graph.flm | R Documentation |

Non-parametric graphical tests of significance in functional general linear model (GLM)

```
graph.flm(
nsim,
formula.full,
formula.reduced,
typeone = c("fwer", "fdr"),
curve_sets,
factors = NULL,
contrasts = FALSE,
savefuns = FALSE,
lm.args = NULL,
GET.args = NULL,
mc.cores = 1L,
mc.args = NULL,
cl = NULL,
fast = TRUE
)
```

`nsim` |
The number of random permutations. |

`formula.full` |
The formula specifying the general linear model,
see |

`formula.reduced` |
The formula of the reduced model with nuisance factors only. This model should be nested within the full model. |

`typeone` |
Character string indicating which type I error rate to control,
either the familywise error rate ('fwer') or false discovery rate ('fdr').
Further arguments to the FWER or FDR envelope can be passed in argument |

`curve_sets` |
A named list of sets of curves giving the dependent variable (Y), and
possibly additionally factors whose values vary across the argument values of the functions.
The dimensions of the elements should match with each other.
Note that factors that are fixed across the functions can be given in the argument |

`factors` |
A data frame of factors. An alternative way to specify factors when they are constant for all argument values of the functions. The number of rows of the data frame should be equal to the number of curves. Each column should specify the values of a factor. |

`contrasts` |
Logical or NULL. FALSE, TRUE and NULL specify the three test functions as described in description part of this help file. |

`savefuns` |
Logical. If TRUE, then the functions from permutations are saved to the attribute simfuns. |

`lm.args` |
A named list of additional arguments to be passed to |

`GET.args` |
A named list of additional arguments to be passed to |

`mc.cores` |
The number of cores to use, i.e. at most how many child processes will be run simultaneously.
Must be at least one, and parallelization requires at least two cores. On a Windows computer mc.cores must be 1
(no parallelization). For details, see |

`mc.args` |
A named list of additional arguments to be passed to |

`cl` |
Allows parallelization through the use of |

`fast` |
Logical. See details. |

The function `graph.flm`

performs the graphical functional GLM of Mrkvička et al. (2021),
described also in Section 3.6 of Myllymäki and Mrkvička (2020) (type `vignette("GET")`

in R).
This is a nonparametric graphical test of significance of a covariate in functional GLM.
The test is able to find not only if the factor of interest is significant, but also which
functional domain is responsible for the potential rejection.
In the case of functional multi-way main effect ANOVA or functional main effect ANCOVA models,
the test is able to find which groups differ (and where they differ).
In the case of functional factorial ANOVA or functional factorial ANCOVA models,
the test is able to find which combination of levels (which interactions) differ (and where they differ).
The described tests are global envelope tests applied in the context of GLMs.
The Freedman-Lane algorithm (Freedman and Lane, 1983) is applied to permute the functions
(to obtain the simulations under the null hypothesis of "no effects");
consequently, the test approximately achieves the desired significance level.

The specification of the full and reduced formulas is important. The reduced model should be
nested within the full model. The full model should include in addition to the reduced
model the interesting factors whose effects are under investigation.
The implementation to find the coefficients of the interesting factors is based on
`dummy.coef`

and the restrictions there apply.

The regression coefficients serve as test functions in the graphical functional GLM.
For a continuous interesting factor, the test function is its regression coefficient across the
functional domain. For a discrete factor, there are three possibilities that are controlled by
the arguments `contrasts`

. If `contrasts = FALSE`

, then the test statistic is
the function/long vector where the coefficients related to all levels of the factor are joined
together. If `contrasts = TRUE`

, then the differences between the same coefficients are
considered instead. Given the coefficients in a specific order that is obtained through the use
of `lm`

and `dummy.coef`

, the differences are taken for couples i and j
where i < j and reducing j from i (e.g. for three groups 1,2,3, the constrasts are 1-2, 1-3, 2-3).
If `contrasts = NULL`

the coefficients given by `lm`

are used directly.

There are different versions of the implementation depending on the application.
Given that the argument `fast`

is TRUE, then

If all the covariates are continuous or

`contrasts = NULL`

and`lm.args = NULL`

the regression coefficients are computed using the normal equation approach instead of using`lm`

.Otherwise, if all the covariates are constant across the functions, i.e. they can be provided in the argument

`factors`

, then a linear model is fitted separately by least-squares estimation to the data at each argument value of the functions fitting a multiple linear model by`lm`

. The possible extra arguments passed in`lm.args`

to`lm`

must be of the form that`lm`

accepts for fitting a multiple linear model. In the basic case, no extra arguments are needed.Otherwise, if some of the covariates vary across the space and there are user specified extra arguments given in

`lm.args`

, then the implementation fits a linear model at each argument value of the functions using`lm`

, which can be rather slow. The arguments`lm.args`

are passed to`lm`

for fitting each linear model.

By setting `fast = FALSE`

, it is possible to use the slow version (third option)
for any case. Usually this is not desired.

A `global_envelope`

or `combined_global_envelope`

object,
which can be printed and plotted directly.

Mrkvička, T., Roskovec, T. and Rost, M. (2021) A nonparametric graphical tests of significance in functional GLM. Methodology and Computing in Applied Probability 23, 593-612. doi: 10.1007/s11009-019-09756-y

Myllymäki, M and Mrkvička, T. (2020). GET: Global envelopes in R. arXiv:1911.06583 [stat.ME]

Freedman, D., & Lane, D. (1983) A nonstochastic interpretation of reported significance levels. Journal of Business & Economic Statistics, 1(4), 292-298. doi:10.2307/1391660

```
data("rimov")
res <- graph.flm(nsim=19, # Increase the number of simulations for serious analysis!
formula.full = Y~Year,
formula.reduced = Y~1,
curve_sets = list(Y=rimov), factors = data.frame(Year = 1979:2014))
plot(res)
# Test if there is a change in the slope in 1994,
# i.e. the full model is T = a + b*year + c*year:group,
res <- graph.flm(nsim = 19, # Increase the number of simulations for serious analysis!
formula.full = Y ~ Year + Year:Group,
formula.reduced = Y ~ Year,
curve_sets = list(Y=rimov),
factors = data.frame(Year = 1979:2014,
Group = factor(c(rep(1,times=24), rep(2,times=12)),
levels=1:2)),
contrasts = FALSE)
plot(res)
# An example of testing the joint effect of a discrete and a continuous variable
nsim <- 999
data("GDPtax")
factors.df <- data.frame(Group = GDPtax$Group, Tax = GDPtax$Profittax)
res.tax_within_group <- graph.flm(nsim = nsim,
formula.full = Y~Group+Tax+Group:Tax,
formula.reduced = Y~Group+Tax,
curve_sets = list(Y=GDPtax$GDP),
factors = factors.df)
plot(res.tax_within_group)
# Image data examples
data("abide_9002_23")
iset <- abide_9002_23$curve_set
# Testing the discrete factor 'group' with contrasts
# (Use contrasts = FALSE for 'means'; and for continuous factors)
res <- graph.flm(nsim = 19, # Increase nsim for serious analysis!
formula.full = Y ~ Group + Sex + Age,
formula.reduced = Y ~ Sex + Age,
curve_sets = list(Y = iset),
factors = abide_9002_23[['factors']],
contrasts = TRUE,
GET.args = list(type = "area"))
plot(res)
# Examples of modifying 2d plots
plot(res, sign.col="white") + ggplot2::scale_fill_viridis_c(option="magma")
plot(res, sign.col="white") + ggplot2::scale_fill_viridis_c(option="magma") +
ggplot2::scale_radius(range = 2*c(1, 6))
plot(res, what=c("obs", "lo", "hi", "lo.sign", "hi.sign"))
plot(res, what=c("obs", "lo", "hi", "lo.sign", "hi.sign"), sign.type="col")
```

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