Berrar D. et al. - Practical Approach to Microarray Data Analysis

steps or components. Three important components are: transformation to normality, centralization, and re-scaling. Transformation to normality refers to the adjustment of data such that it approximates a normal distribution (a prerequisite for many types of statistical analysis). The purpose of centralization is to remove biases from the data. Re-scaling is a final step that may be applied to ensure that data from different hybridizations have equal variances. When considered as three independent steps, the experimenter can apply judgment in selecting and applying (or not applying) an appropriate method in each category. Some variations of this terminology should be expected. In particular, the term scaling has been used by some to refer to what we (and others) call centralization.

5.2Transformation to Normality

There is general agreement that a log transformation of most microarray data provides a good approximation of the normal distribution (e.g., Figure 2.2) with minor exceptions (Hoyle et al., 2002). Different authors have preference for different log bases (usually log10, log2, or loge [=ln]). Although all of these work equally well, the base that is used needs to be explicitly recorded for future reference. The log transformation can be applied to original observations (e.g., median pixel intensities) or to ratios – remembering that log(x) – log(y) is equivalent to log(x/y). Some authors will plot original values on log axes, while others prefer to work directly with log-transformed variables. For simplicity, we now assume that the log transformation is applied to all variables.

5.3Data Centralization

Most published references to normalization deal primarily with what we are calling centralization: the removal of biases in the data. Centralization is particularly important when using ratios to monitor changes in gene expression. Bias arises from a number of sources, including variation within and among arrays, differences in mRNA concentration or quality, unequal dye incorporation, and wavelength-related differences in scanner strength. Without correcting these biases, it may appear as though too many genes are up- (or down-) regulated. This can be especially misleading when one is inspecting the data using a colour scheme3, which is why it is necessary to identify this problem using a graphical aid such as the M vs. A plot.

Most microarray software packages provide an option for global centralization, and this should be considered as a bare minimum. For log ratios (M), the expected mean of a data set is zero, so global centralization is

3 Most microarray software packages provide a feature whereby one variable (e.g., the ratio of medians) is shown in spreadsheet format, with hybridizations in columns, and substances in rows. Each cell is artificially coloured based on the value in the cell (e.g., such that red indicates up-regulation and green indicates down-regulation).

performed by subtracting the overall mean from each data point. One problem with this approach is that the true expected mean may differ from zero due to an abundance of genes that are actually upor down-regulated. This is sometimes addressed by performing global centralization based on the mean of some control substances, whose expression is not expected to change. Technically, this amounts to subtracting the mean of the control substances from each data point. Two types of control substances have been considered: housekeeping genes, and alien sequences (or spiking controls).

Housekeeping genes are those required for basic cellular activity. Unfortunately, the definition of a housekeeping gene is subjective, and the assumption that its expression is not affected by the treatment is not certain. A spiking control is a sequence that is not expected to be present in the sample. It is applied to specific spots on the array, and it is introduced into the sample in a known concentration. This allows for correction of bias related to dyes or wavelengths, but it does not remove bias that may arise due to the quality or concentration of mRNA samples. Furthermore, unless a large number of controls are used, the mean of the controls may actually give a poorer estimate than the mean of the entire data set. One possible alternative is based on systematic identification of classes of genes that do not change (Tseng et al., 2001). Another alternative, called selfnormalization, is based on the assumption that most bias that is related to dye intensity can be removed by averaging the ratios from two reverselabelled hybridizations. Regardless of whether controls are used for centralization, the inclusion of controls on an array should be considered essential. They provide an important point of reference for use in future comparisons, and they may help to identify unusual or erroneous results.

Global centralization cannot correct for biases that are present within specific parts of the data. Two general categories of this are: bias that is due to intensity (e.g., stronger bias at lower intensities) and bias that is spatial (i.e. dependent on the position within the printed array). The former is illustrated in Figure 2.3ii, where bias in measured expression (M) seems to be related to intensity (A) as estimated by the fitted local regression (LOWESS). The use of the LOWESS function to correct this bias has been suggested (Yang et al., 2002), and the result of this correction is shown in Figure 2.3iii. Bias may also relate to position on the microarray. This will not be apparent unless observations from different parts of the array are selectively plotted. When this type of bias is present, it may require centralization that is weighted based on array coordinates or block position. This should not be attempted unless this source of bias is clearly identifiable. Depending on how the arrays were printed, and on other factors that affect the composition of the array (e.g., DNA concentration, source, or purity), spatial bias may take many different forms. Furthermore, if spatial bias is

manifested as changes in spot intensity, then the two types of bias may be entirely related, and spatial bias would be corrected based on the LOWESS function.

5.4Data Re-scaling

Due to technical variability, hybridizations may show differences in the range or variance of response variables. This phenomenon is not corrected by centralization (e.g., after centralization, M values from one hybridization may range from -2 to +2, whereas another may have values ranging from –3 to +3). Ideally, the variance from different hybridizations should be equal to facilitate comparisons among substances. Furthermore, if replicated treatments are to be averaged, re-scaling ensures that each replicate contributes equal information.

A simple method of re-scaling involves dividing a variable by its variance, thus giving the data a standard variance of unity. Algorithms for multivariate analysis (e.g., hierarchical clustering) may provide an option to perform this transformation automatically at the time of analysis. Unfortunately (or fortunately, depending on your viewpoint), the variance within a hybridization is a result of two components: substances and error, and it is only the error variance that should be standardized. Variance due to substances may differ due to treatment effects, which should not be altered. Conceivably, a variance adjustment could be made by dividing a variable by the variance of control substances, similar to the way in which control samples can be used in centralization (see Section 5.4). However, this might be subject to the same uncertainties about whether controls are appropriate. Furthermore, estimates of variance can be unreliable unless they are based on a large set of observations.

Re-scaling is potentially the least necessary component of normalization, and many researchers may choose to avoid the possibility that re-scaling may have unintentional consequences. Consideration can be given to testing whether variances differ prior to re-scaling (Snedecor and Cochran, 1989).

5.5Alternative Approaches

The use of paired samples with data transformation is probably the most common approach to the preparation of microarray data at the time of writing. It is based on the assumption that biases can be removed separately from each hybridization prior to assembling data from multiple hybridizations for analysis. Notable alternatives to this approach involve removing bias from the data after it has been assembled for analysis by using an analysis of variance (ANOVA). The ANOVA is a technique whereby the

total variability (variance) in an experiment is partitioned into components that can be attributed to treatments and to other recognized sources of variability (e.g., developmental stages, sample dyes, or different batches of arrays). After attributing variance to these factors, residual variance (that which remains) is considered to be a measurement of error that can be used to formulate statistical tests of significance among treatment means. Good discussions of this approach are provided by Kerr and Churchill (2001a,b) and Wolfinger et al. (2001). The models that these authors describe would provide more direct comparisons of expression levels among different treatments and would eliminate the need for including a control sample with every hybridization. This would also reduce the need for further data transformation because biases would be removed by fitting their sources as terms in an ANOVA.

In choosing an appropriate approach, a general recommendation is to consider different options and to store data in such a way that options are not excluded. We believe that the ANOVA-based approaches being developed have distinct advantages, particularly for testing specific hypotheses in a planned experiment. Conversely, the approach of normalizing each hybridization may be more conducive to building databases for retrospective exploration. While an ANOVA or mixed model can account for virtually any source of bias, some may find the approach of normalizing each hybridization to be more intuitive, and more readily adaptable to complex sources of bias. In general, an ANOVA can be applied to an experiment that is designed with paired samples, but normalized ratios cannot be constructed without the presence of a common control. Most existing software and databases are oriented toward the paired-sample approach, but data can easily be exported for alternate analysis using statistical analysis packages. In situations where different methods can be applied, it may be useful to determine if different methods give similar results4.

6.DATA INSPECTION

Data inspection is an important step that should be applied at many stages of microarray analysis, but especially during and after transformation. Several tests can be applied to ensure that transformation has been successful, and that additional steps are not required. These inspection steps are aided by software that provides interactive sorting, highlighting groups of substances, and plotting data points in a variety of configurations. As an alternative to commercial software, which some people find inflexible or expensive,

4 Generally, it will be possible to develop an ANOVA that will give similar results to methods where bias is removed through normalization. One technical difference is that prenormalization fails to account for degrees of freedom that are lost through this process.

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various free alternatives are available (e.g., Liao et al., 2000; Breitkreutz et al., 2001). A further alternative is the use of a graphical statistics environment such as “R” (Ihaka and Gentleman, 1996; http://www.r- project.org), which allows infinite variation in methods for data analysis and inspection. These solutions may require a greater amount of training or practice.

One of the data inspection steps that should be considered is the use of an M vs. A plot (Figure 2.3) in which individual substances can be interactively highlighted within the complete distribution. The basic M vs. A plot allows one to visualize whether the entire distribution of “M” values are centred with a mean of zero, and that the mean is not influenced by intensity (“ A”). By highlighting control substances, one can determine if centralization has caused their mean to differ from zero. It will also be clear whether controls represent a range of intensities, thus, whether they can be used to provide intensity-dependent corrections. By highlighting blocks of data from various spatial regions of an array, one can determine whether spatial-dependent centralization is required. For spotted microarrays, it is useful to know the method and sequence by which the array was printed, as this can have more influence than mere proximity on the slide. Some software applications provide direct interaction between graphing tools and individual spots on a microarray image. For example, one could highlight all spots within a certain region of a microarray to see their positions on the M vs. A plot.

Another step that is useful when multiple replicates have been performed is the use of scatter plots and correlation analysis. A scatter plot of variables (e.g., M or A) from one replicate vs. those from another is an informative visual aid that can quickly show whether replications are consistent. This is also an easy verification of whether the ratio of treatment to control has been constructed in the same direction for both arrays (a negative correlation is a good indication that one ratio is inverted). Direct interaction between the scatter plot and the microarray image can be useful to determine whether specific substances or specific parts of the array are contributing to greater or lesser agreement between the arrays. The Pearson correlation coefficient (r) between two hybridizations provides a simple quantification of this agreement. This is also a useful statistic to gauge the success of different transformation methods. The correlation coefficient is not affected by global centralization or global re-scaling, but it might be useful to validate whether improvements have been made based on local centralization or transformation to normality. A matrix of correlation coefficients can be used to compare more than two replicated arrays, and is useful to gauge the results of different types of replication.

7.DATA FILTERING

Data filtering is a term that can take many different meanings. Indeed, like data inspection, it is a concept that can be applied at any stage of preprocessing. Our reason for discussing this topic at the end of this chapter is that there is a general danger of “losing data” before one is certain that it is not useful. Hence, while suspect observations are flagged during image analysis, and annotations about the substances and experimental conditions are stored, all (or most) data are kept intact within a primary database. Thus, filtering can refer to any step that is required to prepare a generic data set for a specific type of analysis.

The concept of filtering can be visualized as taking a large matrix of data (possibly an entire database) and making a smaller matrix. The large matrix contains hybridizations (usually arranged as columns) and substances (usually arranged as rows), and possibly many different types of observations from each hybridization (also arranged as columns). The smaller, filtered matrix probably consists of only one type of observation (e.g., M, a normalized log ratio of medians), a subset of substances, and a subset of the hybridizations.

Filtering involves three general concepts: selection, averaging, and estimation. Examples of each are given in the following paragraphs. It is important to consider that filtering may be a recursive process, and that different types of filtering are appropriate for different types of analysis. Minimal filtering would be required to prepare data for a statistical analysis designed to estimate error and determine significance thresholds. This might be followed by more intense filtering designed to produce reduced data sets that are appropriate for some of the more advanced functional or classifying analyses discussed in later chapters.

Selection is functionally synonymous to elimination, and can be based on substances, hybridizations, or individual observations. Selection of hybridizations would involve choosing those that provide information on specific conditions (e.g., those based on samples collected less than 50 hours after exposure to a pathogen). One might also eliminate entire hybridizations that failed to meet some aspect of quality control. Substances might be selected on the basis of pre-defined subsets or prior annotations (e.g., selecting only the controls, or only those that are known to be transcription factors). Substances can also be selected or eliminated based on observed data. For example, one might eliminate all substances that did not show a change in expression within any hybridization. This might be done based on a rule of thumb (e.g., ratio must be greater than 2 or less than 0.5) or after one had conducted an ANOVA on the complete data set to establish a significance threshold. This type of filtering might be used to prepare data for hierarchical clustering, where unchanging substances would be

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considered uninteresting. Finally the elimination of specific observations based on some measurement of quality (such as those flagged at the image analysis stage) would be considered.

Averaging is used to combine observations from replicated hybridizations (columns) or replicated substances (rows). Some software packages will automatically average all replicated substances, so this step may not be required. Replicated hybridizations should only be averaged if required by the analysis; otherwise, replications should be left un-averaged so that they can be used in the estimation of error.

Estimation of missing values can be considered as a final filtering step that may be required when individual observations have been eliminated, and when empty cells remain after averaging. Depending on the extent of missing values, one may consider another round of selection to remove those rows or columns that contain large proportions of missing values. Or, one may chose to replace missing values with “neutral” averages that are constructed by averaging entire rows and/or columns. Some software packages do this automatically, but this can lead to artificial results if missing values are numerous. Appropriate methods for dealing with missing values are discussed in the next chapter.

Since more than one type of filtering may be required for several types of analysis, and since data analysis may be done at various stages of data collection, it is useful to implement filtering as a set of “rules” that can be saved, modified, and re-applied to the data as needed. Ease of data filtering is one of the advantages of storing data in a relational database, where such rules can be saved as “queries” or “stored procedures”. A further advantage of using a database is that filtering may depend on several types of information that may be stored in different tables (e.g., gene categories in one table, experimental conditions in another). Most relational databases, or microarray software packages that are associated with a database, will provide some type of “query wizard” that can assist the user in defining a query to filter the data. In the absence of a database, filtering will probably be done in spreadsheets, using various manipulations such as copying formulas, sorting, cutting, pasting, inserting, and deleting.

8.CONCLUSIONS

Defined broadly, pre-processing involves many potential steps that are essential for successful microarray experimentation. The need for some steps (e.g., experimental design, image analysis) is unquestionable. Other steps are less dogmatic. Data transformation, inspection, and filtering should occur based on individual analytical goals and data management systems. These steps may take on new meaning as different techniques for analysis become

widely accepted. A complete and perfect recipe for pre-processing and analyzing microarray experiments does not exist. Therefore, each experimenter must develop systems and procedures that are both appropriate and correct. We hope that the concepts introduced in this chapter will help the reader to better understand the detailed presentations found in later chapters.

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