Bioequivalence Docstrings
Bioequivalence.Bioequivalence — ModuleBioequivalence.jlThis module offers a suite of routines for bioequivalence (BE) analysis.
Bioequivalence.EMA_CMAX — ConstantEMA_CMAX = [GeometricMeanRatioBounds(), AverageBioequivalenceWithExpandingLimits(0.76, CV_min = 0.3, CV_max = 0.5)]Criteria for the European Medicines Agency average bioequivalence for Cmax endpoints.
Bioequivalence.EMA_NarrowTherapeuticIndex — ConstantEMA_NarrowTherapeuticIndex = [AverageBioequivalenceWithExpandingLimits(0, lower_bound = 0.9, upper_bound = 1.11)]Criteria for the European Medicines Agency narrow therapeutic index drugs.
Bioequivalence.FDA — ConstantFDA = [GeometricMeanRatioBounds(), AverageBioequivalenceWithExpandingLimits((log(1.25) / 0.25)^2)]Criteria for the U.S. Food and Drug Administration (non-narrow therapeutic index drugs):
If the within subject variability of the reference formulation is less than 30%, the unscaled confidence interval should pass the standard critertia: LB ≥ 80% ^ UB ≤ 125%. Otherwise, the 95% upper confidence bound for (T̄ₜ - Ȳᵣ)² - 𝜃 * σwᵣ² ≤ 0 (numbers should be kept to a minimum of four significant figures for comparison) with 𝜃 ≈ 0.7967 and 80% ≤ GMR ≤ 125%.
Bioequivalence.FDA_NarrowTherapeuticIndex — ConstantFDA_NarrowTherapeuticIndex = [
AverageBioequivalenceWithExpandingLimits(0),
AverageBioequivalenceWithExpandingLimits((log(1 / 0.9) / 0.1)^2, CV_min = 0, CV_max = 0.2142),
WithinSubjectVariabilityRatio()
]Criteria for the U.S. Food and Drug Administration Narrow Therapeutic Index drug:
a) The 95% upper confidence bound for (T̄ₜ - Ȳᵣ)² - 𝜃 * σwᵣ² ≤ 0 (numbers should be kept to a minimum of four significant figures for comparison). 𝜃 ≈ 1.11.
b) Regular unscaled bioequivalence limits of [80.00%, 125.00%] should be passed.
c) The proposed requirement for the upper limit of the 90% equal-tails confidence interval for σwₜ / σwᵣ ≤ 2.5.
Bioequivalence.NoAssessmentBioequivalenceCriterion — ConstantNoAssessmentBioequivalenceCriterionUsed as a default for pumas_be indicating no assessment is needed.
Bioequivalence.StandardBioequivalenceCriterion — ConstantStandardBioequivalenceCriterionStandard bioequivalence criterion: LB ≥ 80% ^ UB ≤ 125%.
Bioequivalence.AverageBioequivalenceWithExpandingLimits — TypeAverageBioequivalenceWithExpandingLimits(
θ::Real = -1;
CV_min::Real = 0.3,
CV_max::Real = Inf,
lower_bound::Real = 0.8,
upper_bound::Real = 1.25
)Criteria for the average bioequivalence acceptable bounds. If θ < 0, use the θ from the BioequivalenceEndpointOutput if available. If θ > 0, overwrite BioequivalenceEndpointOutput parameters in assessment. If θ == 0, use unscaled confidence interval with specified interval.
See also assess_be
Bioequivalence.BioequivalenceCriterion — TypeBioequivalenceCriterionAn abstract bioequivalence criterion.
Bioequivalence.BioequivalenceEndpointOutput — TypeBioequivalenceEndpointOutputA bioequivalence study input object.
See also: pumas_be.
Fields
data::DataFramedata used for the studydata_stats::NamedTupletotal::Intrefers to the number of observations the data passed to the function had.used_for_analysis::Intrefers to the number of observations used for fitting the model (e.g., drop missing values)treatment::DataFramegives a DataFrame with the summary statistics of the statistical model's response by treatmentsequence::DataFramegives a DataFrame with the summary statistics of the statistical model's response by sequenceperiod::DataFramegives a DataFrame with the summary statistics of the statistical model's response by period
design::NamedTuplenumber of subjects in each sequencemodelstatistical models used for the analysismodel_statsstatistics for the modelresult::DataFrameresults for inference
Examples
julia> data = dataset(joinpath("bioequivalence", "RST_RTS_SRT_STR_TRS_TSR", "PJ2017_4_5"))
186×5 DataFrame
Row │ id sequence period AUC Cmax
│ Int64 String3 Int64 Int64? Int64?
─────┼───────────────────────────────────────────
1 │ 1 SRT 1 7260 1633
2 │ 1 SRT 2 6463 1366
3 │ 1 SRT 3 8759 2141
4 │ 2 RTS 1 3457 776
5 │ 2 RTS 2 6556 2387
6 │ 2 RTS 3 4081 1355
7 │ 4 TSR 1 4006 1326
8 │ 4 TSR 2 4879 1028
9 │ 4 TSR 3 3817 1052
10 │ 5 STR 1 4250 945
11 │ 5 STR 2 3487 1041
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮
177 │ 61 RTS 3 3779 1144
178 │ 62 SRT 1 5787 1461
179 │ 62 SRT 2 7069 1995
180 │ 62 SRT 3 6530 1236
181 │ 63 TRS 1 2204 495
182 │ 63 TRS 2 2927 770
183 │ 63 TRS 3 missing missing
184 │ 67 RST 1 4045 1025
185 │ 67 RST 2 7865 2668
186 │ 67 RST 3 missing missing
165 rows omitted
julia> output = pumas_be(data, endpoint = :Cmax)
Design: RST|RTS|SRT|STR|TRS|TSR
Sequences: RST|RTS|SRT|STR|TRS|TSR (6)
Periods: 1:3 (3)
Subjects per Sequence: (RST = 9, RTS = 11, SRT = 11, STR = 10, TRS = 11, TSR = 10)
Average Bioequivalence
───────────────────────────────────────────────────────────────────────────────────
δ SE lnLB lnUB GMR LB UB CV
───────────────────────────────────────────────────────────────────────────────────
S - R 0.466225 0.0525563 0.379094 0.553357 1.59397 1.46096 1.73908 0.296903
T - R 0.261828 0.0525354 0.174731 0.348925 1.2993 1.19093 1.41754 0.296903
───────────────────────────────────────────────────────────────────────────────────
julia> output.data_stats.treatment
3×10 DataFrame
Row │ formulation exp_mean mean std min q25 median q75 max n
│ Cat… Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64 Int64
─────┼──────────────────────────────────────────────────────────────────────────────────────────────
1 │ R 837.478 6.7304 0.466938 5.89715 6.3257 6.66568 7.09589 7.71334 62
2 │ S 1339.96 7.20039 0.419893 6.09131 6.92952 7.20117 7.5251 8.02027 61
3 │ T 1078.08 6.98294 0.473224 5.75574 6.62539 7.00851 7.29641 7.90286 61
julia> output.data_stats.sequence
6×10 DataFrame
Row │ sequence exp_mean mean std min q25 median q75 max n
│ Cat… Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64 Int64
─────┼───────────────────────────────────────────────────────────────────────────────────────────
1 │ RST 997.282 6.90503 0.493722 5.90263 6.6385 6.92755 7.17642 7.88908 26
2 │ RTS 1084.03 6.98844 0.499403 6.09131 6.4677 7.05618 7.32449 7.77779 33
3 │ SRT 1187.43 7.07954 0.482725 5.89715 6.63068 7.11964 7.45124 7.90286 33
4 │ STR 869.299 6.76769 0.404857 6.04501 6.4758 6.8663 6.95607 7.71913 30
5 │ TRS 1180.5 7.07369 0.466733 6.20456 6.71254 7.11698 7.39368 7.96797 32
6 │ TSR 1071.51 6.97682 0.556906 5.75574 6.69448 7.02452 7.28049 8.02027 30
julia> output.data_stats.period
3×10 DataFrame
Row │ period exp_mean mean std min q25 median q75 max n
│ Cat… Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64 Int64
─────┼─────────────────────────────────────────────────────────────────────────────────────────
1 │ 1 1009.89 6.9176 0.498246 5.75574 6.46653 6.97018 7.28186 7.72356 62
2 │ 2 1108.56 7.01082 0.460876 5.89715 6.63035 7.06641 7.31235 8.02027 62
3 │ 3 1076.82 6.98176 0.516518 6.03787 6.63167 6.99805 7.30986 7.96797 60
julia> output.model
StatsModels.TableRegressionModel{GLM.LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}
endpoint ~ 1 + formulation + period + id
Coefficients:
─────────────────────────────────────────────────────────────────────────────────
Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
─────────────────────────────────────────────────────────────────────────────────
(Intercept) 7.18626 0.170519 42.14 <1e-72 6.84859 7.52394
formulation: S 0.466225 0.0525563 8.87 <1e-14 0.362149 0.570301
formulation: T 0.261828 0.0525354 4.98 <1e-05 0.157794 0.365862
period: 2 0.0504953 0.0302861 1.67 0.0981 -0.00947942 0.11047
period: 3 0.00727282 0.0306504 0.24 0.8128 -0.0534233 0.067969
id: 2 -0.214448 0.237319 -0.90 0.3680 -0.684403 0.255508
id: 4 -0.401034 0.237319 -1.69 0.0937 -0.87099 0.0689214
id: 5 -0.606075 0.237319 -2.55 0.0119 -1.07603 -0.136119
id: 6 -0.356888 0.237319 -1.50 0.1353 -0.826843 0.113068
id: 7 -0.349386 0.237319 -1.47 0.1436 -0.819342 0.12057
id: 8 -0.850534 0.237319 -3.58 0.0005 -1.32049 -0.380578
id: 9 -0.237702 0.237319 -1.00 0.3186 -0.707658 0.232254
id: 10 -0.589593 0.237319 -2.48 0.0144 -1.05955 -0.119637
id: 11 -0.109224 0.237319 -0.46 0.6462 -0.57918 0.360732
id: 12 -0.0629286 0.237319 -0.27 0.7913 -0.532884 0.407027
id: 13 -0.829916 0.237319 -3.50 0.0007 -1.29987 -0.35996
id: 14 -0.169207 0.237319 -0.71 0.4773 -0.639163 0.300749
id: 15 -0.85675 0.237319 -3.61 0.0005 -1.32671 -0.386794
id: 16 -0.385301 0.237319 -1.62 0.1071 -0.855257 0.084655
id: 17 0.069506 0.237319 0.29 0.7701 -0.40045 0.539462
id: 18 -0.490339 0.237319 -2.07 0.0410 -0.960295 -0.0203832
id: 19 -0.440248 0.237319 -1.86 0.0661 -0.910203 0.0297083
id: 20 0.292819 0.237319 1.23 0.2197 -0.177137 0.762775
id: 21 -1.02247 0.237319 -4.31 <1e-04 -1.49242 -0.552511
id: 22 -0.975804 0.237319 -4.11 <1e-04 -1.44576 -0.505848
id: 23 -1.14843 0.237319 -4.84 <1e-05 -1.61839 -0.678477
id: 24 -0.620657 0.237319 -2.62 0.0101 -1.09061 -0.150702
id: 25 0.241019 0.237319 1.02 0.3119 -0.228937 0.710974
id: 26 -0.768347 0.237319 -3.24 0.0016 -1.2383 -0.298391
id: 27 -0.837325 0.237319 -3.53 0.0006 -1.30728 -0.367369
id: 28 -0.726773 0.237319 -3.06 0.0027 -1.19673 -0.256817
id: 29 -0.0892112 0.237319 -0.38 0.7077 -0.559167 0.380745
id: 30 -1.01968 0.237319 -4.30 <1e-04 -1.48963 -0.549721
id: 31 -0.425466 0.237319 -1.79 0.0756 -0.895421 0.0444903
id: 32 -0.40831 0.237319 -1.72 0.0880 -0.878266 0.0616457
id: 33 -0.410798 0.237319 -1.73 0.0861 -0.880754 0.0591575
id: 34 -0.817252 0.237319 -3.44 0.0008 -1.28721 -0.347296
id: 35 -0.500386 0.237319 -2.11 0.0371 -0.970342 -0.0304301
id: 36 0.152288 0.237319 0.64 0.5223 -0.317668 0.622244
id: 37 -0.750648 0.237319 -3.16 0.0020 -1.2206 -0.280692
id: 39 -1.38512 0.237319 -5.84 <1e-07 -1.85507 -0.915161
id: 40 -1.17263 0.237319 -4.94 <1e-05 -1.64259 -0.702674
id: 41 0.0824414 0.237319 0.35 0.7289 -0.387514 0.552397
id: 42 -0.305264 0.237319 -1.29 0.2009 -0.77522 0.164691
id: 43 -0.59991 0.237319 -2.53 0.0128 -1.06987 -0.129954
id: 44 -0.841676 0.237319 -3.55 0.0006 -1.31163 -0.371721
id: 45 -0.66548 0.237319 -2.80 0.0059 -1.13544 -0.195524
id: 46 -0.864623 0.237319 -3.64 0.0004 -1.33458 -0.394667
id: 47 -0.283627 0.237319 -1.20 0.2344 -0.753583 0.186329
id: 48 -0.205965 0.237319 -0.87 0.3872 -0.675921 0.26399
id: 49 0.0450347 0.237319 0.19 0.8498 -0.424921 0.514991
id: 50 -0.514194 0.237319 -2.17 0.0323 -0.98415 -0.0442386
id: 51 -0.176366 0.237319 -0.74 0.4589 -0.646322 0.29359
id: 52 -0.99198 0.237319 -4.18 <1e-04 -1.46194 -0.522024
id: 53 -0.494151 0.237319 -2.08 0.0395 -0.964107 -0.0241951
id: 54 -0.0659794 0.237319 -0.28 0.7815 -0.535935 0.403976
id: 55 -0.0876202 0.237319 -0.37 0.7126 -0.557576 0.382336
id: 56 -0.180088 0.237319 -0.76 0.4495 -0.650043 0.289868
id: 57 -0.402746 0.237319 -1.70 0.0923 -0.872702 0.06721
id: 58 0.0550554 0.237319 0.23 0.8169 -0.4149 0.525011
id: 59 -0.178369 0.237319 -0.75 0.4538 -0.648325 0.291587
id: 60 -0.70748 0.237319 -2.98 0.0035 -1.17744 -0.237524
id: 61 -0.721606 0.237319 -3.04 0.0029 -1.19156 -0.25165
id: 62 -0.0939777 0.237319 -0.40 0.6928 -0.563934 0.375978
id: 63 -0.888067 0.266187 -3.34 0.0011 -1.41519 -0.360944
id: 67 -0.00497377 0.266231 -0.02 0.9851 -0.532183 0.522235
─────────────────────────────────────────────────────────────────────────────────Bioequivalence.GeometricMeanRatioBounds — TypeGeometricMeanRatioBounds(lower_bound::Real = 0.8, upper_bound::Real = 1.25)Criteria for the point estimate. Checks that the geometric mean ratio (GMR) lies within the accepted range.
Bioequivalence.ReferenceScaledAverageBioequivalance — TypeReferenceScaledAverageBioequivalance(
data::AbstractDataFrame,
𝜃::Real,
σwᵣ::Real = -1;
k::Union{Real, Missing} = -1,
level::Real = 0.9,
level_y::Real = 0.95,
userepeatedobsonly::Bool = true
) -> ReferenceScaledAverageBioequivalanceReference-scaled average bioequivalence.
Arguments
data::AbstractDataFrame: dataset adhering to the schema frompreprocess_be.𝜃::Real: parameter for reference-scaling.σwᵣ::Real: estimate for the within-subject variability. If negative, the function will compute it.k::Real: degrees of freedom of the within-subject variability. If negative, the function will compute it.level::Real: confidence level for the confidence interval.level_y::Real: confidence level for the upper bound of (Ȳₜ - Ȳᵣ)² - 𝜃 * σwᵣ².userepeatedobsonly: controls the behavior ofwithin_subject_variabilityif used to estimateσwᵣandk.
See also preprocess_be and within_subject_variability
Reference
Food and Drug Administration (2021). Bioequivalence Studies With Pharmacokinetic Endpoints for Drugs Submitted Under an ANDA Guidance for Industry. https://www.fda.gov/media/87219/download.
Examples
julia> data = dataset(joinpath("bioequivalence", "RTTR_TRRT", "SLTGSF2020_DS16"));
julia> pkdata = preprocess_be(data, endpoint = :PK);
julia> rsabe = ReferenceScaledAverageBioequivalance(pkdata, 0.76)
Critical boundary: -0.0416
Regulatory parameter: 0.76
95.0% upper confidence bound with 36.0 degrees of freedom
Bioequivalence.WithinSubjectVariabilityRatio — TypeWithinSubjectVariabilityRatio(upper_bound::Real = 2.5)Requirement for the upper limit of the 90% equal-tails confidence interval for within-subject variability test/reference ratio is less than or equal to the upper limit.
Bioequivalence.assess_be — Functionassess_be(
criterion::BioequivalenceCriterion,
endpoint_output::BioequivalenceEndpointOutput
) -> BioequivalenceEndpointDecision
assess_be(
criteria::AbstractVector{<:BioequivalenceCriterion},
endpoint_output::BioequivalenceEndpointOutput
) -> BioequivalenceEndpointDecisionReturn a BioequivalenceEndpointDecision struct which includes properties crtieria for a vector of criterion. The result of the assessment is accessible through the assessments property which is a BitMatrix. The rows refer to a formulation comparison (e.g., R|T) and the columns whether the criterion was passed or not.
Bioequivalence.assess_be — Methodassess_be(criteria::AbstractVector{<:BioequivalenceCriterion}, endpoint_output::BioequivalenceEndpointOutput)Return the average bioequivalence conclusions based on the specified criteria.
Examples
julia> data = dataset(joinpath("bioequivalence", "RTTR_TRRT", "SLTGSF2020_DS16"))
152×4 DataFrame
Row │ id sequence period PK
│ Int64 String7 Int64 Float64
─────┼──────────────────────────────────
1 │ 1 RTTR 1 0.2813
2 │ 1 RTTR 2 0.2947
3 │ 1 RTTR 3 0.5471
4 │ 1 RTTR 4 0.651
5 │ 2 RTTR 1 0.2024
6 │ 2 RTTR 2 0.1782
7 │ 2 RTTR 3 0.2076
8 │ 2 RTTR 4 0.3604
9 │ 3 TRRT 1 0.4332
10 │ 3 TRRT 2 0.3131
11 │ 3 TRRT 3 0.2132
⋮ │ ⋮ ⋮ ⋮ ⋮
143 │ 38 RTTR 3 2.6522
144 │ 38 RTTR 4 1.2808
145 │ 39 RTTR 1 0.3847
146 │ 39 RTTR 2 0.2991
147 │ 39 RTTR 3 0.3353
148 │ 39 RTTR 4 0.3483
149 │ 40 RTTR 1 1.0289
150 │ 40 RTTR 2 0.753
151 │ 40 RTTR 3 0.7894
152 │ 40 RTTR 4 0.6129
131 rows omitted
julia> output = pumas_be(data, endpoint = :PK)
Design: RTTR|TRRT
Sequences: RTTR|TRRT (2)
Periods: 1:4 (4)
Subjects per Sequence: (RTTR = 20, TRRT = 18)
Reference scaled using 𝜃 = 0.797
Average Bioequivalence
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
δ SE lnLB lnUB GMR LB UB CVᵣ CVₜ σ_ratio σ⁺ cb dof
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
T - R -0.2378 0.07738 -0.3665 -0.1091 0.7883 0.6931 0.8966 0.4972 0.5141 1.031 1.361 -0.04805 86.56
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
julia> assess_be(StandardBioequivalenceCriterion, output)
Formulation: T - R -- Fail
LB ≥ 0.8% ^ UB ≤ 1.25% -- Fail
julia> assess_be(FDA, output)
Formulation: T - R -- Fail
0.8 ≤ GMR ≤ 1.25 -- Fail
Reference Scaled Average Bioequivalance w/ 𝜃: 0.796689, CV_min: 0.3, CV_max: Inf -- Pass
julia> assess_be(FDA_NarrowTherapeuticIndex, output)
Formulation: T - R -- Fail
LB ≥ 0.8% ^ UB ≤ 1.25% -- Fail
Reference Scaled Average Bioequivalance w/ 𝜃: 1.110084, CV_min: 0.0, CV_max: 0.2142 -- Fail
Upper bound of the within-subject variability ratio ≤ 2.5 -- Pass
Bioequivalence.detect_design — Functiondetect_design(sequences::AbstractVector) ->
NamedTuple{(:design, :replicated, :crossover), Tuple{String, ReplicationType, Bool}}
detect_design(
data::AbstractDataFrame,
sequence::Union{AbstractString, Symbol} = :sequence
) -> NamedTuple{(:design, :replicated, :crossover), Tuple{String, ReplicationType, Bool}}Return the design class (Parallel, Nonreplicatedcrossover, Replicatedcrossover) and design.
Bioequivalence.generate_design — Methodgenerate_design(
sequences::AbstractVector{<:AbstractString},
amt::Union{Number,AbstractVector{<:Number}},
subjects_per_sequence::Union{<:Integer,AbstractVector{<:Integer}},
) -> DataFrameReturns a DataFrame with id, sequence, period, amt, evid, cmt, and time. It can be used to quickly set up data for Pumas, NCA, and Bioequivalence. In order to add covariates, use innerjoin to join the result of this function with another DataFrame with covariates.
Examples
julia> skeleton = generate_design(["RT", "TR"], [0, 50], 10)
40×8 DataFrame
Row │ id sequence period formulation amt time evid cmt
│ Int64 Cat… Int64 Char Int64 Int64 Int64 Int64
─────┼──────────────────────────────────────────────────────────────────
1 │ 1 RT 1 R 0 0 4 1
2 │ 1 RT 2 T 50 0 4 1
3 │ 2 RT 1 R 0 0 4 1
4 │ 2 RT 2 T 50 0 4 1
5 │ 3 RT 1 R 0 0 4 1
6 │ 3 RT 2 T 50 0 4 1
7 │ 4 RT 1 R 0 0 4 1
8 │ 4 RT 2 T 50 0 4 1
9 │ 5 RT 1 R 0 0 4 1
10 │ 5 RT 2 T 50 0 4 1
11 │ 6 RT 1 R 0 0 4 1
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
31 │ 16 TR 1 T 50 0 4 1
32 │ 16 TR 2 R 0 0 4 1
33 │ 17 TR 1 T 50 0 4 1
34 │ 17 TR 2 R 0 0 4 1
35 │ 18 TR 1 T 50 0 4 1
36 │ 18 TR 2 R 0 0 4 1
37 │ 19 TR 1 T 50 0 4 1
38 │ 19 TR 2 R 0 0 4 1
39 │ 20 TR 1 T 50 0 4 1
40 │ 20 TR 2 R 0 0 4 1
19 rows omitted
julia> skeleton = generate_design(["RTRT", "TRTR"], [50, 75], [12, 10])
88×8 DataFrame
Row │ id sequence period formulation amt time evid cmt
│ Int64 Cat… Int64 Char Int64 Int64 Int64 Int64
─────┼──────────────────────────────────────────────────────────────────
1 │ 1 RTRT 1 R 50 0 4 1
2 │ 1 RTRT 2 T 75 0 4 1
3 │ 1 RTRT 3 R 50 0 4 1
4 │ 1 RTRT 4 T 75 0 4 1
5 │ 2 RTRT 1 R 50 0 4 1
6 │ 2 RTRT 2 T 75 0 4 1
7 │ 2 RTRT 3 R 50 0 4 1
8 │ 2 RTRT 4 T 75 0 4 1
9 │ 3 RTRT 1 R 50 0 4 1
10 │ 3 RTRT 2 T 75 0 4 1
11 │ 3 RTRT 3 R 50 0 4 1
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
79 │ 20 TRTR 3 T 75 0 4 1
80 │ 20 TRTR 4 R 50 0 4 1
81 │ 21 TRTR 1 T 75 0 4 1
82 │ 21 TRTR 2 R 50 0 4 1
83 │ 21 TRTR 3 T 75 0 4 1
84 │ 21 TRTR 4 R 50 0 4 1
85 │ 22 TRTR 1 T 75 0 4 1
86 │ 22 TRTR 2 R 50 0 4 1
87 │ 22 TRTR 3 T 75 0 4 1
88 │ 22 TRTR 4 R 50 0 4 1
67 rows omittedBioequivalence.geocv2sigma — Methodgeocv2sigma(CV::Union{Real,Missing}) = √(log(1 + CV^2))Transform the the coefficient of variation (CV) to the σ parameter of a log-normal distribution.
Examples
julia> geocv2sigma(0.30)
0.29356037920852396
Bioequivalence.implied_design — Methodimplied_design(data::AbstractDataFrame) -> NamedTupleSequence and number of subjects per sequence
Bioequivalence.inference_model_type — Methodinference_model_type(be_output::BioequivalenceEndpointOutput)::InferenceModelTypeDetermines the inference model type used in a bioequivalence study for an endpoint.
Bioequivalence.preprocess_be — Methodpreprocess_be(
data::AbstractDataFrame,
id::Union{AbstractString, Symbol} = :id,
sequence::Union{AbstractString, Symbol} = :sequence,
period::Union{AbstractString, Symbol} = :period,
endpoint::Union{AbstractString, Symbol} = :AUC,
logtransformed::Bool = false
) -> DataFrameReturn the standardized dataset with id, sequence, period, formulation, and endpoint. The standardized dataset:
- selects only relevant variables
- applies the natural log transformation to the endpoint if in natural scale
- renames variables to their canonical name for the functions
- drops missing observations
- computes the formulation based on the sequence and period variables
- converts the variables into the appropiate format for analysis (e.g., factors)
The sequence/formulation take values RT, RST, or ABCD based on alphabetical order and number of formulations.
Examples
julia> data = dataset(joinpath("bioequivalence", "RT_TR", "SLF2014_1"))
36×4 DataFrame
Row │ id sequence period AUC
│ Int64 String3 Int64 Float64
─────┼──────────────────────────────────
1 │ 1 RT 1 181.09
2 │ 1 RT 2 210.14
3 │ 2 RT 1 114.48
4 │ 2 RT 2 98.72
5 │ 3 TR 1 225.95
6 │ 3 TR 2 241.09
7 │ 4 RT 1 176.91
8 │ 4 RT 2 186.65
9 │ 5 TR 1 147.01
10 │ 5 TR 2 139.56
11 │ 6 TR 1 97.53
⋮ │ ⋮ ⋮ ⋮ ⋮
27 │ 14 TR 1 179.96
28 │ 14 TR 2 181.09
29 │ 15 TR 1 173.86
30 │ 15 TR 2 206.66
31 │ 16 RT 1 144.0
32 │ 16 RT 2 143.25
33 │ 17 RT 1 185.1
34 │ 17 RT 2 192.22
35 │ 18 TR 1 117.99
36 │ 18 TR 2 125.5
15 rows omitted
julia> preprocess_be(data)
36×5 DataFrame
Row │ id sequence period formulation endpoint
│ Cat… Cat… Cat… Cat… Float64
─────┼───────────────────────────────────────────────
1 │ 1 RT 1 R 5.19899
2 │ 1 RT 2 T 5.34777
3 │ 2 RT 1 R 4.7404
4 │ 2 RT 2 T 4.59229
5 │ 3 TR 1 T 5.42031
6 │ 3 TR 2 R 5.48517
7 │ 4 RT 1 R 5.17564
8 │ 4 RT 2 T 5.22924
9 │ 5 TR 1 T 4.9905
10 │ 5 TR 2 R 4.93849
11 │ 6 TR 1 T 4.58016
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮
27 │ 14 TR 1 T 5.19273
28 │ 14 TR 2 R 5.19899
29 │ 15 TR 1 T 5.15825
30 │ 15 TR 2 R 5.33107
31 │ 16 RT 1 R 4.96981
32 │ 16 RT 2 T 4.96459
33 │ 17 RT 1 R 5.2209
34 │ 17 RT 2 T 5.25864
35 │ 18 TR 1 T 4.7706
36 │ 18 TR 2 R 4.83231
15 rows omitted
Bioequivalence.pumas_be — Functionpumas_be(
data::AbstractDataFrame,
criteria::AbstractVector{<:BioequivalenceCriterion} = NoAssessmentBioequivalenceCriterion;
endpoint::Union{AbstractString,Symbol} = "AUC",
logtransformed::Bool = false,
reference_scale::Real = (log(1.25) / 0.25)^2,
cv_max::Real = Inf,
id::Union{AbstractString,Symbol} = "id",
sequence::Union{AbstractString,Symbol} = "sequence",
period::Union{AbstractString,Symbol} = "period",
nonparametric::Bool = occursin(r"(?i)tmax", string(endpoint)),
homogeneity::Union{Bool, Nothing} = nothing,
userepeatedobsonly::Bool = true,
reml::Bool = true,
level::Real = 0.9,
alpha::Real = 0.1,
level_y::Real = 0.95,
sigdigits::Integer = 4
) -> BioequivalenceEndpointOutputBioequivalenceEndpointOutput constructor. See also: BioequivalenceEndpointOutput, preprocess_be, and run_be.
Arguments
data: must haveid,sequence,period, and anendpoint.criteria: by default this argument can be left blank. Yet if it specified,assess_beis also run and a SummaryTables type output is included in the output.endpoint: which variable is the endpoint?logtransformed: has the endpoint been log transformed?id: which variable is the subject identifier?sequence: which variable is the sequence?period: which variable is the period?nonparametric: whether to use a nonparametric (default if endpoint includestmaxignoring case) or parametric model.homogeneity: whether formulation groups should be modeled with equal varianceuserepeatedobsonly: whether estimating the within subject variability should only repeated observationsreference_scale: 𝜃 for reference scale (e.g., FDA ≈ 0.797, FDA/NTI ≈ 1.11, EMA = 0.76)cv_max: maximum within subject variability for reference scaling (FDA = Inf, FDA/NTI = 0.2142, EMA = 0.5)reml: whether the linear mixed model should use restricted maximum likelihood or maximum likelihoodlevel: applies to the confidence intervals for the GMRalpha: applies to the upper bound of the within subject variability ratiolevel_y: applies to the critical boundary for reference-scaled average bioequivalencesigdigits: results given with how many significant digits.
Current designs include: nonparametric, parallel, and various crossover designs | Description | Treatments | Periods | Sequences | Replicated | Crossover | |––––––––––––-|––––––|––––-|–––––-|––––––|–––––-| | R|T | 2 | 1 | 2 | No | No | | RT|TR | 2 | 2 | 2 | No | Yes | | RR|RT|TR|TT | 2 | 2 | 4 | No | Yes | | RTR|TRT | 2 | 3 | 2 | Fully | Yes | | RTR|TRR | 2 | 3 | 2 | Partially | Yes | | RTT|TRR | 2 | 3 | 2 | Fully | Yes | | RRT|RTR|TRR | 2 | 3 | 3 | Partially | Yes | | RTRT|TRTR | 2 | 4 | 2 | Fully | Yes | | RRTT|TTRR | 2 | 4 | 2 | Fully | Yes | | RTTR|TRRT | 2 | 4 | 2 | Fully | Yes | | RRTT|RTTR|TRRT|TTRR | 2 | 4 | 4 | Fully | Yes | | RTRT|RTTR|TRRT|TRTR | 2 | 4 | 4 | Fully | Yes | | RR|TT | 2 | >1 | 2 | Fully | No | | RST|RTS|SRT|STR|TRS|TSR | 3 | 3 | 6 | No | Yes | | ADBC|BACD|CBDA|DCAB | 4 | 4 | 4 | No | Yes |
Examples
julia> data = dataset(joinpath("bioequivalence", "RT_TR", "SLF2014_1"))
36×4 DataFrame
Row │ id sequence period AUC
│ Int64 String3 Int64 Float64
─────┼──────────────────────────────────
1 │ 1 RT 1 181.09
2 │ 1 RT 2 210.14
3 │ 2 RT 1 114.48
4 │ 2 RT 2 98.72
5 │ 3 TR 1 225.95
6 │ 3 TR 2 241.09
7 │ 4 RT 1 176.91
8 │ 4 RT 2 186.65
9 │ 5 TR 1 147.01
10 │ 5 TR 2 139.56
11 │ 6 TR 1 97.53
⋮ │ ⋮ ⋮ ⋮ ⋮
27 │ 14 TR 1 179.96
28 │ 14 TR 2 181.09
29 │ 15 TR 1 173.86
30 │ 15 TR 2 206.66
31 │ 16 RT 1 144.0
32 │ 16 RT 2 143.25
33 │ 17 RT 1 185.1
34 │ 17 RT 2 192.22
35 │ 18 TR 1 117.99
36 │ 18 TR 2 125.5
15 rows omitted
julia> output = pumas_be(data)
Design: RT|TR
Sequences: RT|TR (2)
Periods: 1:2 (2)
Subjects per Sequence: (RT = 9, TR = 9)
Average Bioequivalence
─────────────────────────────────────────────────────────────────────────────────────────────
δ SE lnLB lnUB GMR LB UB CV
─────────────────────────────────────────────────────────────────────────────────────────────
T - R -0.0503868 0.026658 -0.0969286 -0.00384499 0.950862 0.907621 0.996162 0.0801021
─────────────────────────────────────────────────────────────────────────────────────────────
julia> output = pumas_be(data, nonparametric = true)
Design: RT|TR
Sequences: RT|TR (2)
Periods: 1:2 (2)
Subjects per Sequence: (RT = 9, TR = 9)
Average Bioequivalence
────────────────────────────────────────
lnLB lnUB LB UB
────────────────────────────────────────
T - R -0.1064 -0.001216 0.899 0.9988
────────────────────────────────────────
julia> data = dataset(joinpath("bioequivalence", "RTT_TRR", "PJ2017_4_1"))
285×5 DataFrame
Row │ id sequence period AUC Cmax
│ Int64 String3 Int64 Float64? Float64?
─────┼─────────────────────────────────────────────
1 │ 101 TRR 1 12.26 0.511
2 │ 101 TRR 2 16.19 0.688
3 │ 101 TRR 3 11.34 0.533
4 │ 102 TRR 1 397.98 13.27
5 │ 102 TRR 2 267.63 7.933
6 │ 102 TRR 3 487.55 12.952
7 │ 103 TRR 1 243.81 16.771
8 │ 103 TRR 2 141.7 6.926
9 │ 103 TRR 3 198.44 9.257
10 │ 109 TRR 1 182.52 8.816
11 │ 109 TRR 2 112.34 4.921
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮
276 │ 186 RTT 3 87.63 4.87
277 │ 190 RTT 1 82.78 3.88
278 │ 190 RTT 2 164.56 7.37
279 │ 190 RTT 3 213.98 7.01
280 │ 191 RTT 1 98.86 4.59
281 │ 191 RTT 2 99.02 2.96
282 │ 191 RTT 3 75.48 2.38
283 │ 194 RTT 1 21.29 1.51
284 │ 194 RTT 2 46.3 2.74
285 │ 194 RTT 3 15.41 1.41
264 rows omitted
julia> output = pumas_be(data)
Design: RTT|TRR
Sequences: RTT|TRR (2)
Periods: 1:3 (3)
Subjects per Sequence: (RTT = 46, TRR = 48)
Reference scaled using 𝜃 = 0.797
Average Bioequivalence
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
δ SE lnLB lnUB GMR LB UB CVᵣ CVₜ σ_ratio σ⁺ cb dof
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
T - R -0.02654 0.06906 -0.1409 0.08777 0.9738 0.8686 1.092 0.4275 0.6965 1.536 1.983 -0.09135 148.7
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────Bioequivalence.reference_scaled_acceptance_bounds — Functionreference_scaled_acceptance_bounds(σwr::Real, θ::Real)
reference_scaled_acceptance_bounds(obj::BioequivalenceEndpointOutput)Return the lower and upper bounds based on the reference within subject variability and regulatory parameter.
Reference: European Medicines Agency (2010). "GUIDELINE ON THE INVESTIGATION OF BIOEQUIVALENCE": 4.1.10 Highly variable drugs or drug products. Doc. Ref.: CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **
Examples
julia> data = dataset(joinpath("bioequivalence", "RTTR_TRRT", "PJ2017_4_3"));
julia> output = pumas_be(data)
Design: RTTR|TRRT
Sequences: RTTR|TRRT (2)
Periods: 1:4 (4)
Subjects per Sequence: (RTTR = 8, TRRT = 9)
Reference scaled using 𝜃 = 0.797
Average Bioequivalence
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────
δ SE lnLB lnUB GMR LB UB CVᵣ CVₜ σ_ratio σ⁺ cb dof
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────
T - R 0.03568 0.02431 -0.006883 0.07826 1.036 0.9931 1.082 0.08022 0.1084 1.35 2.118 0.001829 15.26
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────
julia> reference_scaled_acceptance_bounds(output)
(0.9381852023891446, 1.0658876280007832)
julia> reference_scaled_acceptance_bounds(geocv2sigma(30), 0.76)
(0.13774539477808087, 7.259770837428591)
julia> reference_scaled_acceptance_bounds(geocv2sigma(21.42), (log(1 / 0.9) / 0.1)^2)
(0.06401589407631314, 15.62112057371101)
Bioequivalence.result_summary_table — Methodresult_summary_table( endpoint_output::BioequivalenceEndpointOutput;
df_output = false)Creates a SummaryTables.Table representing the endpoint output. If str_output is set to true, a rough string output is created and returned instead (useful for debugging and testing).
Bioequivalence.run_be — Methodrun_be(
data::AbstractDataFrame;
reference_scale::Real = (log(1.25) / 0.25)^2,
cv_max::Real = Inf,
nonparametric::Bool = false,
homogeneity::Union{Bool, Nothing} = nothing,
userepeatedobsonly::Bool = true,
level::Real = 0.9,
alpha::Real = 0.1,
level_y::Real = 0.95,
reml::Bool = true,
sigdigits::Integer = 4
) -> BioequivalenceEndpointOutputBioequivalenceEndpointOutput constructor.
See also: BioequivalenceEndpointOutput.
Arguments
data: must haveid,sequence,period, and anendpoint.nonparametric: whether to use a nonparametric analysis (uncommon and usually reserved for Tmax)homogeneity: whether formulation groups should be modeled with equal varianceuserepeatedobsonly: whether estimating the within subject variability should only use repeated observationsreference_scale: 𝜃 for reference scale (FDA ≈ 0.797, FDA/NTI ≈ 1.11, EMA = 0.76)cv_max: maximum within subject variability for reference scaling (FDA = Inf, FDA/NTI = 0.2142, EMA = 0.5)reml: whether the linear mixed model should use restricted maximum likelihood or maximum likelihood.level: applies to the confidence intervals for the GMR.alpha: applies to the upper bound of the within subject variability ratiolevel_y: applies to the critical boundary for reference-scaled average bioequivalencesigdigits: results given with how many significant digits.
Current designs include: nonparametric, parallel, and various crossover designs
| Description | Treatments | Periods | Sequences | Replicated | Crossover |
|---|---|---|---|---|---|
| R | T | 2 | 1 | 2 | No |
| RT | TR | 2 | 2 | 2 | No |
| RR | RT | TR | TT | 2 | 2 |
| RTR | TRT | 2 | 3 | 2 | Fully |
| RTR | TRR | 2 | 3 | 2 | Partially |
| RTT | TRR | 2 | 3 | 2 | Fully |
| RRT | RTR | TRR | 2 | 3 | 3 |
| RTRT | TRTR | 2 | 4 | 2 | Fully |
| RRTT | TTRR | 2 | 4 | 2 | Fully |
| RTTR | TRRT | 2 | 4 | 2 | Fully |
| RRTT | RTTR | TRRT | TTRR | 2 | 4 |
| RTRT | RTTR | TRRT | TRTR | 2 | 4 |
| RR | TT | 2 | >1 | 2 | Fully |
| RST | RTS | SRT | STR | TRS | TSR |
| ADBC | BACD | CBDA | DCAB | 4 | 4 |
Examples
julia> data = dataset(joinpath("bioequivalence", "RT_TR", "SLF2014_1"))
36×4 DataFrame
Row │ id sequence period AUC
│ Int64 String3 Int64 Float64
─────┼──────────────────────────────────
1 │ 1 RT 1 181.09
2 │ 1 RT 2 210.14
3 │ 2 RT 1 114.48
4 │ 2 RT 2 98.72
5 │ 3 TR 1 225.95
6 │ 3 TR 2 241.09
7 │ 4 RT 1 176.91
8 │ 4 RT 2 186.65
9 │ 5 TR 1 147.01
10 │ 5 TR 2 139.56
11 │ 6 TR 1 97.53
⋮ │ ⋮ ⋮ ⋮ ⋮
27 │ 14 TR 1 179.96
28 │ 14 TR 2 181.09
29 │ 15 TR 1 173.86
30 │ 15 TR 2 206.66
31 │ 16 RT 1 144.0
32 │ 16 RT 2 143.25
33 │ 17 RT 1 185.1
34 │ 17 RT 2 192.22
35 │ 18 TR 1 117.99
36 │ 18 TR 2 125.5
15 rows omitted
julia> pkdata = preprocess_be(data)
36×5 DataFrame
Row │ id sequence period formulation endpoint
│ Cat… Cat… Cat… Cat… Float64
─────┼───────────────────────────────────────────────
1 │ 1 RT 1 R 5.19899
2 │ 1 RT 2 T 5.34777
3 │ 2 RT 1 R 4.7404
4 │ 2 RT 2 T 4.59229
5 │ 3 TR 1 T 5.42031
6 │ 3 TR 2 R 5.48517
7 │ 4 RT 1 R 5.17564
8 │ 4 RT 2 T 5.22924
9 │ 5 TR 1 T 4.9905
10 │ 5 TR 2 R 4.93849
11 │ 6 TR 1 T 4.58016
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮
27 │ 14 TR 1 T 5.19273
28 │ 14 TR 2 R 5.19899
29 │ 15 TR 1 T 5.15825
30 │ 15 TR 2 R 5.33107
31 │ 16 RT 1 R 4.96981
32 │ 16 RT 2 T 4.96459
33 │ 17 RT 1 R 5.2209
34 │ 17 RT 2 T 5.25864
35 │ 18 TR 1 T 4.7706
36 │ 18 TR 2 R 4.83231
15 rows omitted
julia> output = run_be(pkdata)
Design: RT|TR
Sequences: RT|TR (2)
Periods: 1:2 (2)
Subjects per Sequence: (RT = 9, TR = 9)
Average Bioequivalence
─────────────────────────────────────────────────────────────────────────────────────────────
δ SE lnLB lnUB GMR LB UB CV
─────────────────────────────────────────────────────────────────────────────────────────────
T - R -0.0503868 0.026658 -0.0969286 -0.00384499 0.950862 0.907621 0.996162 0.0801021
─────────────────────────────────────────────────────────────────────────────────────────────
Bioequivalence.sigma2geocv — Methodsigma2geocv(σw::Union{Real,Missing}) = √(exp(σw^2) - 1)Transform the σ parameter of a log-normal distribution to the coefficient of variation (CV).
Examples
julia> sigma2geocv(0.294)
0.3004689459216001
Bioequivalence.study_paradigm — Methodstudy_paradigm(data::AbstractDataFrame, nonparametric::Bool)::EndpointStudyParadigmDetermines the study paradigm of a dataset.
Bioequivalence.study_paradigm — Methodstudy_paradigm(be_output::BioequivalenceEndpointOutput)::EndpointStudyParadigmDetermines the study paradigm used in a bioequivalence study for an endpoint.
Bioequivalence.update_rsabe_theta! — Methodupdate_rsabe_theta!(obj::ReferenceScaledAverageBioequivalance, 𝜃::Real) -> ReferenceScaledAverageBioequivalanceModifies in-place the 𝜃 of the ReferenceScaledAverageBioequivalance and updates the associated critical boundary.
julia> data = dataset(joinpath("bioequivalence", "RTTR_TRRT", "SLTGSF2020_DS16"));
julia> pkdata = preprocess_be(data, endpoint = :PK);
julia> rsabe = ReferenceScaledAverageBioequivalance(pkdata, 0.76)
Critical boundary: -0.0416
Regulatory parameter: 0.76
95.0% upper confidence bound with 36.0 degrees of freedom
julia> update_rsabe_theta!(rsabe, 1.11)
Critical boundary: -0.1019
Regulatory parameter: 1.11
95.0% upper confidence bound with 36.0 degrees of freedom
Bioequivalence.within_subject_variability — Methodwithin_subject_variability(
data::AbstractDataFrame;
userepeatedobsonly::Bool = true
allownonreplicated::Bool = false,
homogeneity::Union{Bool,Nothing} = nothing,
)Return the within-subject variability estimate and degrees of freedom. It uses a linear model of the log-transformed PK response with fixed effects for subject ID and period. If userepeatedobsonly, data used is a sample with only observations from repeated subject/treatment. If allownonreplicated, the function returns a NamedTuple of missings instead of throwing an error when the passed formulation is not replicated in any sequences. This option is only relevant for partially replicated designs.
The homogeneity argument specifies if equal variances are assumed between formulations. When set to true, multiple formulations are expected in data. When set to false, only a single formulation should be available in data. In this case, design must be replicated. The default value of homogeneity is false for replicated designs and true otherwise.
References:
Schütz H, Tomashevskiy M, Labes D, Shitova A, González-de la Parra M, Fuglsang A. 2020. Reference Datasets for Studies in a Replicate Design Intended for Average Bioequivalence with Expanding Limits. AAPS J. 22(2): Article 44. DOI: 10.1208/s12248-020-0427-6.
Examples
julia> data = dataset(joinpath("bioequivalence", "RTTR_TRRT", "PJ2017_4_3"));
julia> pkdata = preprocess_be(data);
julia> combine(groupby(pkdata, :formulation), within_subject_variability)
2×3 DataFrame
Row │ formulation σw k
│ Cat… Float64 Float64
─────┼─────────────────────────────────
1 │ R 0.0800952 15.0
2 │ T 0.108075 14.0
julia> data = dataset(joinpath("bioequivalence", "RRT_RTR_TRR", "SLTGSF2020_DS02"));
julia> pkdata = preprocess_be(data, endpoint = :PK);
julia> combine(groupby(pkdata, :formulation), t -> within_subject_variability(t; allownonreplicated=true))
2×3 DataFrame
Row │ formulation σw k
│ Cat… Float64? Float64?
─────┼────────────────────────────────────────
1 │ R 0.111361 22.0
2 │ T missing missingBioequivalence.within_subject_variability_ratio — Functionwithin_subject_variability_ratio(
σ::AbstractVector{<:Union{Real, Missing}},
k::AbstractVector{<:Union{Real, Missing}},
level::Real = 0.95
) -> Vector{NamedTuple{(:σ_ratio, :σ⁺), ...}}
within_subject_variability_ratio(
data::AbstractDataFrame,
level::Real = 0.95
) -> DataFrameReturn the within-subject variability ratio and upper bounds. σ contains the estimated within subject variability. k contains the associated degrees of freedom. level determines the confidence level for the upper bound.
Examples
julia> data = dataset(joinpath("bioequivalence", "RTTR_TRRT", "PJ2017_4_3"));
julia> pkdata = preprocess_be(data);
julia> wsv_estimates = combine(groupby(pkdata, :formulation), within_subject_variability)
2×3 DataFrame
Row │ formulation σw k
│ Cat… Float64 Float64
─────┼─────────────────────────────────
1 │ R 0.0800952 15.0
2 │ T 0.108075 14.0
julia> within_subject_variability_ratio(wsv_estimates[!, :σw], wsv_estimates[!, :k])
1-element Vector{NamedTuple{(:σ_ratio, :σ⁺), Tuple{Float64, Float64}}}:
(σ_ratio = 1.349328424089969, σ⁺ = 2.1176303305572595)
julia> within_subject_variability_ratio(wsv_estimates)
2×5 DataFrame
Row │ formulation σw k σ_ratio σ⁺
│ Cat… Float64 Float64 Float64? Float64?
─────┼───────────────────────────────────────────────────────────────
1 │ R 0.0800952 15.0 missing missing
2 │ T 0.108075 14.0 1.34933 2.11763