R: Estimate 90% Confidence Intervals of f2 with Bootstrap...

bootf2 {bootf2}

R Documentation

Estimate 90% Confidence Intervals of `f_2` with Bootstrap Methodology

Description

Main function to estimate 90% confidence intervals of f_2 using bootstrap methodology.

Usage

bootf2(test, ref, path.in, file.in, path.out, file.out,
       n.boots = 10000L, seed = 306L, digits = 2L, alpha = 0.05,
       regulation = c("EMA", "FDA", "WHO","Canada", "ANVISA"),
       min.points = 1L, both.TR.85 = FALSE, print.report = TRUE,
       report.style = c("concise", "intermediate", "detailed"),
       f2.type = c("all", "est.f2", "exp.f2", "bc.f2",
                   "vc.exp.f2", "vc.bc.f2"),
       ci.type = c("all", "normal", "basic", "percentile",
                   "bca.jackknife", "bca.boot"),
       quantile.type = c("all", as.character(1:9), "boot"),
       jackknife.type = c("all", "nt+nr", "nt*nr", "nt=nr"),
       time.unit = c("min", "h"), output.to.screen = FALSE,
       sim.data.out = FALSE)

Arguments

`test`, `ref`	Data frames of dissolution profiles of test and reference product if `path.in` and `file.in` are not specified; otherwise, they should be character strings indicating the worksheet names of the Excel file where the dissolution data is saved. See Input/Output in Details.
`path.in`, `file.in`, `path.out`, `file.out`	Character strings of input and output directories and file names. See Input/Output in Details.
`n.boots`	An integer indicating the number of bootstrap samples.
`seed`	Integer seed value for reproducibility. If missing, a random seed will be generated for reproducibility purpose.
`digits`	An integer indicating the decimal points for the output.
`alpha`	A numeric value between 0 and 1 to estimate `(1-2\times \alpha)\times 100` confidence interval.
`regulation`	Character strings indicating regulatory guidelines. @seealso `calcf2()` for details on regulation rules.
`min.points`	An integer indicating the minimum time points to be used to calculate `f_2`. For conventional `f_2` calculation, the default is 3, however, for bootstrap `f_2`, the value should be lower as there might be less time points available in certain bootstrap samples. The default is 1. @seealso `calcf2()`.
`both.TR.85`	Logical. If `TRUE`, and if `regulation = "FDA"`, all measurements up to the time points at which both test and reference products dissolve more than 85% will be used to calculate `f_2`. This is the conventional, but incorrect, interpretation of the US FDA rule. Therefore, the argument should only be set to `TRUE` for validation purpose such as comparing the results from old literature that use the wrong interpretation to calculate `f_2`. @seealso `calcf2()` for details on regulation rules.
`print.report`	Logical. If `TRUE`, a plain text report will be produced. See Input/Output in Details.
`report.style`	`"concise"` style produces the estimators and their confidence intervals; `"intermediate"` style adds a list of individual `f_2`s for all bootstrap samples in the end of `"concise"` report; `"detailed"` style further adds individual bootstrap samples along with their `f_2`s in the end of `"intermediate"` report. See Input/Output in Details.
`f2.type`	Character strings indicating which type of `f_2` estimator should be calculated. See Types of estimators in Details.
`ci.type`	Character strings indicating which type of confidence interval should be estimated. See Types of confidence intervals in Details.
`quantile.type`	Character strings indicating the type of percentile.
`jackknife.type`	Character strings indicating the type of jackknife method. See Details.
`time.unit`	Character strings indicating the unit of time. It should be either `"min"` for minute or `"h"` for hour. It is mainly used for checking CV rules and making plot. @seealso `calcf2()`.
`output.to.screen`	Logical. If `TRUE`, a `"concise"` style summary report will be printed on screen. See Input/Output in Details.
`sim.data.out`	Logical. If `TRUE`, all individual bootstrap data sets will be included in the output.

Details

Minimum required arguments that must be provided by the user

Arguments test and ref must be provided by the user. They should be R ⁠data frames⁠, with time as the first column, and all individual profiles profiles as the rest columns. The actual names of the columns do not matter since they will be renamed internally.

Input/Output

The dissolution data of test and reference product can either be provided as data frames for test and ref, as explained above, or be read from an Excel file with data of test and reference stored in separate worksheets. In the latter case, the argument path.in, the directory where the Excel file is, and file.in, the name of the Excel file including the file extension .xls or .xlsx, must be provided. In such case, the argument test and ref must be the names of the worksheets in quotation marks. The first column of each Excel worksheet must be time, and the rest columns are individual dissolution profiles. The first row should be column names, such as time, unit01, unit02, ... The actual names of the columns do not matter as they will be renamed internally.

Arguments path.out and file.out are the names of the output directory and file. If they are not provided, but argument print.report is TRUE, then a plain text report will be generated automatically in the current working directory with file name test_vs_ref_TZ_YYYY-MM-DD_HHMMSS.txt, where test and ref are data set names of test and reference, TZ is the time zone such as CEST, YYYY-MM-DD is the numeric date format and HHMMSS is the numeric time format for hour, minute, and second.

For a quick check, set argument output.to.screen = TRUE, a summary report very similar to concise style report will be printed on screen.

Types of Estimators

According to Shah et al, the population f_2 for the inference is

f_2 = 100-25\log\left(1 + \frac{1}{P}\sum_{i=1}^P% \left(\mu_{\mathrm{T},i} - \mu_{\mathrm{R},i}\right)^2 \right)\,,

where P is the number of time points; \mu_{\mathrm{T},i} and \mu_{\mathrm{R},i} are population mean of test and reference product at time point i, respectively; \sum_{i=1}^P is the summation from i = 1 to P.

Five estimators for f_2 are included in the function:

The estimated f_2, denoted by \hat{f}_2, is the one written in various regulatory guidelines. It is expressed differently, but mathematically equivalently, as

\hat{f}_2 = 100-25\log\left(1 + \frac{1}{P}\sum_{i=1}^P\left(% \bar{X}_{\mathrm{T},i} - \bar{X}_{\mathrm{R},i}\right)^2 \right)\:,

where P is the number of time points; \bar{X}_{\mathrm{T},i} and \bar{X}_{\mathrm{R},i} are mean dissolution data at the ith time point of random samples chosen from the test and the reference population, respectively. Compared to the equation of population f_2 above, the only difference is that in the equation of \hat{f}_2 the sample means of dissolution profiles replace the population means for the approximation. In other words, a point estimate is used for the statistical inference in practice.
The Bias-corrected f_2, denoted by \hat{f}_{2,\mathrm{bc}}, was described in the article of Shah et al, as

\hat{f}_{2,\mathrm{bc}} = 100-25\log\left(1 + \frac{1}{P}% \left(\sum_{i=1}^P\left(\bar{X}_{\mathrm{T},i} - % \bar{X}_{\mathrm{R},i}\right)^2 - \frac{1}{n}\sum_{i=1}^P% \left(S_{\mathrm{T},i}^2 + S_{\mathrm{R},i}^2\right)\right)\right)\,,

where S_{\mathrm{T},i}^2 and S_{\mathrm{R},i}^2 are unbiased estimates of variance at the ith time points of random samples chosen from test and reference population, respectively; and n is the sample size.
The variance- and bias-corrected f_2, denoted by \hat{f}_{2,\mathrm{vcbc}}, does not assume equal weight of variance as \hat{f}_{2,\mathrm{bc}} does.

\hat{f}_{2, \mathrm{vcbc}} = 100-25\log\left(1 +% \frac{1}{P}\left(\sum_{i=1}^P \left(\bar{X}_{\mathrm{T},i} -% \bar{X}_{\mathrm{R},i}\right)^2 - \frac{1}{n}\sum_{i=1}^P% \left(w_{\mathrm{T},i}\cdot S_{\mathrm{T},i}^2 +% w_{\mathrm{R},i}\cdot S_{\mathrm{R},i}^2\right)\right)\right)\,,

where w_{\mathrm{T},i} and w_{\mathrm{R},i} are weighting factors for variance of test and reference products, respectively, which can be calculated as follows:

w_{\mathrm{T},i} = 0.5 + \frac{S_{\mathrm{T},i}^2}% {S_{\mathrm{T},i}^2 + S_{\mathrm{R},i}^2}\,,

and

w_{\mathrm{R},i} = 0.5 + \frac{S_{\mathrm{R},i}^2}% {S_{\mathrm{T},i}^2 + S_{\mathrm{R},i}^2}\,.
The expected f_2, denoted by \hat{f}_{2, \mathrm{exp}}, is calculated based on the mathematical expectation of estimated f_2,

\hat{f}_{2, \mathrm{exp}} = 100-25\log\left(1 + \frac{1}{P}% \left(\sum_{i=1}^P\left(\bar{X}_{\mathrm{T},i} - % \bar{X}_{\mathrm{R},i}\right)^2 + \frac{1}{n}\sum_{i=1}^P% \left(S_{\mathrm{T},i}^2 + S_{\mathrm{R},i}^2\right)\right)\right)\,,

using mean dissolution profiles and variance from samples for the approximation of population values.
The variance-corrected expected f_2, denoted by \hat{f}_{2, \mathrm{vcexp}}, is calculated as

\hat{f}_{2, \mathrm{vcexp}} = 100-25\log\left(1 +% \frac{1}{P}\left(\sum_{i=1}^P \left(\bar{X}_{\mathrm{T},i} -% \bar{X}_{\mathrm{R},i}\right)^2 + \frac{1}{n}\sum_{i=1}^P% \left(w_{\mathrm{T},i}\cdot S_{\mathrm{T},i}^2 +% w_{\mathrm{R},i}\cdot S_{\mathrm{R},i}^2\right)\right)\right)\,.

Types of Confidence Interval

The following confidence intervals are included:

The Normal interval with bias correction, denoted by normal in the function, is estimated according to Davison and Hinkley,

\hat{f}_{2, \mathrm{L,U}} = \hat{f}_{2, \mathrm{S}} - E_B \mp % \sqrt{V_B}\cdot Z_{1-\alpha}\,,

where \hat{f}_{2, \mathrm{L,U}} are the lower and upper limit of the confidence interval estimated from bootstrap samples; \hat{f}_{2, \mathrm{S}} denotes the estimators described above; Z_{1-\alpha} represents the inverse of standard normal cumulative distribution function with type I error \alpha; E_B and V_B are the resampling estimates of bias and variance calculated as

E_B = \frac{1}{B}\sum_{b=1}^{B}\hat{f}_{2,b}^\star - % \hat{f}_{2, \mathrm{S}} = \bar{f}_2^\star - \hat{f}_{2,\mathrm{S}}\,,

and

V_B = \frac{1}{B-1}\sum_{b=1}^{B} \left(\hat{f}_{2,b}^\star -\bar{f}_2^\star\right)^2\,,

where B is the number of bootstrap samples; \hat{f}_{2,b}^\star is the f_2 estimate with bth bootstrap sample, and \bar{f}_2^\star is the mean value.
The basic interval, denoted by basic in the function, is estimated according to Davison and Hinkley,

\hat{f}_{2, \mathrm{L}} = 2\hat{f}_{2, \mathrm{S}} -% \hat{f}_{2,(B+1)(1-\alpha)}^\star\,,

and

\hat{f}_{2, \mathrm{U}} = 2\hat{f}_{2, \mathrm{S}} -% \hat{f}_{2,(B+1)\alpha}^\star\,,

where \hat{f}_{2,(B+1)\alpha}^\star and \hat{f}_{2,(B+1)(1-\alpha)}^\star are the (B+1)\alphath and (B+1)(1-\alpha)th ordered resampling estimates of f_2, respectively. When (B+1)\alpha is not an integer, the following equation is used for interpolation,

\hat{f}_{2,(B+1)\alpha}^\star = \hat{f}_{2,k}^\star + % \frac{\Phi^{-1}\left(\alpha\right)-\Phi^{-1}\left(\frac{k}{B+1}\right)}% {\Phi^{-1}\left(\frac{k+1}{B+1}\right)-\Phi^{-1}% \left(\frac{k}{B+1}\right)}\left(\hat{f}_{2,k+1}^\star-% \hat{f}_{2,k}^\star\right),

where k is the integer part of (B+1)\alpha, \hat{f}_{2,k+1}^\star and \hat{f}_{2,k}^\star are the (k+1)th and the kth ordered resampling estimates of f_2, respectively.
The percentile intervals, denoted by percentile in the function, are estimated using nine different types of quantiles, Type 1 to Type 9, as summarized in Hyndman and Fan's article and implemented in R's quantile function. Using R's boot package, program bootf2BCA outputs a percentile interval using the equation above for interpolation. To be able to compare the results among different programs, the same interval, denoted by Percentile (boot) in the function, is also included in the function.
The bias-corrected and accelerated (BCa) intervals are estimated according to Efron and Tibshirani,

\hat{f}_{2, \mathrm{L}} = \hat{f}_{2, \alpha_1}^\star\,,

\hat{f}_{2, \mathrm{U}} = \hat{f}_{2, \alpha_2}^\star\,,

where \hat{f}_{2, \alpha_1}^\star and \hat{f}_{2, \alpha_2}^\star are the 100\alpha_1th and the 100\alpha_2th percentile of the resampling estimates of f_2, respectively. Type I errors \alpha_1 and \alpha_2 are obtained as

\alpha_1 = \Phi\left(\hat{z}_0 + \frac{\hat{z}_0 + \hat{z}_\alpha}% {1-\hat{a}\left(\hat{z}_0 + \hat{z}_\alpha\right)}\right),

and

\alpha_2 = \Phi\left(\hat{z}_0 + \frac{\hat{z}_0 + % \hat{z}_{1-\alpha}}{1-\hat{a}\left(\hat{z}_0 + % \hat{z}_{1-\alpha}\right)}\right),

where \hat{z}_0 and \hat{a} are called bias-correction and acceleration, respectively.

There are different methods to estimate \hat{z}_0 and \hat{a}. Shah et al. used jackknife technique, denoted by bca.jackknife in the function,

\hat{z}_0 = \Phi^{-1}\left(\frac{N\left\{\hat{f}_{2,b}^\star <% \hat{f}_{2,\mathrm{S}} \right\}}{B}\right),

and

\hat{a} = \frac{\sum_{i=1}^{n}\left(\hat{f}_{2,\mathrm{m}} -% \hat{f}_{2, i}\right)^3}{6\left(\sum_{i=1}^{n}\left(% \hat{f}_{2,\mathrm{m}} - \hat{f}_{2, i}\right)^2\right)^{3/2}}\,,

where N\left\{\cdot\right\} denotes the number of element in the set, \hat{f}_{2, i} is the ith jackknife statistic, \hat{f}_{2,\mathrm{m}} is the mean of the jackknife statistics, and \sum is the summation from 1 to sample size n.

Program bootf2BCA gives a slightly different BCa interval with R's boot package. This approach, denoted by bca.boot in the function, is also implemented in the function for estimating the interval.

Notes on the argument `jackknife.type`

For any sample with size n, the jackknife estimator is obtained by estimating the parameter for each subsample omitting the ith observation. However, when two samples (e.g., test and reference) are involved, there are several possible ways to do it. Assuming sample size of test and reference are n_\mathrm{T} and n_\mathrm{R}, the following three possibility are considered:

Estimated by removing one observation from both test and reference samples. In this case, the prerequisite is n_\mathrm{T} = n_\mathrm{R}, denoted by nt=nr in the function. So if there are 12 units in test and reference data sets, there will be 12 jackknife estimators.
Estimate the jackknife for test sample while keeping the reference data unchanged; and then estimate jackknife for reference sample while keeping the test sample unchanged. This is denoted by nt+nr in the function. This is the default method. So if there are 12 units in test and reference data sets, there will be 12 + 12 = 24 jackknife estimators.
For each observation deleted from test sample, estimate jackknife for reference sample. This is denoted by nt*nr in the function. So if there are 12 units in test and reference data sets, there will be 12 \times 12 = 144 jackknife estimators.

Value

A list of 3 or 5 components.

boot.ci: A data frame of bootstrap f_2 confidence intervals.
boot.f2: A data frame of all individual f_2 values for all bootstrap data set.
boot.info: A data frame with detailed information of bootstrap for reproducibility purpose, such as all arguments used in the function, time points used for calculation of f_2, and the number of NAs.
boot.summary: A data frame with descriptive statistics of the bootstrap f_2.
boot.t and boot.r: Lists of individual bootstrap samples for test and reference product if sim.data.out = TRUE.

References

Shah, V. P.; Tsong, Y.; Sathe, P.; Liu, J.-P. In Vitro Dissolution Profile Comparison—Statistics and Analysis of the Similarity Factor, f_2. Pharmaceutical Research 1998, 15 (6), 889–896. DOI: 10.1023/A:1011976615750.

Davison, A. C.; Hinkley, D. V. Bootstrap Methods and Their Application. Cambridge University Press, 1997.

Hyndman, R. J.; Fan, Y. Sample Quantiles in Statistical Packages. The American Statistician 1996, 50 (4), 361–365. DOI: /10.1080/00031305.1996.10473566.

Efron, B.; Tibshirani, R. An Introduction to the Bootstrap. Chapman & Hall, 1993.

Examples

# time points
tp <- c(5, 10, 15, 20, 30, 45, 60)
# model.par for reference with low variability
par.r <- list(fmax = 100, fmax.cv = 3, mdt = 15, mdt.cv = 14,
              tlag = 0, tlag.cv = 0, beta = 1.5, beta.cv = 8)
# simulate reference data
dr <- sim.dp(tp, model.par = par.r, seed = 100, plot = FALSE)
# model.par for test
par.t <- list(fmax = 100, fmax.cv = 3, mdt = 12.29, mdt.cv = 12,
              tlag = 0, tlag.cv = 0, beta = 1.727, beta.cv = 9)
# simulate test data with low variability
dt <- sim.dp(tp, model.par = par.t, seed = 100, plot = FALSE)

# bootstrap. to reduce test run time, n.boots of 100 was used in the example.
# In practice, it is recommended to use n.boots of 5000--10000.
# Set `output.to.screen = TRUE` to view the result on screen
d <- bootf2(dt$sim.disso, dr$sim.disso, n.boots = 100, print.report = FALSE)

[Package bootf2 version 0.4.1 Index]

Estimate 90% Confidence Intervals of f_2 with Bootstrap Methodology