Description Usage Arguments Value References See Also Examples

The function implements the Interval Testing Procedure for testing for significant differences between several functional population evaluated on a uniform grid, in a functional analysis of variance setting. Data are represented by means of the B-spline basis and the significance of each basis coefficient is tested with an interval-wise control of the Family Wise Error Rate. The default parameters of the basis expansion lead to the piece-wise interpolating function.

1 2 3 | ```
ITPaovbspline(formula, order = 2,
nknots = dim(model.response(model.frame(formula)))[2], B = 1000,
method = "residuals")
``` |

`formula` |
An object of class " |

`order` |
Order of the B-spline basis expansion. The default is |

`nknots` |
Number of knots of the B-spline basis expansion. The default is |

`B` |
The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is |

`method` |
Permutation method used to calculate the p-value of permutation tests. Choose " |

`ITPaovbspline`

returns an object of `class`

"`ITPaov`

". The function `summary`

is used to obtain and print a summary of the results.
An object of class "`ITPaov`

" is a list containing at least the following components:

`call` |
The matched call. |

`design.matrix` |
The design matrix of the functional-on-scalar linear model. |

`basis` |
String vector indicating the basis used for the first phase of the algorithm. In this case equal to |

`coeff` |
Matrix of dimensions |

`coeff.regr` |
Matrix of dimensions |

`pval.F` |
Unadjusted p-values of the functional F-test for each basis coefficient. |

`pval.matrix.F` |
Matrix of dimensions |

`adjusted.pval.F` |
Adjusted p-values of the functional F-test for each basis coefficient. |

`pval.factors` |
Unadjusted p-values of the functional F-tests on each factor of the analysis of variance, separately (rows) and each basis coefficient (columns). |

`pval.matrix.factors` |
Array of dimensions |

`adjusted.pval.factors` |
adjusted p-values of the functional F-tests on each factor of the analysis of variance (rows) and each basis coefficient (columns). |

`data.eval` |
Evaluation on a fine uniform grid of the functional data obtained through the basis expansion. |

`coeff.regr.eval` |
Evaluation on a fine uniform grid of the functional regression coefficients. |

`fitted.eval` |
Evaluation on a fine uniform grid of the fitted values of the functional regression. |

`residuals.eval` |
Evaluation on a fine uniform grid of the residuals of the functional regression. |

`R2.eval` |
Evaluation on a fine uniform grid of the functional R-squared of the regression. |

`heatmap.matrix.F` |
Heatmap matrix of p-values of functional F-test (used only for plots). |

`heatmap.matrix.factors` |
Heatmap matrix of p-values of functional F-tests on each factor of the analysis of variance (used only for plots). |

A. Pini and S. Vantini (2017). The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals. Biometrics 73(3): 835–845.

A. Pini and S. Vantini (2017). The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals. Biometrics 73(3): 835–845.

Pini, A., Vantini, S., Colosimo, B. M., & Grasso, M. (2018). Domain‐selective functional analysis of variance for supervised statistical profile monitoring of signal data. *Journal of the Royal Statistical Society: Series C (Applied Statistics)* 67(1), 55-81.

Abramowicz, K., Hager, C. K., Pini, A., Schelin, L., Sjostedt de Luna, S., & Vantini, S. (2018).
Nonparametric inference for functional‐on‐scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament. *Scandinavian Journal of Statistics* 45(4), 1036-1061.

D. Freedman and D. Lane (1983). A Nonstochastic Interpretation of Reported Significance Levels. *Journal of Business & Economic Statistics* 1.4, 292-298.

B. F. J. Manly (2006). Randomization, *Bootstrap and Monte Carlo Methods in Biology*. Vol. 70. CRC Press.

See `summary.ITPaov`

for summaries and `plot.ITPaov`

for plotting the results.
See `IWTaov`

for a functional analysis of variance test that is not based on an a-priori selected basis expansion.
See also `ITPlmbspline`

to fit and test a functional-on-scalar linear model applying the ITP, and `ITP1bspline`

, `ITP2bspline`

, `ITP2fourier`

, `ITP2pafourier`

for one-population and two-population tests.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
# Importing the NASA temperatures data set
data(NASAtemp)
temperature <- rbind(NASAtemp$milan,NASAtemp$paris)
groups <- c(rep(0,22),rep(1,22))
# Performing the ITP
ITP.result <- ITPaovbspline(temperature ~ groups,B=1000,nknots=20,order=3)
# Summary of the ITP results
summary(ITP.result)
# Plot of the ITP results
layout(1)
plot(ITP.result)
# All graphics on the same device
layout(matrix(1:4,nrow=2,byrow=FALSE))
plot(ITP.result,main='NASA data', plot.adjpval = TRUE,xlab='Day',xrange=c(1,365))
``` |

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