Lecture 5 Processing microarray data

Transcription

1 Lecture 5 Processin microarray data (1)Transform the data into a scale suitable for analysis ()Remove the effects of systematic and obfuscatin sources of variation (3)Identify discrepant observations Preprocessin > Quality of downstream analyses Example: roups of 4 arrays from mice, Contr,, Contr4, Trt1,, Trt4, C1 to C4 are controls and bioloical replicates, T! to T4 are treated with one dru. There were 3300 enes arrayed. Loarithmic transformation lo transformation, X lo(x) - The variation of loed intensities may be less dependent on manitude, - los reduces the skewness of hihly skewed distributions. - Takin los improves variance estimation. Other power transformations (X X β for some β>0) - Amaratuna and Cabrera (001) X X and - Tusher et al (001) use a cube root transformation: X X 1/3 Example: Historam and normal probability plot of the data Variance stabilizin transformations X lo(x+c) : Symmetrizin the spot intensity data and stabilizin their variances. Rocke and Durbin (001) arrays with replicate spots. Analoy: models used for estimatin concentration of analyte: X α + µe η + ε α: mean backround, µ:true expression level; η and ε normally distributed error (σ η, σ ε ) - At very low expression levels, where µ is close to zero, the measured spot intensity is dominated by the first term in the model, so that X α, and X is approximately normally distributed with mean α and varianceσ ε. - At very hih expression levels, where µ is lare, the measured spot intensity is dominated by the second term in the model, so that X µe η, and X is approximately lonormally distributed with variance µ S η, where S ση ση η e ( e 1). Thus, the variance of X varies linearly with µ.

2 However, on the lo scale, lo(x) lo(µ)+η, indicatin that the variance of lo(x) is constant. - At moderate expression levels, the measured spot intensity is in between the above two extremes and behaves as a mixture of a normal and lonormal distribution with variance µ + σ, which, aain, varies with µ. S η η - Durbin et al (00) eneralized lo transformation: X lo(( X α) + ( X α) + ( σ ε / S η )) - α, σ η and σ ε must be estimated. - If replicate blanks or neative controls are available, α and σ ε : estimated mean and variance. If not: mean and variance of a set of unexpressed enes. σ η : variance of a set of hihly expressed enes -- Rocke and Durbin (001). - The convenient interpretation of lo ratios as lo fold chanes is lost. - Rocke and Durbin (00) X lo(x+c), with c ( σ / S ) α, is a reasonable compromise. Sources of bias - the concentration and amount of DNA placed on the microarrays, arrayin equipment such as spottin pins that wear out over time, mrna preparation, reverse transcription bias, labelin efficiency, hybridization efficiency, lack of spatial homoeneity of the hybridization on the slide, scanner settins, saturation effects, backround fluorescence, linearity of detection response, ambient conditions. - dye bias : almost all multi-channel experiments: Cy5 hiher than the Cy3 - Manitude of the difference enerally depends on the overall intensity. - Difference between the physico-chemical properties of the dyes, the labelin efficiencies, the scannin properties of the dyes and the scanner settins. Normalization Early microarray researchers noticed substantial differences in intensity measurements even amon microarrays that were treated exactly alike. Differences still persist despite hue improvements in the technoloy normalization remove, by data processin, as much as possible, the effects of any systematic sources of variation. Normalization can be rearded as a sort of calibration process. Example: Fiure Normalization by the sum. In this method, the sums of the intensities of the k microarrays bein normalized are forced to be equal to one another. normalization by the mean, normalization by the median Q3 normalization ε η

3 Intensity dependent normalization nonlinear normalization function: X f(x), Amaratuna and Cabrera (000, 001) normalization could have a profound effect on downstream analysis. reference or baseline microarray: median mock array. If X i denotes the transformed spot intensity measurement for the th ene (1,,) in the ith microarray (i1,,i), the median mock array will have as its th observation: M median{x 1,, X I }. First perform a median or Q3 normalization Selection of an invariant ene set, (1) Their expression levels should remain constant across the arrays bein normalized, so that they can be used to estimate the normalization functions. () Their expression levels should span the entire rane of expression levels observed in the experiment, so that it will not be necessary to extrapolate the estimated normalization functions. (3) The normalization relationship for these enes should be representative of the normalization relationship for the all the enes, so that they can be used to normalize all. The invariant ene set could be: Control enes: Synthetic or cross-species DNA sequences have been used for this purpose. Housekeepin enes: A small number of housekeepin enes could be arrayed onto the microarray. Unchanin enes: Metrics from the raw data could be used to select a subset of enes that appear to be the least likely to be differentially expressed. All the enes on the array: Smooth function normalization In smooth function normalization, each microarray is normalized as follows. First, the inverse, i f i -1, of the monotone normalization function, f i, for the ith microarray, is estimated by fittin the model: X i i (M ) + ε i where ε i is a random error term, to the (X i, M ) data for the invariant ene set. The normalized values for the ith microarray are then obtained from: X i f i (X i ). In spline normalization, the function i is a smooth but flexible function such as a cubic spline with a small number (e.., 7) of derees of freedom; the smaller the derees of freedom, the smoother the fit. In lowess normalization, the function i is estimated by fittin a lowess smooth (Cleveland, 1979) to the invariant ene set. The lowess smooth is essentially a series of locally linear fits, each fitted robustly so as to limit the influence of outliers. A user-specified parameter, span, denotes the fraction of data (e.., span1/3) used for smoothin at any data point; the larer it is, the smoother the fit. Note that neither of these methods is affected by a

4 small percentae of outliers. Alternative smoothers such as a multilinear continuous function, a piecewise runnin median or kernel based methods may also be used. Example: Fiure 5.5 shows the data from microarrays C1B and C5B plotted aainst each other after spline normalization. As these are bioloical replicates and no differential expression was expected between the two, all 3300 enes were used as the invariant ene set. The observations are now in areement. Quantile normalization The objective of quantile normalization is to make the distributions of the transformed spot intensities, {X i }, as similar as possible across the microarrays, or, at least as similar as possible to the spot intensity distribution of the median mock array. Either a subset of quantiles or all the quantiles may be equated. To equate a subset of quantiles, say the percentiles, as in Amaratuna and Cabrera (000,001), calculate the percentiles (Q i0,, Q i100 ) of the ith array and the percentiles (Q M0,, Q M100 ) of the median mock array. For any value X i, find the interval, [Q ih, Q i(h+1) ], to which it belons and obtain its normalized value, X i, by linearly interpolatin between the pair of points: (Q Mh, Q ih ) and (Q M(h+1), Q i(h+1) ). Bolstad et al (00) ive the followin alorithm to equate all the quantiles: Arrane the transformed spot intensity {X i } into a xi matrix X. Sort each column of X to ive X sort. Take the means across the rows of X sort and assin this mean to each element in the row to et X *sort. Obtain the normalized version X of X by rearranin each column of X sort to have the same orderin as the oriinal X. Quantile normalization is useful for normalizin across a series of conditions where it is believed that a small but indeterminate number of enes may be differentially expressed, yet it can be assumed that the distribution of spot intensities does not vary too much. Example: Fiure 5.6 shows the data from microarrays C1B and C5B plotted aainst each other after quantile normalization with, aain, all the enes used as the invariant ene set. As with spline normalization, the observations are in areement. In fact, both methods appear to perform similarly. Normalization of two-channel arrays Consider the lo transformed spot intensities, {X R } and {X }, for the channels of a twochannel array, where the letters R and refer to the colors, red and reen respectively, that are typically used to label the channels. If there is no systematic dye bias, the data points on a scatterplot of X R versus X should enerally lie alon the YX line. If this is not the case, then it is necessary to normalize the two channels. Yan et al (001) arue that it is easier to assess this with an M vs A plot, a scatterplot of M versus A, where M X R -X and A (X R +X )/. Here {A } is analoous to the median mock array that was used as the reference array above. If there is no systematic dye bias, the

5 points on the M vs A plot would be scattered around the M0 line. Otherwise, normalization can be done usin any of the methods described above. For an intensity dependent normalization, the normalization function is fitted to the M vs A plot, the fitted values, Mˆ, which function as the normalization adjustments, are calculated. The normalized values are taken to be X R X R - Mˆ / and X X + Mˆ /. After normalization, the expression ratios, R exp(x R -X ), should be scattered around unity. Example: Fiure 5.7a shows the data from microarrays C10A and C10B plotted aainst each other. Even thouh the arrays appear to reasonably in areement, Fiure 5.7b, the M vs A plot, shows more clearly they are not. After a lowess normalization with span1/3, with all the enes used as the invariant ene set produces normalized arrays that are more in areement with each other, as is clear from Fiures 5.7c and 5.7d. Fiure 5.8 shows historams of the expression ratios before and after the normalization. It can be observed that this distribution is more centralized at the null value of unity after normalization. Spatial normalization Sometimes the arrayin equipment or the experimental conditions can introduce systematic spatial effects within a sinle slide. In such cases, a within-slide normalization should be considered. This can be done by subdividin the slide into a rid. A natural rid is the rid determined by the print tip of the arrayer. Normalization across the subsections of the rid can be done usin any of the methods described above. Staewise normalization When the data include both technical replicates as well as bioloical replicates, it is most effective to carry out the normalization in staes. The technical replicates can be normalized usin smooth function normalization and the bioloical replicates can be normalized usin quantile normalization. If the bioloical replicates fall into roups, such as treatment roups, each roup can be normalized separately usin quantile normalization and then all the arrays in all the roups can be normalized across all the arrays usin quantile normalization. Example: Fiure 5.8 shows the results of a staewise normalization. Fiure 5.8a shows a sideby-side boxplot display of the data before any normalization is done. Fiure 5.8b is the data after normalizin the technical replicates via spline normalization. Fiure 5.8c is the data after normalizin the control bioloical replicates via quantile normalization. Fiure 5.8d is the data after normalizin across all twenty microarrays via another quantile normalization. Judin the success of a normalization {Y 1 } and {Y }. Normalization is truly successful if the arrays are monotonically related to each other. Spearman s rank correlation coefficient:

6 1 1 1 { R1 ( + 1)}{ R ( + 1} 1 ˆ ρs, ( 1) where R i is the rank of Y i when the {Y i } are ranked from 1 to. - Concordance correlation coefficient : s 1 ˆc ρ s + s + Y Y ( ) 1 1 ( ) Y c ( Yc Y ) ( Y c 1 Y1) Y Y 1 1 where Y c and 1 sc (c1,) s1 is the covariance. ρ c is a standardized measure of Ε ( Y ) 1 Y and ρ c1 if and only if {Y 1 } and {Y } are in perfect areement. Otherwise, ρ c <1. s1 - Pearson s correlation coefficient, ˆ ρ, measures linearity rather than areement. ss 1 - Concordance map. 1) If the distributional properties of the values chane substantially durin a normalization (e.., the skewness is decreased), it is possible that the concordance correlation coefficients miht increase, but this may only be an artificial improvement. ) For microarrays that have been normalized by equatin all the quantiles, the concordance correlation coefficient will be equal to Pearson s correlation coefficient. This is because, after such a normalization, the quantiles of both samples are identical and, therefore, both means are equal: Y 1 Y and both variances are equal too: s1 s. 3) Spearman s rank correlation coefficient is equal to (a) Pearson s correlation coefficient calculated on the ranks of the data (b) the concordance correlation coefficient calculated on the ranks of the data. Outlier identification - outlier: an observation, X j, that is different from a majority of values X i for that same ene. Many ways of identifyin outliers: Barnett and Lewis (1994) - The z-score rule (rubbs test): Xi X zi s where X and s are the mean and standard deviation of the th ene. X j is an outlier if z j is lare (> 5) The CV rule: Call the furthest observation X j from the mean, variation, CV s / X exceeds some prespecified cutoff. - maskin, X, an outlier if the coefficient of

7 - resistant - The resistant z-score rule: Calculate a resistant z-score, z * i, for every observation: X * i X! zi! s where X! and s! are the median and MAD of the th ene. X j is an outlier if z * j > 5. In experiments with few replicates, a true relationship, σ f ( µ ), use a smoothed version of MAD, s! as an estimator of scale for the th ene and use the followin revised rule to identify outliers: The revised z-score rule: Calculate a revised z-score, z ** i,, for every observation: X ** i X! zi '! s X j is an outlier if z ** j >5