# Z-score Maximization

### Description

Function to maximize z-scores over subsets of traits or subtypes, with possible restrictions and weights. Should not be called directly. See details.

### Usage

1 2 |

### Arguments

`k` |
Single integer (currently at most 30). The total number of traits or studies or subtypes being analyzed. No default |

`gene.vars` |
Vector of integers or string labels for the SNPs being analyzed. No default. |

`side` |
Either 1 or 2. For two-tailed tests (where absolute values of Z-scores are maximzed), side should be 2. For one-tailed tests side should be 1 (positive tail is assumed). Default is 2, ignored when search is 2. |

`p.vals` |
pvalue matrix. |

`th` |
A vector of thresholds for each SNP, beyond which to stop maximization for that SNP. Default is a threshold of -1 for each SNP , implying no threshold. This argument is for internal use. |

`sub.def` |
A function to restrict subsets, e.g., order restrictions in subtype analysis. Should accept a subset (a logical vector of size k) as its first argument and should return TRUE if the subset satisfies restrictions and FALSE otherwise. Default is NULL implying all (2^k - 1) subsets are considered in the maximum. |

`sub.args` |
Other arguments to be passed to |

`wt.def` |
A function that gives weight of one subset with respect to another. Should accept two subsets as first two argumets and return a single positive weight. Default NULL. Currently this option is not implemented and the argument is ignored. |

`wt.args` |
Other arguments to be passed to |

### Details

This function loops through all possible (2^k - 1) subsets of (k) studies (or traits or subtypes), skips subsets that are not valid (e.g.
that do not satisfy order restrictions), and maximizes the z-scores or re-weighted z-scores if weights are specified.
The function is vectorized to handle blocks of SNPs at a time. Currently weight options are ignored.
This is a helper function that is called internally by `h.traits`

and `h.types`

and should not be called directly. The arguments of this function that have defaults, can be customized using
the argument `zmax.args`

in `h.traits`

and `h.types`

.

### Value

A list with two components. A vector of optimized z-scores (opt.z) and a logical matrix (opt.s) of dimension length(snp.vars) by `k`

.
Each row of (opt.s) has indicators of each trait/subtype being included in the best (optimal) subset.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | ```
set.seed(123)
# Define the function to calculate the z-scores
meta.def <- function(logicalVec, SNP.list, arg.beta, arg.sigma) {
# Get the snps and subset to use
beta <- as.matrix(arg.beta[SNP.list, logicalVec])
se <- as.matrix(arg.sigma[SNP.list, logicalVec])
test <- (beta/se)^2
ret <- apply(test, 1, max)
list(z=ret)
}
# Define the function to determine which subsets to consider
sub.def <- function(logicalVec) {
# Only allow the cummulative subsets:
# TRUE FALSE FALSE FALSE ...
# TRUE TRUE FALSE FALSE ...
# TRUE TRUE TRUE FALSE ...
# etc
sum <- sum(logicalVec)
ret <- all(logicalVec[1:sum])
ret
}
# Assume there are 10 subtypes and 3 SNPs
k <- 10
snp.vars <- 1:3
# Generate some data
nsnp <- length(snp.vars)
beta <- matrix(-0.5 + runif(k*nsnp), nrow=nsnp)
sigma <- matrix(runif(k*nsnp)^2, nrow=nsnp)
meta.args <- list(arg.beta=beta, arg.sigma=sigma)
z.max(k, snp.vars, 2, meta.def, meta.args, sub.def=sub.def)
``` |

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