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* Departments of Medicine and Biochemistry, Stanford University, 269 Campus Drive, Center for Clinical Sciences Research 1115, Stanford, CA
94305-5151; and Communicated by Bradley Efron, Stanford University, Stanford, CA, February 6, 2001 (received for review December 1, 2000)
Microarrays can measure the expression of
thousands of genes to identify changes in expression between different
biological states. Methods are needed to determine the significance of
these changes while accounting for the enormous number of genes. We describe a method, Significance Analysis of Microarrays (SAM), that
assigns a score to each gene on the basis of change in gene expression
relative to the standard deviation of repeated measurements. For genes
with scores greater than an adjustable threshold, SAM uses permutations
of the repeated measurements to estimate the percentage of genes
identified by chance, the false discovery rate (FDR). When the
transcriptional response of human cells to ionizing radiation was
measured by microarrays, SAM identified 34 genes that changed at least
1.5-fold with an estimated FDR of 12%, compared with FDRs of 60 and
84% by using conventional methods of analysis. Of the 34 genes, 19 were involved in cell cycle regulation and 3 in apoptosis.
Surprisingly, four nucleotide excision repair genes were induced,
suggesting that this repair pathway for UV-damaged DNA might play a
previously unrecognized role in repairing DNA damaged by ionizing radiation.
DNA microarrays contain oligonucleotide or cDNA
probes for measuring the expression of thousands of genes in a single
hybridization experiment. Although massive amounts of data are
generated, methods are needed to determine whether changes in gene
expression are experimentally significant. Cluster analysis of
microarray data can find coherent patterns of gene expression (1) but
provides little information about statistical significance. Methods
based on conventional t tests provide the probability
(P) that a difference in gene expression occurred by chance
(2, 3). Although P = 0.01 is significant in the context
of experiments designed to evaluate small numbers of genes, a
microarray experiment for 10,000 genes would identify 100 genes by
chance. This problem led us to develop a statistical method adapted
specifically for microarrays, Significance Analysis of Microarrays
(SAM).
SAM identifies genes with statistically significant changes in
expression by assimilating a set of gene-specific t tests. Each gene is assigned a score on the basis of its change in gene expression relative to the standard deviation of repeated measurements for that gene. Genes with scores greater than a threshold are deemed
potentially significant. The percentage of such genes identified by
chance is the false discovery rate (FDR). To estimate the FDR, nonsense
genes are identified by analyzing permutations of the measurements. The
threshold can be adjusted to identify smaller or larger sets of genes,
and FDRs are calculated for each set. To demonstrate its utility, SAM
was used to analyze a biologically important problem: the
transcriptional response of lymphoblastoid cells to ionizing radiation
(IR).
Preparation of RNA.
Human lymphoblastoid cell lines GM14660 and GM08925 (Coriell Cell
Repositories, Camden, NJ) were seeded at 2.5 × 105 cells/ml and exposed to IR 24 h later.
RNA was isolated, labeled, and hybridized to the
HUGENEFL GENECHIP
microarray according to manufacturer's protocols (Affymetrix, Santa
Clara, CA).
Microarray Hybridization.
Each gene in the microarray was represented by 20 oligonucleotide
pairs, each pair consisting of an oligonucleotide perfectly matched to
the cDNA sequence, and a second oligonucleotide containing a single
base mismatch. Because gene expression was computed from differences in
hybridization to the matched and mismatched probes, expression levels
were sometimes reported by the GENECHIP ANALYSIS SUITE software as negative numbers.
Northern Blot Hybridization.
Total RNA (15 µg) was resolved by agarose gel electrophoresis,
transferred to a nylon membrane, and hybridized to specific radiolabeled DNA probes, which were prepared by PCR amplification.
RNA was harvested from wild-type human lymphoblastoid cell
lines, designated 1 and 2, growing in an unirradiated state (U) or in
an irradiated state (I) 4 h after exposure to a modest dose of 5 Gy of IR. RNA samples were labeled and divided into two identical aliquots for independent hybridizations, A and B. Thus, data for 6,800 genes on the microarray were generated from eight hybridizations (U1A, U1B, U2A, U2B, I1A, I1B, I2A, and I2B).
We scaled the data from different hybridizations as follows. A
reference data set was generated by averaging the expression of each
gene over all eight hybridizations. The data for each hybridization
were compared with the reference data set in a cube root scatter plot.
We chose the cube root scatter plot because it resolved the vast
majority of genes that are expressed at low levels and permitted the
inclusion of negative levels of expression that are sometimes generated
by the GENECHIP software. A linear least-squares fit to the
cube root scatter plot was then used to calibrate each hybridization.
After scaling, a linear scatter plot was generated for average
gene expression in the four A aliquots (U1A, I1A, U2A, and U2A) vs. the
average in the four B aliquots (U1B, I1B, U2B, and U2B), a partitioning
of the data that eliminates biological changes in gene expression (Fig.
1A). The linear scatter plot
confirmed that the data were generally reproducible but failed to
resolve genes expressed at low levels. Better resolution of these genes was achieved by the cube root scatter plot (Fig. 1B), which
revealed three salient features: the large percentage of genes (24%)
assigned negative levels of expression, the large percentage of genes
with low levels of expression, and the low signal-to-noise ratio at low
levels of expression.
Statistics / Genetics
Significance analysis of microarrays applied to the ionizing
radiation response
, and
Department of Health Research and Policy
and Department of Statistics, Stanford University, Stanford, CA 94305
Abstract
Top
Abstract
Introduction
Materials and Methods
Results
Discussion
References
Introduction
Top
Abstract
Introduction
Materials and Methods
Results
Discussion
References
Materials and Methods
Top
Abstract
Introduction
Materials and Methods
Results
Discussion
References
Results
Top
Abstract
Introduction
Materials and Methods
Results
Discussion
References
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Fig. 1.
Gene expression measured by microarrays. (A)
Linear scatter plot of gene expression. Each gene (i) in
the microarray is represented by a point with coordinates consisting of
average gene expression measured from the four A hybridizations (avg
xA) and the average gene expression in the
four B hybridizations (avg xB).
(B) Cube root scatter plot of gene expression. The
average gene expression from the A and B hybridizations have been
plotted on a cube root scale to resolve genes expressed at low levels.
(C) Cube root scatter plot of average gene expression from the four
hybridizations with uninduced cells (avg xU)
and induced cells 4 h after exposure to 5 Gy of IR (avg
xI). Some of the genes that responded to IR
are indicated by arrows.
To assess the biological effect of IR, a scatter plot was generated for average gene expression in the four irradiated states vs. the four unirradiated states (compare Fig. 1 B and C). A few of the potentially significant changes in gene expression are indicated by arrows in Fig. 1C, but the effect was not easily quantified, and a method was needed to identify changes with statistical confidence.
Our approach was based on analysis of random fluctuations in the
data. In general, the signal-to-noise ratio decreased with decreasing
gene expression (Fig. 1). However, even for a given level of
expression, we found that fluctuations were gene specific. To account
for gene-specific fluctuations, we defined a statistic based on the
ratio of change in gene expression to standard deviation in the data
for that gene. The "relative difference"
d(i) in gene expression is:
|
[ 1 ] |
I(i) and
U(i) are defined as
the average levels of expression for gene (i) in states I
and U, respectively. The "gene-specific scatter"
s(i) is the standard deviation of repeated expression measurements:
![s(i)=<RAD><RCD>a<FENCE><LIM><OP>∑</OP></LIM><SUB>m</SUB>[x<SUB>m</SUB>(i)−<A><AC>x</AC><AC>&cjs1171;</AC></A><SUB>I</SUB>(i)]<SUP>2</SUP>+<LIM><OP>∑</OP></LIM><SUB>n</SUB>[x<SUB>n</SUB>(i)−<A><AC>x</AC><AC>&cjs1171;</AC></A><SUB>U</SUB>(i)]<SUP>2</SUP></FENCE></RCD></RAD>](/content/vol98/issue9/fulltext/5116/img002.gif)
|
[ 2 ] |
m and n
are summations of the expression measurements in states I and U,
respectively, a = (1/n1 + 1/n2)/(n1 + n2 
2),
and n1 and
n2 are the numbers of measurements in
states I and U (four in this experiment).
To compare values of d(i) across all genes, the distribution of d(i) should be independent of the level of gene expression. At low expression levels, variance in d(i) can be high because of small values of s(i). To ensure that the variance of d(i) is independent of gene expression, we added a small positive constant s0 to the denominator of Eq. 1. The coefficient of variation of d(i) was computed as a function of s(i) in moving windows across the data. The value for s0 was chosen to minimize the coefficient of variation. For the data in this paper, this computation yielded s0 = 3.3.
Scatter plots of d(i) vs. s(i) are shown in Fig. 2. The scatter plot for relative difference between states I and U is shown in Fig. 2A. By contrast, the scatter plot for relative difference between cell lines 1 and 2 shows more marked changes in Fig. 2B. These relative differences exceeded random fluctuations in the data, as measured by the relative difference between hybridizations A and B in Fig. 2C.
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Although the relative difference computed from hybridizations A and B provided a control for random fluctuations, additional controls were needed to assign statistical significance to the biological effect of IR. Instead of performing more experiments, which are expensive and labor intensive, we generated a large number of controls by computing relative differences from permutations of the hybridizations for the four irradiated and four unirradiated states. To minimize potentially confounding effects from differences between the two cell lines, we analyzed the data by using the 36 permutations that were balanced for cell lines 1 and 2. Permutations were defined as balanced when each group of four experiments contained two experiments from cell line 1 and two experiments from cell line 2. Fig. 2 C and D are examples of balanced permutations.
To find significant changes in gene expression, genes were ranked
by magnitude of their d(i) values, so that
d(1) was the largest relative difference, d(2)
was the second largest relative difference, and
d(i) was the ith largest relative
difference. For each of the 36 balanced permutations, relative
differences dp(i) were also
calculated, and the genes were again ranked such that
dp(i) was the ith
largest relative difference for permutation p. The expected
relative difference, dE(i), was
defined as the average over the 36 balanced permutations,
dE(i) = pdp(i)/36.
To identify potentially significant changes in expression, we
used a scatter plot of the observed relative difference
d(i) vs. the expected relative difference
dE(i) (Fig.
3A). For the vast majority of
genes, d(i) 
dE(i), but some genes are
represented by points displaced from the
d(i) = dE(i) line by a distance
greater than a threshold 
. For example, the threshold = 1.2 illustrated by the broken lines in Fig. 3A yielded 46 genes
that were "called significant." These 46 genes are shown in the
context of the scatter plot for d(i) vs.
s(i) (Fig. 3B) and in the scatter plot
for the cube root of gene expression
I(i) vs.
U(i) (Fig.
3C). Genes identified by d(i) do not
necessarily have the largest changes in gene expression.
|
To determine the number of falsely significant genes generated by
SAM, horizontal cutoffs were defined as the smallest
d(i) among the genes called significantly induced
and the least negative d(i) among the genes
called significantly repressed. The number of falsely significant genes
corresponding to each permutation was computed by counting the number
of genes that exceeded the horizontal cutoffs for induced and repressed
genes. The estimated number of falsely significant genes was the
average of the number of genes called significant from all 36 permutations. For = 1.2, the permuted data sets generated an
average of 8.4 falsely significant genes, compared with 46 genes called
significant, yielding an estimated FDR of 18% (Table
1). As decreased, the number of
genes called significant by SAM increased but at the cost of an
increasing FDR. (Omitting s0 from Eq. 1 produced higher FDRs of 45, 35, and 28% for = 0.6, 0.9, and 1.2.)
|
Our method for setting thresholds provides asymmetric cutoffs for
induced and repressed genes. The alternative is the standard t test, which imposes a symmetric horizontal cutoff,
with d(i) > c for induced genes and
d(i) < c for repressed genes.
However, the asymmetric cutoff is preferred because it allows for the
possibility that d(i) for induced and repressed
genes may behave differently in some biological experiments.
SAM proved to be superior to conventional methods for analyzing
microarrays (Table 1 and Fig.
4A). First, SAM was compared with the approach of identifying genes as significantly changed if an
R-fold change was observed. In this "fold change"
method, r(i) = I(i)/U(i),
and gene (i) was called significantly changed if
r(i) > R or
r(i) < 1/R. To permit
computation of r(i) from negative values for gene
expression, I(i) and
U(i) were converted to 10 when their values were negative or less than 10. The results of
this procedure yielded unacceptably high FDRs of 73-84%.
|
Another approach attempts to account for uncertainty in the data by identifying genes as significantly changed if an R-fold change is observed consistently between paired samples (4). To apply this "pairwise fold change" method to our four data sets before IR and four data sets after IR, changes in gene expression were declared significant if 12 of 16 pairings satisfied the criteria r(i) > R or r(i) < 1/R. Despite the demand for consistent changes between paired samples, this method yielded FDRs of 60-71%.
To understand why fold-change methods fail, note that the vast majority of genes are expressed at low levels where the signal-to-noise ratio is very low (Fig. 3C). Thus, 2-fold changes in gene expression occur at random for a large number of genes. Conversely, for higher levels of expression, smaller changes in gene expression may be real, but these changes are rejected by fold-change methods. The pairwise fold-change method provides modest improvement but remains inferior to SAM.
Of the 46 genes most highly ranked by SAM ( = 1.2), 36 increased or decreased at least 1.5-fold (R = 1.5). The number of falsely significant genes that met these two criteria was 4.5, corresponding to a FDR of 12% (Table 1). Fas was identified three times as alternately spliced forms, leaving 34 independent genes (Table
2). As an indication of biological validity, 10 of the 34 genes have
been reported in the literature as part of the transcriptional response
to IR. TNF-
was reported to be induced by other investigators (5)
but was repressed here. Quantitative reverse transcription-PCR confirmed this result.
|
To test the validity of SAM directly, we performed Northern blots
for genes that were randomly selected from the 46 and 57 genes most
highly ranked by SAM ( = 1.2) and the fold-change method (at
least 3.6-fold change), respectively. Northern blots showed little
correlation with the genes identified by the fold change method (Fig.
4B), but strong correlation with the genes identified by SAM
(Fig. 4C). Indeed, Northern blots contradicted only 1 (maxiK) of 11 genes identified by SAM, consistent with our estimated
FDR.
Nineteen of the 34 genes most highly ranked by SAM appear to be involved in the cell cycle. Three are known to be induced in a p53-dependent manner: p21, cyclin G1, and mdm2 (6-8). Six cell cycle genes were repressed: E2-EPF, p55cdc, cyclin B, ckshs2, cdc25, and wee1 (9, 10). Five genes encoding the mitotic machinery were also repressed: PLK-1, MKLP-1, MCAK, C-TAK1, CENP-E (11-13). Three genes involved in cell proliferation were induced or repressed: PTP(CAAX1), LPAP, and c-myc (14-18). Some responses appeared paradoxical. For example, cdc25 phosphatase and wee1 kinase have antagonistic effects on the phosphorylation state of cdc2, but both genes were repressed. Repression of these genes together with the mitotic genes may represent a damage response that dismantles the cell cycle machinery until the cell has repaired the damaged DNA.
Four of the 34 genes play roles in DNA repair, but none are involved in the repair of IR-induced double-strand breaks. Instead, the genes (p48, XPC, gadd45, PCNA) have roles in nucleotide excision repair, a pathway conventionally associated with UV-induced damage (19-22). We confirmed the induction of these genes by Northern blot (23-25). Fornace et al. reported defective removal of base damage induced by IR in xeroderma pigmentosum cells (26). Leadon et al. reported that a novel DNA repair pathway involving long excision repair patches of at least 150 nucleotides is activated by IR but not UV (27). Our results suggest that this novel pathway might include p48, XPC, gadd45, and PCNA.
Four of the 34 genes play roles in apoptosis (Fas, bbc3,
TNF-, OX40 ligand). The remaining genes may have previously
unsuspected roles in the DNA damage response or may be among the
estimated set of four falsely detected genes.
The 34 genes most highly ranked by SAM are only a subset of all
of the genes that change 1.5-fold with IR. Indeed, we calculated the
difference between the number of genes called significant and the
number of falsely significant genes for decreasing = 0.3, 0.2, and 0.1, and found the differences to be 92, 170, and 184, respectively. Thus, SAM suggests that approximately 180 of the 6,800 genes on the microarray were induced or repressed by 5 Gy
IR.
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Discussion |
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Top
Abstract
Introduction
Materials and Methods
Results
Discussion
References
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SAM is a method for identifying genes on a microarray with statistically significant changes in expression, developed in the context of an actual biological experiment. SAM was successful in analyzing this experiment as well as several other experiments with oligonucleotide and cDNA microarrays (data not shown).
In the statistics of multiple testing (28-30), the family-wise error rate (FWER) is the probability of at least one false positive over the collection of tests. The Bonferroni method, the most basic method for bounding the FWER, assumes independence of the different tests. An acceptable FWER could be achieved for our microarray data only if the corresponding threshold was set so high that no genes were identified. The step-down correction method of Westfall and Young (29), adapted for microarrays by Dudoit et al. (http://www.stat.berkeley.edu/users/terry/zarray/Html/matt.html), allows for dependent tests but still remains too stringent, yielding no genes from our data.
Westfall and Young (29) define "weak control" to be control of the FWER when all of the null hypotheses are true (i.e., when there are no changes in gene expression). "Strong control" is control of the FWER when any subset of the null hypotheses is true. Under certain conditions, weak control implies strong control. In fact, the step-down correction method exerts both weak and strong control.
The method of Benjamini and Hochberg (31) assumes independent tests and guarantees an upper bound for the FDR (with both weak and strong control) by a step-up or step-down procedure applied to the individual P values. For our data, the P value for each gene is calculated from permutations of the eight experiments. Because of the limited number of permutations, the FDR is too "granular", and we identified either zero or 300 significant genes, depending on how the P value was defined. A similar granular result was obtained for the adaptation to dependent tests by Benjamini et al. [The Control of the False Discovery Rate in Multiple Testing Under Dependency (Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv). http://www.math.tau.ac.il/~ybenja/].
SAM does not have strong or weak control of the FWER. Instead,
SAM provides an estimate of the FDR for each value of the tuning parameter . The estimated FDR is computed from permutations of the
data and hence assumes that all null hypotheses are true, allowing for
the possibility of dependent tests. It seems plausible that this
estimated FDR approximates the strongly controlled FDR when any subset
of null hypotheses is true. However, we have not proven this in
general. It is possible for SAM to give an estimate of the FDR that is
greater than 1. However, this has not occurred in our experience.
Indeed, SAM provides a reasonably accurate estimate for the true FDR.
To confirm this, we constructed artificial data sets in which a subset
of genes was induced over a background of noise. SAM successfully
identified the induced genes and estimated the FDR with reasonable accuracy.
Although this paper analyzes a simple two-state experiment, SAM can be generalized to other types of experiments by defining d(i) in a different way. Suppose the data includes gene expression xj(i) and a response parameter yj, in which i = 1, 2, ... , m genes, j = 1, 2, ... , n states. The generalized statistical parameter still takes the form d(i) = r(i)/[s(i) + s0], except that the definitions of r(i) and s(i) change.
To identify genes with changes in expression in an experiment
with three or more states, the parameter d(i) is
defined in terms of the Fisher's linear discriminant. One goal might
be to identify genes whose expression in one type of tumor is different from its expression in other types of tumors. Suppose that a set of
n samples consists of K nonoverlapping subsets,
such that the response parameter yj 
{1, ... , K}. Define
C(k) = {j :
yj = k}. Let
nk = number of observations in
C(k). The average gene expression in each subset
is k(i) = j
C(k)xj(i)/nk and the
average gene expression for all n samples is
(i) = jxj(i)/n. Then
define:
![r(i)={[&Sgr;<SUB>k</SUB>n<SUB>k</SUB>&cjs1134;&Pgr;<SUB>k</SUB>n<SUB>k</SUB>]&Sgr;<SUB>k</SUB>n<SUB>k</SUB>[<A><AC>x</AC><AC>&cjs1171;</AC></A><SUB>k</SUB>(i)−<A><AC>x</AC><AC>&cjs1171;</AC></A>(i)]<SUP>2</SUP>}<SUP>1/2</SUP>](/content/vol98/issue9/fulltext/5116/img003.gif)
|
[ 3 ] |
![s(i)={[&Sgr;<SUB>k</SUB>(1&cjs1134;n<SUB>k</SUB>)&cjs1134;&Sgr;<SUB>k</SUB>(n<SUB>k</SUB>−1)]&Sgr;<SUB>k</SUB>&Sgr;<SUB>j∈C(k)</SUB>[x<SUB>j</SUB>(i)−<A><AC>x</AC><AC>&cjs1171;</AC></A><SUB>k</SUB>(i)]<SUP>2</SUP>}<SUP>1/2</SUP>](/content/vol98/issue9/fulltext/5116/img004.gif)
|
[ 4 ] |
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Acknowledgements |
|---|
We thank Peter Jackson, Ron Davis, James Ferrell, Dean Felsher, Lisa DeFazio, Joe Budman, Jean Tang, Tom Tan, and Kerri Rieger for helpful discussions. This work was supported by the Burroughs Wellcome Clinical Scientist Award and by National Institutes of Health (NIH) Grant CA77302 to G.C., by NIH Small Business Technology Transfer grant CA75675 to G.C. and Affymetrix, and by the Stanford Genome Training Grant to V.T.
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Abbreviations |
|---|
SAM, significance analysis of microarrays; FDR, false discovery rate; IR, ionizing radiation; FWER, family-wise error rate.
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Footnotes |
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To whom reprint requests should be addressed.
E-mail: chu{at}cmgm.stanford.edu.
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References |
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Top
Abstract
Introduction
Materials and Methods
Results
Discussion
References
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