Bootstrapping
The bootstrap is a technique in statistics which consists of resampling the observed data in order to create an empirical distribution of some statistic (Fox and Weisberg, 2012, 2017). This is often done to evaluate the assumptions of the model or to derive empirical distributions which are difficult to approximate. For nonlinear models, bootstrapping can be useful because often questions arise that a typical analysis does not answer. In many instances the distributions of parameters from a nonlinear model are in fact normal and standard approximations work well, in other instances, however, this is not the case.
In the following sections I will illustrate the use of the bootstrap for nonlinear model (nls), generalized nonlinear models (gnls) and nonlinear mixed models (nlme). In another section I will illustrate the ability to simulate from nonlinear (mixed) models
Nonlinear models
The first type of models illustrated here are nonlinear models for which we make standard assumptions for error distributions and there are no random effects.
Simple example
As a simple example, we can use the dataset ‘barley’ in the ‘nlraa’ package. This represents the response of barley yield to different doses of fertilizer over several seasons. We will ignore the effect of ‘year’ in this section.
data(barley, package = "nlraa")
## Quick visualization
ggplot(data = barley, aes(x = NF, y = yield)) + geom_point()
We can fit a very simple model known as the linear-plateau.
## Linear-plateau model
## The function SSlinp is in the 'nlraa' package
fit.lp <- nls(yield ~ SSlinp(NF, a, b, xs), data = barley)
## Visualize data and fit
ggplot(data = barley, aes(x = NF, y = yield)) +
geom_point() +
geom_line(aes(y = fitted(fit.lp)))
At this point several questions might arise:
- What are the confidence intervals for the model parameters?
- How do we describe the uncertainty around the fitted values of the model?
- What is the estimate and confidence interval for the asymptote?
Confidence interval for model parameters
The confidence intervals for the model parameters can be derived using a method called profiling. The generic function ‘confint’ can be used which invokes ‘confint.nls’ from the ‘MASS’ package.
## Waiting for profiling to be done...## 2.5% 97.5%
## a 104.821725 167.73660
## b 19.938746 38.00273
## xs 5.849474 12.23099For nonlinear models this method is preferred to simple confidence intervals that would directly use the standard error of the model parameters. In many cases, there is good agreement between the two, but this is not always the case and it can be informative to compare both methods. Those standard errors can be obtained using the ‘summary’ function, along with hypothesis testing.
##
## Formula: yield ~ SSlinp(NF, a, b, xs)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## a 137.067 16.179 8.472 1.83e-12 ***
## b 27.501 5.269 5.219 1.62e-06 ***
## xs 7.682 1.211 6.342 1.68e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 70.79 on 73 degrees of freedom
##
## Number of iterations to convergence: 2
## Achieved convergence tolerance: 8.76e-16Some interesting plots can be produced which illustrate the symmetry (or lack thereof) of the distribution for the model parameters. Profile plots which show large departures from normality would indicate that the normal approximation for the confidence intervals might not be the best choice.
## This one is fairly symetric and the normal approximation is reasonable
plot(profile(fit.lp, "b"))
Being able to see the whole profile for a parameter can be interesting in terms of understanding its behavior. For example, in this model, which has a ‘break-point’, we would not expect all the parameters to have identical shaped profiles and that is illustrated above for parameter ‘xs’.
In this case, bootstraping is an alternative to computing the confidence intervals and it can be used as a way to double-check the previous results, but it can also be used when profiling fails. A couple of functions which can perform bootstraping for nonlinear models are ‘boot_nls’ in the ‘nlraa’ pacakge or the ‘Boot’ in the ‘car’ package.
## psim = 3 just adds residuals and does not resample parameters
fit.lp.Bt <- boot_nls(fit.lp, psim = 3)## Number of times model fit did not converge 172 out of 999In this case, this function uses the base ‘boot’ package under the hood and it returns an object of that class. This also allows for other options such as defining a function as a combination of model parameters, and the use of more than one core to speed up computation. In this case, since we are also interested in the asymptote, which is ‘a + b * xs’, we can obtain confidence intervals for this parameter by doing the following.
## I'm using Boot in the 'car' package here, but it is similar to 'boot_nls'
fn <- function(x) coef(x)[1] + coef(x)[2] * coef(x)[3]
fit.lp.Bt.asymp <- Boot(fit.lp, f = fn, labels = "asymptote")##
## Number of bootstraps was 819 out of 999 attempted## Bootstrap bca confidence intervals
##
## 2.5 % 97.5 %
## asymptote 323.7295 386.0021
The bootstrap method takes a few seconds here, but it can be computationally much more demanding in larger problems since it re-fits the model many, many times. An alternative is to use the delta method, for which there is a function in the ‘car’ pacakge. The delta method has the advantage of being fast and the disadvantage that it makes the assumption of a normal distribution. In this case the lower bound for the confidence interval is similar to the one obtained with the bootstrap, but the upper bound for the confidence interval is somewhat higher using the bootstrap (381 vs. 370). Since the bootstrap method makes fewer assumptions it is probably the better one to report.
## Estimate SE 2.5 % 97.5 %
## a + b * xs 348.326 11.484 325.817 370.84In order to answer our second question above related to the uncertainty around the fitted values we could plug-in the values from the bootstrap sampling and obtain different regression lines.
## The object 't' in the bootstrap run has
## the parameter estimate values
## First remove missing values
fit.lp.Bt.prms <- na.omit(fit.lp.Bt$t)
nrb <- length(unique(barley$NF))
nrp <- nrow(fit.lp.Bt.prms)
## Set up an empty data.frame
prd.dat <- data.frame(i = as.factor(rep(1:nrp, each = nrb)), NF = rep(unique(barley$NF), nrp), prd = NA)
## A simple loop can be used to run the model multiple times
for(i in 1:nrp){
a.i <- fit.lp.Bt.prms[i,1]
b.i <- fit.lp.Bt.prms[i,2]
xs.i <- fit.lp.Bt.prms[i,3]
prd.dat[c(1 + (nrb*(i - 1))):c(i * nrb),3] <- linp(unique(barley$NF), a.i, b.i, xs.i)
}
## Plot the data with the original fit and the uncertainty
ggplot() +
geom_line(data = prd.dat, aes(x = NF, y = prd, group = i),
color = "gray", alpha = 0.2) +
geom_line(data = barley, aes(x = NF, y = fitted(fit.lp))) +
geom_point(data = barley, aes(x = NF, y = yield)) +
ylab("Yield") + xlab("Nitrogen rate") +
ggtitle("Using results from Boot \n and plug-in into linp")
The previous graph shows the black line with the original fit and the variability in the fitted values due to the resampling generated during the bootstrap process. The function ‘predict.nls’ at the moment ignores the arguments ‘se.fit’ and ‘interval’ which means that this functionality is not available in the base ‘stats’ package. (There is an alternative approach in the ‘propagate’ package - see references.)
So we could use the quantiles of ‘prd.dat’ object above to derive confidence intervals for the regression line. The previous example is an attempt to make it clear how the uncertainty could be displayed. Equivalently, we can use ‘Boot’ for this purpose, but with some effort manipulating the data.
##
## Number of bootstraps was 822 out of 999 attemptedfttd <- na.omit(fit.lp.Bt2$t)
prds <- c(t(fttd))
ndat <- data.frame(i = as.factor(rep(1:nrow(fttd), each = ncol(fttd))),
NF = rep(0:14, nrow(fttd)))
ndat$prd <- prds
## Essentially the same graph as the one above
ggplot() +
geom_line(data = ndat, aes(x = NF, y = prd, group = i),
color = "gray", alpha = 0.2) +
geom_line(data = barley, aes(x = NF, y = fitted(fit.lp))) +
geom_point(data = barley, aes(x = NF, y = yield)) +
ylab("Yield") + xlab("Nitrogen rate")
Bootstrapping generalized nonlinear models
Implementing bootstrap for more complex models takes extra work. For this, I’m taking the approach of sampling from the vector of fixed parameters and also bootstrapping the standardized residuals for ‘gnls’ and ‘nlme’ objects. (This methodology can be improved in the future.) This takes advantage that we assume that the residuals are normally distributed.
As a first example, we can compare the bootstrapped confidence intervals with the ones obtained by ‘intervals’. Note: I’m running this only a few hundred times in the examples below for efficiency, but it is a good practice to run these a few thousand times.
set.seed(101)
## Simplify the dataset to make the set up simpler
barley2 <- subset(barley, year < 1974)
fit.lp.gnls2 <- gnls(yield ~ SSlinp(NF, a, b, xs), data = barley2)
intervals(fit.lp.gnls2)## Approximate 95% confidence intervals
##
## Coefficients:
## lower est. upper
## a 138.79981 176.800000 214.800186
## b 13.75248 27.603093 41.453706
## xs 3.64251 6.174127 8.705744
##
## Residual standard error:
## lower est. upper
## 25.50341 35.17935 56.67543## Compare this to the bootstrapping approach
## R = 200 is too low for bootstrap, this is for illustration only
fit.lp.gnls2.bt <- boot_nlme(fit.lp.gnls2, R = 200)## Number of times model fit did not converge 24 out of 200##
## Number of bootstrap replications R = 176
## original bootBias bootSE bootMed
## 1 176.8000 -3.29560 21.2417 173.591
## 2 27.6031 1.16721 7.7877 28.061
## 3 6.1741 0.01468 1.5194 5.879## Bootstrap percent confidence intervals
##
## 2.5 % 97.5 %
## 1 126.591966 215.57081
## 2 14.731698 45.83062
## 3 4.206532 10.14666The confidence intervals obtained by bootstrap are wider (as expected) than the ones obtained using intervals because they consider the uncertainty in the parameters of the nonlinear model. The next example shows a slightly more complex model in which we introduce the (fixed) effect of year for a subset of the data.
set.seed(101)
barley2$year.f <- as.factor(barley2$year)
cfs <- coef(fit.lp.gnls2)
fit.lp.gnls3 <- update(fit.lp.gnls2,
params = list(a + b + xs ~ year.f),
start = c(cfs[1], 0, 0, 0,
cfs[2], 0, 0, 0,
cfs[3], 0, 0, 0))
## This bootstraps the vector of parameters
fit.lp.gnls3.bt <- boot_nlme(fit.lp.gnls3, R = 300)## Number of times model fit did not converge 152 out of 300## Bootstrap percent confidence intervals
##
## 2.5 % 97.5 %
## 1 111.1375192 196.119143
## 2 -45.3383112 86.644389
## 3 -52.9212287 65.101962
## 4 -0.5059473 114.993794
## 5 14.3750448 41.574802
## 6 -20.9463834 15.697041
## 7 -17.7796470 16.955886
## 8 -18.1083851 16.965825
## 9 4.4735481 9.074686
## 10 -2.6628309 4.960814
## 11 -4.1799869 1.682101
## 12 -2.8113050 3.862596
This is simply to illustrate the use of bootstrapping for a ‘gnls’ object, which is something that function ‘car::Boot’ does not seem to be able to handle (the deltaMethod function also fails for this type of model, because it cannot handle the names in the vector of coefficients which uses periods).
Bootstrapping nonlinear mixed models
For illustration, I will continue to use the barley example, but this time the model is fitted to each year individually and then a nonlinear mixed model which assumes a diagonal matrix for the random effects (for simplicity). In this case we want an esimtate of the confidence interval for the asymptote which is not an explicit parameter but rather a combination ‘a + b * xs’ of the three parameters.
set.seed(101)
barley$year.f <- as.factor(barley$year)
barleyG <- groupedData(yield ~ NF | year.f, data = barley)
fitL.bar <- nlsList(yield ~ SSlinp(NF, a, b, xs), data = barleyG)
fit.bar.nlme <- nlme(fitL.bar, random = pdDiag(a + b + xs ~ 1))
## Confidence intervals of the model fixed parameters
intervals(fit.bar.nlme, which = "fixed")## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## a 109.212647 134.663569 160.11449
## b 23.970656 27.939486 31.90832
## xs 6.823868 7.671139 8.51841## Function which computes the asymptote
fna <- function(x) fixef(x)[1] + fixef(x)[2] * fixef(x)[3]
## Bootstrap the model for the asymptote
fit.bar.nlme.bt <- boot_nlme(fit.bar.nlme, f = fna, R = 200)## Number of times model fit did not converge 90 out of 200## Bootstrap percent confidence intervals
##
## 2.5 % 97.5 %
## 1 306.707 382.6985
For this model the estimate for the asymptote is 349, but from the model fit we do not get a direct confidence interval. Bootstrapping makes this easy (but it takes a bit of time) and it results in 307, 383.
Confidence bands for generalized nonlinear models
For this example I will use the Orange dataset. The goal here is to place confidence bands around the mean response function.
data(Orange)
## This fits a model which considers the fact that
## the variance typically increases as the fitted
## values increase
fitg <- gnls(circumference ~ SSlogis(age, Asym, xmid, scal),
data = Orange, weights = varPower())
## Here we use bootstrapping to investigate
## the uncertainty around the fitted values
fitg.bt1 <- boot_nlme(fitg, fitted, psim = 1, R = 300)## Number of times model fit did not converge 3 out of 300## Compute 90% quantile confidence bands
lwr1.q <- apply(t(fitg.bt1$t), 1, quantile, probs = 0.05, na.rm = TRUE)
upr1.q <- apply(t(fitg.bt1$t), 1, quantile, probs = 0.95, na.rm = TRUE)
ggplot() +
geom_point(data = Orange, aes(x = age, y = circumference)) +
geom_line(data = Orange, aes(x = age, y = fitted(fitg))) +
geom_ribbon(aes(x = Orange$age, ymin = lwr1.q, ymax = upr1.q),
fill = "purple", alpha = 0.2) +
ggtitle("Orange fit using the logistic: \n 90% confidence band for the mean function")
