Statistical Analysis of Microarray Time Course Data
Yu Chuan Tai and Terence P. Speed University of California, Berkeley, USA
Walter and Eliza Hall Institute of Medical Research, Australia {yuchuan, terry}@stat.Berkeley.edu
March 31, 2005
6.4.1
Introduction
This chapter discusses the statistical analysis of microarray time course data, with a focus on developmental time course experiments. The methods reviewed here are generally suit-able for experiments based on the most widely used kinds of microarray platforms, including single-color, fluorescently labeled, high-density short oligonucleotide arrays on silicon chips (Lockhart et al., 1996) , radiolabeled cDNA arrays on nylon membranes, or two-color, fluores-cently labeled cDNA (DeRisi et al., 1996; Brown and Botstein, 1999) or long oligonucleotide arrays on glass slides.
Microarray time course experiments typically involve gene expression measurements for thousands of genes over relatively few time points, under one (e.g. wildtype) or more bio-logical conditions (e.g. mutant 1, mutant 2,...). The number of time points can be 3-10 for shorter and 11-20 for longer time courses. The time points at which mRNA samples are taken are usually determined by the investigator’s judgement concerning the biological events of interest and are frequently irregularly spaced, although for periodic time course experiments, equally-spaced times are standard. Measurements of mRNA abundance will be based on mRNA extracted from cell lines, tissue samples or whole organisms, and in what follows we will use the general term units for the source of the mRNA. The major advantage of microar-ray time course studies is that they give us the ability to monitor the temporal behaviour of a biological process of interest through the measurement of expression levels of thousands of genes simultaneously. This can be a powerful experimental design for identifying patterns of gene expression in the units of interest.
Time course experiments can be classified into two main categories which we term periodicand developmental. Periodic time courses include natural biological processes whose
temporal profiles follow regular patterns. Examples are cell cycles, Cho et al. (1998), Spell-man et al. (1998), Cho et al. (2001), and circadian rhythms, Storch et al. (2002), and we expect regulated genes to have periodic expression patterns. In the literature, periodic time course experiments are frequently unreplicated, that is, they arise as a single series of mi-croarray experiments, experimenters perhaps preferring to use scarce resources obtaining a finer temporal resolution, rather than repeating measurements at times already observed. In what we term developmental time course experiments, gene expression levels are measured at successive times during a developing process, for example, during the natural growth and development of, or following a treatment applied to the units. In such cases there are usually few prior expectations concerning the form of the temporal profiles. Here there are commonly 2-5 replicate series, but sometimes there is no replication.
We now summarize several microarray time course experiments. Tomancak et al. (2002) conducted a study of Drosophila embryogenesis using microarray time course experiments as the control of RNA in situ hybridization. Canton S fly embryos were collected, transferred to an incubator and aged. At hours 1 to 12 post egg laying, the embryos were dechorionated and quick-frozen, yielding 12 time point samples. The same procedure was repeated on 3 different days, producing 3 replicates. Himanen et al. (2004) reported a study on a lateral root induction system of Arabidopsis thaliana to characterize the early molecular regulation induced by auxin. Seeds of A. thaliana were germinated on media containing auxin transport inhibitor N-1-naphthylphthalamic acid (NPA). After the germination, the seeds were moved to media with auxin 1-naphthalene acetic acid (NAA) only and the samples of the root segments were collected at four time points: 0, 2, 4, and 6 hours after the transfer from NPA to NAA. There were two biological replicates at each time point and cDNA microarray experiments with the reference design were performed. They identified 906 differentially expressed genes over time and grouped these genes into six major clusters. In Qi et al. (2003), bone marrow-derived mesodermal progenitor cells (MPCs) were obtained from three donors and the gene expression profiles for MPCs or MPCs induced to the osteoblast or chondroblast lineage for 1, 2, and 7 days were monitored using cDNA microarrays. They identified 41 transcription factors differentially expressed over time, in addition to some known signaling genes, hormones, and growth factors involved in osteogenesis. A fourth example of a developmental time course study is Schwamborn et al. (2003). These authors studied the transcriptional response of the human astrocytoma cells U373 to tumor necrosis factor α (TNFα). Again, cDNA microarray experiments were performed. Samples from both TNFα treated and untreated U373 cells were collected at 1, 2, 4, 8, and 12 hours post treatment, and each time point had three biological replicates. The temporal profiles between these two treatments were compared. More than 880 genes were shown to be responsive to TNFα. In Tepperman et al. (2004), gene expression samples were collected at 6 time points,
and the gene profiles of wildtype (wt) and the phytochrome B (phyB) null mutant A. thaliana were compared to identify genes regulated by phyB in response to continuous monochromatic red light (Rc) during the induction of seedling de-etiolation. The study of transcriptional response to corticotrophin-releasing factor (CRF) in Peeters et al. (2004) provides an example of unreplicated time course experiments with four different treatments: DMSO, ovine CRF in DMSO, R121919 in DMSO, and CRF plus R121919 in DMSO. Samples were collected at 0, 0.5, 1, 2, 4, 8, and 24 hours after these four treatments were applied to mouse AtT-20 cells, and were hybridized to Affymetrix chips.
Following standard practice in statistics, Diggle et al. (2002), we further categorize time course experiments into longitudinal and cross-sectional. Longitudinal experiments are those in which the mRNA samples at different times are extracted from the same unit, be it cell line, tissue or individual. This allows joining of ordinate values corresponding to observations on the same unit at different times, either by straight lines or fitted curves, to give the unit’s time course for each gene, which will also be called the temporal pattern or profile. By contrast, cross-sectional time course experiments are those in which the mRNA samples at different times are extracted from different sources (units). With cross-sectional data, the individual data points can also be joined across time, using averages when there are replicate measurements, but the interpretation of the resulting curve is different. It will not correspond to any particular unit, but will be thought of as a population curve. In practice there will be experiments exhibiting features of both longitudinal and cross-sectional types, e.g. Tomancak et al. (2002). There are more cross-sectional microarray time course experiments published to date than longitudinal ones, e.g. Tepperman et al. (2004) and Himanen et al. (2004) cited above. This is probably because it is often infeasible to carry out longitudinal experiments because of the limited availability of mRNA from individual organisms such as laboratory mice. However, Qi et al. (2003) is an example of longitudinal studies.
In this chapter we review methods for the design and analysis of microarray time course experiments. After discussing design issues in section 6.4.2, we turn to methods for identifying the genes of interest to the experimenter in section 6.4.3, be they genes which change over time, or genes which change differently over time between two or more biological conditions. Depending on one’s perspective, this task can be viewed as a ”filtering” of the genes to remove those which are not of interest, before turning to a different kind of analysis such as clustering, or it can be seen as identifying a small to moderate list of genes for validation and further characterization. We use the microarray time course data of the study in Drosophila embryogenesis in Tomancak et al. (2002) to illustrate the concept of moderation and compare some of the statistics we describe below. We then review the literatures on clustering gene expression microarray time course data in section 6.4.4, and end with a few comments about alignment of time series in section 6.4.5.
We close this introduction with a few remarks on why the analysis of microarray time course data is special, and not adequately covered by the enormous literature that already exists on the analysis of time series, see e.g. Diggle (1990) and references therein. There are three principal reasons. One is the fact that microarray time series are usually so short that we cannot consider applying methods typically used to analyze time series data, e.g. ARMA, Fourier or wavelet methods, as in Diggle’s and other time series books. A second reason is that there are typically thousands of genes and hence thousands of short time series, all sharing the common experimental conditions. It is natural to think of analyses which have elements in common for all genes, such as the empirical Bayes (EB) methods described below. While there is some literature on EB methods in time series, we know of none involving thousands of series, as is the case here. Finally, an important aspect of microarray gene expression data is the clustering of genes, here based on their temporal profiles. As far as we know, this problem has not arisen in the traditional time series literature, at least not in the form we meet it here. A far more relevant body of literature is that on longitudinal data analysis, for when the data are longitudinal, this is precisely the right context. When the data are not longitudinal, our context is that of many related regression models.
6.4.2
Design
The choice of design for microarray time course experiments will depend on several factors, principally the questions the researcher wishes to address, and the available resources, in-cluding mRNA, microarrays, and related reagents. We refer to Yang and Speed (2003) for a general discussion of design issues for microarray experiments.
The first and most important microarray time course design question for an investigator will be whether to carry out a longitudinal or cross-sectional study. As explained in Diggle et al. (2002), while it is often possible to address the same biological question using either longitudinal or cross-sectional experiments, the major merit of longitudinal time course stud-ies is they provide information about the temporal changes in gene expression levels within units, something that is not possible with cross-sectional studies. In statistical terms, the difference between longitudinal and cross-sectional experiments comes from the fact that gene expression measurements are typically correlated over time within units, and these correla-tions can be estimated and used to advantage in longitudinal studies. On the other hand, such biological correlations cannot be detected in cross-sectional time course studies, since mRNA is extracted from different sources at different times.
It follows from what has just been said that if temporal changes in gene expression over time are of primary interest to the experimenter, an effort should be made to carry out a longitudinal study wherever possible. We appreciate that in many, perhaps most cases, this
may be infeasible, because of the impossibility of repeatedly sampling the mRNA from the same units. Nevertheless, a good approximation to a longitudinal study can be often realized by creating parallel, identically treated units, and sampling from different ones at different times. The difference between this kind of design and true longitudinal design depends on just how similar are the parallel, identically treated units. In some cases, they can be very similar indeed, and we observe the correlations characteristic of a genuinely longitudinal design, although their origins may be different. In such cases, these designs can be more powerful than cross-sectional studies for detecting changes.
Although we might give the impression that for design purposes, the distinction between longitudinal and cross-sectional studies is straightforward, this is not really the case. There are many contexts in which hybrid studies pose more challenging design problems. For ex-ample, we might run replicate time course experiments on plants grown in a growth chamber, where each replicate consists of a series of successively sampled plants grown together un-der controlled conditions. This study is cross-sectional from the perspective of plants, but longitudinal from the point of view of growth chambers. The appropriate number of full replicates, and of plants at each time within replicates given fixed resources, will depend on the relative magnitude of the different components of variation.
The number of time points is usually decided by the biological background and the cost of the study. More time points permits a finer analysis of temporal patterns, e.g. a more accurate determination of the time of onset or decay of a gene’s expression, but in some experiments, accuracy here of this kind is not required.
Questions of interest to an investigator might concern the temporal profile of genes for one biological condition, such as a desire to identify cell-cycle regulated genes. Alternatively, interest might focus on comparison between gene profiles across two or more conditions. We might want to identify those genes which change over time in a wild-type organism, and similarly those genes which change over time in a mutant organism, and identify those genes whose temporal profiles for the wild-type and mutant are different. The latter may include many genes which are unchanging in one or the other of the two biological conditions. The way in which such questions can affect the choice of design is explained in Yang and Speed (2003), and we will not revisit that here in detail.
A short time course experiment can be regarded as a single factor experiment with time as a factor, see Searle (1997) and Neter et al. (1996). What makes it different from other single factor experiments is the additional information from the natural ordering of time course samples. This natural ordering of levels will lead to certain comparisons being of greater interest to the researcher, and others of lesser interest. For example, comparisons of each time point mRNA sample with the baseline or differences between consecutive time points are likely to be of greater interest than comparisons between widely separated times.
Interest might focus on specific aspects of the temporal patterns of gene responses, such as monotonicity, convexity, and linearity (Yang and Speed, 2003).
The design of time course experiments can be considerably more complicated in the two-color comparative experiments (e.g. cDNA arrays) in comparison with single-channel experiments (e.g. Affymetrix chips), although if a common reference design is used, the two cases are fairly similar. For short two-color comparative time courses, it is possible to enumerate all the possibilities to find the optimal design. However, for those with much larger number of time points like the yeast cell cycle data in Spellman et al. (1998), this is not feasible. There is not much literature on the design of time course experiments, but recently, Glonek and Solomon (2004) described a method for designing short cDNA time course experiments. They optimized statistical efficiency and identified so-called admissible designs, and selected efficient designs based on the effects of most interest to the biologists, the number of arrays available, and other resources. Their approach was shown to give designs better than the popular common reference design and those incorporating all possible pairwise comparisons. Optimal design for microarray time course experiments is a research topic for the future.
Replication
Replication is an important aspect of all statistical experimental design. As described in Yang and Speed (2003), there can be three types of microarray replicates: biological repli-cates where mRNA samples are taken from different units; technical replirepli-cates, where mRNA samples are taken from the same unit and are split and hybridized onto different arrays; and within-array replicates, where probes are spotted in replicate on the same array. These types apply equally to longitudinal and cross-sectional time course experiments. The variation between gene expression measurements taken on these three types of replicates will be dif-ferent, and in general is gene-specific. Replication is a good thing, as it provides estimates of variability relative to which temporal changes and/or condition differences can be assessed, making analysis much more straighforward. Biological replicates are generally best, as they permit the conclusions from the experiment to be extrapolated to the wider population of units from which the experimental units were obtained, something which is not possible with only technical replicates. With unreplicated experiments, the inference to a wider population is not possible, and the analysis is less straightforward, being more dependent on unverifiable assumptions, as there is no estimate of pure error which can be used. We suggest at least 3 replicates at every time point. When replicates are available, it is better to use the variation between them in any analysis, rather that just average across the replicates and proceed as with a single time course experiment. Many of the methods we describe below require
replicates, and are designed to be effective with the small numbers of replicates common in this context.
6.4.3
Identifying the genes of interest
There have been many published studies involving developmental time course experiments. As indicated above, experimenters’ aims vary, but we can categorize them broadly as : a) one-sample, where the aim is to identify genes which change over time, perhaps in some specific way; b) two-sample, where in addition to identifying temporally varying gene expression, interest is in comparisons across two biological conditions, and c) D > 2-sample experiments, which are as in b), with D ≥ 3 biological conditions. These categories apply equally to longitudinal and cross-sectional studies. Before we go on to the analysis methods, we illustrate the foregoing by assigning some of the case studies cited above to their category.
The study in Himanen et al. (2004) on a lateral root induction system of A. thaliana to characterize the early molecular regulation induced by auxin is an example of one-sample problem: only a single treatment (i.e. NPA followed by NAA) was applied to the same type of cells, and the genes whose expression levels change over time were of interest. Examples in the two-sample category include the study of Schwamborn et al. (2003), who compared the temporal profiles between the TNFα treated and untreated human astrocytoma cells U373, in order to elucidate the post-treatment transcriptional response. Similarly, Tepperman et al. (2004) compared the temporal profiles of genes in wildtype (wt) and phytochrome B (phyB) null mutant A. thaliana, to identify genes regulated by phyB in response to continuous monochromatic red light (Rc) during the induction of seedling de-etiolation. Both of these studies involved just two different biological conditions: treated versus control cells, and wildtype versus mutant organisms. In this type of problem, identifying all the genes with different temporal profiles between the two biological conditions was usually of interest, although sometimes only genes with different shapes were of interest, with those having similar shapes but different magnitude across the two conditions not being of interest. An example in category c) is the study of transcriptional response to corticotrophin-releasing factor (CRF) described in Peeters et al. (2004). There gene expression levels were measured at 0, 0.5, 1, 2, 4, 8, 24 hours after four different treatments are applied to mouse AtT-20 cells. As in the two-sample problem, genes of interest are those which have different gene expression profiles over time, either in shape and magnitude, or in magnitude only, between the four treatments.
The identification of temporally changing or differentially changing genes not only gives insight into the biological processes under study, it also provides a way of selecting a subset of genes from the entire gene set for further analysis such as clustering. As yet there are
relatively few methods available for identifying the genes of interest in this context. The approaches most widely used are those for identifying differentially expressed genes for repli-cated microarray experiments across two or more independent sample groups (Baldi and Long, 2001; Efron et al., 2001; Tusher et al., 2001; Dudoit et al., 2002; L¨onnstedt and Speed, 2002; Broberg, 2003; Ge et al., 2003; Kendziorski et al., 2003; Reiner et al., 2003; Smyth, 2004). The idea here is the simple one of testing whether there are changes in gene’s ex-pression across time by making comparisons between times, e.g. between all consecutive pairs of time points, or all possible pairs of time points, see e.g. Maratou et al (2003). It is reasonable though not ideal to analyze time course data with these approaches, as they assume independence of the samples across different times, which is not true for longitudinal time course data. We will give some solutions to these problems for both longitudinal and cross-sectional data in the following subsections.
ANOVA and the F-statistic
An intuitive way to select differentially expressed genes from time course data is to use the classical ANOVA (cross-sectional) or mixed-effect ANOVA (longitudinal) model, see e.g. Neter et al. (1996) and Diggle et al. (2002). In the one-sample case, one includes time as a factor and possibly replicate as another factor, and calculates the F -statistic corresponding to time. Similarly, for two- and D > 2-sample case, one includes the factors time and biological condition, and their interaction term in the model, and possibly replicate as another factor, and computes the F -statistic for the interaction term. Wang and Kim (2003) has a one-sample example using classical one-way ANOVA. In order to obtain approximately valid p-values, care needs to be taken to deal with multiple testing, see e.g. Ge et al. (2003) and references therein.
Park et al. (2003) proposed a modified ANOVA approach and some variants without the normality assumption. Their basic model is like the usual two-way ANOVA with time, biological condition, and their interaction as effects. Genes which are not significant (after p-value adjustments) in the time × condition interaction term will be fitted with the second model, removing the interaction term. Then genes with significant time effect after p-value adjustments are selected. However, their method ignores the potential correlations among times from longitudinal time course data. Even if there are no biological correlations, the fact that there are usually only very few replicates makes the estimation of gene-specific variances unstable. Romagnolo et al. (2002) and Wang and Kim (2003) used mixed-effect ANOVA to identify genes with different temporal profiles between the heat-shock treated let-60 and wildtype, and dauer exit and L1 starvation c. elegans, respectively. Himanen et al. (2004) also used mixed-effect ANOVA for their one-sample problem. Chapter 6 of Diggle
et al. (2002) discusses some standard ANOVA methods for longitudinal data, and we refer the reader there.
A number of questions are not adequately addressed by classical ANOVA methods or the variants of Park et al. (2003). First, to obtain an F -distribution for the F -statistic, we require that the samples at different time points are independent, an assumption easily violated by longitudinal time course data, and we also require normality of the observations. The robustness of the F -distribution to deviations from normality is well-studied, see e.g. Scheff´e (1959), but for gene expression measurements on the log scale this may not be a great concern. Secondly, as a result of the very large number of genes, and relatively small number of replicates, the F -statistic may lead to more false positives and false negatives than would normally be the case, because of poorly estimated variances in the denominator. See Tai and Speed (2004) for the results of a simulation study. This issue can be addressed using the notion of moderation, see below.
Despite these reservations concerning the F -statistic, it should be understood that it has been and will continue to be effective for identifying genes of interest to researchers. However, we believe that we can do this job better with alternative statistics.
Moderation
Typically, genes with large changes in gene expression levels over time relative to their repli-cate variances are best candidates for following up. However, given the thousands of genes in a microarray time course experiment, and the small number of replicates, the variances (in the case of cross-sectional data) or variance-covariance matrices (in the case of longitudinal data), are usually very poorly estimated. As a result, some genes which exhibit relatively small amounts of change over time and small replicate variances, may have large between to within time F -statistics because of these under-estimated denominators. We may conclude that such genes are changing over time, but if they are not, they will be false positives. For example, Jiang et al. (2001) (page 219) mentioned such genes. Figure 6.4.1 gives an exam-ple of such a gene from Tomancak et al. (2002). The F -statistic for this gene has a higher ranking than many other genes exhibiting greater change in expression levels over time, so we consider it a false positive. On the other hand, some genes with large amounts of changes over time, but also large replicate variances, may have small F -statistics because of over estimated denominators. Such genes may be false negatives, i.e. may be changing over time, but not be identified as such. The gene in figure 6.4.2 is clearly changing over time, however, the expression level of experiment B at 10 hours is much lower than those of experiments A and C. This single outlier leads to lower rankings for F -statistic compared to all the other statistics mentioned below, however, this gene is very likely to be of interest. By moving
(shrinking) gene-specific variances or variance-covariance matrices towards a common value estimated from the whole gene set, the total number of false positives and false negatives can usually be reduced. This is what we term moderation.
The idea of moderation has entered into the analysis of microarray data in different forms. Efron et al. (2001) and Tusher et al. (2001) tuned the t-statistic by adding a suitable constant to the standard deviation, the constant being estimated by a percentile of sample standard deviations, by minimizing a coefficient of variation, respectively. Their approaches are not based on any distributional theory. L¨onnstedt and Speed (2002) brought the idea of mod-eration in the univariate hierarchical Bayesian mixture model for two-channel comparative experiments by smoothing gene-specific residual sample variances toward a common value. Smyth (2004) formally introduced the moderated t-statistic in the univariate general linear model setting by substituting the denominator of t-statistic with a moderated denominator. The gene-specific moderated sample variance in Smyth (2004) is defined based on some nice distributional theory and the hyperparameter estimates derived there are shown to perform better than those in L¨onnstedt and Speed (2002). Tai and Speed (2004) further extended the univariate model in L¨onnstedt and Speed (2002) and Smyth (2004) into multivariate settings and introduced the M B-statistic (multivariate empirical Bayes statistic) and the eT2 statistic to rank genes in the order of differential expression in the one- and two- sample cases in the longitudinal time course context. In addition, Tai and Speed (2005a,b) derived M B-statistics for D > 2 samples for longitudinal and cross-sectional data.
Likelihood-based approach
Before we go on to discuss our multivariate empirical Bayes methods, here we briefly comment on the likelihood-based approaches for longitudinal time course data. Given the very few replications in this context, the gene-specific sample variance-covariance matrix or its variant may be singular and the resulting analysis can be unstable. Guo et al. (2003) proposed the gene-specific score based on the robust Wald statistic for one-sample longitudinal data, using an approach similar to Tusher et al. (2001), adding a small positive number times the identity matrix in the denominator.
w(i) = [L ˆβ(i)]T[L ˆVS(i)LT + λwIr×r]−1
[L ˆβ(i)], (6.4.1) where L is an r × p matrix of rank r, ˆβ is the estimated p × 1 vector of unknown regression parameters β, ˆVS is the estimated covariance matrix for ˆβ, and λw is a positive scalar. The way they estimated λw is exploratory. Moreover, their approach is for one-sample problem
only, and the fact that the number of subjects is usually very small makes asymptotic theory inappropriate.
Storey et al. (2004) also proposed a likelihood-ratio based approach, assuming gene ex-pression values are composed of population mean and individual deviates. They constructed the F -statistics for both longitudinal and cross-sectional data in a standard way, and pre-sented a careful treatment of the multiple testing issue. In contrast, Tai and Speed (2004) suggested the moderated LR and Hotelling T2 statistics, with the smoothing of gene-specific sample variance-covariance matrix making use of the replicate variability information across the whole gene set, and then they simply ranked genes according to one or the other statis-tic. These two statistics were shown in a simulation study to perform about as well as the M B-statistic described below.
Empirical Bayes
The existence of thousands of genes in the microarray time course context brings to mind the empirical Bayes (EB) approach to inference. This is a model-based way of introducing moderation into the analysis.
In Tai and Speed (2004), a multivariate hierarchical normal model with conjugate priors is proposed to derive the posterior odds for differential expression in the one- and two- sample problems. This is designed for longitudinal data, and takes into account correlations across times. When all genes have the same number of replicates, the M B-statistic for the null hypothesis that the expected profile equals to 0, or the paired two-sample problem with the null hypothesis that the expected profiles are the same, is a monotonic increasing function of the statistic eT2 = ˜t0˜t, where
˜t = n1 2eS−
1
2X, (6.4.2)
is just the traditional multivariate t-statistic with the denominator replaced by a moderated covariance matrix eS. This expression incorporates the gene-specific covariances but also shares the covariance information across genes:
e
S= (n − 1)S + νΛ
n− 1 + ν . (6.4.3)
Here S is the gene-specific sample variance-covariance matrix; X is the gene-specific sample average time course vector; n is the number of replicates; ν and Λ are hyperparameters estimated from the whole gene set, see Tai and Speed (2004) for details. The M B-statistic or
e
T2 statistic for the independent two-sample problem is also derived in Tai and Speed (2004), where differential expression now means that the expected profiles are different between the two biological conditions. It is shown there that the M B-statistic achieves the lowest number of false positives and false negatives, and performs about as well as the moderated Hotelling T2 statistic. One of the values of the multivariate empirical Bayes is that it provides a natural
way to estimate the gene-specific moderated sample covariance matrix, while the likelihood ratio based approach (moderated Hotelling T2 statistic) does not.
Let Ydi be the i-th replicate for the d-th condition, and Y and Ydbe the overall sample
average, and average for the d-th condition only, respectively. In the case that there are more than 2 biological conditions and all genes have the same number n of replicates within each condition, the posterior odds for difference between conditions for a conjugate normal model are proportional to ¯ |TSSP + M + νΛ| ¯ ¯PDd=1WSSPd+ PD d=1Md+ νΛ ¯ ¯ ¯ 1 2(nD+ν) , (6.4.4)
where TSSP =PDd=1Pni=1(Ydi− Y)(Ydi− Y)
0
and WSSPd=Pni=1(Ydi− Yd)(Ydi− Yd)
0
are the total and within sums of squares and products, Md, M, and νΛ are matrices involving
the (condition-specific) prior means and variance-covariance matrices, respectively. This is our EB analogue of Wilks’ likelihood-based Λ from MANOVA, see Tai and Speed (2005a) for details.
For cross-sectional data across D ≥ 2 biological conditions, Tai and Speed (2005b) de-rive the posterior odds that the expected temporal profiles are different among biological conditions versus they are the same. Let Ydji be the log2 intensity value or log2 ratio of this
gene for the d-th biological condition, j-th time point, and i-th replicate. Ydji are independent
across times (j = 1, ..., k), biological replicates within conditions (i = 1, ..., ndj) and biological
conditions (d = 1, ..., D). The sampling times need not to be the same within and across biological conditions. For the simplest case, we assume they are, and that all genes have the same number of replicates n for all conditions and times. Again, under a conjugate normal model with unstructured means, the posterior odds are proportional to
à T SS+ m + νλ2 PD d=1W SSd+ PD d=1md+ νλ2 !1 2(nkD+ν) , (6.4.5)
where Ydj = n−1Pni=1Ydjiand Yj = D−1PDd=1Ydj denote the average log2(relative)
expres-sion level at the j-th time point for the d-th condition only and all the conditions, respectively; T SS=PDd=1Pkj=1Pni=1(Ydji− Yj)2 and W SSd=Pkj=1Pni=1(Ydji− Ydj)2 are the total and
within sums of squares, respectively; md, m, and νλ2 are quantities involving
(condition-specific) prior means and variances. This is a special case of our fully moderated F -statistic, the EB analogue of traditional F -statistic. Gene selection using either the M B-statistic or the eT2 statistic can be based on rankings.
The multivariate EB procedure in Tai and Speed (2004) focuses on moderating the denominator of the multivariate t-statistic t, and ranks genes according to the moderated
statistic eT2, to reduce the number of false positives and false negatives resulting from very small or very large replicate variances or covariances. Alternatively, one could replace the numerator of the multivariate t-statistic with a robust estimate, to avoid the problem of very large eT2 resulting from outliers. Such outliers issue can be common in the microarray time course context, when the sample sizes are typically very small (2 or 3). Incorporating robust methods into the analysis of microarray time course is a research topic of interest here. Figures 6.4.3 – 6.4.5 gives the profiles of the top-ranked genes from Tomancak et al. (2002) using the one-sample longitudinal M B-statistic (Tai and Speed, 2004), the one-sample cross-sectional M B-statistic with a fifth-degree polynomial model for the means (Tai and Speed, 2005b) , the moderated F -statistic (Smyth, 2004), and the usual F -statistic with unstructured means.
Regression approaches, including B-splines
To date these methods have been used mainly for unreplicated time course data under a single biological condition. Zhao et al. (2001) outlined a regression model to search for genes with transcriptional response to a stimulus. Their regression function was built to relate each gene’s profile to a vector of covariates including dummies for the stimulus categories, time, and other characteristics of the sample. Their model included gene-specific parameters and parameters to model the heterogeneity across arrays. The mean vector was estimated using the technique described in Liang and Zeger (1986). They further focused on the single-pulse model (SPM), which is specific for the setting when cells are released from cell cycle arrest. Xu et al. (2002) described an application of the same kind of regression model to a time course study involving Huntington’s disease.
Several researchers have suggested the use of B-splines to model gene profiles. In Bar-Joseph et al. (2003b) the expression profiles for each gene and each of two biological conditions were represented by continuous curves fitted using B-splines. A global difference between the two continuous curves and an ad hoc likelihood based p-value was calculated for each gene. Other papers using B-splines to model profiles are Luan and Li (2004) and Hong and Li (2004). Luan and Li (2004) adopt the shape-invariant model (Lawton et al., 1972; Wang and Brown, 1996) for guide genes, and model the common periodic function shared by all periodically expressed genes using a B-spline basis. Such genes were identified using a false discovery rate (FDR) procedure. Both the B-spline based approach in Bar-Joseph et al. (2003b) and Luan and Li (2004) were illustrated on the yeast cell cycle datasets. They do not seem suitable for short time courses. Similarly, Hong and Li (2004) proposed a B-spline based approach to identify differentially expressed genes in the two-sample case. There they modeled the expected profile as linear combinations of B-spline basis functions, and used a Markov chain Monte Carlo EM algoritm (MCEM) to estimate the gene-specific parameters
and hyperparameters from the hierarchical model. They selected differentially expressed genes using empirical Bayes log posterior odds, and the posterior probability based FDR. They showed their method performed better than the traditional ANOVA model. As above, the approach in Hong and Li (2004) seems more suited to longer time course data.
Contrasts
A simple but powerful tool for extracting temporal patterns is found in contrasts: linear combinations of gene expression measurements over time. Contrasts usually but not always have their coefficients summing to zero. An example of the use of contrasts can be seen in L¨onnstedt et al. (2003) where samples were taken from cells at 0.5, 1, 4, and 24 hours after stimulation with a growth factor. Genes were regarded as early responders if they had large values of < c, E >= PtctEt where ct = (t − 24.5)2 and Et is the gene expression value
at time t, while those having large values when ct = t2 were termed late responding genes.
Smyth (2004) used contrasts in the univariate linear model setting, and derived a partly moderated F -statistic for testing whether there is any change in gene expression levels over time. This approach assumes the samples are independent, and so would be appropriate for cross-sectional data. Fleury et al. (2002) described a valuable multi-criterion optimization method called Pareto front analysis, for ranking and selecting genes of interest. In their paper they made use of contrasts to select genes with many predefined patterns. In essence, Pareto fronts and their variants (Hero and Fleury, 2002; Fleury et al., 2002; Hero and Fleury, 2004) seek to identify genes with large values for all of a set of competing contrasts of interest.
Hidden Markov Models
Yuan et al. (2003) presented a hidden Markov model approach for selecting differentially expressed genes from replicated time course experiments with multiple biological conditions. They considered all possible equality and inequality relations among means across biological conditions as states, and the expression pattern process was modeled as a Markov chain, with either time-homogeneous or non-homogeneous transition matrices. The observations were conditionally independent given the state of the chain. In this approach, dependence between gene expression values at different times was completely described by the pattern process (i.e., the hidden Markov chain), and genes were selected based on the posterior probabilities of states of interest. This is an example of using an HMM to model time-dependence in microarray time course data, while many others have used HMMs in this context for clustering.
6.4.4
Clustering
The identification of differentially expressed genes narrows down the number of genes for further analysis. Clustering genes with similar temporal profiles is commonly the next phase of the initial analysis. This is done in the belief that genes with similar temporal profiles may well be involved in similar biological processes, e.g. in the same aspect of response to a treatment. Frequently there is a further hope that genes in the same cluster share common sequence motifs in their regulatory region. In clustering, the focus might be on grouping genes with particular temporal profiles, say early induced, monotonic increasing, up first and then down and so on, or it may simply be a way of partitioning all genes into automatically defined groups.
Below we briefly summarize the literature on clustering, referring to the review of M¨oller-Levet et al. (2003) for a fuller treatment of the issues. One of the earliest examples was hierar-chical clustering of the yeast cell cycle data by Spellman et al. (1998) and Eisen et al. (1998). Shortly afterwards, self-organizing maps (SOM) were applied to the same data, as well as a human dataset concerning hematopoietic differentiation by Tamayo et al. (1999), while the k-means algorithm was used in Tavazoie et al. (1999). This early literature used rather arbitrary criteria for reducing the number of genes prior to clustering. In the GENECLUSTER package, genes are filtered by a simple variation criteria, see Reich et al. (2004) and GENECLUSTER 2 Reference Guide for details http://www.broad.mit.edu/cancer/software/genecluster2/ gc_ref.html. In producing a 6 × 4 grid SOM, Saban et al. (2001) started with 588 genes which had to be induced at least three-fold over the initial time-point in one replicate at some time-point, and induced at least two-fold over the initial time-point in the second replicate at that time-point. Such filtering rules are not uncommon in the literature, see also Gurok et al. (2004). A more recent example was the hierarchical clustering of 906 genes into six main groups representing three major patterns in Himanen et al. (2004). A significance test within a mixed-model analysis was used to select 906 genes.
Different clustering algorithms and distance measures can lead to very different results. A perennial challenge with cluster analysis is the determination of the number of clusters. In recent years methods have been developed to deal with this issue, e.g. the gap statistic in Hastie et al. (2000), see also Chipman et al. (2003).
As well as these classical clustering approaches, a number of model-based clustering al-gorithms have been proposed, e.g. Fraley and Raftery (2002). Ramoni et al. (2002) gave a Bayesian model-based clustering algorithm, which represents temporal profiles by autoregres-sive models and used an agglomerative procedure to determine the number of clusters. Yeung et al. (2001) used Gaussian mixture models in which each component corresponds to a cluster, the number of clusters being determined by the Bayesian Information Criterion (BIC). HMM
clustering can be found in Schliep et al. (2003) and Schliep et al. (2004). There, each clus-ter was represented as one HMM. The method started with a collection HMMs with typical qualitative behavior, and an iterative algorithm was used to fit these models and assign genes to clusters in such a way as to maximize the joint likelihood. This method also dealt with missing data, and was illustrated on yeast cell cycle data of Spellman et al. (1998), and on the fibroblast serum response data of Iyer et al. (1999). Similarly, Ji et al. (2003) and Zeng and Garcia-Frias (2004) also used HMM approaches to cluster microarray time course data. Bar-Joseph (2003a) and Luan and Li (2003) did likewise, but first represented the profile for each gene by a continuous curve fitted by B-splines with gene-specific and class-specific parameters. Both papers illustrated their methods on the yeast cell cycle and fibroblasts serum response datasets. Zhang et al. (2004) proposed a biclustering algorithm to discover genes which are co-regulated in only part of the time course. They illustrated their algorithm on the yeast cell cycle data of Cho et al. (1998). Other noteworthy approaches were outlined in Peddada et al. (2003) and Wakefield et al. (2003).
Based on the above examples and our experience, we note that most model-based cluster-ing algorithms have been effective for periodic time courses, but their satisfactory performance on short time-course experiments is not so clear. For developmental time-course data, tra-ditional algorithms such as hierarchical clustering, SOM, or their variants based on distance measures have been more popular, probably because there are usually too few time points to allow the fitting of models. However, these approaches have the drawbacks of ignoring possible dependency across times in longitudinal studies, and generally ignoring the ordered nature of the time index. We feel that clustering methods which combine features from both the traditional and model-based approaches are urgently needed, ones which recognize the time ordering, and will deal with few time points, as well as temporal dependence and replicates where appropriate.
We end this clustering section by briefly mentioning a couple of other exploratory ap-proaches like clustering that have used to analyze microarray time course data. These are correspondance analysis, Fellenberg et al. (2001) and Tan et al. (2004), and singular value de-composition (SVD), see Holter et al. (2000) and Alter et al. (2000). Such graphical methods can be quite powerful.
6.4.5
Curve alignment
The occasional need to align gene expression profiles comes from the fact that the rates of biological processes may be different across biological or environmental conditions, or the sampling times are different between two time course datasets to be compared. Aach and Church (2001) suggested a time-warping algorithm with or without interpolation, while
Bar-Joseph et al. (2003a) gave a B-spline approach also for the same task, assuming that each gene’s profile is fitted with gene-specific and class-specific parameter. As with so much we have mentioned, these approaches have only been illustrated on the yeast cell-cycle data. Again, this idea is well-suited to longer time series, but may not be suitable for shorter time series. There is clearly room for more work here.
6.4.6
Software
Many of the algorithms described in this chapter are implemented in open source software R (Ihaka and Gentleman, 1996), which can be downloaded from http://cran.r-project.org. The Bioconductor project (http://www.bioconductor.org) provides many software tools for the analysis of microarray data. For the algorithms described in this chapter but not listed here, the reader should consult with the authors for software availability.
Differential expression.
• The CyberT program in Baldi and Long (2001) with a web interface can be downloaded from visitor.ics.uci.edu/genex/cybert/. It is also available as R codes hdarray. • The M B-statistic and eT2 statistic (Tai and Speed, 2004) are implemented in the
Bio-conductor package timecourse
www.stat.berkeley.edu/users/terry/Group/research/timecourse.html.
• The B-statistic (L¨onnstedt and Speed, 2002; Smyth, 2004) and moderated F -statistic (Smyth, 2004) are implemented in the Bioconductor package limma. The latest limma can be downloaded from bioinf.wehi.edu.au/limma/. LimmaGUI and affylmGUI (Wettenhall and Smyth, 2004) are two nice graphical user interfaces to limma for two-color arrays and Affymetrix chips, respectively.
• The empirical Bayes method across multiple independent groups in Kendziorski et al. (2003) is available through the Bioconductor package EBarrays.
Clustering.
• SOM: The package GeneSOM written by Jun Yan is available in R. GENECLUSTER 2.0 in Tamayo et al. (1999) can be downloaded from
• hierarchical clustering: The R function hclust implements bottom-up hierarchical clus-tering with several types of linkages.
• k-means: The R function kmeans is based on the algorithm in Hartigan and Wong (1979).
• QT clust proposed in Heyer et al. (1999) is implemented by Witold Wolski. It can be ac-cessed from www.molgen.mpg.de/~wolski/downloads/clustering/clustering.html. • CLARITY proposed in Balasubramaniyan et al. (2004) is available upon request from
authors.
• The model-based clustering approach in Fraley and Raftery (2002) is available as a R package mclust: www.stat.washington.edu/fraley/mclust/
• The HMM clustering software GHMM in Schliep et al. (2003) and Schliep et al. (2004) is available from ghmm.org/.
Curve Alignment.
• Aach and Church’s (2001) time warping programs genewarp and genewarpi are available as DOS executables under Win 32. The download page is
arep.med.harvard.edu/timewarp/pgmlicense.html.
6.4.7
Remarks
We have discussed some statistical issues in the analysis of microarray time course experi-ments, touching on their design, the identification of genes of interest, clustering, and align-ment. For the practioner, we have tried to offer some ways of addressing these issues, partic-ularly the second.
As will be apparent from our discussion, many, perhaps most of the methods in the literature available for choosing or clustering genes in time course experiments have been devised and tested on the yeast and human cell-cycle datasets. There is room for much more research on the analysis of what we have called developmental time course data, especially their clustering.
A less obvious bias in our coverage is the fact that almost all of the methods we have discussed have been for data generated in an experimental setting, with mRNA from cell lines, tissue samples or experimental organisms such as whole Drosophila embryos, or tissue from inbred strains of mice or Arabidopsis plants. Recently microarrays have moved to the
wider clinical setting, with microarray data now being collected on human subjects over time. Such longitudinal studies present novel analytical challenges, as subject-to-subject variation, even within the same treatment group, can be substantial. The methods we have reviewed here will not be appropriate in the clinical context without modifications, for example, by including fixed or random effects for subjects. This is an important area for future research, but we can refer to Storey et al. (2004) for a promising start.
6.4.8
Acknowledgments
We would like to thank all the biologists who have challenged us with their microarray time course data, specifically Suzie Grant, Jason Dugas, Mary Wildermuth, Moriah Szpara, Steve Perrin, and we particularly thank Pavel Tomancak and his colleagues for creating the Drosophila data set we have used in this chapter. Finally, thanks are due to Peter Diggle and Christina Kendziorski for reading and commenting on an earlier version of this chapter.
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