===== Population Attributable Risk Model =====
Under this model, case control status are simulated based on difference in population attributable risk between case control groups. Power and sample size calculations can be performed using a number of theoretical and empirical methods.

==== Command Interface ====
<code bash>
spower PAR -h
</code>\\
<hidden initialState="visible" -noprint>
  usage: spower PAR [-h] [-a GAMMA] [-b GAMMA] [-c GAMMA] [-d GAMMA]
                    [--PAR_variable] [-f P] [--def_rare P]
                    [--def_neutral VALUE VALUE] [--def_protective VALUE VALUE]
                    [-P P] [-Q P] [--sample_size N] [--p1 P]
                    [--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 GAMMA, --PAR_rare_detrimental GAMMA
                          Population attributable risk for detrimental rare
                          variants (default set to 0.0)
    -b GAMMA, --PAR_rare_protective GAMMA
                          Population attributable risk for protective rare
                          variants (default set to 0.0)
    -c GAMMA, --PAR_common_detrimental GAMMA
                          Population attributable risk for detrimental common
                          variants (default set to 0.0)
    -d GAMMA, --PAR_common_protective GAMMA
                          Population attributable risk for protective common
                          variants (default set to 0.0)
    --PAR_variable        use variable population attributable risks: the
                          smaller the MAF the larger the PAR
    -f P, --baseline_effect P
                          penetrance of wildtype genotypes (default set to 0.01)
    --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 avaliable)
  
  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
    --p1 P                proportion of affected individuals (default set to
                          0.5), or individuals with high extreme QT values
                          sampled from infinite population (default set to None,
                          meaning to sample from finite population speficied by
                          --sample_size option).
  
  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 ===
These options specify the **total PAR** for variants in the region of interest. ''-a'' is PAR for detrimental rare variants. When used without ''%%--%%PAR_variable'' option (see below) all detrimental rare variants will be assigned a fixed PAR equals \(\frac{\gamma}{M}\) where \(\gamma\) is the specified total PAR and \(M\) is total number of detrimental rare variants. Similarly ''-b'' specifies PAR for protective variants. Unlike for detrimental variant, protective variants will decrease PAR for samples having a mutation. ''-c'' and ''-d'' specifies PAR for common variants.

<box 80% round blue|**__Note__**>
 Reasonable assumption of PAR for rare variants associated with complex diseases is 0.5% to 5%.
</box>
=== --PAR_variable ===
With this switch on, the total PAR will be distributed to each variant based on the inverse of their MAF, i.e., the rarer the variant the higher the PAR value.

=== --moi ===
Mode of inheritance. Most aggregated analysis method implicitly assumes an "additive" effect of rare variants, thus simulating and analyzing data under additive model represents a best case scenario for many methods, which is also the default value for both the simulation and the analysis. This option only controls on how the data is simulated. In some ''%%--%%method'' option there are sub-options to specify the mode of inheritance under which the data will be analyzed.

==== Simulation via Repeated Sampling ====
Motivation of the ''%%--%%resampling'' option was previously discussed for the [[http://bioinformatics.org/spower/mlogit|LOGIT model]]. In PAR model it is implemented as follows:

	*  A sample genotype be generated either from the population MAF or drawn directly from given haplotype pools
	*  A penetrance value \(f\) for the genotype be calculated from input parameters
		*  PAR -> Odds ratio -> penetrance
	*  Sample be labeled as a //case// with probability \(f\); control otherwise

==== Examples ====
	*  [[http://bioinformatics.org/spower/analytic-tutorial|Analytic power analysis]] for case control studies
	*  [[http://bioinformatics.org/spower/empirical-tutorial|Empirical power analysis]] for case control studies