The npROCRegression package
This document explains the usage
of the npROCRegression package. The package allows the user
to apply in practice the nonparametric induced and direct ROC regression
approaches presented in Rodríguez-Álvarez,
Roca-Pardiñas, and Cadarso-Suárez (2011b) and Rodríguez-Álvarez, Roca-Pardiñas, and Cadarso-Suárez
(2011a) respectively.
R functions to estimate induced nonparametric ROC regression models
The main R function is INPROCreg() which
estimates the covariate-specific ROC curve in the presence of a
one-dimensional continuous covariate based on the induced nonparametric
ROC regression approach presented in Rodríguez-Álvarez, Roca-Pardiñas, and Cadarso-Suárez
(2011b), and creates an object of class INPROCreg. A
brief summary of this class can be obtained by using the functions
print.INPROCreg() and summary.INPROCreg().
Finally, the function plot.INPROCreg() provides
automatically several plots of interest. A brief description of these
functions is shown in the following Table.
| Function | Description |
|---|---|
INPROCreg |
Fits an induced nonparametric ROC regression model for a continuous covariate. |
controlINPROCreg |
Function used to set several parameters controlling the ROC regression fitting process. |
print.INPROCreg |
Default print method for objects fitted with
INPROCreg(). |
summary.INPROCreg |
Produces a summary of an INPROCreg object. |
plot.INPROCreg |
Plots (a) the estimated regression and variance functions in both the healthy and diseased populations, (b) the covariate-specific ROC curve and AUC, (c) the covariate-adjusted ROC curve (AROC); and, optionally, (d) the Youden Index (YI) or the value for which the TPF and the TNF coincides (EQ); and/or (e) the optimal thresholds based on these criteria (TH)). |
INPROCreg() function
The function INPROCreg() estimates the
covariate-specific ROC curve in the presence of a one-dimensional
continuous covariate based on the induced nonparametric ROC regression
approach presented in Rodríguez-Álvarez,
Roca-Pardiñas, and Cadarso-Suárez (2011b). As a result, this
function returns an object of class INPROCreg. The
following Table shows a description of the arguments of this
function.
| Argument | Description |
|---|---|
marker |
A character string with the name of the diagnostic test variable. |
covariate |
A character string with the name of the continuous covariate. |
group |
A character string with the name of the variable that distinguishes healthy from diseased individuals. |
tag.healthy |
The value codifying the healthy individuals in the variable
group. |
data |
Data frame representing the data and containing all needed variables. |
ci.fit |
A logical value. If TRUE, confidence intervals are computed. |
test |
A logical value. If TRUE, the bootstrap-based test for detecting covariate effect is performed. |
accuracy |
A character vector indicating if the Youden index (“YI”), the value for which the TPF and the TNF coincides (“EQ”), and/or optimal threshold (“TH”) based on these two criteria should be computed. |
accuracy.cal |
A character string indicating if the accuracy measures (argument
accuracy) should be calculated based on the
covariate-specific ROC curve (“ROC”) or on the covariate-adjusted ROC
curve (“AROC”). |
newdata |
A data frame containing the values of the covariate at which predictions are required |
control |
Output of the controlINROCreg() function. |
weights |
An optional vector of “prior weights” to be used in the fitting process. |
Usage is as follows:
INPROCreg (marker, covariate, group, tag.healthy, data, ci.fit=FALSE, test=FALSE, accuracy = NULL, accuracy.cal = c("ROC","AROC"), newdata = NULL, control = controlINPROCreg(), weights=NULL)Through marker and covariate arguments,
users indicate the diagnostic test variable and the continuous covariate
of interest, respectively. In group and
tag.healthy arguments, we have to indicate respectively the
name of the variable that distinguishes healthy from diseased
individuals, and the value codifying healthy individuals in that
variable. The data argument is a data frame representing
the data and containing all needed variables. Bootstrap confidence
intervals for the regression and variance functions, as well as for
several accuracy measures, are obtained by setting the argument
ci.fit to TRUE. Argument test
should be set to TRUE in order to evaluate the effect of
the continuous covariate on the ROC curve by means of the test presented
in Rodríguez-Álvarez et al. (2016). By
default, the INPROCreg() function returns the estimated
regression and variance functions in both healthy and diseased
populations. As far as accuracy measures is concerned, the function
provides the estimated covariate-specific ROC curve, the associated
covariate-specific AUCs (with the integral being approximated by
numerical integration methods), and the covariate-adjusted ROC curve
(AROC) (Janes and Pepe 2009). In addition,
it is also possible to obtain the Youden index (“YI”), the value for
which the TPF and the TNF coincides (“EQ”); and/or the optimal
thresholds (“TH”) based on these two criteria (argument
accuracy). Both the YI and the EQ values (and thus the
optimal threshold) can be calculated based on the covariate-specific ROC
curve or the AROC curve (argument accuracy.cal). It should
be noted that, when a diagnostic test’s discriminatory capacity is not
affected by a covariate, this does not necessarily mean that the
threshold value for which optimal operational characteristics are
attained will not vary with the covariate values. In such cases, the
AROC curve should be used to choose the optimal TPF and TNF pairing
Rodríguez-Álvarez, Roca-Pardiñas, and
Cadarso-Suárez (2011b). An optional data frame containing the
values of the covariate at which predictions are required can be
specified in argument newdata. If this dataset is not
specified, an adequate set of points from the data used in the fit is
selected. A finer control of the fitting process can be achieved by the
argument control, which should be the output of the
function controlINPROCreg(). This function will be
described later on.
We now turn to illustrate the usage of function
INPROCreg() by presenting the code used in the analyses
discussed in Rodríguez-Álvarez, Roca-Pardiñas,
and Cadarso-Suárez (2011b) and Pardo-Fernández, Rodríguez-Álvarez, and van Keilegom
(2014). For confidentiality reasons, we use here a simulated data
set that resemble the original data. Specifically, in that papers we
aimed at assessing the performance of the body mass index (BMI) for
predicting clusters of cardiovascular disease (CVD) risk factors.
Diseased subjects were defined as those having two or more CVD risk
factors (raised triglycerides, reduced high-density lipoprotein
cholesterol, raised blood pressure and raised fasting plasma glucose),
following the International Diabetes Federation criteria (International Diabetes Federation 2006). It is
well known that anthropometric measures behave differently according to
both age and gender, and thus it is advisable to incorporate both
covariates into the ROC analysis. Since the proposal implemented in the
package only admits one continuous covariate, separate analyses were
conducted on men and women.
library(npROCRegression)
data(endosim)
summary(endosim)
#> age gender idf_status bmi
#> Min. :18.25 Men :1317 Min. :0.0000 Min. :11.53
#> 1st Qu.:29.57 Women:1523 1st Qu.:0.0000 1st Qu.:23.43
#> Median :39.28 Median :0.0000 Median :26.59
#> Mean :41.43 Mean :0.2433 Mean :26.75
#> 3rd Qu.:50.84 3rd Qu.:0.0000 3rd Qu.:29.79
#> Max. :84.66 Max. :1.0000 Max. :51.21
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Analysis for males
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fit.men <- INPROCreg(marker = "bmi", covariate = "age", group = "idf_status",
tag.healthy = 0,
data = subset(endosim, gender == "Men"),
ci.fit = TRUE, test = TRUE,
accuracy = c("EQ","TH"),
accuracy.cal="AROC",
control=controlINPROCreg(p=1,kbin=30,step.p=0.01),
newdata = data.frame(age = seq(18,85,l=50)))
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Analysis for females
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fit.women <- INPROCreg(marker = "bmi", covariate = "age", group = "idf_status",
tag.healthy = 0,
data = subset(endosim, gender == "Women"),
ci.fit = TRUE, test = TRUE,
accuracy = c("EQ","TH"),
accuracy.cal="ROC",
control=controlINPROCreg(p=1,kbin=30,step.p=0.01),
newdata = data.frame(age = seq(18,85,l=50)))As a result, the function INPROCreg() provides a list
with the following components:
names(fit.men)
#> [1] "call" "marker" "covariate" "group" "ci.fit" "X"
#> [7] "fpf" "h" "d" "ROC" "AUC" "AROC"
#> [13] "EQ" "TH" "pvalue"where
| Component | Description |
|---|---|
call |
The matched call. |
X |
The data frame used in the predictions. |
fpf |
Set of false positive fractions at which the covariate-specific ROC curve has been estimated. |
h |
Estimated regression and variance functions in healthy population. |
d |
Estimated regression and variance functions in diseased population. |
ROC |
Estimated covariate-specific ROC curve. |
AUC |
Estimated covariate-specific AUC, and corresponding confidence intervals if required. |
AROC |
Estimated covariate-adjusted ROC curve. |
YI/EQ |
If required, estimated covariate-specific YI (or values at which the true positive fraction (TPF) and the true negative fraction (TNF) coincide), and corresponding bootstrap confidence intervals. |
TH |
If required, estimated optimal threshold values based on either the YI or the criterion of equality of TPF and TNF, and corresponding bootstrap confidence intervals. |
pvalue |
If required, p-value obtained with the test for checking the effect of the continuous covariate on the ROC curve. |
A numerical summary of the results can be obtained by calling up the
print.INPROCreg() or INPROCreg() functions,
which can be abbreviated by print() and
summary():
summary(fit.men)
#>
#> Call:
#> INPROCreg(marker = "bmi", covariate = "age", group = "idf_status",
#> tag.healthy = 0, data = subset(endosim, gender == "Men"),
#> ci.fit = TRUE, test = TRUE, accuracy = c("EQ", "TH"), accuracy.cal = "AROC",
#> newdata = data.frame(age = seq(18, 85, l = 50)), control = controlINPROCreg(p = 1,
#> kbin = 30, step.p = 0.01))
#>
#> *************************************************
#> Induced non-parametric ROC regression
#> *************************************************
#> MARKER: bmi
#> COVARIATE: age
#> STATUS: idf_status
#>
#> ----------------------------------------------
#> Test for continuous covariate effect (p-value): 0.095
#>
summary(fit.women)
#>
#> Call:
#> INPROCreg(marker = "bmi", covariate = "age", group = "idf_status",
#> tag.healthy = 0, data = subset(endosim, gender == "Women"),
#> ci.fit = TRUE, test = TRUE, accuracy = c("EQ", "TH"), accuracy.cal = "ROC",
#> newdata = data.frame(age = seq(18, 85, l = 50)), control = controlINPROCreg(p = 1,
#> kbin = 30, step.p = 0.01))
#>
#> *************************************************
#> Induced non-parametric ROC regression
#> *************************************************
#> MARKER: bmi
#> COVARIATE: age
#> STATUS: idf_status
#>
#> ----------------------------------------------
#> Test for continuous covariate effect (p-value): 0
#> controlINPROCreg() function
The argument control of the function
INPROCreg() can be used to set several parameters
controlling the ROC regression fitting process. This argument is the
output of the function controlINPROCreg(). This function
has the following arguments:
| Argument | Description |
|---|---|
step.p |
a numeric value, defaulting to 0.02. ROC curves are calculated at a
regular sequence of false positive fractions with step.p
increment. |
kbin |
an integer value specifying the number of binning knots. By default 30. |
p |
an integer value specifying the order of the local polynomial kernel estimator for the regression functions. By default 1. |
h |
a vector of length 4 specifying the bandwidths to be used for the estimation of the regression and variance functions in healthy population and the regression and variance functions in diseased populations (in this order). By default -1 (selected using cross-validation.). A value of 0 would indicate a linear fit. |
seed |
an integer value specifying the seed for the bootstrap resamples. If NULL it is initialized randomly. |
nboot |
an integer value specifying the number of bootstrap resamples for the construction of the confidence intervals. By default 500. |
level |
a real value specifying the confidence level for the confidence intervals. By default 0.95. |
resample.m |
a character string specifying if bootstrap resampling (for the confidence intervals) should be done with or without regard to the disease status (“coutcome” or “noutcome”). When the resampling method is done conditionally on the disease status, the resampling is based on the residuals of the regression models in healthy and diseased populations. However, when the bootstrap resampling is done without regard to the disease status, a naive bootstrap is used. By default, the resampling is done conditionally on the disease status. |
plot.INPROCreg() function
The function plot.INPROCreg() takes as input argument an
INPROCreg object, and returns several plots of interest. By
default, the function returns the plot of the estimated regression and
variance functions in both healthy and diseased populations, the
covariate-specific ROC curve and AUC, and the AROC. When required in the
call to the INPROCreg(), this plot function also returns
the YI or EQ values, and corresponding optimal thresholds.
layout(matrix(c(1,1,2,2,3,3,4,4,5,5,6,6,0,7,7,0),4,4, byrow = TRUE), widths = c(1.75,1.75,1.75,1.75), heights = c(3.5,3.5,3.5,3.5))
plot(fit.men, ask = FALSE)
Male population. Top row: Nonparametric estimates of BMI by age (years), along with 95% pointwise bootstrap confidence bands. Solid line: diseased population. Dashed line: healthy population. Left: Regression function. Right: Variance function. Second row: Estimated covariate-specific ROC curves and AUC along with 95% pointwise bootstrap confidence bands. Third and bottom rows: Estimated covariate-adjusted ROC curve (AROC) and estimated threshold curves for BMI (bottom panel), with the sensitivity and specificity linked to these values (right panel in third row), along with 95% pointwise bootstrap confidence bands
layout(matrix(c(1,1,2,2,3,3,4,4,5,5,6,6,0,7,7,0),4,4, byrow = TRUE), widths = c(1.75,1.75,1.75,1.75), heights = c(3.5,3.5,3.5,3.5))
plot(fit.women, ask = FALSE)
Female population. Top row: Nonparametric estimates of BMI by age (years), along with 95% pointwise bootstrap confidence bands. Solid line: diseased population. Dashed line: healthy population. Left: Regression function. Right: Variance function. Second row: Estimated covariate-specific ROC curves and AUC along with 95% pointwise bootstrap confidence bands. Third and bottom rows: Estimated covariate-adjusted ROC curve (AROC) and estimated threshold curves for BMI (bottom panel), with the sensitivity and specificity linked to these values (right panel in third row), along with 95% pointwise bootstrap confidence bands
R functions to estimate direct nonparametric ROC regression models
The main R function is DNPROCreg() which
estimates the covariate-specific ROC curve in the presence of
multidimensional covariates by means of the ROC-GAM regression model
presented in Rodríguez-Álvarez, Roca-Pardiñas,
and Cadarso-Suárez (2011a). Once the model is fitted, a brief
numerical summary can be obtained by using the functions
print.DNPROCreg() and summary.DNPROCreg. A
plot of the estimated covariate-specific ROC curve and corresponding AUC
can be obtained through the function plot.DNPROCreg(). A
summary of these functions is shown in the following Table.
| Function | Description |
|---|---|
DNPROCreg |
Fits a direct nonparametric ROC regression model for a set of continuous and categorical covariates. |
controlDNPROCreg |
Function used to set several parameters controlling the ROC regression fitting process. |
print.DNPROCreg |
Default print method for objects fitted with
DNPROCreg(). |
summary.DNPROCreg |
Produces a summary of a DNPROCreg object. |
plot.DNPROCreg |
Plots the covariate-specific ROC curve and AUC. |
DNROCregData |
Selects an adequate set of points from the original data to be used as a default dataset for obtaining predictions or plots. |
DNPROCreg() function
The following Table shows a description of the arguments of the
DNPROCreg() function
| Argument | Description |
|---|---|
marker |
A character string with the name of the diagnostic test variable. |
formula.h |
Right-hand formula(s) giving the mean and variance model(s) to be fitted in healthy population. Atomic values are also valid, being recycled. |
formula.ROC |
Right-hand formula giving the ROC regression model to be fitted (ROC-GAM model). |
group |
A character string with the name of the variable that distinguishes healthy from diseased individuals. |
tag.healthy |
The value codifying the healthy individuals in the variable
group. |
data |
Data frame representing the data and containing all needed variables. |
ci.fit |
A logical value. If TRUE, confidence intervals are computed. |
test.partial |
A numeric vector containing the covariate components in the ROC-GAM formula to be tested for a possible effect. If NULL, no test is performed. |
newdata |
A data frame containing the values of the covariate at which predictions are required. |
control |
Output of the controlDNROCreg() function. |
weights |
An optional vector of `prior weights’ to be used in the fitting process. |
Usage is as follows:
DNPROCreg(marker, formula.h=~1, formula.ROC=~1, group, tag.healthy, data, ci.fit=FALSE, test.partial=NULL, newdata=NULL, control=controlDNPROCreg(), weights=NULL)The diagnostic test variable is indicated by the argument
marker. The nonparametric location-scale regression model
for the healthy population is specified by formula.h. This
argument should be a vector (of length 2) of right-hand formulas (atomic values are
also valid, because they are recycled). The first right-hand formula is
the model for the conditional mean function, and the second one is the
model for the (logarithm) of the conditional variance function. These
formulas are similar to that used for the glm() function,
except that nonparametric functions can be added to the additive
predictor by means of function s(). For instance,
specification ~ x1 + s(x2) would assume a linear effect of
x1 and a nonparametric effect of x2.
Categorical variables (factors) can be also incorporated, as well as
factor-by-curve interaction terms. For example, to include the
interaction between age and gender we need to
specify ~ gender + s(age) + s(age, by = gender). Note that,
for identifiability purposes, the “main” effects of the continuous and
categorical covariates need to be included into the formula. All these
considerations also apply to the argument formula.ROC,
where the ROC-GAM regression model is specified. The name of the
variable that distinguishes healthy from diseased individuals is
specified in argument group, and in
tag.healthy the value codifying the healthy individuals in
this variable. The data argument is a data frame
representing the data and containing all needed variables. Pointwise
bootstrap confidence intervals for each component of the additive
predictor of the ROC-GAM, as well as the covariate-specific AUCs (with
the integral being approximated by numerical integration methods), are
obtained by setting the argument ci.fit to
TRUE. The components of the ROC-GAM to be tested for their
possible effect are indicated in test.partial. In this
argument, we pass the position of the components as specified in the
formula.ROC argument. An optional data frame containing the
covariate values at which predictions are required can be specified in
argument newdata. If missing, an adequate set of points
from the dataset used in the fit is selected. To that end, the function
DNPROCregdata() is used. Argument control
allows to modify some default parameters that control the fitting
process, and should be the output of the function
controlDNPROCreg(). This function will be described later
on.
To illustrate the usage of this function, we now analyse the endocrine data presented above and discussed in Rodríguez-Álvarez, Roca-Pardiñas, and Cadarso-Suárez (2011a). For the sake of illustration, in the following we will show only the statistical analysis conducted with the Body Mass Index (BMI). Since it is well established that anthropometric measures perform differently according to gender, the age-by-gender interaction was included in both the location-scale regression model for healthy population and the ROC-GAM.
library(npROCRegression)
data(endosim)
fit.endo <- DNPROCreg(marker = "bmi", formula.h = "~ gender + s(age) + s(age, by = gender)",
formula.ROC = "~ gender + s(age) + s(age, by = gender)",
group = "idf_status",
tag.healthy = 0,
data = endosim,
control = list(card.P=50, kbin=30, step.p=0.02),
ci.fit = TRUE, test.partial = 3)As a result, the function DNPROCreg() provides a list
with the following components:
names(fit.endo)
#> [1] "call" "marker" "group" "formula.h" "formula.ROC"
#> [6] "ci.fit" "model" "fpf" "newdata" "pfunctions"
#> [11] "coefficients" "ROC" "AUC" "pvalue"where
| Component | Description |
|---|---|
call |
The matched call. |
model |
Data frame containing all variables and observations used in the fitting process. |
fpf |
Set of false positive fractions at which the covariate-specific ROC curve has been estimated. |
newdata |
Data frame containing the values of the covariates at which the covariate-specific ROC curve has been estimated. |
pfunctions |
Matrices containing the estimates of each component of the additive predictor of the ROC-GAM. One matrix contains the effects of the covariates, the other the effect of the FPF. Confidence intervals are returned if required). |
coefficients |
Vector of parametric coefficient of the fitted ROC-GAM. |
ROC |
Estimated covariate-specific ROC curve. |
AUC |
Estimated covariate-specific AUC, and corresponding confidence intervals if required. |
pvalue |
If required, p-values are obtained - with two different
bootstrap-based tests (Rodríguez-Álvarez et al.
2016) - for each model term indicated in argument
test.partial (T2: L2-based test; and T1:
L1-based
test). |
As before, a numerical summary of the results can be obtained by
calling up the print() and summary()
functions:
summary(fit.endo)
#>
#> Call:
#> DNPROCreg(marker = "bmi", formula.h = "~ gender + s(age) + s(age, by = gender)",
#> formula.ROC = "~ gender + s(age) + s(age, by = gender)",
#> group = "idf_status", tag.healthy = 0, data = endosim, ci.fit = TRUE,
#> test.partial = 3, control = list(card.P = 50, kbin = 30,
#> step.p = 0.02))
#>
#> *************************************************
#> Direct non-parametric ROC regression
#> *************************************************
#> Marker: bmi
#> Group (status): idf_status
#> Healthy regression model (mean): ~gender + s(age) + s(age, by = gender)
#> Healthy regression model (variance): ~gender + s(age) + s(age, by = gender)
#> ROC regression model: ~gender + s(age) + s(age, by = gender)
#>
#> ----------------------------------------------
#> Parametric coefficients (ROC curve):
#> (Intercept) gender_Men gender_Women
#> 0.63972784 -0.06940575 0.06940575
#>
#> ----------------------------------------------
#> Tests for effects (p-values)
#> s(age, by = gender)
#> T2 0
#> T1 0controlDNPROCreg() function
The argument control of the function
DNPROCreg() can be used to set several parameters
controlling the ROC regression fitting process. This argument is the
output of the function controlDNPROCreg(). This function
has the following arguments:
| Argument | Description |
|---|---|
step.p |
a numeric value, defaulting to 0.02. ROC curves are calculated at a
regular sequence of false positive fractions with step.p
increment. |
kbin |
an integer value specifying the number of binning knots. By default 30. |
card.P |
an integer value specifying the cardinality of the set of false positive fractions used in the estimation process. By default 50. |
p |
an integer value specifying the order of the local polynomial kernel estimator. By default 1. |
seed |
an integer value specifying the seed for the bootstrap resamples. If NULL it is initialized randomly. |
nboot |
an integer value specifying the number of bootstrap resamples for the construction of the confidence intervals. By default 500. |
level |
a real value specifying the confidence level for the confidence intervals. By default 0.95. |
resample.m |
a character string specifying if bootstrap resampling (for the confidence intervals) should be done with or without regard to the disease status (“coutcome” or “noutcome”). In both cases, a naive bootstrap is used. By default, the resampling is done conditionally on the disease status. |
link |
a character string specifying the link function (“probit”, “logit” or “cloglog”). By default the link is the probit function. |
plot.DNPROCreg() function
The function plot.DNPROCreg() takes as input argument an
`DNPROCreg’ object, and returns the plots of the covariate-specific ROC
curve and AUC.
layout(matrix(c(1,3,2,4),2,2, byrow = FALSE), widths = c(3.5,3.5), heights = c(3.5,3.5))
plot(fit.endo, ask = FALSE)
Estimated covariate-specific ROC curves and AUCs, along with 95% pointwise bootstrap confidence interval, in male and female populations.
The function plot.DNPROCreg() does not plot the partial
effects of the covariates on the ROC curve. However, these plots can be
obtained from the element pfunctions of the
DNPROCreg object.
names(fit.endo$pfunctions)
#> [1] "covariates" "fpf"
names(fit.endo$pfunctions$covariates)
#> [1] "gender" "genderll" "genderul"
#> [4] "s(age)" "s(age)ll" "s(age)ul"
#> [7] "s(age, by = gender)" "s(age, by = gender)ll" "s(age, by = gender)ul"
names(fit.endo$pfunctions$fpf)
#> [1] "s(fpf)" "s(fpf)ll" "s(fpf)ul"
layout(matrix(c(1,3,2,4),2,2, byrow = FALSE), widths = c(3.5,3.5), heights = c(3.5,3.5))
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Main effect of age
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
sel.row <- fit.endo$newdata$gender == "Women" # Same effect for both genders
plot(fit.endo$newdata$age[sel.row],fit.endo$pfunctions$covariates[sel.row, "s(age)"], xlab="age", ylab="s(age)", type="l", main = "Main effect of age", ylim=c(-1,1))
lines(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age)ul"], lty=2)
lines(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age)ll"], lty=2)
abline(h = 0, col="grey")
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Effect of age: deviation for males
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
sel.row <- fit.endo$newdata$gender == "Women"
plot(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age, by = gender)"], xlab="age", ylab = "s(age, by=gender)", type = "l", main = " Age effect: Deviation for males", ylim = c(-1.2, 0.8))
lines(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age, by = gender)ul"], lty=2)
lines(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age, by = gender)ll"], lty=2)
abline(h = 0, col="grey")
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Effect of age: deviation for females
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
sel.row <- fit.endo$newdata$gender == "Men"
plot(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age, by = gender)"], xlab="age", ylab = "s(age, by=gender)", type = "l", main = " Age effect: Deviation for females", ylim = c(-0.8, 1.2))
lines(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age, by = gender)ul"], lty=2)
lines(fit.endo$newdata$age[sel.row], fit.endo$pfunctions$covariates[sel.row, "s(age, by = gender)ll"], lty=2)
abline(h = 0, col="grey")
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# Effect of FPF
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
plot(fit.endo$fpf, fit.endo$pfunctions$fpf[,1], xlab = "fpf", ylab = "s(fpf)", main = "False positive fraction", type="l")
lines(fit.endo$fpf, fit.endo$pfunctions$fpf[,2], lty=2)
lines(fit.endo$fpf, fit.endo$pfunctions$fpf[,3], lty=2)
abline(h = 0, col="grey")
Nonparametric estimates of partial functions (solid lines), along with 95% pointwise bootstrap confidence interval (dashed lines).