QTL mapping in mice, completed.

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QTL mapping in mice, completed.

• Lecture 12, Statistics 246
• March 2, 2004

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REVIEW

• n backcross mice M markers in all, with at most
  a handful expected to be near QTL
• xij genotype (0/1) of mouse i at marker j
• yi phenotype (trait value) of mouse i
• Yi ? ?j1M ?jxij ?j
  Which ?j ? 0 ?
• ?Variable selection in linear models (regression)

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Variable selection in linear models
a few generalities

• There is a huge literature on the selection
  of variables in linear models. Why? First, if we
  leave out variables which should be in our model,
  we risk biasing the parameter estimates of those
  we include, and our error has systematic
  components, reducing our ability to identify
  those which should be there. Second, when we
  includes variables in a model which do not need
  to be there, we reduce the precision of the
  estimates of the ones which are there.
• We need to include all the relevant (perhaps
  important is a better word), and no irrelevant
  variables in our model. This is an instance of
  classic statistical trade-off of bias and
  variance too few variables, higher bias too
  many, higher variance. The literature contains a
  host of methods of addressing this problem, known
  as model or variable selection methods. Some are
  general, i.e. not just for linear models, e.g.
  the Akaike information criterion (AIC), the Bayes
  information criterion (BIC), or cross-validation
  (CV). Others are specifically designed for linear
  models, e.g. ridge regression, Mallows Cp, and
  the lasso. We dont have the time to offer a
  thorough review of this topic, but well comment
  on and compare a few approaches.

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Special features of our problem

• We have a known joint distribution of the
  covariates (marker genotypes), and these are
  (approximately) Markovian
• We may have lots of missing covariate
  information, but it can be dealt with.that is,
  we can effectively use all our data
• Our aim is to identify the key players (markers),
  not to minimize prediction error or find the
  true model.
• Conclusion Our problem is not a generic linear
  model selection problem, and this offers the hope
  that we can do better than we might in general.
• END REVIEW

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Finding QTL as model selection(the loci are part
of the model)

• Select class of models
• Additive models
• Additive plus pairwise interactions
• Regression trees
• Compare models (?)
• BIC?(?) logRSS(?) ?(?log n/n)
• Sequential permutation tests

• Search model space
• Forward selection (FS)
• Backward elimination (BE)
• FS followed by BE
• MCMC
• Assess performance
• Maximize no QTL found
• control false positive rate

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Model class additive (linear) models

• n backcross mice M markers
• xij genotype (0/1) of mouse i at marker j
• yi phenotype (trait value) of mouse i
• Yi ? ?j1M ?jxij ?j
  
• The model is characterized by its variable subset
  ? j ?j ? 0.
• We could also include some pairwise interactions
  of variables (loci) in the model.

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Another model class regression trees
Tree model here ? would include splits on
genotypes at specified loci (q1 etc) and
average QT values at tips.
Regression trees are good for capturing
nonlinearities.
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Model comparison criteria

• Within a model class, we want to compare models.
  One obvious criterion is the residual sum of
  squares, RSS(?).
• Exercise. Prove that for a normal linear model,
  with unknown regression coefficients and error
  variance, the maximized log-likelihood lmax is
  essentially -(n/2)logRSS(?).
• However, we know that including more variables
  reduces the RSS, without necessarily improving
  the quality of our model we need a penalty for
  model size or model complexity. There are many
  such, and we choose to minimize criteria of the
  form
• ?(?) logRSS(?) ?n-1D(n)
• for suitable D(n). The case D(n) n is called
  AIC, while D(n) logn is BIC, and D(n) loglogn
  is due to Hannan and Quinn.
• Under certain assumptions, the last two are
  consistent estimators of a true model, while AIC
  tends to overfit under the same assumptions. It
  has different optimality properties. For finding
  QTL, we prefer BIC, but use D(n) ?logn, which
  we call BIC?, with ? gt 1.

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Model comparison a Bayesian approach

• We could place prior probabilities on each of
  the possible models, as well as the model
  parameters, and use Bayes theorem to calculate
  the posterior probability of the models given the
  data. It would make good sense (for a
  non-Bayesian!) to choose the model with the
  maximum posterior probability.
• Assume y ?, ?? ,?2 N(X? ?? , ?2 ), and
  use the priors ?? N(0, c?2(X? X? )-1 ), and
  p(?2 ?) ? 1/?2 , while for the model ? itself,
  assume p(?) ? (c/d)?/2 . Now let c ?? (a
  diffuse improper prior), and integrate out ?? and
  ?2. The resulting posterior distribution for ?
  satisfies
• -(2/n) log p(?y)
  logRSS(?) ?n-1 logd.
• This is just our general criterion with D(n)
  d, which provides further support for it.
  However, the only real justification for it is
  its performance.
• Exercise. Derive the expression involving p(?y)
  just given.

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Sequential permutation tests

• Lets begin by seeing how permutation tests
  are applied in this context, an idea first put
  forward by Churchill Doerge.
• Mouse i has phenotype yi and marker data mi.
  To test the null hypothesis of no association
  between phenotype and marker data, we can simply
  de-couple the two create many sets of
  pseudo-data by choosing a random permutation ? of
  the labels 1, .., n, and forming pairs (yi ,
  m?(i)), i 1,n.
• To assign p-values to LOD scores, say, we
  would compare the value for the observed data to
  the set of values obtained for (say) 1,000 sets
  of pseudo-data obtained in this way. As always,
  the question of significance levels arises, as
  does the issue of multiple testing. At least for
  the genome-wide calculation of LOD scores,
  permutation testing gives a neat solution simply
  repeat on the pseudo-data whatever multiple
  testing one does with the actual data (e.g.
  taking the genome wide maximum LOD).
• For model selection, permutation tests are
  carried out to assess the significance associate
  with adding or dropping a variable. Fixed
  significance levels are typically used, e.g.
  0.05, with no corrections for multiple testing.

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Some methods for model search

• Forward selection (FS) selects the variable not
  in the model which provides the greatest
  reduction in the penalty function (with a rule
  for making no change). Rejecting at a fixed
  significance level is usually adopted. Called
  greedy by computer scientists.
• Backwards elimination (BE) eliminates the
  variable whose removal makes the smallest
  increase in the penalty function (with a rule for
  making no change). Again, the use of a fixed
  significance level is standard.
• FS followed by BE deliberately overfit, then
  prune back. We need suitable parameters to
  implement this approach.
• Markov chain Monte Carlo (MCMC) provides a way
  of moving around the class of all models,
  selecting new or eliminating existing variables
  according to a stochastic rule. Eventually, we
  can choose as the preferred model, the one which
  minimizes the criterion function among all those
  models visited. A Bayesian would compute
  empirical posterior probabilities using the
  series of models visited.

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We like BIC? why?

• For a fixed number of markers, letting n ??,
  BIC? is consistent
• There exists a prior (on models coefficients)
  for which BIC? is the -log posterior
• BIC? is essentially equivalent to use of a
  threshold on the conditional LOD score
• It performs well

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Choice of ?

• Larger ? include more loci, higher false
  positive rate
• Smaller ? include fewer loci lower false
  positive rate
• Let L 95 genome-wide LOD threshold
• (comparing single QTL models to the null
  model)
• Choose ? 2 L/log10n .
• With this choice of ?, in the absence of QTL,
  well include at least one extraneous locus, 5
  of the time.

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Simulations comparing strategies

• Backcross with n 250
• No crossover interference
• 9 chromosomes, each 100cM
• Markers at 10cM spacing complete genotype
  data
• 7 QTL, equal effects 0.76
• -One pair in coupling (same sign)
• -One pair in repulsion (opp. sign)
• -Three unlinked QTL
• Heritability 50 (env var 1.0)
• 2,000 simulated replicates

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Methods compared

• ANOVA at marker loci
• Composite interval mapping (CIM)
• Forward selection with permutation tests
• Forward selection with BIC?
• Backward elimination with BIC?
• FS followed by BE with BIC?
• MCMC with BIC?

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Methods, some detail

• ANOVA at marker loci essentially 2 sample
  t-tests, with marker genotype defined groups,
  used simulated genome-wide threshold
• Composite interval mapping (CIM) described on
  next slide
• Forward selection with permutation tests (1,000
  perms, ?0.05)
• Forward selection to 25 markers, with BIC?
• Backward elimination, starting from the full
  model, with BIC?
• FS to 25 markers, followed by BE, with BIC?
• MCMC for 1,000 moves with BIC?
• More details and results for other sample sizes
  can be found in Broman Speed, JRSSB, 2002.

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Composite interval mapping

• Here an initial subset S of markers is
  chosen to control for background variation.
• Then, as in conventional interval mapping, a
  genome-wide scan is carried out, at each locus
  testing the hypothesis that a QTL at that locus
  need not be added to the existing model, against
  the alternative that it should be added.
  Markers in S in close proximity to the current
  locus (e.g. 10cM) are excluded.
• As in standard interval mapping, LOD scores
  are computed, and the result plotted. Regions
  where the LOD exceeds a suitable genome-wide
  threshold C are thought to contain QTL. Here we
  simulated to get C.
• A key question is how many markers should
  there be in the initial subset S, and how should
  they be chosen. There is no sharp answer to this
  question. We used 3, 5, 7, 9, and 11.

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Conclusions from the simulation

• The BIC? methods performed pretty well, best with
  MCMC, though only slightly better than FS. FS
  with permutation tests had fewer extraneous than
  FS with BIC.
• The CIM ones didnt do so well, except when the
  of variables was 7, though their rate of
  extraneous loci was quite low.
• ANOVA was quite poor, in relative terms.

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Selective genotyping

• The idea here is to genotype only animlas in
  the high and low extremes of the phenotypic
  distribution, but including the phenotypes
  (without genotypes) of the remainder in the
  analysis. This involves some loss of power,
  though not much if the not genotyped isnt
  large. Two issues why would we include
  phenotypes of animals not genotyped, and how
  would we do that.
• To make the point, we compare
• Strategy A Analyse all genotypes and all
  phenotypes
• Strategy B Analyse all phenotypes, but
  include genotypes only for individuals in the
  extremes of the phenotypic distribution
• Strategy C Analyse genotypes and phenotypes
  only for individuals in the extremes of the
  phenotypic distribution
• Exercise. Write down the likelihoods in all
  three cases.

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Selective genotyping in a BC
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Some simulation results

• The slides that follow are for a hypothetical F2
  intercross with 150 individual mice, a set of 150
  markers quite similar to those of our NOD B6
  cross, comparing 3 analysis strategies
• A analyse both genotypes and phenotypes for all
  mice
• B analyse phenotypes for all mice, but only
  genotypes for the phenotypically extreme mice,
  correctly incorporating the other genotypes as
  missing
• C analyse genotypes and phenotypes for
  phenotypically extreme mice only, and not saying
  so.

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Type I error for given genome-wide LOD
thresholds
Conclusion strategy C needs a different
genome-wide threshold.
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Power for a given QTL effect ?
Solid strategy A Dashed strategy B, with 50
genotyped
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Power for a given proportion not genotyped
Proportion not genotyped
Solid strategy A, Dashed strategy B with ?
0.25.
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Summary

• QTL mapping is a model selection problem
• Key issue the comparison of models
• Large scale simulations are important
• More refined procedures do not necessarily give
  improved results
• BIC? with forward selection followed by backward
  elimination works quite well
• Real savings to be made by genotyping only
  extremes phenotypically

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Acknowledgements

• Karl Broman, Johns Hopkins
• Nusrat Rabbee, UCB