===== Linear Model Quantitative Traits =====
Simulation of quantitative trait (QT) values follows from a simple linear model of the additive effect of mutations. A mutation contributes to the change in mean of QT value by a specific amount. For protective variants, mutations decrease QT; for detrimental variants, mutations increase QT. Under this model both analytic and empirical power and sample size calculations are available for quantitative traits analysis methods.

==== Command Interface ====
<code bash>
spower LNR -h
</code>\\
<hidden initialState="visible" -noprint>
  usage: spower LNR [-h] [-a MULTIPLIER] [-b MULTIPLIER] [-A MULTIPLIER]
                    [-B MULTIPLIER] [-c MULTIPLIER] [-d MULTIPLIER]
                    [--def_rare P] [--def_neutral VALUE VALUE]
                    [--def_protective VALUE VALUE] [-P P] [-Q P]
                    [--sample_size N] [--def_valid_locus VALUE VALUE]
                    [--rare_only] [--missing_as_wt] [--missing_low_maf P]
                    [--missing_sites P] [--missing_sites_deleterious P]
                    [--missing_sites_protective P] [--missing_sites_neutral P]
                    [--missing_sites_synonymous P] [--missing_calls P]
                    [--missing_calls_deleterious P]
                    [--missing_calls_protective P] [--missing_calls_neutral P]
                    [--missing_calls_synonymous P] [--error_calls P]
                    [--error_calls_deleterious P] [--error_calls_protective P]
                    [--error_calls_neutral P] [--error_calls_synonymous P]
                    [--power P] [-r N] [--alpha ALPHA] [--moi {A,D,R,M}]
                    [--resampling] [-l N] [-o file] [-t NAME] [-v {0,1,2,3}]
                    [-s N] [-j N] [-m METHODS [METHODS ...]]
                    [--discard_samples [EXPR [EXPR ...]]]
                    [--discard_variants [EXPR [EXPR ...]]]
                    DATA
  
  positional arguments:
    DATA                  name of input data or prefix of input data bundle (see
                          the documentation for details)
  
  optional arguments:
    -h, --help            show this help message and exit
  
  model parameters:
    -a MULTIPLIER, --meanshift_rare_detrimental MULTIPLIER
                          mean shift in quantitative value w.r.t standard
                          deviation due to detrimental rare variants i.e., by
                          "MULTIPLIER * sigma" (default set to 0.0)
    -b MULTIPLIER, --meanshift_rare_protective MULTIPLIER
                          mean shift in quantitative value w.r.t. standard
                          deviation due to protective rare variants i.e., by
                          "MULTIPLIER * sigma" (default set to 0.0)
    -A MULTIPLIER, --meanshiftmax_rare_detrimental MULTIPLIER
                          maximum mean shift in quantitative value w.r.t
                          standard deviation due to detrimental rare variants
                          i.e., by "MULTIPLIER * sigma", applicable to variable
                          effects model (default set to None)
    -B MULTIPLIER, --meanshiftmax_rare_protective MULTIPLIER
                          maximum mean shift in quantitative value w.r.t
                          standard deviation due to protective rare variants
                          i.e., by "MULTIPLIER * sigma", applicable to variable
                          effects model (default set to None)
    -c MULTIPLIER, --meanshift_common_detrimental MULTIPLIER
                          mean shift in quantitative value w.r.t standard
                          deviation due to detrimental common variants i.e., by
                          "MULTIPLIER * sigma" (default set to 0.0)
    -d MULTIPLIER, --meanshift_common_protective MULTIPLIER
                          mean shift in quantitative value w.r.t standard
                          deviation due to protective common variants i.e., by
                          "MULTIPLIER * sigma" (default set to 0.0)
    --moi {A,D,R,M}       mode of inheritance: "A", additive (default); "D",
                          dominant; "R", recessive; "M", multiplicative (does
                          not apply to quantitative traits model)
    --resampling          directly draw sample genotypes from given haplotype
                          pools (sample genotypes will be simulated on the fly
                          if haplotype pools are not available)
  
  variants functionality:
    --def_rare P          definition of rare variants: variant having "MAF <=
                          frequency" will be considered a "rare" variant; the
                          opposite set is considered "common" (default set to
                          0.01)
    --def_neutral VALUE VALUE
                          annotation value cut-offs that defines a variant to be
                          "neutral" (e.g. synonymous, non-coding etc. that will
                          not contribute to any phenotype); any variant with
                          "function_score" X falling in this range will be
                          considered neutral (default set to None)
    --def_protective VALUE VALUE
                          annotation value cut-offs that defines a variant to be
                          "protective" (i.e., decrease disease risk or decrease
                          quantitative traits value); any variant with
                          "function_score" X falling in this range will be
                          considered protective (default set to None)
    -P P, --proportion_detrimental P
                          proportion of deleterious variants associated with the
                          trait of interest, i.e., the random set of the rest (1
                          - p) x 100% deleterious variants are non-causal: they
                          do not contribute to the phenotype in simulations yet
                          will present as noise in analysis (default set to
                          None)
    -Q P, --proportion_protective P
                          proportion of protective variants associated with the
                          trait of interest, i.e., the random set of the rest (1
                          - p) x 100% protective variants are non-causal: they
                          do not contribute to the phenotype in simulations yet
                          will present as noise in analysis (default set to
                          None)
  
  sample population:
    --sample_size N       total sample size
  
  quality control:
    --def_valid_locus VALUE VALUE
                          upper and lower bounds of variant counts that defines
                          if a locus is "valid", i.e., locus having number of
                          variants falling out of this range will be ignored
                          from power calculation (default set to None)
    --rare_only           remove from analysis common variant sites in the
                          population, i.e., those in the haplotype pool having
                          MAF > $def_rare
    --missing_as_wt       label missing genotype calls as wildtype genotypes
  
  sequencing / genotyping artifact:
    --missing_low_maf P   variant sites having population MAF < P are set to
                          missing
    --missing_sites P     proportion of missing variant sites
    --missing_sites_deleterious P
                          proportion of missing deleterious sites
    --missing_sites_protective P
                          proportion of missing protective sites
    --missing_sites_neutral P
                          proportion of missing neutral sites
    --missing_sites_synonymous P
                          proportion of missing synonymous sites
    --missing_calls P     proportion of missing genotype calls
    --missing_calls_deleterious P
                          proportion of missing genotype calls at deleterious
                          sites
    --missing_calls_protective P
                          proportion of missing genotype calls at protective
                          sites
    --missing_calls_neutral P
                          proportion of missing genotype calls at neutral sites
    --missing_calls_synonymous P
                          proportion of missing genotype calls at synonymous
                          sites
    --error_calls P       proportion of error genotype calls
    --error_calls_deleterious P
                          proportion of error genotype calls at deleterious
                          sites
    --error_calls_protective P
                          proportion of error genotype calls at protective sites
    --error_calls_neutral P
                          proportion of error genotype calls at neutral sites
    --error_calls_synonymous P
                          proportion of error genotype calls at synonymous sites
  
  power calculation:
    --power P             power for which total sample size is calculated (this
                          option is mutually exclusive with option '--
                          sample_size')
    -r N, --replicates N  number of replicates for power evaluation (default set
                          to 1)
    --alpha ALPHA         significance level at which power will be evaluated
                          (default set to 0.05)
  
  input/output specifications:
    -l N, --limit N       if specified, will limit calculations to the first N
                          groups in data (default set to None)
    -o file, --output file
                          output filename
  
  runtime options:
    -t NAME, --title NAME
                          unique identifier of a single command run (default to
                          output filename prefix)
    -v {0,1,2,3}, --verbosity {0,1,2,3}
                          verbosity level: 0 for absolutely quiet, 1 for less
                          verbose, 2 for verbose, 3 for more debug information
                          (default set to 2)
    -s N, --seed N        seed for random number generator, 0 for random seed
                          (default set to 0)
    -j N, --jobs N        number of CPUs to use when multiple replicates are
                          required via "-r" option (default set to 2)
  
  association tests:
    -m METHODS [METHODS ...], --methods METHODS [METHODS ...]
                          Method of one or more association tests. Parameters
                          for each method should be specified together as a
                          quoted long argument (e.g. --methods "m --alternative
                          2" "m1 --permute 1000"), although the common method
                          parameters can be specified separately, as long as
                          they do not conflict with command arguments. (e.g.
                          --methods m1 m2 -p 1000 is equivalent to --methods "m1
                          -p 1000" "m2 -p 1000".). You can use command 'spower
                          show tests' for a list of association tests, and
                          'spower show test TST' for details about a test.
  
  samples and genotypes filtering:
    --discard_samples [EXPR [EXPR ...]]
                          Discard samples that match specified conditions within
                          each test group. Currently only expressions in the
                          form of "%(NA)>p" is provided to remove samples that
                          have more 100*p percent of missing values.
    --discard_variants [EXPR [EXPR ...]]
                          Discard variant sites based on specified conditions
                          within each test group. Currently only expressions in
                          the form of '%(NA)>p' is provided to remove variant
                          sites that have more than 100*p percent of missing
                          genotypes. Note that this filter will be applied after
                          "--discard_samples" is applied, if the latter also is
                          specified.
</hidden>\\
==== Details ====
Model specific options are documented in details below. You should find the rest of the options [[http://bioinformatics.org/spower/options|otherwise documented]].

=== -a/b/c/d/A/B ===
These options specify the effect of quantitative trait loci in the region of interest, modeled by the shift in mean QT value due to these mutations. The unit of mean shift is the standard deviation. For example ''-a 0.1'' means a detrimental mutation increase the mean value of the QT by \(0.1\sigma\). ''-a'' is QT shift due to detrimental rare variants. When used by itself, all detrimental rare variants will be assigned a fixed effect size as specified. With the ''-A'' option, they together model "variable effects" with ''-a'' being the minimum effect size and ''-A'' being the maximum effect size for detrimental rare variants. In variable effects model the maximum effect will be assigned to the variant having smallest MAF, and the minimum effect to the one having largest MAF. Values in between are interpolated based on these specified max. and min. values. Similarly ''-b'' and ''-B'' are for fixed and variable effects of protective variants, which decrease the mean QT as oppose to increasing it. ''-c'' and ''-d'' are effects for common detrimental and protective variants respectively. No variable effects model for common variants is available for this model.

<box 80% round blue|**__Note__**>
 We recommend using values ranging from 0.1 to 4.0 for rare variants. For example In the context of a trait such as human height, \(0.25\sigma\) would correspond to a change in mean height of 0.5 inch.
</box>
==== Examples ====
	*  [[http://bioinformatics.org/spower/analytic-tutorial|Analytic power analysis]] for quantitative traits
	*  [[http://bioinformatics.org/spower/empirical-tutorial|Empirical power analysis]] for quantitative traits