This document describes how to estimate PoD curve parameters using PoDBAY package. This process can be applied when user doesnâ€™t have individual level data about vaccinated and control populations, but only summary statistics data and corresponding estimatedcase-count vaccine efficacy.

The goal of this document is to show how to estimate point estimate of PoD curve parameters in two steps

- \(et_{50}\) and \(\gamma\) estimation

Required input:

- vaccinated population - mean, standard deviation of titers
- control population - mean, standard deviation of titers
- reference efficacy - vaccine case-count efficacy estimate from large clinical trial (converging to the true value of efficacy)

- \(p_{max}\) estimation

Required input:

- \(et_{50}\) and \(\gamma\) estimates from the step 1
- incidence rate for low titer population - we assume it is represented by incidence rate of control population
- control population - mean, standard deviation of titers

Function `PoDEfficacySquaredError()`

is used to estimate \(et_{50}\) and \(\gamma\). As the inputs to the function we use `vaccinated`

and `control`

mock-up population class objects together with artificially chosen `TrueEfficacy`

parameter.

Note: To convert your data in to the `population`

class object use `generatePopulation()`

function from PoDBAY package. See vignette `vignette("population", package = "PoDBAY")`

for further details.

```
# Mockup vaccinated and control population class objects
data(vaccinated)
data(control)
# Observed vaccine efficacy
TrueEfficacy <- 0.53
# PoD curve parameter estimation
params_et50_slope <- PoDEfficacySquaredError(TrueEfficacy,
vaccinated,
control,
initialSlope = 6)
params_et50_slope
#> et50 slope
#> 5.268031 6.179620
```

**NOTE**

- Estimated \(et_{50}\) and \(\gamma\) parameters highly depends on the initial setup of slope parameter
- \(p_{max}\) parameter is not part of the optimization as CoP-based (PoDBAY) efficacy is not dependent on the \(p_{max}\) value. Hence it needs to be estimated separately.

Once we have \(et_{50}\) and \(\gamma\) estimated we can proceed with \(p_{max}\) estimation using `PmaxEstimation`

. As the inputs to the function we use estimated \(et_{50}\) and \(\gamma\), `control`

mock-up population class object together with artificially chosen `IncidenceRate`

parameter.