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<h1 class="title toc-ignore">Chapter 7</h1>
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<div id="exploratory-analysis-of-the-survival-data-in-nhanes"
class="section level2">
<h2>Exploratory Analysis of the Survival Data in NHANES</h2>
<p>We first illustrate the survival data analysis concepts using
examples from the NHANES data collected between 2011-2014.</p>
<pre class="r"><code>library(tidyverse)
library(survival)
library(mgcv)
library(refund)
library(cowplot)
df_cleaned <- readRDS("./data/nhanes_fda_with_r.rds")
df_fig71 <- df_cleaned %>%
mutate(time_yr = round(time / 12, 2)) %>%
select(SEQN, time = time_yr, event, age, BMI, race, gender, CHD) %>%
filter(SEQN %in% c(64759, 70962, 73561, 75630, 77244, 82820))
#get survival data with functional predictors
nhanes_df_surv <- df_cleaned %>%
mutate(time_yr = round(time / 12, 2)) %>%
filter(!is.na(event)) %>%
select(SEQN, time = time_yr, event, age, BMI, race, gender, CHD, TMIMS, MIMS)</code></pre>
<p>Below is the code to obtain a sample of six participants in NHANES
2011-2014. The code then reproduce Figure 7.2 of the book.</p>
<pre class="r"><code>df_sample <- df_fig71
df_sample$MIMS <- df_cleaned$MIMS[which(df_cleaned$SEQN %in% df_sample$SEQN),]
str(df_sample)
## 'data.frame': 6 obs. of 9 variables:
## $ SEQN : num 64759 70962 73561 75630 77244 ...
## $ time : num 3.33 8.17 0.75 6.33 6.58 1.5
## $ event : int 1 0 1 0 0 1
## $ age : num 80 40 73 59 42 78
## $ BMI : num 24.8 45 19.7 28.3 25.9 26.9
## $ race : Factor w/ 6 levels "Mexican American",..: 3 4 3 1 4 3
## $ gender: Factor w/ 2 levels "Male","Female": 1 1 2 2 1 1
## $ CHD : Factor w/ 4 levels "No","Yes","Refused",..: 1 1 1 1 1 1
## $ MIMS : 'AsIs' num [1:6, 1:1440] 0.0969 1.7742 0.7577 10.6671 4.459 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:6] "64759" "70962" "73561" "75630" ...
## .. ..$ : chr [1:1440] "MIN0001" "MIN0002" "MIN0003" "MIN0004" ...
#plot NHANES sample data
layout(matrix(c(1:21), 7, 3, byrow = FALSE), widths = c(1, 2, 4))
#title of other info
par(mar = c(0,0,0,0))
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
text(x = 0.45, y = 0.5, paste("SEQN"), cex = 1.8, col = "black", font = 2)
#other info
par(mar = c(1,0,1,1))
for(i in 1:6){
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
mtext(side = 4, text = df_sample$SEQN[i], line = -6.5, cex = 1.2, las = 1)
}
#title of predictors
par(mar = c(0,0,0,0))
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
text(x = 0.45, y = 0.5, paste("Predictors"), cex = 1.8, col = "black", font = 2)
#other info
par(mar = c(1,0,1,1))
for(i in 1:6){
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
mtext(side = 4, text = paste0("age: ", df_sample$age[i],
", gender: ", df_sample$gender[i],
", BMI: ", df_sample$BMI[i],
", \n", df_sample$race[i],
", CHD: ", df_sample$CHD[i]), line = -17, cex = 0.8, las = 1)
}
#title of survival data
par(mar = c(0,0,0,0))
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
text(x = 0.45, y = 0.5, paste("Time to event"), cex = 1.8, col = "black", font = 2)
#survival data
par(mar = c(0.2,0.5,0.2,0.5))
for(j in 1:6){
if(df_sample$event[j] == 1){
plot(x = c(0, df_sample$time[j]), y = c(0, 0), type = "l", xlim = c(0, 8.7), xaxt = "n", yaxt = "n", bty = "n", col = "red", lwd = 2)
points(x = df_sample$time[j], y = 0, pch = 4, col = "red", cex = 2)
text(df_sample$time[j], 0, labels = paste0(format(round(df_sample$time[j], 2), nsmall = 2), " years"), cex = 1.5, pos = 3, col = "red")
}else{
plot(x = c(0, df_sample$time[j]), y = c(0, 0), type = "l", xlim = c(0, 8.7), xaxt = "n", yaxt = "n", bty = "n", lwd = 2)
points(x = df_sample$time[j], y = 0, pch = 16, cex = 2)
text(df_sample$time[j], 0, labels = paste0(format(round(df_sample$time[j], 2), nsmall = 2), " years"), cex = 1.5, pos = 3)
}
if(j == 1){
legend(4.85, 1.3, legend=c("Censored", "Deceased"),
col=c("black", "red"), lty=1, lwd = 2, cex=1.5, box.lty=0, y.intersp=1.3)
}
}</code></pre>
<p><img src="chapter_07_files/figure-html/unnamed-chunk-2-1.png" width="90%" /></p>
<p>Instead of only focusing on the survival outcomes, we now incorporate
the functional predictors into the data set. The code below shows how to
reproduce Figure 7.4 of the book.</p>
<pre class="r"><code>layout(matrix(c(1:21), 7, 3, byrow = FALSE), widths = c(1, 2, 4))
#title of other info
par(mar = c(0,0,0,0))
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
text(x = 0.45, y = 0.5, paste("SEQN"), cex = 1.8, col = "black", font = 2)
#other info
par(mar = c(1,0,1,1))
for(i in 1:6){
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
mtext(side = 4, text = df_sample$SEQN[i], line = -6.5, cex = 1.2, las = 1)
}
#title of predictors
par(mar = c(0,0,0,0))
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
text(x = 0.45, y = 0.5, paste("Predictors"), cex = 1.8, col = "black", font = 2)
#other info
par(mar = c(2,0,1,1))
for(i in 1:6){
plot(df_sample$MIMS[i,], type = "l", xaxt = "n", xlab = "", bty = "n", yaxt = "n")
axis(side = 1, at = c(1, 72, 144, 216, 288)*5, labels = c("00:00", "06:00", "12:00", "18:00", "24:00"), padj = -1)
mtext(side = 4, text = paste0("age: ", df_sample$age[i], ", ...\n"), line = -20, cex = 0.8, las = 1)
}
#title of survival data
par(mar = c(0,0,0,0))
plot(c(0, 1), c(0, 1), ann = F, bty = 'n', type = 'n', xaxt = 'n', yaxt = 'n')
text(x = 0.5, y = 0.5, paste("Time to event"), cex = 2, col = "black", font = 2)
#survival data
par(mar = c(0.2,0.5,0.2,0.5))
for(j in 1:6){
if(df_sample$event[j] == 1){
plot(x = c(0, df_sample$time[j]), y = c(0, 0), type = "l", xlim = c(0, 8.7), xaxt = "n", yaxt = "n", bty = "n", col = "red", lwd = 2)
points(x = df_sample$time[j], y = 0, pch = 4, col = "red", cex = 2)
text(df_sample$time[j], 0, labels = paste0(format(round(df_sample$time[j], 2), nsmall = 2), " years"), cex = 1.5, pos = 3, col = "red")
}else{
plot(x = c(0, df_sample$time[j]), y = c(0, 0), type = "l", xlim = c(0, 8.7), xaxt = "n", yaxt = "n", bty = "n", lwd = 2)
points(x = df_sample$time[j], y = 0, pch = 16, cex = 2)
text(df_sample$time[j], 0, labels = paste0(format(round(df_sample$time[j], 2), nsmall = 2), " years"), cex = 1.5, pos = 3)
}
if(j == 1){
legend(4.85, 1.3, legend=c("Censored", "Deceased"),
col=c("black", "red"), lty=1, lwd = 2, cex=1.5, box.lty=0, y.intersp=1.3)
}
}</code></pre>
<p><img src="chapter_07_files/figure-html/unnamed-chunk-3-1.png" width="90%" /></p>
<div id="kaplan-meier-estimators" class="section level3">
<h3>Kaplan-Meier Estimators</h3>
<p>We further explore the NHANES 2011-2014 survival data by
investigating the Kaplan-Meyer (KM) estimators of the survival function
for specific subgroups. The code below reproduce Figure 7.5 of the book
which displays results for different age groups at baseline.</p>
<pre class="r"><code>library(survival)
library(survminer)
library(gridExtra)
nhanes_df_surv$Age_cat <- cut(nhanes_df_surv$age,
breaks = c(20, 50, 80),
include.lowest = TRUE)
#set PA group within each age group
TMIMS_cat_age <- rep(NA, nrow(nhanes_df_surv))
for(age_group in c("[20,50]", "(50,80]")){
TMIMS_cat_age[which(nhanes_df_surv$Age_cat == age_group)] <-
cut(nhanes_df_surv$TMIMS[which(nhanes_df_surv$Age_cat == age_group)],
breaks = quantile(nhanes_df_surv$TMIMS[which(nhanes_df_surv$Age_cat == age_group)], c(0, 0.333, 0.667, 1)),
labels = c("low TMIMS", "medium TMIMS", "high TMIMS"),
include.lowest = TRUE)
}
nhanes_df_surv$TMIMS_cat_age <- factor(TMIMS_cat_age, labels = c("low TMIMS", "medium TMIMS", "high TMIMS"))
rm(age_group, TMIMS_cat_age)
#draw KM curves by TMIMS category
##create survival object
nhanes_df_surv$SurvObj <- with(nhanes_df_surv, Surv(time, event))
##TMIMS by age
TMIMS.age.km.plot <- list()
for(age_group in c("[20,50]", "(50,80]")){
TMIMS.age.km <- survfit(SurvObj ~ TMIMS_cat_age, data = nhanes_df_surv %>% filter(Age_cat == age_group))
TMIMS.age.km.plot[[age_group]] <- ggsurvplot(fit = TMIMS.age.km,
data = nhanes_df_surv,
risk.table = TRUE,
pval = TRUE,
conf.int = FALSE,
break.time.by = 1,
ggtheme = theme_minimal(),
risk.table.y.text.col = TRUE,
risk.table.y.text = FALSE,
legend.title = "PA Level",
legend.labs = c("low", "medium", "high"),
title = paste0("Age: ", age_group),
xlim = c(0, 8.2),
ylim = c(0.5, 1),
pval.size = 6,
pval.coord = c(0.58, 0.57),
fontsize = 4.3)
TMIMS.age.km.plot[[age_group]]$plot <- TMIMS.age.km.plot[[age_group]]$plot +
theme(legend.text = element_text(size = 15, color = "black"),
legend.title = element_text(size = 15, color = "black"),
title = element_text(size = 14, color = "black"),
axis.title.x = element_text(size = 15)) +
xlim(0, 8.2)
TMIMS.age.km.plot[[age_group]]$table <- TMIMS.age.km.plot[[age_group]]$table +
theme(axis.title.x = element_text(size = 15),
axis.title.y = element_text(size = 15),
title = element_text(size = 14))
}
grid.arrange(TMIMS.age.km.plot$`[20,50]`$plot,
TMIMS.age.km.plot$`(50,80]`$plot,
TMIMS.age.km.plot$`[20,50]`$table,
TMIMS.age.km.plot$`(50,80]`$table,
heights = c(3, 1),
nrow = 2)</code></pre>
<p><img src="chapter_07_files/figure-html/unnamed-chunk-4-1.png" width="90%" /></p>
</div>
<div id="results-for-the-standard-cox-models" class="section level3">
<h3>Results for the Standard Cox Models</h3>
<p>We now show how to fit standard Cox models. The results are shown in
Table 7.1 of the book.</p>
<pre class="r"><code>df_cleaned <- nhanes_df_surv %>%
filter(age >= 50 & !is.na(BMI) & !is.na(race) & !is.na(gender) & !is.na(CHD) & !is.na(TMIMS)
& CHD %in% c("Yes", "No"))
fit_M1 <- coxph(Surv(time, event) ~ age + gender + race, data = df_cleaned)
summary(fit_M1, conf.int = FALSE)
## Call:
## coxph(formula = Surv(time, event) ~ age + gender + race, data = df_cleaned)
##
## n= 4180, number of events= 698
##
## coef exp(coef) se(coef) z Pr(>|z|)
## age 0.102206 1.107612 0.004884 20.926 < 2e-16 ***
## genderFemale -0.272688 0.761330 0.076073 -3.585 0.000338 ***
## raceOther Hispanic -0.169661 0.843951 0.216029 -0.785 0.432242
## raceNon-Hispanic White 0.210762 1.234619 0.163065 1.293 0.196181
## raceNon-Hispanic Black 0.180265 1.197534 0.171762 1.050 0.293947
## raceNon-Hispanic Asian -0.389520 0.677382 0.229697 -1.696 0.089925 .
## raceOther Race 0.543402 1.721854 0.300130 1.811 0.070210 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Concordance= 0.754 (se = 0.01 )
## Likelihood ratio test= 626.7 on 7 df, p=<2e-16
## Wald test = 529.1 on 7 df, p=<2e-16
## Score (logrank) test = 630.8 on 7 df, p=<2e-16
fit_M2 <- coxph(Surv(time, event) ~ age + gender + race + BMI + CHD, data = df_cleaned)
summary(fit_M2, conf.int = FALSE)
## Call:
## coxph(formula = Surv(time, event) ~ age + gender + race + BMI +
## CHD, data = df_cleaned)
##
## n= 4180, number of events= 698
##
## coef exp(coef) se(coef) z Pr(>|z|)
## age 0.098356 1.103356 0.004982 19.743 < 2e-16 ***
## genderFemale -0.214727 0.806762 0.077161 -2.783 0.00539 **
## raceOther Hispanic -0.163071 0.849531 0.216077 -0.755 0.45044
## raceNon-Hispanic White 0.182709 1.200465 0.163155 1.120 0.26278
## raceNon-Hispanic Black 0.209753 1.233373 0.171802 1.221 0.22212
## raceNon-Hispanic Asian -0.412425 0.662043 0.231816 -1.779 0.07522 .
## raceOther Race 0.519289 1.680832 0.300163 1.730 0.08363 .
## BMI -0.005780 0.994237 0.006638 -0.871 0.38391
## CHDYes 0.567735 1.764266 0.103012 5.511 3.56e-08 ***
## CHDRefused NA NA 0.000000 NA NA
## CHDDon't know NA NA 0.000000 NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Concordance= 0.76 (se = 0.01 )
## Likelihood ratio test= 654.3 on 9 df, p=<2e-16
## Wald test = 573.6 on 9 df, p=<2e-16
## Score (logrank) test = 690.5 on 9 df, p=<2e-16
fit_M3 <- coxph(Surv(time, event) ~ age + gender + race + BMI + CHD + TMIMS, data = df_cleaned)
summary(fit_M3, conf.int = FALSE)
## Call:
## coxph(formula = Surv(time, event) ~ age + gender + race + BMI +
## CHD + TMIMS, data = df_cleaned)
##
## n= 4180, number of events= 698
##
## coef exp(coef) se(coef) z Pr(>|z|)
## age 0.0709039 1.0734781 0.0052525 13.499 < 2e-16 ***
## genderFemale -0.0599678 0.9417948 0.0780030 -0.769 0.442019
## raceOther Hispanic -0.1494232 0.8612046 0.2160182 -0.692 0.489116
## raceNon-Hispanic White 0.0717456 1.0743820 0.1626740 0.441 0.659185
## raceNon-Hispanic Black 0.0837017 1.0873045 0.1718321 0.487 0.626178
## raceNon-Hispanic Asian -0.4691413 0.6255392 0.2316202 -2.025 0.042818 *
## raceOther Race 0.4365582 1.5473723 0.3000154 1.455 0.145636
## BMI -0.0248345 0.9754714 0.0067596 -3.674 0.000239 ***
## CHDYes 0.4405216 1.5535173 0.1033900 4.261 2.04e-05 ***
## CHDRefused NA NA 0.0000000 NA NA
## CHDDon't know NA NA 0.0000000 NA NA
## TMIMS -0.0001674 0.9998326 0.0000135 -12.393 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Concordance= 0.783 (se = 0.009 )
## Likelihood ratio test= 811.6 on 10 df, p=<2e-16
## Wald test = 704.6 on 10 df, p=<2e-16
## Score (logrank) test = 826.3 on 10 df, p=<2e-16</code></pre>
</div>
</div>
<div id="cox-regression-with-baseline-functional-predictors"
class="section level2">
<h2>Cox Regression with Baseline Functional Predictors</h2>
<div id="linear-functional-cox-model" class="section level3">
<h3>Linear Functional Cox Model</h3>
<p>As discussed in Chapter 7.3 of the book, we consider the extension of
the linear functional model to Cox regression, the Linear Functional Cox
Model (LFCM). We motivate the model with an application to the NHANES
2011-2014 accelerometry data. As in previous chapters, the functional
predictor, 24-hour activity profiles, is denoted as <span
class="math inline">\(W_i:S\rightarrow \mathbb{R}\)</span> for each
study participant, <span class="math inline">\(i=1,\ldots,n\)</span>.
The linear Cox model is then</p>
<p><span class="math display">\[\begin{equation}
\log \lambda_i\{t|\mathbf{Z}_i,W_i(\cdot)\} =
\log\{\lambda_0(t)\} + \mathbf{Z}_i^t \boldsymbol{\gamma} + \int_S
W_i(s)\beta(s)ds\;.
\label{eq:ch7_n61}
\end{equation}\]</span></p>
<p>Before estimating this model, we first subset our data to those
participants aged 50-80 (recall, data has been subset already to ages 50
and over). Additionally, we drop factor variables in the models (in this
case just CHD) with levels “Don’t know” and “refused”.</p>
<pre class="r"><code>## get a set of variables for subjects using pair-wise available data
## who meet our age criteria
## and re-code dont know or refused as missing
nhanes_df_surv <-
df_cleaned %>%
filter(age < 80) %>%
## replace don't know and refused levels with NA
mutate_if(.predicate=is.factor, .funs=function(x) recode_factor(x, "Don't know" = NA_character_, "Refused" = NA_character_)) %>%
## create mortality time in years
mutate(time = time/12) </code></pre>
<p>With the data set up, we first estimate the model using
<strong>refund::pfr</strong> by using the <strong>refund::lf</strong>
constructor for linear functional terms, applying a cyclic cubic
regression spline basis to estimate the functional effect, <span
class="math inline">\(\beta\)</span>.</p>
<pre class="r"><code>nS <- ncol(nhanes_df_surv$MIMS) # dimension of the functional predictor
svec <- seq(0, 1, len = nS) # observed points on the functional domain
fit_lfcm_pfr <- pfr(time ~ age + BMI + gender + CHD +
lf(MIMS, bs="cc",k=30,argvals=svec),
method="REML", data=nhanes_df_surv,weights=nhanes_df_surv$event,
family=cox.ph)
## Warning in newton(lsp = lsp, X = G$X, y = G$y, Eb = G$Eb, UrS = G$UrS, L = G$L,
## : gam.fit5 step failed: max magnitude relative grad = 2.08629722778414e-08</code></pre>
<p>As in the book, we now show how to fit the same model using
<strong>mgcv::gam</strong> directly. To do this, we need to set up the
relevant data matrices used for fitting the LFCM using our spline basis
expansion combined with numerical apprxoimation of the integral term.
Note that for simplicity of presentation we use Riemann integration to
determine our quadrature weights, while <strong>refund::pfr</strong>
uses Simpson’s rule by default. The results from the two models will
thus be very similar, but not identical. If we used Simpson’s rule for
assigning quadrature weights, the results would indeed by identical.</p>
<pre class="r"><code>N <- nrow(nhanes_df_surv) # number of samples
#Quadrature weights for Riemann integration
lvec <- matrix(1 / nS, 1440, 1)
# matrix containing quadrature weights for all participants
L <- kronecker(matrix(1, N, 1),t(lvec))
# matrix containing functional domain values
S <- kronecker(matrix(1, N, 1), t(svec))
nhanes_df_surv$S <- I(S)
#Pointwise product of w i(s) and the quadrature weights, "L"
nhanes_df_surv$W_L <- I(nhanes_df_surv$MIMS * L)</code></pre>
<p>We then fit the model using the corresponding
<strong>mgcv::gam</strong> syntax.</p>
<pre class="r"><code>fit_lfcm_mgcv <- gam(time ~ age + BMI + gender + CHD +
s(S, by=W_L, bs="cc",k=30), method="REML",
data=nhanes_df_surv, weights=nhanes_df_surv$event,
family=cox.ph)</code></pre>
<p>Finally, we plot the results from the two estimation procedures
(point estimates and point-wise confidence intervals) using the plot
method associated with the <strong>mgcv::gam</strong> and
<strong>refund::pfr</strong> functions. The way to obtain estimated
coefficient values and their corresponding standard errors is shown in
the next subsection of this vignette.</p>
<pre class="r"><code>
xinx <- (c(1,6,12,18,23)*60+1)/1440
xinx_lab <- c("01:00","06:00","12:00","18:00","23:00")
par(mfrow=c(1,2))
plot(fit_lfcm_pfr, rug=FALSE, shade=TRUE,xlab="Time of Day", ylab=expression(hat(beta)(s)), main="Estimate from refund::pfr()",
xaxt='n')
abline(h=0,col='red',lty=2)
axis(1, at=xinx, labels=xinx_lab)
plot(fit_lfcm_mgcv, rug=FALSE, shade=TRUE,xlab="Time of Day", ylab=expression(hat(beta)(s)), main="Estimate from mgcv::gam()",
xaxt='n')
abline(h=0,col='red',lty=2)
axis(1, at=xinx, labels=xinx_lab)</code></pre>
<p><img src="chapter_07_files/figure-html/lfcm_plot_mgcv_vs_pfr-1.png" width="90%" /></p>
</div>
<div id="smooth-effects-of-traditional-and-functional-predictors"
class="section level3">
<h3>Smooth Effects of Traditional and Functional Predictors</h3>
<p>We not consider the scenario where we want to estimate a linear
functional term and a non-linear effect of a scalar covariate in a Cox
model framework. Suppose that an individual’s covariate vector of
non-functional predictors is <span class="math inline">\(\mathbf{Z}_i =
[\mathbf{Z}^t_{i,1}, \mathbf{Z}^t_{i,2} ]^t\)</span>, where <span
class="math inline">\(\mathbf{Z}_{i,1}\)</span> is a vector of length
<span class="math inline">\(Q_1\)</span> containing the covariates for
which we assume a linear effect, and <span
class="math inline">\(\mathbf{Z}_{i,2}\)</span> is a vector of length
<span class="math inline">\(Q_2\)</span> containing the covariates for
which we assume a non-linear (additive) effect. Here <span
class="math inline">\(Q=Q_1+Q_2\)</span> is the total number of scalar
covariates. The LFCM then becomes <span
class="math display">\[\begin{align}
\log \lambda_i\{t|\mathbf{Z}_i, W_i(\cdot)\} &=
\log\{\lambda_0(t)\} + \sum_{q=1}^{Q_1} \gamma_q Z_{iq} +
\sum_{q=Q_1+1}^Q f_q(Z_{iq}) + \int_S W_i(s)\beta(s)ds
\label{eq:lfcm_am}\;.
\end{align}\]</span></p>
<p>Returning to our data appliaction, suppose we want to let <span
class="math inline">\(Z_{i,1} = [\text{age}_i, \text{BMI}_i]^t\)</span>
and <span class="math inline">\(Z_{i,2} = [\text{gender}_i,
\text{CHD}_i]^t\)</span>. That is, we want to estimate non-linear
effects of age and BMI on mortality, with linear effects for our
categorical (binary) predictors of gender and CHD.</p>
<p>We first show how to fit a model with non-linear effects for age and
BMI using the same syntax presented in previous chapters.</p>
<pre class="r"><code>fit_acm <- gam(time ~ s(age,k=30,bs="cr") + s(BMI,k=30,bs="cr")+
gender+CHD,
method="REML",
data=nhanes_df_surv, weights=nhanes_df_surv$mortstat,
family=cox.ph)</code></pre>
<p>We can add a linear functional term easily by moving to the
<strong>refund::pfr</strong> function (alternatively, use
<strong>mgcv::gam</strong> directly using the approach described
above).</p>
<pre class="r"><code>fit_lfcm_acm <- pfr(time ~ s(age,k=30,bs="cr") + s(BMI,k=30,bs="cr") +
gender + CHD +
lf(MIMS, bs="cc",k=30,argvals=svec),
method="REML", data=nhanes_df_surv,weights=nhanes_df_surv$mortstat,
family=cox.ph)</code></pre>
<p>The estimated coefficients can then be obtained using a call the the
predict method for <strong>refund::pfr</strong> with type=“terms”.</p>
<pre class="r"><code>aind <- seq(50,80,len=100)
bind <- seq(15,80,len=100)
df_pred <- data.frame("age" = aind,
"BMI" = bind,
gender="Male",CHD="No","MIMS"=I(matrix(nhanes_df_surv$MIMS[1,], 100, 1440, byrow=TRUE)))
coefs_lfcm_acm <- predict(fit_lfcm_acm, newdata=df_pred,type='terms',se.fit=TRUE)
coefs_lfcm_acm <- data.frame("est_age" = coefs_lfcm_acm$fit[,"s(age)"],
"se_age" = coefs_lfcm_acm$se.fit[,"s(age)"],
"est_bmi" = coefs_lfcm_acm$fit[,"s(BMI)"],
"se_bmi" = coefs_lfcm_acm$se.fit[,"s(BMI)"])
coefs_lfcm_acm <-
coefs_lfcm_acm %>%
mutate("LB_age" = est_age - qnorm(0.975)*se_age,
"UB_age" = est_age + qnorm(0.975)*se_age,
"LB_bmi" = est_bmi - qnorm(0.975)*se_bmi,
"UB_bmi" = est_bmi + qnorm(0.975)*se_bmi)
beta_lfcm_acm <- coef(fit_lfcm_acm, n=100, select=3)
coefs_lfcm_acm[["est_beta"]] <- beta_lfcm_acm$value[,1]
coefs_lfcm_acm[["sind"]] <- beta_lfcm_acm$MIMS.argvals
coefs_lfcm_acm[["LB_beta"]] <- beta_lfcm_acm$value[,1] - qnorm(0.975)*beta_lfcm_acm$se
coefs_lfcm_acm[["UB_beta"]] <- beta_lfcm_acm$value[,1] + qnorm(0.975)*beta_lfcm_acm$se
coefs_acm <- predict(fit_acm, newdata=df_pred,type='terms',se.fit=TRUE)
coefs_acm <- data.frame("est_age" = coefs_acm$fit[,"s(age)"],
"se_age" = coefs_acm$se.fit[,"s(age)"],
"est_bmi" = coefs_acm$fit[,"s(BMI)"],
"se_bmi" = coefs_acm$se.fit[,"s(BMI)"])
coefs_acm <-
coefs_acm %>%
mutate("LB_age" = est_age - qnorm(0.975)*se_age,
"UB_age" = est_age + qnorm(0.975)*se_age,
"LB_bmi" = est_bmi - qnorm(0.975)*se_bmi,
"UB_bmi" = est_bmi + qnorm(0.975)*se_bmi)
coefs_acm <- data.frame(dplyr::select(df_pred, -MIMS), coefs_acm)
coefs_lfcm_acm <- data.frame(dplyr::select(df_pred, -MIMS), coefs_lfcm_acm)</code></pre>
<p>We can then create Figure 7.10 in the textbook using the code
below.</p>
<pre class="r"><code>lims_bmi <- max(abs(c(as.vector(unlist(coefs_acm[,c("LB_bmi","UB_bmi")])), as.vector(unlist(coefs_lfcm_acm[,c("LB_bmi","UB_bmi")]) ))))*c(-1,1)
lims_age <- max(abs(c(as.vector(unlist(coefs_acm[,c("LB_age","UB_age")])), as.vector(unlist(coefs_lfcm_acm[,c("LB_age","UB_age")]) ))))*c(-1,1)
lims_beta <- max(abs(as.vector(unlist(coefs_lfcm_acm[,c("LB_beta","UB_beta")] ))))*c(-1,1)
plt_acm_age_coef <-
coefs_acm %>%
ggplot() +
geom_ribbon(aes(x=age, ymin=LB_age,ymax=UB_age),alpha=0.2) +
geom_line(aes(x=age,y=est_age)) +
geom_line(aes(x=age,y=est_age)) +
geom_hline(yintercept=0, lty=2, color='red') +
xlab("Age (years)") +
ylab("")+
theme_classic(base_size=18)+ ylim(lims_age)
plt_acm_bmi_coef <-
coefs_acm %>%
ggplot() +
geom_ribbon(aes(x=BMI, ymin=LB_bmi,ymax=UB_bmi),alpha=0.2) +
geom_line(aes(x=BMI,y=est_bmi)) +
geom_line(aes(x=BMI,y=est_bmi)) +
geom_hline(yintercept=0, lty=2, color='red') +
xlab("BMI") +
ylab("")+
theme_classic(base_size=18)+ ylim(lims_bmi)
plt_lfcm_acm_age_coef <-
coefs_lfcm_acm %>%
ggplot() +
geom_ribbon(aes(x=age, ymin=LB_age,ymax=UB_age),alpha=0.2) +
geom_line(aes(x=age,y=est_age)) +
geom_line(aes(x=age,y=est_age)) +
theme_classic(base_size=18)+ ylim(lims_age)+
geom_hline(yintercept=0, lty=2, color='red') +
xlab("Age (years)") +
ylab("")
plt_lfcm_acm_bmi_coef <-
coefs_lfcm_acm %>%
ggplot() +
geom_ribbon(aes(x=BMI, ymin=LB_bmi,ymax=UB_bmi),alpha=0.2) +
geom_line(aes(x=BMI,y=est_bmi)) +
geom_line(aes(x=BMI,y=est_bmi)) +
theme_classic(base_size=18) + ylim(lims_bmi) +
geom_hline(yintercept=0, lty=2, color='red') +
xlab("BMI") +
ylab("")
plt_lfcm_acm_beta_coef <-
coefs_lfcm_acm %>%
ggplot() +
geom_ribbon(aes(x=sind, ymin=LB_beta,ymax=UB_beta),alpha=0.2) +
geom_line(aes(x=sind,y=est_beta)) +
geom_hline(yintercept=0, lty=2, color='red') +
theme_classic(base_size=18) +
ylab("")+
scale_x_continuous(breaks=(c(1,6,12,18,23)*60+1)/1440, labels=c("01:00","06:00","12:00","18:00","23:00")) +
xlab("Time of Day (s)") +ylim(lims_beta)
title_coef_lfcm_v2 <- ggplot() +
theme(panel.background = element_rect(fill = "black"),plot.title = element_text(size=18))+
labs(title = expression(bold("Functional Effect: " ~ hat(bold(beta))(s) %+-% bold("1.96") %*% SE(hat(bold(beta))(s)) ~ phantom(hat(bold(f))))))
title_coef_acm_age <- ggplot() +
theme(panel.background = element_rect(fill = "black"),plot.title = element_text(size=18))+
labs(title = expression(bold("Age Effect: " ~ hat(bold(f))(Age) %+-% bold("1.96") %*% SE(hat(bold(f))(Age)))))
title_coef_acm_bmi <- ggplot() +
theme(panel.background = element_rect(fill = "black"),plot.title = element_text(size=18))+
labs(title = expression(bold("BMI Effect: " ~ hat(bold(f))(BMI) %+-% bold("1.96") %*% SE(hat(bold(f))(BMI)))) )
title_acm <- ggplot() +
theme(panel.background = element_rect(fill = "black"),plot.title = element_text(size=18))+
labs(title = expression(bold("Additive Cox Model")))
title_lfcm_acm <- ggplot() +
theme(panel.background = element_rect(fill = "black"),plot.title = element_text(size=18))+
labs(title = expression(bold("(B) Functional Linear Additive Cox Model: Estimated Coefficients")))
title_acm <- ggplot() +
theme(panel.background = element_rect(fill = "black"),plot.title = element_text(size=18))+
labs(title = expression(bold("(A) Additive Cox Model: Estimated Coefficients")))
title_coef_lfcm_acm <- plot_grid(NULL, title_coef_acm_age, title_coef_acm_bmi, title_coef_lfcm_v2, ncol=4, rel_widths = c(0.075,1,1,1))
title_coef_acm <- plot_grid(NULL, title_coef_acm_age, title_coef_acm_bmi, NULL, ncol=4, rel_widths = c(0.075,1,1,1))
plt_coef_acm_lfcm <-
plot_grid(NULL, plt_lfcm_acm_age_coef, plt_lfcm_acm_bmi_coef, plt_lfcm_acm_beta_coef, ncol=4,
label_size = 12, rel_widths = c(0.05,1,1,1))+
draw_label(x=0.03,y=0.53, label=expression("Lower Risk" %<-% "No Risk" %->% "Higher Risk"),
angle=90,size=16)
plt_coef_acm <-
plot_grid(NULL, plt_acm_age_coef, plt_acm_bmi_coef, NULL, ncol=4,
label_size = 12, rel_widths = c(0.05,1,1,1))+
draw_label(x=0.03,y=0.53, label=expression("Lower Risk" %<-% "No Risk" %->% "Higher Risk"),
angle=90,size=16)
plt_coef_acm_lfcm_comp <- plot_grid(title_lfcm_acm, title_coef_lfcm_acm,plt_coef_acm_lfcm, rel_heights = c(0.115,0.1385, 1),ncol=1 )
plt_coef_acm_comp <- plot_grid(title_acm, title_coef_acm,plt_coef_acm, rel_heights = c(0.115,0.1385, 1),ncol=1 )