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Copyright Ó 2007 by the Genetics Society of America
DOI: 10.1534/genetics.106.063982
Genetic Mapping in the Presence of Genotyping Errors
Dustin A. Cartwright,*
,†,1
Michela Troggio,
†
Riccardo Velasco
†
and Alexander Gutin*
*Myriad Genetics, Salt Lake City, Utah 84108 and
†
Genetics and Molecular Biology Department, IASMA Research Center,
San Michele a/Adige (TN) 38010, Italy
Manuscript received July 27, 2006
Accepted for publication January 18, 2007
ABSTRACT
Genetic maps are built using the genotypes of many related individuals. Genotyping errors in these data
sets can distort genetic maps, especially by inflating the distances. We have extended the traditional
likelihood model used for genetic mapping to include the possibility of genotyping errors. Each individual
marker is assigned an error rate, which is inferred from the data, just as the genetic distances are. We have
developed a software package, called TMAP, which uses this model to find maximum-likelihood maps for
phase-known pedigrees. We have tested our methods using a data set in Vitis and on simulated data and
confirmed that our method dramatically reduces the inflationary effect caused by increasing the number of
markers and leads to more accurate orders.
G
ENETIC mapping uses the genotypes of many
related individuals at selected markers to deter-
mine the relative locations of these markers. The geno-
type data allow us to infer where recombinations have
occurred, which is directly related to the genetic dis-
tance. The purpose of a genetic mapping algorithm is
to reconstruct as accurately as possible the order of the
markers on the chromosomes and the genetic distances
between them.
Genetic mapping algorithms fall into two categories:
those that use multipoint-likelihood maximization and
those that rely only on two-point statistics. MapMaker
(Lander et al. 1987), CRI-MAP (Green et al. 1990),
CarthaGe`ne (de Givry et al. 2005), and R/qtl (Broman
et al. 2003) fall into the former category, while GMendel
(Echt et al. 1992), JoinMap (Stam 1993), and RECORD
(van Os et al. 2005b) fall into the latter. Multipoint-
likelihood maximization has theoretical advantages, but
is slower than two-point methods.
We use multipoint-likelihood maximization, because
it is more robust in the presence of missing data. Two-
point statistics derive no information when an individual’s
genotype is missing for one of the markers. However,
multipoint analysis uses nearby markers to approxi-
mate the missing genotypes, appropriately discounted
because of possible recombinations. For the same rea-
son, multipoint analysis is more powerful with markers
that are not fully informative. In backcross and in-
tercross pedigrees, this advantage is less apparent,
but in outbred pedigrees, the markers will generally
have many different segregation types, and two-point
analysis between these will not incorporate all the
information.
Without accounting for genotyping errors, each error
in a nonterminal marker causes two apparent recombi-
nations in the data set. Thus, every 1% error rate in a
marker adds 2 cM of inflated distance to the map. If
there is an average of one marker every 2 cM, then an
average of a 1% error rate will double the size of the
map. Markers with very high error rates will have large
distances to the adjacent markers. These cases can be
detected, either manually or automatically, and the
markers removed. However, markers with low error
levels will not be detected and, furthermore, may rep-
resent too large a portion of the data set to eliminate
completely.
Apparent double recombinations may also be due
to biological phenomena such as gene conversion or
mutation and not laboratory errors. Nevertheless, as
with laboratory genotyping errors, these phenomena
are not indicative of recombination and treating
them as recombinations inflates the map distances
(Castiglione et al. 1998). For the purpose of this article,
we use the term error to refer to any process that causes
changes to single genotypes at a time, as opposed to
recombination, which also affects all subsequent
genotypes.
Previous work has presented methods for detecting
errors in genotype data once the marker order has been
decided (Lincoln and Lander 1992; Douglas et al.
2000; van Os et al. 2005a). The suspect genotypes can
be checked and corrected if necessary. However, this
verification procedure can be time consuming and
not necessarily fully effective because some combina-
tions of markers and individuals may consistently pro-
duce the same erroneous genotypes. Alternatively, the
1
Corresponding author: Myriad Genetics, 320 Wakara Way, Salt Lake City,
UT 84108. E-mail: dcartwri@myriad.com
Genetics 176: 2521–2527 (August 2007)
verification step may be skipped and the markers
recoded solely on the basis of the error detection algo-
rithm. This method may itself introduce errors, unless
the parameters are chosen very conservatively, in which
case it may miss errors. Finally, since the map itself has
been built using the error-containing data set, those
errors may be less apparent with that map.
In contrast, our approach integrates error detection
and compensation into the map-building procedure.
Furthermore, we use a likelihood model that does not
force a dichotomy between correcting or not correcting
particular genotypes. Instead, we have a probability dis-
tribution over the possible genotypes, which depends
on both the observed genotype and the estimated prob-
ability of error. Thus, even genotypes that are only pos-
sibly erroneous can be correctly utilized in constructing
the map.
Previous work modeling errors within the map-
ordering process has not incorporated both indepen-
dent error probabilities for the markers and estimation
of the parameters from the data. MapMaker 3.0 includes
an optional genotyping error rate for the entire linkage
group but has no provisions for estimating this param-
eter from the data (Lincoln and Lander 1992). R/qtl
is a software package that primarily performs QTL
analysis, but includes a model for building maps with a
fixed, uniform error rate, similar to MapMaker (Broman
et al. 2003). Thallman et al. (2001) presented a model
with independent error rates for each marker, but without
provisions for estimating these from the data. On the
other hand, Rosa et al. (2002) presented a method that
estimates a global error rate from the data while ordering,
but they use Gibbs sampling and not the EM algorithm,
and thus their approach requires many more iterations to
converge to a solution.
In the context of linkage analysis, the notion of
complex-valued recombination fractions has been in-
troduced (Go¨ ring and Terwilliger 2000; see also
Abkevich et al. 2001). The purpose was to account for
errors in the phenotype models. Our approach is simi-
lar, except that our errors are in the genotypes, not in
the model, and we account for errors at every locus, not
just at the disease locus.
We have developed a software package that uses
the error-compensating likelihood model to find the
maximum-likelihood map under that model. We have
named the package TMAP after the tlod statistic of
Abkevich et al. (2001). Although this method could
apply to any pedigree type, TMAP works only with pedi-
grees where all parents are completely genotyped and
phase known. This includes backcross, intercross, and
phase-known outbred pedigrees. For phase-unknown
outbred pedigrees, it is possible to determine the phases
with sufficiently many offspring, as was done with
the Vitis data used in this article (D. A. Cartwright,
unpublished results). TMAP is freely available from
http://math.berkeley.edu/
dustin/tmap/.
METHODS
Likelihood model: In our likelihood model, each
marker has both an observed genotype, which is speci-
fied in a data file, and a true genotype, which is not
observed directly and can only be inferred. The re-
lationship between the two genotypes is parameterized
by an error rate e. In each haplotype, the true and ob-
served genotypes coincide with probability 1 e. Thus,
the overall genotypes coincide with probability (1 e)
2
and differ only in the maternal haplotype with proba-
bility (1 e )e, only in the paternal haplotype also with
probability (1 e)e, and in both haplotypes with prob-
ability e
2
. This error model is completely analogous to
the probability distribution of recombinations between
a pair of markers. Of course, the true genotype cannot
be known a priori, and in many cases the observed geno-
types are not fully known either. Thus when computing
the likelihood, we sum over the likelihoods of all pos-
sible values for these genotypes.
Explicitly, the equation is as follows. Let n and m
denote the number of individuals and markers, respec-
tively. Let u
i
denote the recombination rate between
markers i and i 1 1, and let e
i
denote the error rate for
marker i. Then, the likelihood is a function of these two
sets of parameters,
X
g 2G
g 9 2G9
Y
m1
i¼1
‘ðrðg
i
; g
i11
Þ; u
i
Þ
Y
m
i¼1
‘ðrðg
i
; g 9
i
Þ; e
i
Þ
!
; ð1Þ
where G is the set of all possible genotypes, G9 is the set of
all genotypes that are consistent with the observations,
each element g consists of the true genotypes g
i
, each
element g 9 consists of the observed genotypes g 9
i
, r(g
1
,
g
2
) is the number of recombinations between genotypes
g
1
and g
2
, and
‘ðr; uÞ¼u
r
ð1 uÞ
2nr
is the likelihood of having exactly r recombinations
between two markers separated by a recombination
fraction u (or equivalently, exactly r errors in a marker
with error rate u).
We can represent this model visually as shown in
Figure 1. Each node represents an abstract marker, i.e.,
genotypes for all individuals in the pedigree. The leaf
nodes are the known, observed, possibly erroneous
markers, and the internal nodes are the inferred, un-
observed, error-free markers. Thus, except for the
terminal markers, each physical marker corresponds
to two nodes, one error free and one observed. Each arc
represents separation between two markers, either
because of recombination (vertical) or because of errors
(horizontal).
As shown in the graph (Figure 1), there is no point
in computing an error rate for the markers at either
end. For these markers, errors and recombinations are
2522 D. A. Cartwright et al.
indistinguishable in the model, so we conservatively
assume that all the apparent recombinations are true
recombinations and not errors.
Thus, the error rates effectively add m 2 parameters
to each linkage group of m markers. The maximum-
likelihood values of these additional parameters can be
estimated along with the genetic distances using the EM
algorithm (Lander and Green 1987). In the notation of
Equation 1, we can use approximate values of u
i
and e
i
to compute the joint probability distribution over G and
G9 (E step), which can then be used to compute better
approximations of u
i
and e
i
(M step). Iterating these two
steps typically converges to the maximum-likelihood
solution.
Finally, the recombination rates are translated into
map distances using the Kosambi map function. The
Kosambi map function models recombination interfer-
ence, even though the model assumes that each of the u
i
is independent of the others, meaning that recombina-
tion events separated by markers have independent
probabilities.
Since errors are defined in a way that is mathemati-
cally equivalent to recombinations, the position at one
end of the map is equivalent to the neighboring posi-
tion in this model. Any pair of maps that differs only by
switching these two markers will have the same likeli-
hood. Therefore, any likelihood maximization of the
order will leave each of these two pairs in an arbitrary
order. These symmetries are analogous to the equiva-
lence of any given order and the reverse order, except
that reversing a map is a physical as well as a mathemat-
ical symmetry, but reversing the final two markers is not
a physical symmetry. For the final map, we can pick the
order that minimizes the error, again assuming that
recombinations are more likely than errors, all else
being equal. However, while building the map, it is
useful to explicitly acknowledge these symmetries.
Marker order: We begin building our maps by trying
all possible orders of s seed markers. Because of the
additional symmetries, there are only s!/8 unique or-
ders. Then, we provisionally insert the next marker in all
possible positions, keeping the t highest likelihoods.
Each additional marker is added in the same way. On
the basis of our experiments, we have chosen s ¼ 6 and
t ¼ 3 to provide a good balance between speed and
accuracy.
When inserting a new marker near either end of the
map, the symmetries described above complicate the
possibilities. When adding a marker C to a map that
begins AB ..., there would seem to be three places to
add it: ABC ..., ACB ..., CAB . . . . However, the last two
are equivalent orders. Furthermore, the order of A
and B was arbitrary, so the orders BAC ..., BCA ..., and
CBA ...are just as plausible. In fact, these six orders
consist of three pairs of equivalent orders, where each
equivalent pair is defined by the marker in the third
position. Thus we try each of the three equivalent pairs
of orders only once.
After building an initial order, we use a simple Monte
Carlo algorithm to find the maximum-likelihood order.
At each iteration, a random permutation from the
neighborhood is applied to the marker order, and the
log likelihood is computed. If the new log likelihood is
less than the old one, the new order is accepted. If the
new is greater then the old, it is nonetheless accepted
with probability e
dL/T
, where dL is the difference in log
10
likelihood, and T, known as the temperature, is a pa-
rameter of the algorithm. This is similar to simulated
annealing but with a fixed temperature (Kirkpatrick
et al. 1983). We use two phases of Monte Carlo optimiza-
tion, first with T ¼ 0.5 and then with T ¼ 0.05.
We define our neighborhood to have two different
kinds of permutations, which we call flips and moves. A
flip consists of taking a stretch of the map consisting of
two or more markers and reversing its orientation in
place, which is equivalent to a 2-change from the theory
of the traveling salesman problem (Schiex and Gaspin
1997). A move consists of removing a marker from one
location and inserting it in another. These are illus-
trated in Figure 2. Rather than consider each permuta-
tion equally, we bias the neighborhood toward the more
local, smaller-scale alterations, which are more likely to
Figure 1.—Graphical representation of the error model.
Each node represents an abstract marker, i.e., genotypes for
all individuals in the pedigree. The leaf nodes are the known,
observed, possibly erroneous markers, and the internal nodes
are the inferred, unobserved, error-free markers. Thus, except
for the terminal markers, each physical marker corresponds
to two nodes, one error free and one observed. Each arc rep-
resents separation between two markers, either because of
recombination (vertical) or because of errors (horizontal).
Genetic Mapping With Data Errors 2523
have similar likelihoods. Within each family of permu-
tations, each permutation has probability C
r
r
‘
, where ‘
represents the size of the subsection in a flip and the
length of the move, and C
r
is a constant to make the
total probability 1. We use a value of r ¼ 0.9 for both sets
of permutations.
Implementation: The core algorithms in TMAP are
implemented in C. There is a command-line interface
for Unix and a Java graphical interface that has been
tested on Solaris, Linux, Windows, and Mac OS X.
Validation: We tested TMAP using data from 94 pro-
geny of a cross in Vitis vinifera, which were genotyped
at 1006 markers (Troggio et al. 2007, accompanying
article in this issue), as well as simulated data sets. Two
facets of the program were assessed: first, the likelihood
model for compensating for genotyping errors; second,
the Monte Carlo search algorithm for finding optimal
solutions.
To test the ability of the error model to counteract the
inflationary effect of genotyping errors, we performed
the simple experiment of removing every other marker
in each linkage group and measuring the change in the
linkage group’s size. In the presence of uncompensated
errors, removing markers will cause the distances to
shrink because there will be fewer apparent double
recombinations, but not if the errors are properly
compensated. First, we used the Monte Carlo algorithm
to determine the maximum-likelihood order of each
group. Then, we computed the size of each group
and the size of each group after removing every other
marker. We modified TMAP to not take errors into ac-
count and repeated the last step.
In some cases, we observed that error compensation
also improved the ordering. Both with and without
compensation, markers with many errors tend to be
placed at the ends of the linkage groups, because they
do not fit well anywhere in the middle. However, with
error compensation, this effect is less pronounced.
To verify this phenomenon, we simulated a backcross
pedigree consisting of 19 markers and 94 individuals
with a distance of 5 cM between adjacent markers and
5% of the genotypes missing. We added a varying amount
of simulated errors to the 10th marker. Then, we ordered
the markers using both TMAP, the modified version that
didnot compensatefor errors, and a versionthat assumed
a fixed error rate of 2%, similar to MapMaker and R/qtl
(Lincoln and Lander 1992; Broman et al. 2003).
To validate the parameters in the Monte Carlo
iterative improvement algorithm we experimented with
many variant parameters. First, we used a long run of the
improving algorithm to determine the maximum likeli-
hood, or at least a close approximation of it, for each
linkage group of the grapevine data. Then, for a variety
of parameters, the Monte Carlo improvement algorithm
was applied to each linkage group until the log
10
like-
lihood was within 0.1 of the optimum or until a maxi-
mum number of iterations was reached. This operation
was repeated 10 times for each set of parameters, and we
recorded the average number of iterations required.
RESULTS
Error model: The results of removing every other
marker from linkage groups in the Vitis data set are
shown in Figure 3. Without error compensation, the
linkage groups always decreased in size when markers
were removed, and, furthermore, there is not a lot of
correlation between the sizes, but with error compen-
sation the sizes typically remained very consistent.
Figure 4 shows the proportion of incorrect place-
ments of a marker with a varying error rate. The results
show that the error compensation method helps cor-
rectly position markers with significant error rates. Fur-
thermore, the plot underestimates the relative accuracy
of error compensation, because, with error compensa-
tion, many of the incorrect placements were only one
or two positions away from the correct position, but
without error compensation most of the incorrect place-
ments were at the ends of the group.
Monte Carlo parameters: Figure 5 shows the effect of
removing one class of permutations on the time to
converge to an optimal solution. Each point represents
a single linkage group. On the x-axis is the average
number of steps needed to converge using the standard
parameter set, and on the y-axis is the average number of
Figure 2.—Illustration of the two types of permutations
used in the marker-ordering algorithm: moves (left) and flips
(right). Each square represents a single marker.
2524 D. A. Cartwright et al.
steps needed to converge for a variant that had one
of the two permutation types (flips or moves) disabled.
On some linkage groups, the optimization performed
poorly with only one of the permutation types, justifying
the inclusion of both. Note that in some of these cases
the maximum number of iterations was reached before
convergence, so this plot underestimates the difference
between the parameter choices.
Similarly, we experimented with varying the parameter
r for one or both permutation types and the temperature
of T, to arrive at our choices for these parameters, al-
though the differences are less dramatic. In particular,
convergence was slower with r ¼ 1, justifying the non-
uniform distribution of permutations.
Error rate distribution: The distribution of the non-
zero error rates in the Vitis data set is shown in Figure 6.
Among the markers with nonzero errors, most have an
error rate of ,5%. Without error compensation, the
cumulative effect of these markers would be to inflate
the map distances, but to remove all of them would
significantly reduce the usefulness of the map. Fur-
thermore, an additional 67% of the markers had an
estimated error rate of exactly 0%. In these cases, the
error-compensating likelihood model reduces to the
traditional one, and there is no loss of information.
Finally, the distribution clearly shows that the error rate
is not the same for all markers, which has been the
assumption in all previous models of genotyping errors.
There are a handful of markers with error rates in the
range 15–35%. Their presence did not significantly
affect the other markers in their linkage groups, so
we did not remove them from the map. These markers
with high error rates are analogous to phenotypes with
Figure 4.—Simulation of the effect of errors on marker or-
dering. In a linkage group of 19 markers, the 10th marker was
simulated with errors, and the markers were ordered, using
three different likelihood models. The first uses TMAP with
the error model described in this article. The second uses a
version of TMAP that assumes a fixed error rate of 2% for
every marker. The third does not model any error at all.
Figure 5.—Effect of removing one of the two permutation
types on the speed of convergence to the correct order.
Figure 3.—Effect on linkage group size of removing every
other marker both with and without compensation for errors.
Error compensation leads to more consistent genetic distances.
Figure 6.—Distribution of nonzero error rates in the Vitis
data set. In addition, 625 markers (67%) had an estimated
error rate of exactly 0%.
Genetic Mapping With Data Errors 2525
incomplete penetrance. The error rate reduces the
informativeness of the markers, but it is still possible
to localize them to a specific area of the linkage group.
We extended the analogy between markers with high
error rates and phenotypes in linkage analysis to esti-
mate the accuracy of the positions of these markers. In
linkage analysis, the range of positions with log
10
likeli-
hood 1 unit less than the maximum log
10
likelihood
measures the uncertainty in a marker’s position. For
each marker, a similar analysis was performed by hold-
ing the rest of the linkage group fixed and computing
the log likelihood with the marker positioned every
0.1 cM along the length of the linkage group. The error
rate and the size of the 1-unit-down interval for each
marker are plotted in Figure 7. In general, markers with
higher error rates are localized less precisely in the
linkage group. However, even for the markers with the
largest error rates, the 1-unit-down interval was never
.21 cM.
DISCUSSION
We have defined our error model to be the same as
the recombination model. This means that we treat the
correct genotyping of the haplotype from the mother
and of the haplotype from the father as independent
events. An alternative error model would be to treat
each individual’s genotype as a whole as either correct
or incorrect. However, a different error model would
remove the symmetry between the recombination frac-
tion of a terminal marker and the error of the adjacent
marker for many, but not all, segregation types. Thus,
the relative position of these two markers would be
decided by the likelihoods and not by the error-
minimizing rule above. Furthermore, the processes that
cause genotyping errors are more likely to produce
errors in only one haplotype than in both. For example,
it is more likely to misread an AA genotype as AB than
as BB.
More complex classes of genotyping errors are not
detected by this model. For example, in one linkage
group of the Vitis data, there was a pair of markers that
each had the same set of errors in their genotype data.
Because the genotypes from each marker seemed to
confirm the genotypes from the other, the method did
not detect the errors. However, there were large gaps on
either side of the pair, and removing either one caused
the gaps to disappear and be absorbed in the error rate
of the remaining marker. This linkage group gave rise to
one of the outliers in Figure 3.
CarthaGe`ne and GMendel have both previously ap-
plied Monte Carlo techniques to the marker ordering
problem. CarthaGe`ne uses a neighborhood consisting
of flips and a permutation based on a 3-change that
moves whole blocks of markers at a time, but does
not bias either permutation toward smaller changes.
GMendel only swaps pairs of markers and does include a
bias toward nearby markers that is active only during the
later phases of the improvement. However, as our results
show, both a richer neighborhood and a bias toward
small-scale permutations improve convergence.
We have used only two temperatures in our Monte
Carlo improving algorithm, rather than the more com-
mon steady decrease in temperature used in simulated
annealing. Simulated annealing starts with a high initial
temperature that effectively randomizes the marker
order. Thus, it is not possible to take advantage of the
result of the incremental ordering algorithm as a starting
point. However, we found that the incremental algorithm
can often quickly find good approximate solutions, so
we chose a Monte Carlo algorithm that could take
advantage of this.
We have shown that genotyping errors can be ac-
commodated by a simple extension to the mapping-
likelihood model, which gives a more accurate marker
order and especially distances.
This work was supported by the ‘‘Grapevine Physical Mapping’’ and
‘‘A.M.I.CA. Vitis’’ projects funded by the Provincia Autonoma di
Trento.
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Communicating editor: R. W. Doerge
Genetic Mapping With Data Errors 2527
