# Comparing Quantitative Trait Loci and Gene Expression Data

Authors: Bing Han; Naomi S. Altman; Jessica A. Mong; Laura Cousino Klein; Donald W. Pfaff; David J. Vandenbergh
Journal: Advances in Bioinformatics
 (2008)

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## Abstract

We develop methods to compare the positions of quantitative trait loci (QTL) with a set of genes selected by other methods, such as microarray experiments, from a sequenced genome. We apply our methods to QTL for addictive behavior in mouse, and a set of genes upregulated in a region of the brain associated with addictive behavior, the nucleus accumbens (NA). The association between the QTL and NA genes is not significantly stronger than expected by chance. However, chromosomes 2 and 16 do show strong associations suggesting that genes on these chromosomes might be associated with addictive behavior. The statistical methodology developed for this study can be applied to similar studies to assess the mutual information in microarray and QTL analyses.


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## Body

## 1. Introduction

The association  between a complex
phenotypic trait and genetic markers on the chromosomes can be detected through
statistical analysis, leading to the identification of quantitative trait loci
(QTL)—regions of the chromosomes
that appear to be associated with the phenotype. Quantitative trait loci (QTL) are
expected to be associated with the genes controlling some aspects of the
phenotype. One mechanism by which a gene might be associated with the trait is
through altered transcription which is easily measured by microarray analysis.
Microarrays have the ability to measure a large percentage of the genes in the genome,
and this assessment parallels the genome-wide scan performed by QTL methods.Several investigators have considered
combining QTL and microarray data for studying a genetic trait. For example, Wayne
and McIntyre [1] proposed a way of identifying candidate genes based on both
QTL mapping and microarray data, where loci for an interesting quantitative
trait were primarily used for prescreening genes. A parallel microarray study
focused on the filtered gene list and identified differentially expressed genes
related to the same trait. When type I error is particularly emphasized, a QTL
analysis prescreen can be used to greatly reduce the number of genes under
consideration, and hence reduce the effect of multiple testing. When the
objective of the research is gene discovery, a combined analysis can focus
attention on genes and QTL most likely to be associated with the trait. Fischer
et al. [2] developed a web-based software tool for combined visualization and
exploration of gene expression data and QTL. The methodology developed in this
work is complementary to the analyses that can be performed on the GeneNetwork
website (WebQTL, http://www.genenetwork.org/), which allows assessment of the
relationship between gene expressions and QTL in recombinant in bred mice [3].Comparing QTL and microarray data is not completely
straightforward. The estimated range of QTL positions is generally wide, containing
many possibly interesting genes. In addition, QTL analysis may also miss some
interesting genes [1]. The high level of experimental errors and limitations in
microarray data analysis inevitably introduce mistakes in the identification of
relevant genes. Finally, QTL studies include the entire genome including
noncoding regions, while microarray studies seldom include the entire genome.Further problems arise when we try to
associate phenotypes with gene expressions in specific tissues. While the association
is direct if the tissue from which transcription is assayed defines the
phenotype, unanticipated associations can arise if the tissue indirectly regulates
the phenotype—for example, bone
strength may be regulated through physical activities regulated by the brain.
Alternatively, association can arise through pleiotropic expression of the gene
in a tissue not included in the expression study but in which the gene plays a
role in the phenotype. In addition, the association between a phenotype and a
tissue may depend on ephemeral conditions that may not be present when the
tissue was collected for the microarray study or on a small percentage of cells
in the organism, which may be masked by bulk tissue preparation.In this paper, we suggest methods to
examine the strength of association between a set of genes and a set of
intervals along the genome. The methods identify genomic intervals with
unusually high numbers of genes in the set, and hence genes and genomic
intervals which are more likely to be associated. The methodology is not tied
to the data source. Typically, the genes will be chosen from a gene expression
study, while the genomic intervals will be QTL. The methods by
which the regions and genes are selected affect the biological interpretation
of the results, but not the statistical assessment of association.


## 2. Methods

Our objective is to determine whether genomic intervals on the chromosome
(usually QTL) and gene loci (often selected from gene expression studies) are collocated
on the chromosome. This requires matching of gene loci to genomic intervals and
then measuring the strength of association. Quantitative trait
loci (QTL) are often measured in recombination distance, while
gene loci are usually reported as physical distances. In this section, we first
discuss conversion between recombination and physical distances. We then
discuss 2 approaches to measure strength of association.


### 2.1. Converting Recombination Distance to Physical Distance

Quantitative trait loci (QTL) are
reported in centimorgans (cM), which measure recombination frequencies between
markers on a chromosome. Gene locations are usually measured by the physical distance
in base pairs (bp) or megabase pairs (1 Mb = 106 bp). To match QTL
sets and gene sets, we adopted the embedded conversion tool in expressionview [2]
to estimate physical distances from cM, using a subset of genes for which both
distances are available. The “smoothing window” technique used in expressionview
essentially applies piecewise regression. In regions in which expressionview
appeared to give poor estimates, we also used polynomial regression to estimate
physical distance from cM by using genes for which both measures are available.
Any QTL with a resulting span that extends beyond the end of a chromosome is
truncated.


### 2.2. Measures of Association Based on Completeness and Accuracy

For
convenience, we  denote a set of QTLs, such as drug abuse QTLs, byQ, and a set of genes, such as the NA genes,
by
G. A natural approach is to consider the percentage of genes in G covered by
at least one QTL in Q. The association between Q and G is strong if this number
is big. This quantification reflects the “completeness” of Q in terms of
covering G. A complementary approach is to consider whether each QTL in Q covers
at least one gene in G. If a QTL in Q covers no genes in G, it is called
“empty”; otherwise it is “nonempty.” The association between Q and G is strong
when the percentage of empty QTL is small. This quantification reflects the
“accuracy” of Q in terms of covering G.IfQ is strongly associated with G, we expect both completeness and accuracy to be
high. However, the two methods do not necessarily give the same result because
they are measuring complementary aspects of an association. As quantitative
trait loci (QTL) are added to Q, we expect higher completeness because the QTL in Q cover more segments of each chromosome.
However, if these quantitative trait loci (QTL) are unrelated to G, we expect
many of them to be empty. Similarly, as genes are added to G, we expect higher
accuracy because selected genes are found in more locations. But if the
additional genes are unrelated to Q, we expect few of them to be covered by Q.To
account for the effect of increasing the size ofQ or G, we need to develop a combined
measure on both completeness and accuracy together to answer the question: is
the overall association strong? We propose an appropriately weighted average of
completeness and accuracy which penalizes for adding spurious QTL or genes to
the sets.LetN be the number of genes in G, M be the number of QTL in Q, n be the number genes in G covered by Q,
and m be the nonempty QTL in Q, then
completeness C=n/N, and accuracy A=m/M. We define the combined
measure of association as(1)S=CM+AN. The weight is
chosen to diminish the effect of matching by chance. When M increases, more of the genome will be covered by Q; we compensate
by dividing by M. We weight accuracy by 1/N to penalize for increasing the size of G in the same fashion.The limiting behaviors of the combined
measureS satisfy the need to
differentiate a strong association from a noisy one in which matching results
by chance. Let s be the number of
genes in G which really match some QTL in Q. Correspondingly, let r be the number of QTL in Q that really
match some genes in G. Note r usually
is not equal to s. Besides the true
matching relationship, every gene has probability p=p(M) of being covered
by Q leading to u genes which are
covered by chance. On the other hand, every QTL has a probability 1−q=1−q(N) of being empty with respect to G, so that it has a probability q of being “nonempty” leading
to v QTL which are nonempty by chance.
Completeness can be written as(2)C=s+uN. Thus, the
expectation of C is straightforward:(3)EC=s+(N−s)pN. Similarly,(4)A=r+vM,EA=r+(M−r)qM. Then,(5)ES=r+s+(N−s)p+(M−r)qMN. Consider
the following limiting circumstances: (1) (perfect match) when s→N and r→M, ES monotonically
increases to the limit (M+N)/MN; (2) (totally random) when s→0 and r→0, ES monotonically decreases to the limit (Np+Mq)/MN; (3) (G has
spurious genes) when N→∞ and M is fixed, notice q=q(N)→1 in this case, ES will converge
to p/M; (4) (Q has spurious QTL) when M→∞ and N is fixed, notice p=p(M)→1 in this case, ES will converge to q/N. From the above, it can be concluded
that the combined measure S will
approach its maximum when a perfect match arises and decrease when the
association weakens in some respect.


## 2.1. Converting Recombination Distance to Physical Distance

Quantitative trait loci (QTL) are
reported in centimorgans (cM), which measure recombination frequencies between
markers on a chromosome. Gene locations are usually measured by the physical distance
in base pairs (bp) or megabase pairs (1 Mb = 106 bp). To match QTL
sets and gene sets, we adopted the embedded conversion tool in expressionview [2]
to estimate physical distances from cM, using a subset of genes for which both
distances are available. The “smoothing window” technique used in expressionview
essentially applies piecewise regression. In regions in which expressionview
appeared to give poor estimates, we also used polynomial regression to estimate
physical distance from cM by using genes for which both measures are available.
Any QTL with a resulting span that extends beyond the end of a chromosome is
truncated.


## 2.2. Measures of Association Based on Completeness and Accuracy

For
convenience, we  denote a set of QTLs, such as drug abuse QTLs, byQ, and a set of genes, such as the NA genes,
by
G. A natural approach is to consider the percentage of genes in G covered by
at least one QTL in Q. The association between Q and G is strong if this number
is big. This quantification reflects the “completeness” of Q in terms of
covering G. A complementary approach is to consider whether each QTL in Q covers
at least one gene in G. If a QTL in Q covers no genes in G, it is called
“empty”; otherwise it is “nonempty.” The association between Q and G is strong
when the percentage of empty QTL is small. This quantification reflects the
“accuracy” of Q in terms of covering G.IfQ is strongly associated with G, we expect both completeness and accuracy to be
high. However, the two methods do not necessarily give the same result because
they are measuring complementary aspects of an association. As quantitative
trait loci (QTL) are added to Q, we expect higher completeness because the QTL in Q cover more segments of each chromosome.
However, if these quantitative trait loci (QTL) are unrelated to G, we expect
many of them to be empty. Similarly, as genes are added to G, we expect higher
accuracy because selected genes are found in more locations. But if the
additional genes are unrelated to Q, we expect few of them to be covered by Q.To
account for the effect of increasing the size ofQ or G, we need to develop a combined
measure on both completeness and accuracy together to answer the question: is
the overall association strong? We propose an appropriately weighted average of
completeness and accuracy which penalizes for adding spurious QTL or genes to
the sets.LetN be the number of genes in G, M be the number of QTL in Q, n be the number genes in G covered by Q,
and m be the nonempty QTL in Q, then
completeness C=n/N, and accuracy A=m/M. We define the combined
measure of association as(1)S=CM+AN. The weight is
chosen to diminish the effect of matching by chance. When M increases, more of the genome will be covered by Q; we compensate
by dividing by M. We weight accuracy by 1/N to penalize for increasing the size of G in the same fashion.The limiting behaviors of the combined
measureS satisfy the need to
differentiate a strong association from a noisy one in which matching results
by chance. Let s be the number of
genes in G which really match some QTL in Q. Correspondingly, let r be the number of QTL in Q that really
match some genes in G. Note r usually
is not equal to s. Besides the true
matching relationship, every gene has probability p=p(M) of being covered
by Q leading to u genes which are
covered by chance. On the other hand, every QTL has a probability 1−q=1−q(N) of being empty with respect to G, so that it has a probability q of being “nonempty” leading
to v QTL which are nonempty by chance.
Completeness can be written as(2)C=s+uN. Thus, the
expectation of C is straightforward:(3)EC=s+(N−s)pN. Similarly,(4)A=r+vM,EA=r+(M−r)qM. Then,(5)ES=r+s+(N−s)p+(M−r)qMN. Consider
the following limiting circumstances: (1) (perfect match) when s→N and r→M, ES monotonically
increases to the limit (M+N)/MN; (2) (totally random) when s→0 and r→0, ES monotonically decreases to the limit (Np+Mq)/MN; (3) (G has
spurious genes) when N→∞ and M is fixed, notice q=q(N)→1 in this case, ES will converge
to p/M; (4) (Q has spurious QTL) when M→∞ and N is fixed, notice p=p(M)→1 in this case, ES will converge to q/N. From the above, it can be concluded
that the combined measure S will
approach its maximum when a perfect match arises and decrease when the
association weakens in some respect.


## 3. Statistical Tests for Accuracy and Completeness

Until  the biology is fully understood, we
cannot be certain if the association betweenQ and G is due to chance. We,
therefore, want to study the pair of hypotheses H0: there is no stronger association
between QTL and genes than expected by chance, that is, we cannot benefit from
combining the results; Ha: the association is stronger than expected
by chance. In this section, we determine the statistical significance
of the observed levels of association by comparing with the null distribution of
completely random association determined by simulation. Random selection of QTL
is not readily done as selection of random intervals along the chromosomes because
we do not know the location of all possible QTL. Hence random selected
intervals are unlikely to model the true distribution of QTL. However, since
the physical locations of all genes on the microarray are known, random sets of
genes are readily created by choosing genes at random and considering the null
distributions of completeness or accuracy of the QTL sets with respect to these
randomly chosen genes.To assess the strength of association
between a QTL setQ and a gene set G of size N, we compute the completeness and accuracy of Q. We then select
genes at random from all the genes represented on the microarray. The simplest
way to do this is to select N genes
at random from the array. However, since there is considerable variability in
the percentage of tissue-specific genes on each chromosome and since the QTL
may not be randomly distributed among chromosomes, we can also consider selecting Ni genes from the ith chromosome, where Ni is the number of genes in
the gene set on the chromosome. We call the latter method the conditional
method because the random selection strategy is conditional on the number of
genes in G on each chromosome. By contrast, the former method, which selects
genes completely at random, is called the unconditional method. It is not
entirely clear when the conditional and unconditional methods are more
appropriate or powerful. When the gene set and quantitative trait loci (QTL)
are distributed across all of the chromosomes, we might expect that the
conditional method will be more powerful as well as more precise. However, when
there are chromosomes with no genes in G, the unconditional method should be
more powerful, as it takes into account the probability that QTL and genes in
the gene set may not be on the same chromosome, whereas the conditional method
uses only information about the chromosomes which include genes in G.
Conversely, when there are chromosomes with no QTL in G, the conditional method
may be more powerful because genes selected from these chromosomes will not
contribute to completeness or accuracy under conditional random sampling
strategy.By
repeatedly selecting gene sets at random either conditionally or
unconditionally and computing the completeness and accuracy forQ, an unconditional
or conditional null distribution is then simulated. The P-values for the
observed completeness C, accuracy A, and combined measure S are the percentages of simulated
datasets for which the simulated C, A,
and S are as strong as or stronger
than the corresponding observed values. The estimated P-values are
displayed in Table 1 based on 10000 sets of randomly selected genes. Since the P-values
are based on count data, we also consider 3 continuity corrections which differ
in how the rejection region includes the observed counts. The simulations took
about 296 seconds on a 2.8 GHz computer with 2 GB of RAM running Windows XP.Table 1
Estimated limiting extrema of combined measureS.
Limiting case defined in Section2ConditionalUnconditionalRandom selection0.004060.00389Spurious genes0.00690.00160Spurious QTL0.002370.00229Theoretical maximumS.0144ObservedS.00758As
part of the simulation, we can also compute the distribution of number of genes
covered by each QTL. Percentiles of this distribution can be used to identify quantitative
trait loci (QTL) that have unusually large coverage on the observed data and
are thus more likely to be associated with genes implicated in the QTL
condition.


### 3.1. Chi-Square Tests of Association

The count
of nonempty QTLmi and
covered genes ni of each
chromosome can be used to construct a chi-square-type of test for either
accuracy or completeness. Using chromosomes as the natural category, we define
the chi-square (𝒳2) test
statistic as(6)𝒳2=∑i=1T(Xi−EXi)2EXi∼𝒳T−12underH0:the link is no different from random,Xi=ni,mi,where EXi under H0 can be estimated by random
sampling genes, and T is the total
number of chromosomes. Under
H0: the link is no different from random; EXi is the same as expected counts when genes are
selected at random. For a given set of QTL, we can repeatedly sample random
genes. The average of observed counts of nonempty QTL or covered genes is a
consistent estimator for EXi and is quite accurate since we repeat sampling many times. A true association
between QTL and genes will increase the observed counts which result in a larger
chi-square test statistic. As well, the chi-square test provides us with
additional insight in identifying potential candidate differentially expressed
genes, if quantitative trait loci (QTL) are mapping the same or very similar
quantitative traits as the partner microarray study. On the one hand, a
chromosome that has a large positive value suggests a region of strong
association with the gene set, and hence lends support to the hypotheses that
the QTL and the covered genes are associated with the trait of interest.
Conversely, a chromosome for which Xi−EXi is a small positive value
or a larger negative value suggests that the QTL and the covered genes may not
be associated with the trait. Thus,
association between the Q and G can also be used to select QTL and genes which
are more likely to be of interest.


### 3.2. Materials

We apply our methods to a set of mouse QTL identified from the literature
and a set of mouse genes identified from a microarray study. First, we
identified a set of 120 QTL associated with drug abuse behaviors in mice [4] from
the Mouse Genome Informatics (MGIs) database (http://www.informatics.jax.org).
A set of 166 genes that are preferentially  expressed in the nucleus
accumbens (NA) region of the mouse brain (the NA genes) was determined from a microarray study of brain tissues [5]
using Affymetrix MG-U74Av2 arrays. This array contains about
1/3 of the coding genes in the mouse genome. Briefly, the NA genes were
identified as being expressed at least 1.5 fold higher in the nucleus accumbens
compared to two other brain regions, the medial basal hypothalamus and preoptic
area, in one-day-old C57BL/6J mouse pups. The study did not include replication, so the statistical significance
of the expression differences cannot readily be assessed; association of these
genes with QTL, therefore, becomes an important tool to help assess the
biological significance of the observed differential expression.The
two brain regions used for comparison are physically close to the NA on the
ventral surface of the brain but are largely derived from a different embryonic
region of the brain (diencephalon compared to telencephalon for the NA). The NA
plays an important role in drug abuse-related behavior. Our primary objective
was to determine if the QTL and gene expression studies are detecting a set of
genes in common that might be related to drug abuse behaviors. However, because the NA genes are selected
based on their expression in a selected region of the brain, rather than their
direct involvement in drug abuse behaviors, we need to be cautious about the
biological interpretation of an association between the QTL and  the gene set.


### 3.3. Results

Figure1 shows the correspondence between the set of QTL and the set of NA genes. The
long horizontal dashed lines are numbered to represent the mouse chromosomes. No
data were available regarding gene expression or QTL on the very short Y
chromosome. The positions of the NA genes were determined using the Affymetrix
metadata for the MG-U74Av2 (version 1.10.0) array provided in the Bioconductor
suite in R [6].Figure 1
Combined
visualization of QTL and NA genes. The short discrete horizontal segments are
the spans of the QTL defined as +/−5 centimorgans (cM) from the peak QTL
position. The small circles in the center of every segment are the peak
positions of the QTL. Finally, the vertical lines are the NA genes.No
obvious matches between the QTL set and the NA genes can be seen in Figure1.
The visual impression does not support a strong association between QTL and
genes. The observed
completeness is 44.6%, and the observed accuracy is 64.2%.
The observed S is .0076, compared
with the theoretical maximum for S of 0.014. p(M), and q(N)
can be estimated from the simulation and hence we can estimate the three local
minima under limiting circumstances 2, 3, and 4 discussed in Section 2.Table1 displays the comparison of S values
under both randomization and limiting circumstances. Although it is far from
the maximum, the observed S is above
all the estimated local minima for genes selected at random.TheP-values for completeness, accuracy, and S are in Table 2. The P-values for the conditional tests are
smaller than those for the unconditional tests, which is expected since there
are chromosomes with NA genes but no QTL. However, there is no indication that
there is significant association between the NA genes and the QTL set.Table 2
Simulated
one-sidedP-value for the hypothesis H0:
the association is not stronger than expected by chance.
MeasureObservedDefinition ofP-valueConditionalUnconditionalC (Completeness)0.446p (# >observed)0.1440.3291/2p (# =observed) + p (# >observed)0.1680.357p (
# ≥
observed)0.1920.385A (Accuracy)0.642p (# >observed)0.3280.5441/2p (# =observed) + p (# >observed)0.3610.576p (
# ≥
observed)0.3950.608S (Combined).0076p (# >observed)0.2130.4411/2p (# =observed) + p (# >observed)0.2290.459p (
# ≥
observed)0.2450.478The results of the chi-square test are in Table3. The unconditional
chi-square test supports the hypothesis that the completeness is higher than
expected by chance, but accuracy is only marginally higher than expected by
chance—that is, more genes are covered than expected,
but the number of QTL containing genes in G is about what is expected by
chance. For example, chromosomes 2 and 16 have much higher positive values than
expected under unconditional sampling which suggests that the genes on these
two chromosomes are more likely to be associated with the drug abuse trait, and
the QTL on these two chromosomes are more likely to be associated with the NA
region than genome-wide average.Table 3
P-value
from the chi-square test for the hypothesis H0:
the association is not different from expected by chance.
Chi-square testConditionalUnconditionalP-value (Completeness)0.9020.041*P-value (Accuracy)0.6080.246*Significant at 5%
level.Two genes on chromosome 2, Pax6 (paired
box gene 6) and Pcna (proliferating cell nuclear antigen) and one on chromosome
16, Tiam1 (T-cell lymphoma invasion and  metastasis 1) lie under QTL and have
some relationship to addictive behavior traits. Given that differences in gene
expression were used as the criterion for selecting genes, it is not unexpected that two of these genes, Pax6 and Tiam1, are transcription factors. Pax6
plays a role in differentiation of precursor cells into neurons and glia and is
altered in models of fetal alcohol syndrome [7]. Tiam1 is known to regulate
growth cone morphology, a process that can be altered by drugs of abuse [8].We note that chromosome 18 includes a
gene that is just outside a QTL region. To better understand the effect of a
small change in the QTL definition, we increased the length of the QTL by 0.5 Mb,
which is less than a 5% increase in length for most QTL. Although most of the
measures of association and their statistical significance are scarcely
affected, theP-value for the conditional chi-square test of
completeness changed from 0.910 to 0.182. This finding further supports the conjecture
that the completeness in our study may be higher than expected by chance. On
the other hand, this also demonstrates that the proposed chi-square test is not
robust against the disturbance in data than the previous simulation-based
tests.A similar
chi-square test can be used to determine if a particular QTL covers more genes
than expected by chance. For example, the 3 QTL on chromosome 16 and the QTL on
chromosome 18 all cover more genes than expected by chance, but none of the QTL
on chromosome 19 are more accurate than expected by chance.


## 3.1. Chi-Square Tests of Association

The count
of nonempty QTLmi and
covered genes ni of each
chromosome can be used to construct a chi-square-type of test for either
accuracy or completeness. Using chromosomes as the natural category, we define
the chi-square (𝒳2) test
statistic as(6)𝒳2=∑i=1T(Xi−EXi)2EXi∼𝒳T−12underH0:the link is no different from random,Xi=ni,mi,where EXi under H0 can be estimated by random
sampling genes, and T is the total
number of chromosomes. Under
H0: the link is no different from random; EXi is the same as expected counts when genes are
selected at random. For a given set of QTL, we can repeatedly sample random
genes. The average of observed counts of nonempty QTL or covered genes is a
consistent estimator for EXi and is quite accurate since we repeat sampling many times. A true association
between QTL and genes will increase the observed counts which result in a larger
chi-square test statistic. As well, the chi-square test provides us with
additional insight in identifying potential candidate differentially expressed
genes, if quantitative trait loci (QTL) are mapping the same or very similar
quantitative traits as the partner microarray study. On the one hand, a
chromosome that has a large positive value suggests a region of strong
association with the gene set, and hence lends support to the hypotheses that
the QTL and the covered genes are associated with the trait of interest.
Conversely, a chromosome for which Xi−EXi is a small positive value
or a larger negative value suggests that the QTL and the covered genes may not
be associated with the trait. Thus,
association between the Q and G can also be used to select QTL and genes which
are more likely to be of interest.


## 3.2. Materials

We apply our methods to a set of mouse QTL identified from the literature
and a set of mouse genes identified from a microarray study. First, we
identified a set of 120 QTL associated with drug abuse behaviors in mice [4] from
the Mouse Genome Informatics (MGIs) database (http://www.informatics.jax.org).
A set of 166 genes that are preferentially  expressed in the nucleus
accumbens (NA) region of the mouse brain (the NA genes) was determined from a microarray study of brain tissues [5]
using Affymetrix MG-U74Av2 arrays. This array contains about
1/3 of the coding genes in the mouse genome. Briefly, the NA genes were
identified as being expressed at least 1.5 fold higher in the nucleus accumbens
compared to two other brain regions, the medial basal hypothalamus and preoptic
area, in one-day-old C57BL/6J mouse pups. The study did not include replication, so the statistical significance
of the expression differences cannot readily be assessed; association of these
genes with QTL, therefore, becomes an important tool to help assess the
biological significance of the observed differential expression.The
two brain regions used for comparison are physically close to the NA on the
ventral surface of the brain but are largely derived from a different embryonic
region of the brain (diencephalon compared to telencephalon for the NA). The NA
plays an important role in drug abuse-related behavior. Our primary objective
was to determine if the QTL and gene expression studies are detecting a set of
genes in common that might be related to drug abuse behaviors. However, because the NA genes are selected
based on their expression in a selected region of the brain, rather than their
direct involvement in drug abuse behaviors, we need to be cautious about the
biological interpretation of an association between the QTL and  the gene set.


## 3.3. Results

Figure1 shows the correspondence between the set of QTL and the set of NA genes. The
long horizontal dashed lines are numbered to represent the mouse chromosomes. No
data were available regarding gene expression or QTL on the very short Y
chromosome. The positions of the NA genes were determined using the Affymetrix
metadata for the MG-U74Av2 (version 1.10.0) array provided in the Bioconductor
suite in R [6].Figure 1
Combined
visualization of QTL and NA genes. The short discrete horizontal segments are
the spans of the QTL defined as +/−5 centimorgans (cM) from the peak QTL
position. The small circles in the center of every segment are the peak
positions of the QTL. Finally, the vertical lines are the NA genes.No
obvious matches between the QTL set and the NA genes can be seen in Figure1.
The visual impression does not support a strong association between QTL and
genes. The observed
completeness is 44.6%, and the observed accuracy is 64.2%.
The observed S is .0076, compared
with the theoretical maximum for S of 0.014. p(M), and q(N)
can be estimated from the simulation and hence we can estimate the three local
minima under limiting circumstances 2, 3, and 4 discussed in Section 2.Table1 displays the comparison of S values
under both randomization and limiting circumstances. Although it is far from
the maximum, the observed S is above
all the estimated local minima for genes selected at random.TheP-values for completeness, accuracy, and S are in Table 2. The P-values for the conditional tests are
smaller than those for the unconditional tests, which is expected since there
are chromosomes with NA genes but no QTL. However, there is no indication that
there is significant association between the NA genes and the QTL set.Table 2
Simulated
one-sidedP-value for the hypothesis H0:
the association is not stronger than expected by chance.
MeasureObservedDefinition ofP-valueConditionalUnconditionalC (Completeness)0.446p (# >observed)0.1440.3291/2p (# =observed) + p (# >observed)0.1680.357p (
# ≥
observed)0.1920.385A (Accuracy)0.642p (# >observed)0.3280.5441/2p (# =observed) + p (# >observed)0.3610.576p (
# ≥
observed)0.3950.608S (Combined).0076p (# >observed)0.2130.4411/2p (# =observed) + p (# >observed)0.2290.459p (
# ≥
observed)0.2450.478The results of the chi-square test are in Table3. The unconditional
chi-square test supports the hypothesis that the completeness is higher than
expected by chance, but accuracy is only marginally higher than expected by
chance—that is, more genes are covered than expected,
but the number of QTL containing genes in G is about what is expected by
chance. For example, chromosomes 2 and 16 have much higher positive values than
expected under unconditional sampling which suggests that the genes on these
two chromosomes are more likely to be associated with the drug abuse trait, and
the QTL on these two chromosomes are more likely to be associated with the NA
region than genome-wide average.Table 3
P-value
from the chi-square test for the hypothesis H0:
the association is not different from expected by chance.
Chi-square testConditionalUnconditionalP-value (Completeness)0.9020.041*P-value (Accuracy)0.6080.246*Significant at 5%
level.Two genes on chromosome 2, Pax6 (paired
box gene 6) and Pcna (proliferating cell nuclear antigen) and one on chromosome
16, Tiam1 (T-cell lymphoma invasion and  metastasis 1) lie under QTL and have
some relationship to addictive behavior traits. Given that differences in gene
expression were used as the criterion for selecting genes, it is not unexpected that two of these genes, Pax6 and Tiam1, are transcription factors. Pax6
plays a role in differentiation of precursor cells into neurons and glia and is
altered in models of fetal alcohol syndrome [7]. Tiam1 is known to regulate
growth cone morphology, a process that can be altered by drugs of abuse [8].We note that chromosome 18 includes a
gene that is just outside a QTL region. To better understand the effect of a
small change in the QTL definition, we increased the length of the QTL by 0.5 Mb,
which is less than a 5% increase in length for most QTL. Although most of the
measures of association and their statistical significance are scarcely
affected, theP-value for the conditional chi-square test of
completeness changed from 0.910 to 0.182. This finding further supports the conjecture
that the completeness in our study may be higher than expected by chance. On
the other hand, this also demonstrates that the proposed chi-square test is not
robust against the disturbance in data than the previous simulation-based
tests.A similar
chi-square test can be used to determine if a particular QTL covers more genes
than expected by chance. For example, the 3 QTL on chromosome 16 and the QTL on
chromosome 18 all cover more genes than expected by chance, but none of the QTL
on chromosome 19 are more accurate than expected by chance.


## 4. Conclusion and Discussion

A strong
association was expected between the NA genes and the drug abuse QTL, but this
hypothesis was not fully supported by the data. A possible reason is that there
are a number of QTL not associated with NA genes. In addition, this large set
of QTL is associated with diverse drug abuse traits, including both
physiological and psychological factors, and hence may be associated with
multiple brain regions. As well, because the genes were selected from an
unreplicated study based on fold change, we can expect large false detection
and nondetection rates. A large false detection rate implies that the NA genes
likely include several genes which are not associated with the NA and are,
therefore, similar to genes selected at random. We can also expect a large
number of false nondetections which should induce a correspondingly large
impact on the power to detect the accuracy of the QTL.Completeness, accuracy, and the combined
measure have been proposed as methods to determine whether a set of QTL and a
set of genes are associated. The statistical significance of the association can
be estimated by selecting sets of genes at random from the population of genes
from which the gene set was determined. WhenP-values or other measures
of strength of association between the trait of interest and the QTL are
available or when some QTL are of particular interest a priori, we might
consider weighting the measures of accuracy so that a penalty is incurred if a
QTL highly associated with the trait is empty, and a gain is incurred if a gene
is covered by a highly associated QTL. Weights on the QTL can readily be
incorporated in the simulations required to estimate the P-values
because the QTL are fixed in the simulation. Weights on the genes are less
readily handled because weights are not available for genes selected at random.
In this study, we did not use weights—however, several
QTL are represented more than once in the QTL set, because the same QTL was
identified from multiple sources. For example, on chromosome 1, there are 3 QTL
identified in 3 studies investigating different phenotypes and located
identically. These QTL may be caused by the same gene or by nearby genes. The
chromosome locations covered by these QTL have higher weight in the computation
of both accuracy and completeness. Removing duplicate QTL did not alter the
conclusions for this set of genes and QTL.The chi-square test provides a simple
method to identify QTL and genes from the gene set that are most highly
associated. A QTL that covers more genes than expected by chance is likely to
include a cluster of genes from the gene set, which lends credence to the
hypotheses that the QTL and the covered genes are associated with the trait of
interest. Some examples of this have been detected in these data, for example,
the QTL on chromosomes 16 and 18 are more accurate than expected under
unconditional sampling. The chi-square tests appear to be more sensitive than
the other suggested tests to small changes in the definition of the QTL,
particularly when a chromosome has only a small number of QTL, and the change
in definition changesmi or ni. The other tests of accuracy and completeness
or the combined measure appeared to be more robust against changes of the same
magnitude.The data and R code can be accessed fromhttp://www.stat.psu.edu/~naomi/QTLsoftware/.



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*Source: 719818-2008-09-16.xml*