# Compute univariate MDMR effect sizes

### Description

`delta`

computes permutation-based effect sizes on individual items
comprising the distance matrix outcome used in multivariate distance matrix
regression. It returns the omnibus estimates of delta (i.e. effect size of
the entire design matrix on each outcome) as well as estimates of each
pair-wise effect size (i.e. the effect of each predictor on each outcome
variable, conditional on the rest of the predictors).

### Usage

1 2 3 |

### Arguments

`X` |
A |

`Y` |
Outcome data: |

`dtype` |
Measure of dissimilarity that will be used by |

`niter` |
Number of times to permute each outcome item in the procedure
to compute delta. The final result is the average of all |

`x.inds` |
Vector indicating which columns of X should have their
conditional effect sizes computed. Default value of |

`y.inds` |
Vector indicating which columns of Y effect sizes should be
computed on. Default value of |

`G` |
Gower's centered similarity matrix computed from |

`G.list` |
List of length |

`ncores` |
Integer; if |

`seed` |
Integer; sets seed for the permutations of each variable comprising Y so that results can be replicated. |

`plot.res` |
Logical; Indicates whether or not a heat-map of the results should be plotted. |

`grayscale` |
Logical; Indicates whether or not the heat-map should be plotted in grayscale. |

`cex` |
Multiplier for cex.axis, cex.lab, cex.main, and cex that are passed to the plotted result. |

`y.las` |
Orientation of labels for the outcome items. Defaults to vertical (2). Value of 1 prints horizontal labels, and is only recommended if the multivariate outcome is comprised of few variables. |

### Details

See McArtor et al. (second revision under review) for a detailed description of how delta is computed. Note that it is a relative measure of effect, quantifying which effects are strong (high values of delta) and weak (low values of delta) within a single analysis, but estimates of delta cannot be directly compared across different datasets.

There are two options for using this function. The first option is to
specify the predictor matrix `X`

, the outcome matrix `Y`

, the
distance type `dtype`

(supported by "dist" in R), and number of
iterations `niter`

. This option conducts the permutation of each Y-item
`niter`

times (to average out random association in each permutation)
and reports the median estimates of delta over the `niter`

reps.

The second option is to specify `X`

, `G`

, and `G.list`

, a
list of G matrices where the permutation has already been done for each item
comprising Y. The names of the elements in `G.list`

should correspond
to the names of the variables that were permuted. This option is implemented
so that delta can be computed when MDMR is being used in conjunction with
distance metrics not supported by `dist`

.

### Value

A data frame whose rows correspond to the omnibus effects and the
effect of each individual predictor (conditional on the rest), and whose
columns correspond to each outcome variable whose effect sizes are being
quantified. If `plot.res = TRUE`

, a heat-map is plotted of this data
frame to easily identify the strongest effects. Note that the heatmap is
partitioned into the omnibus effect (first row) and pair-wise effects
(remaining rows), because otherwise the omnibus effect would dominate the
heatmap.

### Author(s)

Daniel B. McArtor (dmcartor@nd.edu) [aut, cre]

### References

McArtor, D. B., Lubke, G. H., & Bergeman, C. S. (second revision under review). Extending multivariate distance matrix regression with an effect size measure and the distribution of the test statistic.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
data(mdmrdata)
# --- Method 1 --- #
delta(X.mdmr, Y = Y.mdmr, dtype = 'euclidean', niter = 1, seed = 12345)
# --- Method 2 --- #
D <- dist(Y.mdmr, method = 'euclidean')
G <- gower(D)
q <- ncol(Y.mdmr)
G.list <- vector(mode = 'list', length = q)
names(G.list) <- names(Y.mdmr)
for(i in 1:q){
Y.shuf <- Y.mdmr
Y.shuf[,i] <- sample(Y.shuf[,i])
G.list[[i]] <- gower(dist(Y.shuf, method = 'euclidean'))
}
delta(X.mdmr, G = G, G.list = G.list)
``` |