pcrboot {qpcR}

Bootstrapping and jackknifing qPCR data

Description

Confidence intervals for the estimated parameters and goodness-of-fit measures are calculated for a nonlinear qPCR data fit by either
a) boostrapping the residuals of the fit or
b) jackknifing and refitting the data.

Confidence intervals can also be calculated for all parameters obtained from the efficiency analysis.

Usage

pcrboot(object, type = c("boot", "jack"), B = 100, njack = 1,
        plot = TRUE, do.eff = TRUE, conf = 0.95, verbose = TRUE, ...)

Arguments

object

an object of class 'pcrfit'.

type

either bootstrapping or jackknifing.

B

numeric. The number of iterations.

njack

numeric. In case of type = "jack", how many datapoints to exclude. Defaults to leave-one-out.

plot

should the fitting and final results be displayed as a plot?

do.eff

logical. If TRUE, efficiency analysis will be performed.

conf

the confidence level.

verbose

logical. If TRUE, the iterations will be printed on the console.

...

other parameters to be passed on to the plotting functions.

Details

Non-parametric bootstrapping is applied using the centered residuals.
1) Obtain the residuals from the fit:

\hat{\varepsilon}_t = y_t - f(x_t, \hat{\theta})

2) Draw bootstrap pseudodata:

y_{t}^{\ast} = f(x_t, \hat{\theta}) + \epsilon_{t}^{\ast}

where \epsilon_{t}^{\ast} are i.i.d. from distribution \hat{F}, where the residuals from the original fit are centered at zero.
3) Fit \hat\theta^\ast by nonlinear least-squares.
4) Repeat B times, yielding bootstrap replications

\hat\theta^{\ast 1}, \hat\theta^{\ast 2}, \ldots, \hat\theta^{\ast B}

One can then characterize the EDF and calculate confidence intervals for each parameter:

\theta \in [EDF^{-1}(\alpha/2), EDF^{-1}(1-\alpha/2)]

The jackknife alternative is to perform the bootstrap on the data-predictor vector, i.e. eliminating a certain number of datapoints.
If the residuals are correlated or have non-constant variance the latter is recommended. This may be the case in qPCR data, as the variance in the low fluorescence region (ground phase) is usually much higher than in the rest of the curve.

Value

A list containing the following items:

ITER

a list containing each of the results from the iterations.

CONF

a list containing the confidence intervals for each item in ITER.

Each item contains subitems for the coefficients (coef), root-mean-squared error (rmse), residual sum-of-squares (rss), goodness-of-fit measures (gof) and the efficiency analysis (eff). If plot = TRUE, all data is plotted as boxplots including confidence intervals.

Author(s)

Andrej-Nikolai Spiess

References

Nonlinear regression analysis and its applications.
Bates DM & Watts DG.
Wiley, Chichester, UK, 1988.

Nonlinear regression.
Seber GAF & Wild CJ.
Wiley, New York, 1989.

Boostrap accuracy for non-linear regression models.
Roy T.
J Chemometics (1994), 8: 37-44.

Examples

## Simple bootstrapping with
## too less iterations...
par(ask = FALSE)
m1 <- pcrfit(reps, 1, 2, l4)
pcrboot(m1, B = 20)

## Jackknifing with leaving
## 5 datapoints out.
m2 <- pcrfit(reps, 1, 2, l4)
pcrboot(m2, type = "jack", njack = 5, B = 20)