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

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

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)?ControlTest

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

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

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) group1 group2 cy3 cy5 group1 group2 cy5 cy3

ControlTest y = µ+F1+F2+...+error log transform, normalization

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 Control Test

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)

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

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: MF no treatmenty111, y112, y113, y114 y121, y122, y123, y124 treatment Ay211, y212, y213, y214 y221, y222, y223, y224 treatment By311, y312, y313, y314 y321, y322, y323, y324

Analysis of variance ANOVA can be used to analyse factorial designs y = µ + drug + sex + drug.sex + error MF no treatment1.0, 1.1, 0.9, , 0.5, 0.6, 0.8 treatment A1.1, 1.2, 0.8, , 0.8, 0.6, 0.9 treatment B2.1, 1.9, 1.7, , 1.3, 1.4, 1.1  summary(aov(y~drug*sex,data=data)) Df Sum Sq Mean Sq F value Pr(>F) drug e-08 *** sex e-06 *** drug:sex Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

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: MF no treatment1.0, 1.1, 0.9, , 0.5, 0.6, 0.8 treatment A1.1, 1.2, 0.8, , 0.8, 0.6, 0.9 treatment B2.1, 1.9, 1.7, , 1.3, 1.4, 1.1  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 B B-A

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 simultaneouslyControlTest y=µ+F1+F2+...+error

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 dye1dye2 controly111, y112, y113 y121, y122, y123 treatment Ay211, y212, y213 y221, y222, y223 treatment By311, y312, y313 y321, y322, y323

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

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):

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

actual situation drug has effect actual situation drug has no effect our conclusion drug has effect correct conlusion true positive probability  type I error false positive probability  our conclusion drug has no effect type II error false negative probability  correct conclusion true negative probability 

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 = delta = 1 sd = 1 sig.level = 0.05 power = 0.95 alternative = two.sided NOTE: n is number in *each* group

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 = alternative = two.sided NOTE: n is number in *each* group

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 = sd = 1 sig.level = 0.05 power = 0.95 alternative = two.sided NOTE: n is number in *each* group

Two kinds of replicates biological replicates: biological variability technical replicates: measurement accuracy most statistical programs assume independent samples A3A2A1B3B2B1C3C2C1D3D2D1

Pooling A1 B1 A2 B2 A3 B3 A1 A2 A3 B1 B2 B3

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) inference for most genes was not affected by pooling

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? A B C D

Example of four-array experiment array cy3cy5log(cy5/cy3) 1ABlog(B) – log(A) 2AB 3BAlog(A) – log(B) 4BA B A cy5 cy3 cy5 cy3

Reference design Ref A B C D array cy3cy5log(cy5/cy3) 1RefAlog(A) – log(Ref) 2RefBlog(B) – log(Ref) 3RefClog(C) – log(Ref) 4RefDlog(D) – log(Ref) 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)

Loop design A B C D array cy3cy5log(cy5/cy3) 1ABlog(B) – log(A) 2BClog(C) – log(B) 3CDlog(D) – log(C) 4DAlog(A) – log(D) 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

Comparing the designs Ref A B C A B C A B C reference designreference design with replicates loop design number of arrays363 amount of RNA required per sample 1+Ref2+Ref2 error

Design with all direct pairwise comparisons

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

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

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

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 Google