This vignette documents the implementation of NBR 0.1.3 for linear models.

We will analyze the `frontal3D`

dataset, which contains a 3D volume of 48 matrices, each matrix representing the functional connectivity between 28 nodes (in the frontal lobe). Phenotypic information (`frontal_phen`

) includes diagnostic GROUP (patient or control), sex, and age. We will test for a GROUP effect.

```
library(NBR)
cmx <- NBR:::frontal3D # Load 3D array
brain_labs <- NBR:::frontal_roi # Load node labels
phen <- NBR:::frontal_phen # Load phenotypic info
dim(cmx) # Show 3D array dimensions
#> [1] 28 28 48
```

We can plot the sample average matrix, with `lattice::levelplot`

.

```
library(lattice)
avg_mx <- apply(cmx, 1:2, mean)
# Set max-absolute value in order to set a color range centered in zero.
flim <- max(abs(avg_mx)[is.finite(avg_mx)])
levelplot(avg_mx, main = "Average", ylab = "ROI", xlab = "ROI",
at = seq(-flim, flim, length.out = 100))
```

As we can observe, this is a symmetric matrix with the pairwise connections of the 28 regions of interest (ROI) `brain_labs`

. The next step is to check the phenotypic information (stored in `phen`

) to perform statistic inferences edgewise. Before applying the NBR-LM, we check that the number of matrices (3rd dimension in the dataset) matches the number of observations in the `phen`

data.frame.

```
head(phen)
#> Group Sex Age
#> 1 Control F 8.52
#> 2 Control M 16.16
#> 3 Patient M 17.75
#> 4 Control M 12.27
#> 5 Control F 12.07
#> 6 Patient F 8.71
nrow(phen)
#> [1] 48
identical(nrow(phen), dim(cmx)[3])
#> [1] TRUE
```

The data.frame contains the individual information for diagnostic group, sex, and chronological age. So, we are all set to perform an NBR-LM. We are going to test the effect of diagnostic group with a minimal number of permutations to check that we have no errors.

```
set.seed(18900217) # Because R. Fisher is my hero
before <- Sys.time()
nbr_group <- nbr_lm_aov(net = cmx, nnodes = 28, idata = phen,
mod = "~ Group", thrP = 0.01, nperm = 10)
#> Computing observed stats.
#> Computing permutated stats.
#> Permutation progress: ....
after <- Sys.time()
show(after-before)
#> Time difference of 6.696129 secs
```

Although ten permutations is quite low to obtain a proper null distribution, we can see that they take several seconds to be performed. So we suggest to parallelizing to multiple CPU cores with `cores`

argument.

```
set.seed(18900217)
library(parallel)
before <- Sys.time()
nbr_group <- nbr_lm_aov(net = cmx, nnodes = 28, idata = phen,
mod = "~ Group", thrP = 0.01, nperm = 100, cores = detectCores())
after <- Sys.time()
length(nbr_group)
```

NBR functions return a nested list of at least two lists. The first list encompasses all the individual significant edges, their corresponding component and statistical inference (p < 0.01, in this example). In this case all the significant edges belong to a single component.

```
# Plot significant component
edge_mat <- array(0, dim(avg_mx))
edge_mat[nbr_group$components$Group[,2:3]] <- 1
levelplot(edge_mat, col.regions = rev(heat.colors(100)),
main = "Component", ylab = "ROI", xlab = "ROI")
```

As we can observe, significant edges are displayed in the upper triangle of the matrix, and the second list (`fwe`

) contains, for each term of the equation, the probability of the observed values to occur by chance, based on the null distribution.