fgam: Functional Generalized Additive Models

Description

Implements functional generalized additive models for functional and scalar covariates and scalar responses. Additionally implements functional linear models. This function is a wrapper for mgcv's gam and its siblings to fit models of the general form $g (E (Y_{i})) = β_{0} + \int_{T_{1}} F (X_{i 1}, t) d t + \int_{T_{2}} β (t) X_{i 2} d t + f (z_{i 1}) + f (z_{i 2}, z_{i 3}) + \dots$ with a scalar (but not necessarily continuous) response Y, and link function g

Usage

fgam(formula, fitter = NA, tensortype = c("te", "t2"), ...)

Value

a fitted fgam-object, which is a gam-object with some additional information in a fgam-entry. If fitter is "gamm" or "gamm4", only the $gam part of the returned list is modified in this way.

Arguments

formula: a formula with special terms as for gam, with additional special terms af(), lf(), re().
fitter: the name of the function used to estimate the model. Defaults to gam if the matrix of functional responses has less than 2e5 data points and to bam if not. "gamm" (see gamm) and "gamm4" (see gamm4) are valid options as well.
tensortype: defaults to te, other valid option is t2
...: additional arguments that are valid for gam or bam; for example, specify a gamma > 1 to increase amount of smoothing when using GCV to choose smoothing parameters or method="REML" to change to REML for estimation of smoothing parameters (default is GCV).

Warning

Binomial responses should be specified as a numeric vector rather than as a matrix or a factor.

Author

Mathew W. McLean mathew.w.mclean@gmail.com and Fabian Scheipl

References

McLean, M. W., Hooker, G., Staicu, A.-M., Scheipl, F., and Ruppert, D. (2014). Functional generalized additive models. Journal of Computational and Graphical Statistics, 23 (1), pp. 249-269. Available at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3982924/.

Examples

Run this code

data(DTI)
## only consider first visit and cases (no PASAT scores for controls)
y <- DTI$pasat[DTI$visit==1 & DTI$case==1]
X <- DTI$cca[DTI$visit==1 & DTI$case==1, ]
X_2 <- DTI$rcst[DTI$visit==1 & DTI$case==1, ]

## remove samples containing missing data
ind <- rowSums(is.na(X)) > 0
ind2 <- rowSums(is.na(X_2)) > 0

y <- y[!(ind | ind2)]
X <- X[!(ind | ind2), ]
X_2 <- X_2[!(ind | ind2), ]

N <- length(y)

## fit fgam using FA measurements along corpus callosum
## as functional predictor with PASAT as response
## using 8 cubic B-splines for marginal bases with third
## order marginal difference penalties
## specifying gamma > 1 enforces more smoothing when using
## GCV to choose smoothing parameters
#fit <- fgam(y ~ af(X, k = c(8, 8), m = list(c(2, 3), c(2, 3))), gamma = 1.2)


## fgam term for the cca measurements plus an flm term for the rcst measurements
## leave out 10 samples for prediction
test <- sample(N, 10)
#fit <- fgam(y ~ af(X, k = c(7, 7), m = list(c(2, 2), c(2, 2))) +
 #      lf(X_2, k=7, m = c(2, 2)), subset=(1:N)[-test])
#plot(fit)
## predict the ten left outs samples
#pred <- predict(fit, newdata = list(X=X[test, ], X_2 = X_2[test, ]), type='response',
 #               PredOutOfRange = TRUE)
#sqrt(mean((y[test] - pred)^2))
## Try to predict the binary response disease status (case or control)
##   using the quantile transformed measurements from the rcst tract
##   with a smooth component for a scalar covariate that is pure noise
y <- DTI$case[DTI$visit==1]
X <- DTI$cca[DTI$visit==1, ]
X_2 <- DTI$rcst[DTI$visit==1, ]

ind <- rowSums(is.na(X)) > 0
ind2 <- rowSums(is.na(X_2)) > 0

y <- y[!(ind | ind2)]
X <- X[!(ind | ind2), ]
X_2 <- X_2[!(ind | ind2), ]
z1 <- rnorm(length(y))

## select=TRUE allows terms to be zeroed out of model completely
#fit <- fgam(y ~ s(z1, k = 10) + af(X_2, k=c(7,7), m = list(c(2, 1), c(2, 1)),
 #           Qtransform=TRUE), family=binomial(), select=TRUE)
#plot(fit)

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