/ Computational Genomics. Normalization

Transcription

1 0-80 /02-70 Computatonal Genomcs Normalzaton

2 Gene Expresson Analyss Model Computatonal nformaton fuson Bologcal regulatory networks Pattern Recognton Data Analyss clusterng, classfcaton normalzaton, mss. value estmaton functonal assgnment, response programs dff. expressed genes Expermental Desgn array desgn, number of repeats experment selecton

3 Experment desgn A number of computatonal ssues should be addressed: Selectng short subsequences for olgo arrays to mnmze cross hybrdzatons Determnng the number of replcates for each sample Samplng rates for tme seres experments

4 Typcal experment: replcates healthy cancer Techncal replcates: same sample usng multple arrays Dye swap: reverse the color code between arrays Clncal replcates: samples from dfferent ndvduals Many experments have all three knds of replcates

5 Data analyss Normalzaton Combnng results from replcates Identfyng dfferentally expressed genes Dealng wth mssng values Statc vs. tme seres

6 Data analyss Normalzaton Combnng results from replcates Identfyng dfferentally expressed genes Dealng wth mssng values Statc vs. tme seres

7 Normalzng across arrays Consder the followng two sets of values:

8 Lets put them together

9 Normalzng between arrays The frst step n the analyss of mcroarray data n a gven experment s to normalze between the dfferent arrays. Smple assumpton: mrna quantty s the same for all arrays M n n y Where n and T are the total number of genes and arrays, respectfully. M s known as the sample mean M Next we transform each value to make all arrays have the same mean: yˆ y M + M T T M

10 Normalzng the mean

11 Varance normalzaton In many cases normalzng the mean s not enough. We may further assume that the varance should be the same for each array Implctly we assume that the expresson dstrbuton s the same for all arrays though dfferent genes may change n each of the arrays Here V s the sample varance. Next, we transform each value as follows: T n V T V M y n V 2 V V M M y y + ˆ

12 Normalzng mean and varance

13 Typcal experment: ratos healthy cancer In many experments we are nterested n the rato between two samples For example - Cancer vs. healthy - Progresson of dsease rato to tme pont 0

14 Transformaton Whle ratos are useful, they are not symmetrc. If R 2*G then R/G 2 whle G/R ½ Ths makes t hard to vsualze the dfferent changes Instead, we use a log transform, and focus on the log rato: y log R G log R logg Emprcal studes have also shown that n mcroarray experments the log rato of most genes tends to be normally dstrbuted

15 Normalzng between array: Locally weghted lnear regresson Normalzng the mean and the varance works well f the varance s ndependent of the measured value. However, ths s not the case n gene expresson. For mcroarrays t turns out that the varance s value dependent.

16 Locally weghted lnear regresson Instead of computng a sngle mean and varance for each array, we can compute dfferent means and varances for dfferent expresson values. Gven two arrays, R and G we plot on the x axs the log of ther ntensty and on the y axs ther rato We are nterested n normalzng the average log expresson rato for the dfferent ntensty values

17 Computng local mean and varance Settng may work, however, t requres that many genes have the same x value, whch s usually not the case Instead, we can use a weghted sum where the weght s propotonal to the dstance of the pont from x: x x x x x m y k x v y k x m 2 x m y x w x w x v y x w x w x m 2 ˆ x v V M x m x y x y +

18 Determnng the weghts There are a number of ways to determne the weghts Here we wll use a Gaussan centered at x, such that w x x x σ 2πσ e σ 2 s a parameter that should be selected by the user

19 Locally weghted regresson: Results Orgnal values normalzed values

20 Other normalzaton methods If you are not comfortable wth the equal mrna assumpton, there are other possble normalzaton methods: We can use genes known as house keepng genes. These genes are assumed to be expressed at smlar levels regardless of the condton the cell s n. Alternatvely, we can use controls. These are sequences that are manually nserted nto the sample wth known quanttes ths s manly useful for olgo arrays.

21 Usng spke controls Suppose we have m raw measurements of spked controls per chp and T chp experments altogether We need to construct a model over these observatons that dsentangles the experment dependent scalng and the underlyng supposedly fxed control levels x. x T x m x m T

22 Determnng the underlyng expresson We can try to learn the parameters of a model that attempts to dsentangles the experment dependent scalng and the underlyng fxed control levels : x.. m r e x m r Here: x s the th measurement for control m s the fxed control amount r s the unknown experment dependent scalng e s random multplcatve nose T T e T

23 Log transform Log-transform all the varables x m + r + log e log log log y µ ρ, ε log x, log m, log r log e After the transformaton we can express the model n the smple form Observaton Model + nose y µ + ρ + ε

24 Nose model We make some addtonal assumptons about the model y µ + ρ + ε, ε ~ N0, σ Nose ε s ndependent across controls / experments The nose s Gaussan orgnal multplcatve nose s log-normal The nose varance does not depend on the experment but may depend on the specfc spked control 2

25 Maxmum lkelhood estmate Maxmum lkelhood estmate MLE s a general and powerful technques for fttng parameters of a probablstc model. Gven a parametrc model for example, Gaussan nose and observed data, we look for the set of parameters n our case, mean and varance that maxmze the lkelhood of the model. If we observe data D, then we look for parameters that wll maxmze pd M where M s the model we assume

26 Maxmum lkelhood estmate: Example Assume a unform dstrbuton model X~U0,N. For such a model we have parameter to ft N We now observe the values:.2, 0.5, 3.4, 2.4,.5, 0.8, 2.2, 3.2 what value should we use for N? Recall that n a unform model, px /N for 0<x<N and px 0 for x>n The lkelhood of the data gven N s thus: N x < N x > N 0

27 Maxmum lkelhood estmate: Example.2, 0.5, 3.4, 2.4,.5, 0.8, 2.2, 3.2 Recall that n a unform model, px /N for 0<x<N and px 0 for x>n The lkelhood of the data gven N s thus: It s easy to see that to maxmze ths value we must pck an N that s at least as large as the maxmum value we observed. N On the other hand, the larger N the smaller /N Thus, the value that maxmzes the lkelhood s N 3.4, the largest value we observed. x < N x > N 0

28 Back to our model We want to ft our model to the log transformed raw data We frst wrte log lkelhood term for the observed expresson values: n L Y p y µ, ρ, σ 2 y y.. y m ~ N µ + ρ, σ y T.. y m T 2 y µ ρ 2 0.5log2πσ 2 2σ

29 Iteratve soluton n y n ρ µ n y n 2 2 ρ µ σ M m y 2 2 σ µ σ ρ We terate untl convergence

30 Normalzng usng the estmated parameters Once we obtan the estmate for the scalng parameter we rescale each measured value as follows: so that all genes n all arrays wll have a scalng factor of log scalng of 0 ρ y y ρ ˆ y y ρ ˆ

31 Some addtonal notes The maxmum lkelhood estmates of the nose varances may become too small; would need MAP or Bayesan estmates for the varances n practce. The smple log-normal nose model may not be adequate σ 2 n n y µ ρ 2

32 Olgo arrays: Negatve values In many cases olgo array can return values that are less than 0 Why? There are a number of ways to handle these values The most common s to threshold at a certan postve value A more sophstcated way s to use the negatve values to learn somethng about the varance of the specfc gene

33 Data analyss Normalzaton Combnng results from replcates Identfyng dfferentally expressed genes Dealng wth mssng values Statc vs. tme seres

34 Motvaton In many cases, ths s the goal of the experment. Such genes can be key to understandng what goes wrong / or get fxed under certan condton cancer, stress etc.. In other cases, these genes can be used as features for a classfer. These genes can also serve as a startng pont for a model for the system beng studed e.g. cell cycle, phermone response etc..

35 Problems As mentoned n the prevous lecture, dfferences n expresson values can result from many dfferent nose sources. Our goal s to dentfy the real dfferences, that s, dfferences that can be explaned by the varous errors ntroduced durng the expermental phase. Need to understand both the expermental protocol and take nto account the underlyng bology / chemstry

36 The wrong way Durng the early days though some contnue to do ths today the common method was to select genes based on ther fold change between experments. The common value was 2 or absolute log of. Obvously ths method s not perfect

37 Sgnfcance bands for Affy arrays

38 Value dependent varance

39 Typcal experment: replcates healthy cancer Techncal replcates: same sample usng multple arrays Dye swap: reverse the color code between arrays Clncal replcates: samples from dfferent ndvduals Many experments have all three knds of replcates

40 What you should know The dfferent nose factors that nfluence mcroarray results The two maor normalzaton methods: - Assumng the same mrna quantty - Usng spke controls or house keepng genes Maxmum lkelhood estmaton MLE prncpal