mztia: Martinez & Zhao Tolerance Interval Approach
In disprofas: Non-Parametric Dissolution Profile Analysis
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
The Martinez & Zhao Tolerance Interval Approach (mztia) is a
simple approach for the comparison of dissolution profiles. The
mztia() function calculates tolerance intervals (TI) at each time
point of the dissolution profiles of a set of reference batches. By aid
of a graphical display the test batches are checked to lie within the TI
boundaries or within certain limits exceeding the TI boundaries by a
specified percentage.
Usage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Arguments
data
A data frame with the dissolution profile data in wide or in
long format (see parameter shape). If the data frame is in wide
format, it is tried to extract the information on the time points of
dissolution testing from the column names of the columns specified by
the tcol parameter. Thus, they must contain extractable numeric
information, e.g., (t_0, t_5, t_10). If the data frame is in long
format, it must have a column of time points (column specified via the
tcol parameter).
shape
A character string specifying if the data frame is in long or
in wide format.
tcol
If shape is "wide" an integer vector of indices,
if shape is "long" an integer, specifying the column(s)
containing the profile time points. If the data frame is in wide
format it is reshaped using the function reshape()
from the ‘stats’ package.
grouping
A character string specifying the column in data
that contains the group names (i.e. a factorial variable, e.g., for the
differentiation of batches or formulations of a drug product).
reference
A character string specifying the name of the reference
group from the grouping variable.
response
A character string that is expected if data is
provided in long format in order to specify the column with the % drug
release values. The default is NULL.
alpha
A numeric value between 0 and 1 specifying the probability
level. The default is 0.05.
P
A numeric value between 0 and 1 specifying the proportion of the
population being enclosed by the tolerance interval boundaries. The
default is 0.99.
cap
A logical variable specifying if the calculated tolerance limits
should be limited (i.e. capped). The default is TRUE.
bounds
A numeric vector of the form c(lower, upper) specifying
the “lower” and “upper” limits, respectively, for the %
drug release at which the calculated tolerance interval limits should
be capped (see parameter cap. This parameter is only relevant if
cap = TRUE. The default is c(0, 100).
QS
A numeric vector of the form c(Q S1, Q S2) that specifies
the allowable deviations from the specifications in percent according to
the S1 and S2 acceptance criteria of USP chapter <711> on
dissolution. The default is c(5, 15).
...
Further arguments passed on to the reshape()
from the ‘stats’ package.
Details
The tolerance interval approach proposed by Martinez & Zhao (2018)
is a simple approach for the comparison of dissolution profiles. The authors
propose to calculate for each time point of a set of reference dissolution
profiles a tolerance interval (TI), i.e. intervals containing P%
of the population of potential values for reference product at a probability
level of alpha / 2 per tail (i.e., (1 - alpha) 100% confidence).
Based on these TIs the dissolution profiles of the test batch(es) is
(are) compared, i.e. the corresponding data points should lie within the
TIs. The TIs are calculated as
Y_{utl,ltl} = Y.bar +- k*s
where Y.bar is the average, s is the sample standard
deviation, and the factor k is calculated according to Hahn (Hahn &
Meeker (1991)), as proposed in Martinez & Zhao (2018).
Since the goal of the comparison is not to confirm simply
“statistical sameness” but “product comparability”,
Martinez & Zhao propose allowing acceptable deviations by utilizing the
concepts described by the United States Pharmacopoeia (USP), chapter <711>
on dissolution, defining allowable deviations from a set of product
specifications (Q). The TIs serve as the target value Q
at each sampling time. The allowable deviations about Q are defined
by the S1 and S2 acceptance criteria of USP chapter <711> on
dissolution:
The S1 level boundary is defined by Q +- 5%
at each time point. For every 12 profiles tested, only one profile is
allowed to exceed the S1 bounds.
The S2 level boundary is defined by Q +- 15%
at each time point. No observation from any of the test dissolution
profiles is allowed to exceed the S2 bounds.
In situations where the reference formulation itself has more than one of
twelve observations (profiles) exceeding S1 at one or more time points,
additional runs of the reference product must be performed. It is deemed
appropriate to use the same values of S1 and S2 across all time
points because the high variability associated with the early sampling times
is already factored into the TIs.
TI calculation according to Hahn is proposed because it appeared to be
more numerically stable and gave more consistent TIs than the TI
calculation method proposed by Howe (Howe 1969) when samples were very
variable. The reason might be due to the less stringent requirements
imposed by Hahn's method with respect to the normality of the data.
Value
An object of class ‘mztia’ is returned, containing the
following elements:
Variables
A list of the variables and the corresponding values.
Limits
A data frame of the limits calculated for each time point.
Data
A data frame consisting of the provided data, complemented by
the calculated tolerance interval results.
Profile.TP
If shape is "wide" a named numeric vector
of the columns in data specified by tcol. Given that the
column names contain extractable numeric information, e.g., specifying
the testing time points of the dissolution profile, it contains the
corresponding values. Elements where no numeric information could be
extracted are NA. If shape is "long" it is a numeric
value, specifying the column containing the % release values.
References
Martinez, M.N., and Zhao, X. A simple approach for comparing the
in vitro dissolution profiles of highly variable drug products: a
proposal. AAPS Journal. (2018); 20: 78.
doi: 10.1208/s12248-018-0238-1
Howe, W.G. Two-sided tolerance limits for normal populations - some
improvements. J Am Stat Assoc. (1969); 64: 610-620.
doi: 10.1080/01621459.1969.10500999
Hahn, G.J., and Meeker, W. Q. Statistical intervals: A guide for
practitioners. (1991); John Wiley & Sons, New York.
Hahn's method is also described in: SAS/QC 13.1: User's Guide. Chapter 5,
sub-chapter “Details: INTERVALS Statement”, pp 421-424. SAS Institute
Inc. 2013. Cary, NC.
https://support.sas.com/documentation/cdl/en/qcug/66857/PDF/default/qcug.pdf
U.S. Pharmacopoeia. 2016 U.S. Pharmacopoeia-National Formulary
(USP 39 NF 34). Volume 1. Rockville, Md: United States Pharmacopeial
Convention, Inc; 2015. <711> Dissolution.
See Also
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
# Fluid weights of 100 drink cans were measured in ounces:
str(dip5)
# 'data.frame': 100 obs. of 3 variables:
# $ type : Factor w/ 1 level "reference": 1 1 1 1 1 1 1 1 1 1 ...
# $ batch : Factor w/ 100 levels "b1","b10","b100",..: 1 13 24 35 46 57 68 ...
# $ weight: num 12.1 12 12 12 12 ...
# Calculation of tolerance intervals
m_alpha_P <- matrix(c(rep(c(0.01, 0.05, 0.1), each = 3),
1 - rep(c(0.1, 0.05, 0.01), times = 3)),
ncol = 2, byrow = FALSE)
ll <-
apply(m_alpha_P, MARGIN = 1, FUN = function(x)
mztia(data = dip5, shape = "long", tcol = 1, grouping = "type",
reference = "reference", response = "weight", alpha = x[1],
P = x[2], cap = FALSE)[["Data"]][102, "weight"])
ul <-
apply(m_alpha_P, MARGIN = 1, FUN = function(x)
mztia(data = dip5, shape = "long", tcol = 1, grouping = "type",
reference = "reference", response = "weight", alpha = x[1],
P = x[2], cap = FALSE)[["Data"]][103, "weight"])
# Expected results in ll and ul
rbind(ll, ul)
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
# ll 11.91648 11.8987 11.86395 11.92132 11.90446 11.87152 11.92373 11.90734
# ul 12.10212 12.1199 12.15465 12.09728 12.11414 12.14708 12.09487 12.11126
# [,9]
# ll 11.8753
# ul 12.1433
# Use a data frame in wide format
# Dissolution data of one reference batch and one test batch of n = 6
# tablets each:
str(dip1)
# 'data.frame': 12 obs. of 10 variables:
# $ type : Factor w/ 2 levels "R","T": 1 1 1 1 1 1 2 2 2 2 ...
# $ tablet: Factor w/ 6 levels "1","2","3","4",..: 1 2 3 4 5 6 1 2 3 4 ...
# $ t.5 : num 42.1 44.2 45.6 48.5 50.5 ...
# $ t.10 : num 59.9 60.2 55.8 60.4 61.8 ...
# $ t.15 : num 65.6 67.2 65.6 66.5 69.1 ...
# $ t.20 : num 71.8 70.8 70.5 73.1 72.8 ...
# $ t.30 : num 77.8 76.1 76.9 78.5 79 ...
# $ t.60 : num 85.7 83.3 83.9 85 86.9 ...
# $ t.90 : num 93.1 88 86.8 88 89.7 ...
# $ t.120 : num 94.2 89.6 90.1 93.4 90.8 ...
# Using the defaults; Limits are capped to the range specified by 'bounds'
res1 <- mztia(data = dip1, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R")
res1$Limits
# Expected results in res1$Limits
# Time Mean LTL UTL S1.LTL S1.UTL S2.LTL S2.UTL
# 1 5 46.77167 27.22641 66.31693 22.22641 71.31693 12.22641 81.31693
# 2 10 60.13333 46.15483 74.11184 41.15483 79.11184 31.15483 89.11184
# 3 15 67.27500 56.90417 77.64583 51.90417 82.64583 41.90417 92.64583
# 4 20 71.98667 65.44354 78.52979 60.44354 83.52979 50.44354 93.52979
# 5 30 78.07000 69.54259 86.59741 64.54259 91.59741 54.54259 101.59741
# 6 60 84.81667 77.20275 92.43058 72.20275 97.43058 62.20275 107.43058
# 7 90 89.09333 76.24588 100.00000 71.24588 105.00000 61.24588 115.00000
# 8 120 91.43833 80.29321 100.00000 75.29321 105.00000 65.29321 115.00000
# Without capping of limits to 105%
res2 <- mztia(data = dip1, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R", cap = FALSE)
res2$Limits
# Expected results in res1$Limits
# Time Mean LTL UTL S1.LTL S1.UTL S2.LTL S2.UTL
# 1 5 46.77167 27.22641 66.31693 22.22641 71.31693 12.22641 81.31693
# 2 10 60.13333 46.15483 74.11184 41.15483 79.11184 31.15483 89.11184
# 3 15 67.27500 56.90417 77.64583 51.90417 82.64583 41.90417 92.64583
# 4 20 71.98667 65.44354 78.52979 60.44354 83.52979 50.44354 93.52979
# 5 30 78.07000 69.54259 86.59741 64.54259 91.59741 54.54259 101.59741
# 6 60 84.81667 77.20275 92.43058 72.20275 97.43058 62.20275 107.43058
# 7 90 89.09333 76.24588 101.94079 71.24588 106.94079 61.24588 116.94079
# 8 120 91.43833 80.29321 102.58346 75.29321 107.58346 65.29321 117.58346
# Tolerance intervals are calculated exclusively for the level of the
# grouping variable that is specified by the reference variable. Therefore,
# the following code produces the same limits summary as in res2$Limits.
tmp <- rbind(dip1,
data.frame(type = "T2",
tablet = as.factor(1:6),
dip1[7:12, 3:10]))
res2 <- mztia(data = dip1, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R", cap = FALSE)
res3 <- mztia(data = tmp, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R", cap = FALSE)
isTRUE(all.equal(res2$Limits, res3$Limits))
# [1] TRUE
disprofas documentation built on Dec. 8, 2021, 5:10 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Description Usage Arguments Details Value References See Also Examples
Description
The Martinez & Zhao Tolerance Interval Approach (mztia) is a
simple approach for the comparison of dissolution profiles. The
mztia() function calculates tolerance intervals (TI) at each time
point of the dissolution profiles of a set of reference batches. By aid
of a graphical display the test batches are checked to lie within the TI
boundaries or within certain limits exceeding the TI boundaries by a
specified percentage.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 |
Arguments
data |
A data frame with the dissolution profile data in wide or in
long format (see parameter |
shape |
A character string specifying if the data frame is in long or in wide format. |
tcol |
If |
grouping |
A character string specifying the column in |
reference |
A character string specifying the name of the reference
group from the |
response |
A character string that is expected if |
alpha |
A numeric value between 0 and 1 specifying the probability
level. The default is |
P |
A numeric value between 0 and 1 specifying the proportion of the
population being enclosed by the tolerance interval boundaries. The
default is |
cap |
A logical variable specifying if the calculated tolerance limits
should be limited (i.e. capped). The default is |
bounds |
A numeric vector of the form |
QS |
A numeric vector of the form |
... |
Further arguments passed on to the |
Details
The tolerance interval approach proposed by Martinez & Zhao (2018) is a simple approach for the comparison of dissolution profiles. The authors propose to calculate for each time point of a set of reference dissolution profiles a tolerance interval (TI), i.e. intervals containing P% of the population of potential values for reference product at a probability level of alpha / 2 per tail (i.e., (1 - alpha) 100% confidence). Based on these TIs the dissolution profiles of the test batch(es) is (are) compared, i.e. the corresponding data points should lie within the TIs. The TIs are calculated as
Y_{utl,ltl} = Y.bar +- k*s
where Y.bar is the average, s is the sample standard deviation, and the factor k is calculated according to Hahn (Hahn & Meeker (1991)), as proposed in Martinez & Zhao (2018).
Since the goal of the comparison is not to confirm simply “statistical sameness” but “product comparability”, Martinez & Zhao propose allowing acceptable deviations by utilizing the concepts described by the United States Pharmacopoeia (USP), chapter <711> on dissolution, defining allowable deviations from a set of product specifications (Q). The TIs serve as the target value Q at each sampling time. The allowable deviations about Q are defined by the S1 and S2 acceptance criteria of USP chapter <711> on dissolution:
The S1 level boundary is defined by Q +- 5% at each time point. For every 12 profiles tested, only one profile is allowed to exceed the S1 bounds.
The S2 level boundary is defined by Q +- 15% at each time point. No observation from any of the test dissolution profiles is allowed to exceed the S2 bounds.
In situations where the reference formulation itself has more than one of twelve observations (profiles) exceeding S1 at one or more time points, additional runs of the reference product must be performed. It is deemed appropriate to use the same values of S1 and S2 across all time points because the high variability associated with the early sampling times is already factored into the TIs.
TI calculation according to Hahn is proposed because it appeared to be more numerically stable and gave more consistent TIs than the TI calculation method proposed by Howe (Howe 1969) when samples were very variable. The reason might be due to the less stringent requirements imposed by Hahn's method with respect to the normality of the data.
Value
An object of class ‘mztia’ is returned, containing the
following elements:
Variables |
A list of the variables and the corresponding values. |
Limits |
A data frame of the limits calculated for each time point. |
Data |
A data frame consisting of the provided data, complemented by the calculated tolerance interval results. |
Profile.TP |
If |
References
Martinez, M.N., and Zhao, X. A simple approach for comparing the
in vitro dissolution profiles of highly variable drug products: a
proposal. AAPS Journal. (2018); 20: 78.
doi: 10.1208/s12248-018-0238-1
Howe, W.G. Two-sided tolerance limits for normal populations - some
improvements. J Am Stat Assoc. (1969); 64: 610-620.
doi: 10.1080/01621459.1969.10500999
Hahn, G.J., and Meeker, W. Q. Statistical intervals: A guide for
practitioners. (1991); John Wiley & Sons, New York.
Hahn's method is also described in: SAS/QC 13.1: User's Guide. Chapter 5,
sub-chapter “Details: INTERVALS Statement”, pp 421-424. SAS Institute
Inc. 2013. Cary, NC.
https://support.sas.com/documentation/cdl/en/qcug/66857/PDF/default/qcug.pdf
U.S. Pharmacopoeia. 2016 U.S. Pharmacopoeia-National Formulary (USP 39 NF 34). Volume 1. Rockville, Md: United States Pharmacopeial Convention, Inc; 2015. <711> Dissolution.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | # Fluid weights of 100 drink cans were measured in ounces:
str(dip5)
# 'data.frame': 100 obs. of 3 variables:
# $ type : Factor w/ 1 level "reference": 1 1 1 1 1 1 1 1 1 1 ...
# $ batch : Factor w/ 100 levels "b1","b10","b100",..: 1 13 24 35 46 57 68 ...
# $ weight: num 12.1 12 12 12 12 ...
# Calculation of tolerance intervals
m_alpha_P <- matrix(c(rep(c(0.01, 0.05, 0.1), each = 3),
1 - rep(c(0.1, 0.05, 0.01), times = 3)),
ncol = 2, byrow = FALSE)
ll <-
apply(m_alpha_P, MARGIN = 1, FUN = function(x)
mztia(data = dip5, shape = "long", tcol = 1, grouping = "type",
reference = "reference", response = "weight", alpha = x[1],
P = x[2], cap = FALSE)[["Data"]][102, "weight"])
ul <-
apply(m_alpha_P, MARGIN = 1, FUN = function(x)
mztia(data = dip5, shape = "long", tcol = 1, grouping = "type",
reference = "reference", response = "weight", alpha = x[1],
P = x[2], cap = FALSE)[["Data"]][103, "weight"])
# Expected results in ll and ul
rbind(ll, ul)
# [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
# ll 11.91648 11.8987 11.86395 11.92132 11.90446 11.87152 11.92373 11.90734
# ul 12.10212 12.1199 12.15465 12.09728 12.11414 12.14708 12.09487 12.11126
# [,9]
# ll 11.8753
# ul 12.1433
# Use a data frame in wide format
# Dissolution data of one reference batch and one test batch of n = 6
# tablets each:
str(dip1)
# 'data.frame': 12 obs. of 10 variables:
# $ type : Factor w/ 2 levels "R","T": 1 1 1 1 1 1 2 2 2 2 ...
# $ tablet: Factor w/ 6 levels "1","2","3","4",..: 1 2 3 4 5 6 1 2 3 4 ...
# $ t.5 : num 42.1 44.2 45.6 48.5 50.5 ...
# $ t.10 : num 59.9 60.2 55.8 60.4 61.8 ...
# $ t.15 : num 65.6 67.2 65.6 66.5 69.1 ...
# $ t.20 : num 71.8 70.8 70.5 73.1 72.8 ...
# $ t.30 : num 77.8 76.1 76.9 78.5 79 ...
# $ t.60 : num 85.7 83.3 83.9 85 86.9 ...
# $ t.90 : num 93.1 88 86.8 88 89.7 ...
# $ t.120 : num 94.2 89.6 90.1 93.4 90.8 ...
# Using the defaults; Limits are capped to the range specified by 'bounds'
res1 <- mztia(data = dip1, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R")
res1$Limits
# Expected results in res1$Limits
# Time Mean LTL UTL S1.LTL S1.UTL S2.LTL S2.UTL
# 1 5 46.77167 27.22641 66.31693 22.22641 71.31693 12.22641 81.31693
# 2 10 60.13333 46.15483 74.11184 41.15483 79.11184 31.15483 89.11184
# 3 15 67.27500 56.90417 77.64583 51.90417 82.64583 41.90417 92.64583
# 4 20 71.98667 65.44354 78.52979 60.44354 83.52979 50.44354 93.52979
# 5 30 78.07000 69.54259 86.59741 64.54259 91.59741 54.54259 101.59741
# 6 60 84.81667 77.20275 92.43058 72.20275 97.43058 62.20275 107.43058
# 7 90 89.09333 76.24588 100.00000 71.24588 105.00000 61.24588 115.00000
# 8 120 91.43833 80.29321 100.00000 75.29321 105.00000 65.29321 115.00000
# Without capping of limits to 105%
res2 <- mztia(data = dip1, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R", cap = FALSE)
res2$Limits
# Expected results in res1$Limits
# Time Mean LTL UTL S1.LTL S1.UTL S2.LTL S2.UTL
# 1 5 46.77167 27.22641 66.31693 22.22641 71.31693 12.22641 81.31693
# 2 10 60.13333 46.15483 74.11184 41.15483 79.11184 31.15483 89.11184
# 3 15 67.27500 56.90417 77.64583 51.90417 82.64583 41.90417 92.64583
# 4 20 71.98667 65.44354 78.52979 60.44354 83.52979 50.44354 93.52979
# 5 30 78.07000 69.54259 86.59741 64.54259 91.59741 54.54259 101.59741
# 6 60 84.81667 77.20275 92.43058 72.20275 97.43058 62.20275 107.43058
# 7 90 89.09333 76.24588 101.94079 71.24588 106.94079 61.24588 116.94079
# 8 120 91.43833 80.29321 102.58346 75.29321 107.58346 65.29321 117.58346
# Tolerance intervals are calculated exclusively for the level of the
# grouping variable that is specified by the reference variable. Therefore,
# the following code produces the same limits summary as in res2$Limits.
tmp <- rbind(dip1,
data.frame(type = "T2",
tablet = as.factor(1:6),
dip1[7:12, 3:10]))
res2 <- mztia(data = dip1, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R", cap = FALSE)
res3 <- mztia(data = tmp, shape = "wide", tcol = 3:10, grouping = "type",
reference = "R", cap = FALSE)
isTRUE(all.equal(res2$Limits, res3$Limits))
# [1] TRUE
|
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.