • simple vs complex model y=µ+F1+F2+F3+...+error • there should be balance between model complexity and the amount of data

  16. How the number of samples affects the confidence of our results? • measurement error is always present, see the example self-self hybridization:

  17. How the number of samples affects the confidence of our results? • let’s compute the mean average of expression level of a gene • how accurate is this value? • variance(mean) = variance(error)/number of samples • samples from normal distribution (mean 0, sd 1):

  18. Theoretical sample size calculations • for each statistical test, there is a (test-specific) relation between: • power of a test: 1 – probability(type I error) • significance level: probability(type II error) • error variance • mean difference needed to be detected • number of samples

  19. How many samples are needed to detect sample mean difference of 1 unit ? R function power.t.test: > power.t.test(delta=1,power=0.95,sd=1,sig.level=0.05) Two-sample t test power calculation n = 26.98922 delta = 1 sd = 1 sig.level = 0.05 power = 0.95 alternative = two.sided NOTE: n is number in *each* group

  20. What is the power of test when using 10 samples ? R function power.t.test: > power.t.test(n=10,delta=1,sd=1,sig.level=0.05) Two-sample t test power calculation n = 10 delta = 1 sd = 1 sig.level = 0.05 power = 0.5619846 alternative = two.sided NOTE: n is number in *each* group

  21. How small difference between sample means we are able to detect using 10 samples ? R function power.t.test: > power.t.test(n=10,power=0.95,sd=1,sig.level=0.05) Two-sample t test power calculation n = 10 delta = 1.706224 sd = 1 sig.level = 0.05 power = 0.95 alternative = two.sided NOTE: n is number in *each* group

  22. Two kinds of replicates • biological replicates: biological variability • technical replicates: measurement accuracy • most statistical programs assume independent samples A3 A2 A1 B3 B2 B1 C3 C2 C1 D3 D2 D1

  23. A1 B1 A2 B2 A3 B3 Pooling A1 A2 A3 B1 B2 B3

  24. Pooling • ok when the interest is not on the individual, but on common patterns across individuals (population characteristics) • results in averaging  reduces variability  substantive features are easier to find • recommended when fewer than 3 arrays are used in each condition • beneficial when many subjects are pooled • one pool vs independent samples in multiple pools C. Kendziorski, R. A. Irizarry, K.-S. Chen, J. D. Haag, and M. N. Gould, "On the utility of pooling biological samples in microarray experiments", PNAS March 2005, 102(12) 4252-4257 inference for most genes was not affected by pooling

  25. A B C D How to allocate the samples to microarrays? • which samples should be hybridized on the same slide? • different experimental designs • reference design, loop design • what is the optimal design?

  26. Example of four-array experiment B cy5 cy3 1 2 3 4 cy3 cy5 A

  27. Reference design A 1 Ref B 2 3 C 4 log(C/A) = log(C) - log(A) = log(C) - log(Ref) + log(Ref) - log(A) = log(C) - log(Ref) – (log(A) - log(Ref)) = logratio(array3) - logratio(array1) D

  28. Loop design A 1 4 B D 2 C 3 log(C/A) = log(C) – log(B) + log(B) – log(A) = logratio(array2) + logratio(array1) log(C/A) = log(C) – log(D) + log(D) – log(A) = - logratio(array3) - logratio(array4) log(C/A)=(logratio1 + logratio2)/2

  29. A A A B Ref Ref B B C C C Comparing the designs

  30. Design with all direct pairwise comparisons 2 3 1 4 6 5

  31. Genotype Environment Reference Sample Assay Variation Example: examining genotype, phenotype, and environment Parental - stressed Derived - stressed Parental - unstressed Derived - unstressed

  32. Optimal design • maximize the accuracy of parameters of interest • procedure: enumerate all possible designs, calculate the parameter accuracy for each of them and select the best design • optimal design is model specific

  33. About the nature of microarray data • Microarray data can give hypothesis to be tested further • Results from microarray analysis should be cerified by other means (qPCR,...) • quality of microarray data depends on samples, probes, hybridization, lab work • data pre-processing, normalization, and outlier detection are as important as good experimental design

  34. More about statistics • M.J. Crawley: ”Statistics – An Introduction using R”, John Wiley&Sons, 2005 • S.A. Glantz: ”Primer of Biostatistics”, McGraw-Hill, 5th ed., 2002 • D.C. Montgomery: ”Design and Analysis of Experiments”, John Wiley&Sons, 5th ed. 2001 • Google

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