Department of Statistics, Seoul National University, Seoul, Korea

Department of Applied Mathematics, Sejong University, Seoul, Korea

Department of Biochemistry, Hanyang University College of Medicine, Seoul, Korea

Biometric Research Branch, Division of Cancer Treatment & Diagnosis National Cancer Institute, Bethesda MD, USA

Abstract

Background

Microarray technology allows the monitoring of expression levels for thousands of genes simultaneously. This novel technique helps us to understand gene regulation as well as gene by gene interactions more systematically. In the microarray experiment, however, many undesirable systematic variations are observed. Even in replicated experiment, some variations are commonly observed. Normalization is the process of removing some sources of variation which affect the measured gene expression levels. Although a number of normalization methods have been proposed, it has been difficult to decide which methods perform best. Normalization plays an important role in the earlier stage of microarray data analysis. The subsequent analysis results are highly dependent on normalization.

Results

In this paper, we use the variability among the replicated slides to compare performance of normalization methods. We also compare normalization methods with regard to bias and mean square error using simulated data.

Conclusions

Our results show that intensity-dependent normalization often performs better than global normalization methods, and that linear and nonlinear normalization methods perform similarly. These conclusions are based on analysis of 36 cDNA microarrays of 3,840 genes obtained in an experiment to search for changes in gene expression profiles during neuronal differentiation of cortical stem cells. Simulation studies confirm our findings.

Background

Biological processes depend on complex interactions between many genes and gene products. To understand the role of a single gene or gene product in this network, many different types of information, such as genome-wide knowledge of gene expression, will be needed. Microarray technology is a useful tool to understand gene regulation and interactions

In microarray experiments, there are many sources of systematic variation. Normalization attempts to remove such variation which affects the measured gene expression levels. Yang

Several normalization methods have been proposed using statistical models (Kerr

We focus on comparing the normalization methods for cDNA microarrays. A detailed description on normalization methods considered in our study is given in the next section. Complex methods do not necessarily perform better than simpler methods. Complex methods may add noise to the normalized adjustment and may even add bias if the assumptions are incorrect. The fact that a non-linear method linearizes a graph of red intensity versus green intensity or a plot of

We first compare the methods using data from cDNA microarrays of 3,840 genes obtained in an experiment to search for changes in gene expression profiles during neuronal differentiation of cortical stem cells. Then, we perform simulation studies to compare these normalization methods systematically. Bolstad

The paper is organized as follows. Section 2 describes normalization methods. Section 3 describes the measures for variability to compare normalization methods. Section 4 shows the comparison results for cDNA microarrays obtained from a cortical stem cells experiment along with some simulation results. Finally, Section 5 summarizes the concluding remarks.

Normalization methods

As pointed out by Yang

As a first step, one needs to decide which set of genes to use for normalization. Yang

Let (

(1) Global normalization (G) using the global median of log intensity ratios

(2) Intensity dependent linear normalization (L)

(3) Intensity dependent nonlinear normalization (N) using a LOWESS curve.

Under ideal experimental conditions, we expect

For global normalization α_{0 }is estimated by the median of _{0}, β_{1}) by least squares estimation, and

_{j}) = _{j }- _{j}). (1)

Thus, the normalization process transforms (

Flowchart of normalization

Flowchart of normalization

The detailed descriptions for each normalization approach considered in this study are given in Tables

Abbreviation for Normalization Methods

Abbreviation

Description

O

Original data

G

Global median normalization

L

Intensity dependent linear normalization

N

Intensity dependent nonlinear normalization (LOWESS)

P

Print-tip normalization

S

Print-tip scale normalization

.s

Between-slide scale normalization

List of Normalization Methods

List of normalization methods including print-tip scale normalization

Method

Notation

Description

O

Original data

Global

G

Global median normalization

GP

Global median normalization on each print-tip

GPS

Global median normalization on each print-tip with scale normalization

G.s

Global median normalization and between-slide scale normalization

GP.s

Global median normalization on each print-tip and between-slide scale normalization

GPS.s

Global median normalization on print-tip with scale normalization and between-slide scale normalization

Linear

L

Intensity dependent linear regression normalization

LP

Intensity dependent linear regression normalization on each print-tip

LPS

Intensity dependent linear regression normalization on each print-tip with scale normalization

L.s

Intensity dependent linear regression normalization and between-slide scale normalization

LP.s

Intensity dependent linear regression normalization on each print-tip and between-slide scale normalization

LPS.s

Intensity dependent linear regression normalization on each print-tip with scale normalization and between-slide scale normalization

Nonlinear

N

Intensity dependent nonlinear regression normalization (LOWESS)

NP

Intensity dependent nonlinear regression normalization (LOWESS) on each print-tip

NPS

Intensity dependent nonlinear regression normalization (LOWESS) on each print-tip with scale normalization

N.s

Intensity dependent nonlinear regression normalization (LOWESS) and between-slide scale normalization

NP.s

Intensity dependent nonlinear regression normalization (LOWESS) on each print-tip and between-slide scale normalization

NPS.s

Intensity dependent nonlinear regression normalization (LOWESS) on each print-tip with scale normalization and between-slide scale normalization

Measures of variation

In order to derive measures of variation, we now use

Also suppose that there are

Let _{ijkl }be the logarithm of the red to green intensity ratio from group _{ijkl}. It is expected that the better the normalization method, the smaller the variation among the replicated observations. Let σ_{l }be the variability measure for the

Method 1. Pooled variance estimators

For gene

where

Method 2. Variance estimator using analysis of variance models

Consider the following two-way analysis of variance(ANOVA) model with interactions for each gene.

_{ijkl }= μ_{l }+ α_{il }+ β_{jl }+ (αβ)_{ijl }+ ε_{ijkl}, (3)

where _{l }capture the overall mean intensity in fluorescent signals for genes across the arrays, groups, and time points. The α_{il }terms account for gene specific group effects representing overall differences between two groups. The β_{jl }account for time effects that capture differences in the overall concentration of mRNA in the samples from the different time points. The terms (αβ)_{ijl }account for the interaction effects between group and time representing the signal contribution due to the combination of group and time.

From this ANOVA model, an unbiased estimate of

Results

Data

The data studied here are from a study of cortical stem rat cells. The goal of the experiment is to identify genes that are associated with neuronal differentiation of cortical stem cells. A detailed description of data is given by Park

(A) The original slide with a non-linear pattern

(A) The original slide with a non-linear pattern. (B-D) Three normalized slides (global median, intensity dependent linear regression, intensity dependent non-linear regression.

For this dataset, Park

Figure

Dot Plots of Log-transformed Variance Estimates for Cortical Stem Cells Data

Dot Plots of Log-transformed Variance Estimates for Cortical Stem Cells Data. The Y-axis represents normalization methods and the X-axis represents the mean values of log-transformed variance estimates. (A) Dot plots for O, G, G.s, L, L.s, N, and N.s, (B) Dot plots for Global Normalization Methods, (C) Dot plots for Intensity-dependent Linear Normalization Methods, (D) Dot plots for Intensity-dependent Nonlinear Normalization Methods.

As shown in Figure

Figure

Figure

Based on these figures, we conclude that intensity dependent normalization methods perform better than global normalization methods. Small differences are observed between the linear and nonlinear normalization methods. In addition, small effects are observed for scale normalization methods.

Simulation studies

In order to compare the normalization methods more systematically, we performed a simulation study by generating typical patterns of microarray data. Using simulated data we could compare the normalization methods with regard to bias and mean squared error as well as variance. In general, mean square error (MSE) is defined by the sum of variance and bias^{2}. MSE has been used as a criterion to compare normalization methods for the cases when they can be computed. Bolstad et al.

Four types of (

Four types of (

Type I

Figure

Type II

Figure

Type III

Figure

Type IV

Figure ^{I}, ^{I}), we assume that they are observed from the bivariate normal distribution with mean vector **β **and covariance matrix Σ, where

For simplicity, let the random vector (_{G}, _{R})^{T }represent the log-transformed intensity values, say (^{I}, ^{I}). To generate bivariate normal random variables, it is convenient to generate _{G }first and then generate _{R }from the conditional distribution of _{R}| _{G }=

where μ_{R|G }= μ_{R }- ρσ_{R}/σ_{G}(_{G}) and _{G }generated from Type I. For _{R}, however, we use the transformed mean and variance to add a pattern like that shown in Figure _{1 }and _{2 }are found by trial and error. For Type III, we use a similar approach. That is, we use the same values of _{G }generated from Type I. For _{R}, we use the transformed variance to allow larger variability for the lower intensity observations. Thus, _{2 }used for Type II. Similarly, for Type IV we use the same values of _{G }generated from Type I. For _{R}, we use the transformed mean and variance to add a specific pattern. That is, _{3 }and _{4 }are also found by trial and error. For each type, we generate ten replicated samples. For these replicated samples, we apply the normalization methods and draw the distributions for the variation measures. For simplicity, we do not consider the print-tip variation in this simulation. We compare four graphs: original data(O), globally normalized data(G), linear normalized data(L), and non-linear normalized data(N) using LOWESS.

In order to compare these normalization methods, we compute the mean square error(MSE). For each gene, the MSE is computed as the average of the distances between normalized data and true expected value. Figure

Dot Plots of Log-transformed Variance Estimates for Simulated Data

Dot Plots of Log-transformed Variance Estimates for Simulated Data. The Y-axis represents normalization methods and the X-axis represents the mean values of log-transformed variance estimates.

We summarize our findings from the simulation studies. First, the normalization methods performed similarly when the original data has a linear pattern such as Type I, and when there are high variabilities for observations with lower intensities such as Types III. Second, when there is a specific pattern such as Type II, all normalization methods tend to reduce MSEs. Third, in this case, MSEs are dominated by biases. Fourth, only small differences are observed between the intensity-dependent linear and non-linear normalization methods.

Discussion

In this article, we compare normalization methods commonly used to analyze microarray data. The comparison is based on the variability measures derived from the replicated microarray samples. These variability measures can be easily derived from any replicated microarray experiment. As pointed out by many researchers, many undesirable systematic variations are observed in the microarray experiment. Normalization becomes a standard process for removing some of the variation which affects the measured gene expression levels. Although a number of normalization methods have been proposed, it has been difficult to decide which method performs better than the others. Thus, the evaluation of normalization methods in microarray data analysis is indeed an important issue. In this article, we show that the intensity dependent normalization method performs better than the simpler global normalization methods in many cases. We have not been sure about whether apparent nonlinearity of an M-A scatter plot or a scatter plot of red vs green is sufficient basis for feeling confident that a non-linear normalization is useful. Although we have studied only a limited number of data sets, our findings can provide some guidance on the selection of normalization methods. There are clearly cases where intensity dependent normalization performs substantially better than global normalization methods. In most of the cases that we considered, the non-linear intensity dependent procedures did not perform substantially better than a linear intensity dependent method, although there may be datasets where this is not the case. For the cases we considered, we did not see large benefits to separate normalization by print-tip grids or for scale normalization. None of the normalization methods effectively addressing the dependence of the variance of measurements on intensity level. Recently, some other non-linear normalization methods have been employed such as B-splines and Gaussian-kernel fitting

Acknowledgement

The authors wish to thank two anonymous referees whose comments were extremely helpful. The work was supported by the IMT2000 contribution from Korea Ministry of Health and Welfare, and Ministry of Information and Communication (01-PJ11-PG9-01BT-00A-0045).