In a frequent case when serum samples at baseline and after vaccination are collected and assayed only in a subset of subjects (“immunogenicity sample/ subset”) and the assay value of titer is obtained also for all disease cases at the same time points, the general method for PoD curve estimation described above can be extended. Further details can be found in the publication Appendix A.
Non-diseased immunogenicity sample
Assume hypothetical case where we have clinical trial data of 2,000 subjects from which only 200 subjects’ plasma samples are collected and examined in the immunogenicity study. Further, out of these 2,000 we identify 35 disease cases to which we measure titers from the same time point. In the end we have titer information about 200 subjects from the immunogenicity study and 35 diseased subjects.
| Whole Trial |
|
| All subjects |
2,000 |
| Diseased |
35 |
| Non-diseased |
1,965 |
|
|
| Measured titers |
|
| Diseased |
35 |
| Immunogenicity sample |
200 |
Note that in the immunogenicity sample the disease status is unknown as the sample is created before the clinical study. However, vaccination status is known.
In our example the steps would be following:
Titers of all non-diseased subjects (N = 1,965) are generated by random sampling with replacement from immunogenicity subset (N = 200).
Titers of all diseased (N = 35) and all generated non-diseased (N = 1,965) subjects are used for estimation of PoD curve parameters.
Titers of all diseased (N = 35) and all generated non-diseased (N = 1,965) subjects are put together and bootstrapped (N = 2,000).
New immunogenicity subset is selected from all new non-diseased (\(N^`\) = 2000 - X), such that the ratio of all new diseased (\(N^`\) = X) versus new non-diseased in immunogenicity subset in new data match the ratio in original data (ratio = 200:35).
| New diseased |
\(X\) |
| New non-diseased |
\(2000 - X\) |
| New Immunogenicity sample |
\(X * \frac{200}{35}\) |
Titers of all new non-diseased subjects are generated by random sampling with replacement from new immunogenicity subset (\(N^` = X * \frac{200}{35}\))
Titers of all new diseased (\(N^` = X\)) and all new generated non-diseased subjects (\(N^` = 2000 - X\)) are used for second estimation of PoD curve parameters.
Data preparation
Diseased and non-diseased subject level data are required. We’ll use PoDBAY::diseased and PoDBAY::nondiseased mock-up data. Both datasets contain population summary statistics (N, mean, sd) and individual subject level data (log2 titers, diseases status (DS))
Only the individual subject level data (log2 titers, DS) are used for the PoD curve estimation as described above.
We create the immunogenicity sample from our mock-up data as described above - We start with the titer information about 200 subjects from the immunogenicity study and 35 diseased subjects.
data(diseased)
data(nondiseased)
# Immunogenicity sample created
ImmunogenicitySample <- BlindSampling(diseased, nondiseased, method = list(name = "Fixed", value = 200))
nondiseasedImmunogenicitySample <- ImmunogenicitySample$ImmunogenicityNondiseased
str(diseased)
#> Reference class 'Population' [package ".GlobalEnv"] with 8 fields
#> $ N : int 35
#> $ mean : num 3.83
#> $ stdDev : num 1.66
#> $ unknownDistribution: logi FALSE
#> $ UDFunction :function ()
#> $ titers : Named num [1:35] 5.59 6.07 2.43 5.84 6.29 ...
#> ..- attr(*, "names")= chr [1:35] "vacc" "vacc" "vacc" "vacc" ...
#> $ PoDs : num(0)
#> $ diseaseStatus : logi [1:35] TRUE TRUE TRUE TRUE TRUE TRUE ...
#> and 24 methods, of which 10 are possibly relevant:
#> assignPoD, getDiseasedCount, getDiseasedTiters, getNondiseasedCount,
#> getNondiseasedTiters, getTiters, getUnknown, initialize, popFun, popX
str(nondiseasedImmunogenicitySample)
#> Reference class 'Population' [package "PoDBAY"] with 8 fields
#> $ N : int 196
#> $ mean : num 5.93
#> $ stdDev : num 2.45
#> $ unknownDistribution: logi FALSE
#> $ UDFunction :function ()
#> $ titers : Named num [1:196] 7.7 8.59 8.15 9.1 11.89 ...
#> ..- attr(*, "names")= chr [1:196] "vacc" "vacc" "vacc" "vacc" ...
#> $ PoDs : num(0)
#> $ diseaseStatus : logi [1:196] FALSE FALSE FALSE FALSE FALSE FALSE ...
#> and 24 methods, of which 10 are possibly relevant:
#> assignPoD, getDiseasedCount, getDiseasedTiters, getNondiseasedCount,
#> getNondiseasedTiters, getTiters, getUnknown, initialize, popFun, popX
PoD curve estimation
Note: From now on the analysis and used functions are the same as in general case. Only the input variable change from unifected to NondiseasedImmunogenicitySample. The nondiseasedGenerationCount remains the same as the total number of nondiseased remains the same in the whole trial.
Once we have our data prepared, function PoDParamEstimation is used to estimate PoD curve parameters in six steps as described above. For more details about the usage of the function see examples in ?PoDParamEstimation().
estimatedParametersAP <- PoDParamEstimation(diseasedTiters = diseased$titers,
nondiseasedTiters = nondiseasedImmunogenicitySample$titers,
nondiseasedGenerationCount = nondiseased$N,
repeatCount = 50)
Step 1: \(p_{max}^`\), \(et_{50}^`\) and \(\gamma^`\)
Results corresponding to the first step of Estimation of PoD-titer relationship can be obtained via estimatedParametersAP$resultsPriorReset.
#> # A tibble: 49 x 3
#> pmax slope et50
#> <dbl> <dbl> <dbl>
#> 1 0.0324 34.3 6.05
#> 2 0.0318 34.3 6.08
#> 3 0.0322 34.3 6.08
#> 4 0.0324 34.3 6.08
#> 5 0.0315 34.3 6.10
#> 6 0.0316 34.3 6.07
#> 7 0.0325 34.3 6.09
#> 8 0.0322 34.3 6.10
#> 9 0.0316 34.3 6.09
#> 10 0.0323 34.3 6.07
#> # … with 39 more rows
Note that parameter estimates are now different for each repeatCount iteration. This is according to our expectations as titers of all non-diseased subjects are generated by random sampling with replacement from immunogenicity subset in every iteration in step 1 of this example.
Step 2: Data generation and re-assignment of disease status
Step 3: \(p^{``}_{max}\), \(et^{``}_{50}\) and \(\gamma^{``}\)
Results corresponding to the sixth step of estimation of PoD-titer relationship can be obtained via estimatedParametersAP$results.
#> # A tibble: 49 x 3
#> pmax slope et50
#> <dbl> <dbl> <dbl>
#> 1 0.0347 39.8 5.99
#> 2 0.0231 16.0 5.89
#> 3 0.0351 21.4 5.77
#> 4 0.0419 12.1 5.46
#> 5 0.0301 39.8 6.04
#> 6 0.0483 4.59 3.92
#> 7 0.0393 43.7 5.94
#> 8 0.0386 36.8 6.06
#> 9 0.0314 40.5 6.35
#> 10 0.0263 32.7 6.00
#> # … with 39 more rows
Non-parametric bootstrap described in step 3 together with creation of new immunogenicity sample in step 4-5is applied inside the function.
PoD curve point estimate
Parameters of PoD curve point estimate representing the PoD-titer relationship are estimated using results from ‘step 1’ - estimatedParametersAP$resultsPriorReset.
PoDParamsPointEst <- PoDParamPointEstimation(estimatedParametersAP$resultsPriorReset)
PoDParamsPointEst
#> pmax slope et50
#> pmax 0.03218197 33.85696 6.085131
PoD curve confidence intervals
Confidence intervals (80%, 90% and 95% level of significance) of PoD curve parameters are calculated using results from ‘step 6’ - estimatedParametersAP$results.
PoDParametersCI <- PoDParamsCI(estimatedParametersAP$results)
unlist(PoDParametersCI)
#> PmaxCILow PmaxCIHigh Et50CILow Et50CIHigh SlopeCILow SlopeCIHigh
#> 0.02140353 0.04420208 5.36664708 6.54030103 6.63482304 43.47149473
PoD curve plot
Using PoD curve point estimate and results from step 6 - estimatedParametersAP PoD curve can be plotted.
titers <- seq(from = 0, to = 15, by = 0.01)
PoDCurve <- PoDCurvePlot(titers,
estimatedParametersAP,
ci = 0.95)
PoDCurve
2. PoDBAY efficacy estimation - Trial B
There are two possible situations:
- Trial A = Trial B. Therefore both trials are the same and we have immunogenicity subset information only for both trials as described above.
- Only change would be in the data availability as for Trial B we do not have the titer information from the whole trial (N = 2,000) but only from the immunogenicity sample (N = 200). However, the PoDBAY efficacy estimation approach and analysis steps remain the same as in general case.
- Trial A - we have immunogenicity subset as described above. Trial B is the same as in general case.
- No change in the PoDBAY efficacy estimation approach. Analysis steps remain the same as in general case.
We will describe the approach in the situation where Trial A = Trial B.
Data preparation
As stated above the only difference is in the data availability. The fact that vaccinated and control population summary statistics (N, mean, sd) are required remains the same. Therefore, we calculate summary statistics for both populations using immunogenicity subset data - created in the PoD-titer relationship estimation step.
# Immunogenicity sample - vaccinated
str(ImmunogenicitySample$ImmunogenicityVaccinated)
#> Reference class 'Population' [package "PoDBAY"] with 8 fields
#> $ N : int 97
#> $ mean : num 7.03
#> $ stdDev : num 2.34
#> $ unknownDistribution: logi FALSE
#> $ UDFunction :function ()
#> $ titers : Named num [1:97] 7.7 8.59 8.15 9.1 11.89 ...
#> ..- attr(*, "names")= chr [1:97] "vacc_FALSE" "vacc_FALSE" "vacc_FALSE" "vacc_FALSE" ...
#> $ PoDs : num(0)
#> $ diseaseStatus : logi [1:97] FALSE FALSE FALSE FALSE FALSE FALSE ...
#> and 24 methods, of which 10 are possibly relevant:
#> assignPoD, getDiseasedCount, getDiseasedTiters, getNondiseasedCount,
#> getNondiseasedTiters, getTiters, getUnknown, initialize, popFun, popX
# Immunogenicity sample - control
str(ImmunogenicitySample$ImmunogenicityControl)
#> Reference class 'Population' [package "PoDBAY"] with 8 fields
#> $ N : int 103
#> $ mean : num 4.73
#> $ stdDev : num 2.13
#> $ unknownDistribution: logi FALSE
#> $ UDFunction :function ()
#> $ titers : Named num [1:103] 2.17 4.53 2.51 6.08 4.22 ...
#> ..- attr(*, "names")= chr [1:103] "control_FALSE" "control_FALSE" "control_FALSE" "control_FALSE" ...
#> $ PoDs : num(0)
#> $ diseaseStatus : logi [1:103] FALSE FALSE FALSE FALSE FALSE FALSE ...
#> and 24 methods, of which 10 are possibly relevant:
#> assignPoD, getDiseasedCount, getDiseasedTiters, getNondiseasedCount,
#> getNondiseasedTiters, getTiters, getUnknown, initialize, popFun, popX
Efficacy point estimate
means <- list("vaccinated" = ImmunogenicitySample$ImmunogenicityVaccinated$mean,
"control" = ImmunogenicitySample$ImmunogenicityControl$mean)
standardDeviations <- list("vaccinated" = ImmunogenicitySample$ImmunogenicityVaccinated$stdDev,
"control" = ImmunogenicitySample$ImmunogenicityControl$stdDev)
EfficacyPointEst <- efficacyComputation(PoDParamsPointEst,
means,
standardDeviations)
EfficacyPointEst
#> [1] 0.5298448
Efficacy set \(e^{``}\)
efficacySet <- PoDBAYEfficacy(estimatedParametersAP$results,
ImmunogenicitySample$ImmunogenicityVaccinated,
ImmunogenicitySample$ImmunogenicityControl)
Efficacy confidence intervals
CI <- EfficacyCICoverage(efficacySet)
unlist(CI)
#> mean median CILow95 CIHigh95 CILow90 CIHigh90 CILow80 CIHigh80
#> 0.5375903 0.5419903 0.4432219 0.6231131 0.4527044 0.6106371 0.4608834 0.5999084
PoDBAY efficacy - output summary
Analysis provides following results:
- PoDBAY efficacy point estimate with its confidence intervals (80%, 90% and 95% level of significance)
- \(p_{max}\), \(et_{50}\) and \(\gamma\) parameters of the PoD curve point estimate with its confidence intervals (80%, 90% and 95% level of significance)
- Plot of the PoD curve with its confidence intervals
result <- list(
EfficacyPointEst = EfficacyPointEst,
efficacyCI = unlist(CI),
PoDParamsPointEst = PoDParamsPointEst,
PoDParametersCI = unlist(PoDParametersCI),
PoDCurve = PoDCurve
)
result
#> $EfficacyPointEst
#> [1] 0.5298448
#>
#> $efficacyCI
#> mean median CILow95 CIHigh95 CILow90 CIHigh90 CILow80 CIHigh80
#> 0.5375903 0.5419903 0.4432219 0.6231131 0.4527044 0.6106371 0.4608834 0.5999084
#>
#> $PoDParamsPointEst
#> pmax slope et50
#> pmax 0.03218197 33.85696 6.085131
#>
#> $PoDParametersCI
#> PmaxCILow PmaxCIHigh Et50CILow Et50CIHigh SlopeCILow SlopeCIHigh
#> 0.02140353 0.04420208 5.36664708 6.54030103 6.63482304 43.47149473
#>
#> $PoDCurve
