Signal Selection in Microarray Data Analysis

The paper describes the use of statistical signal selection in DNA microarray data analysis. The experiments considered produce time-series data. A commonly used method of thresholding is compared to using the ‘run test’, ‘turning point test’,‘Kendall’s tau test’ and ‘autocorrelation function’. We conclude that the use of such techniques, developed for time-series analysis, can help to distinguish between non-informative signals or noise and signals which show either a trend or are correlated over time. 1 DNA Microarray Data For the data considered in this paper we assume that gene expression profiles were obtained from time course DNA microarray experiments. To illustrate the idea and without loss of generality, we use the yeast data set described by Eisen et. al [2] and choose the 18 measurements obtained for the cell division cycle after synchronization by alpha factor arrest. In general, a gene expression data matrix X has i = 1, . , n rows for the genes and j = 1, . , r columns for samples. Here we have n = 2467 yeast genes and r = 18 time-points. Each row vector xi = [xi1, . , xij , . , xir] represents a particular gene expression profile. Throughout the paper we refer to a gene by its (row) number in X. xij is the gene expression level at time j of gene i. xij ∈ xi is the normalized log2 ratio Eij/Rij where Eij is the expression level or state at time j of the gene i, and Rij is the reference state of the gene, which is a constant value throughout the experiment [1]: xij = log2 ( Eij Rij ) √∑18 k=1 ( log2 ( Eij Rij ))2 i = 1, . , 2467 j = 1, . , 18 (1) With the normalization of (1), xij is positive when Eij ≥ Rij and we say the gene is induced or “up-regulated”. When Eij ≤ Rij , xij is negative and it is said that the gene is repressed or “down-regulated”. In this paper we assume that there are no missing values, i.e., measurements have been obtained for all sampling points of the time course or missing values have been replaced by a suitable method. Assuming that not all genes are involved in the dynamic response to an experiment, the problem considered here is one of signal selection. We investigate test for randomness, stationarity and trend to identify informative signals and to disregard genes from the analysis for which signals appear non-informative or as noise. 2 Signal Selection and Preprocessing Problems The principal aim of microarray experiments is to study gene activity levels through the measurement of mRNA abundance under varying conditions. Knowledge of the functional role of a ∗Author to whom correspondence should be addressed. 2 4 6 8 10 12 14 16 18 -2 -1 0 1 Fig. 1: Three “similar” genes expression profiles (1274, 1281 and 1745). gene can be used to classify other genes by comparing their expression profiles. Genes involved in the same processes are expected to have “similar” expression profiles in a microarray experiment. Two difficult problems arise from such analysis: • Conditions of the experiment may not affect all genes. Frequently, a large number of genes are not involved in the process under consideration leading to a “flat” signal with little or no change from the reference state. Those signals can be eliminated from the matrix. This preprocessing can lead to a substantial reduction of the number of genes which makes the analysis of the remaining easier. The criteria for such as selection are not clear as we will discuss later. • What do we understand by “similar” and how do we decide when two signals are similar? Fig. 1 shows three gene expression profiles. Two of the three profiles are clearly similar to each other. A common approach to discuss similarity is to view the vector xi as describing a point in a r-dimensional space where r is the size of the vector. Using the Euclidean distance between the three points, the two signals which are virtually identical would be separated from the third which is time shifted. On the other hand, if we are to view the time course as a dynamic response and fit an autoregressive (AR) model to the data, we find that for all these signals the distance in the parameter space of the AR model is small and all three genes appear related. In this paper, we will discuss the first problem using statistical test for randomness and trends. 3 Statistical Test for Signal Selection This section reviews some techniques used to decide whether a signal has been produced by a process with a “deterministic” component or whether it is just “noise”. The condition for a sample of data to be regarded as noise is that, if we assume that each data point is a random variable, these have to be independent identically distributed. The distribution that they follow, depends on the process that generates the data. Purely random data such as a lottery, may follow an uniform distribution but very often, processes in life, such as the error of a machine which is taking measurements, follows asymptotically a normal distribution i.e. the random variable which represent the process is a normal distributed when the same experiment is carried out a large number of times. This can be the case for microarray data where noise comes from the apparatus involved in the measurement process. The gene expression ratio xij in these cases fluctuates around zero if the gene does not responds to the experiment, Eij ≈ Rij . This generates a “flat” signal on the horizontal axis, xi ≈ 0. This suggest that signals around zero, with little variations may be considered as noise.