Interval Testing Procedure for testing Functional-on-Scalar Linear Models with B-spline basis

The function is used to fit and test functional linear models. It can be used to carry out regression, and analysis of variance. It implements the Interval Testing Procedure for testing the significance of the effects of scalar covariates on a functional population evaluated on a uniform grid. 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.

Usage

ITPlmbspline(
  formula,
  order = 2,
  nknots = dim(stats::model.response(stats::model.frame(formula)))[2],
  B = 10000,
  method = "residuals"
)

Arguments

formula

An object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

order

Order of the B-spline basis expansion. The default is order=2.

nknots

Number of knots of the B-spline basis expansion. The default is nknots=dim(data1)[2].

B

The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. The defualt is B=1000.

method

Permutation method used to calculate the p-value of permutation tests. Choose "residuals" for the permutations of residuals under the reduced model, according to the Freedman and Lane scheme, and "responses" for the permutation of the responses, according to the Manly scheme.

Value

ITPlmbspline returns an object of class "ITPlm". The function summary is used to obtain and print a summary of the results. An object of class "ITPlm" 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 "B-spline".

coeff

Matrix of dimensions c(n,p) of the p coefficients of the B-spline basis expansion. Rows are associated to units and columns to the basis index.

coeff_regr

Matrix of dimensions c(L+1,p) of the p coefficients of the B-spline basis expansion of the intercept (first row) and the L effects of the covariates specified in formula. Columns are associated to the basis index.

pval_F

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

pval_matrix_F

Matrix of dimensions c(p,p) of the p-values of the multivariate F-tests. The element (i,j) of matrix pval_matrix contains the p-value of the joint NPC test of the components (j,j+1,...,j+(p-i)).

adjusted_pval_F

Adjusted p-values of the functional F-test for each basis coefficient.

pval_t

Unadjusted p-values of the functional t-tests for each partial regression coefficient including the intercept (rows) and each basis coefficient (columns).

pval_matrix_t

Array of dimensions c(L+1,p,p) of the p-values of the multivariate t-tests. The element (l,i,j) of array pval_matrix contains the p-value of the joint NPC test on covariate l of the components (j,j+1,...,j+(p-i)).

adjusted_pval_t

adjusted p-values of the functional t-tests for each partial regression coefficient including the intercept (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_t

Heatmap matrix of p-values of functional t-tests (used only for plots).

References

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.

Examples

# Defining the covariates
temperature <- rbind(NASAtemp$milan,NASAtemp$paris)
groups <- c(rep(0,22),rep(1,22))

# Performing the ITP
#ITP_result <- ITPlmbspline(temperature ~ groups,B=100,nknots=20)
# Summary of the ITP results
#summary(ITP_result)

# Plot of the ITP results
#plot(ITP_result,main='NASA data', plot_adjpval = TRUE,xlab='Day',xrange=c(1,365))

#plot(ITP_result,main='NASA data', plot_adjpval = TRUE,xlab='Day',xrange=c(1,365))