# PoD curve point estimation using vaccine efficacy and population summary statistics

#### 2021-09-20

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

1. $$et_{50}$$ 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)
1. $$p_{max}$$ 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

## 1. $$et_{50}$$ and $$\gamma$$ estimation

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

1. Estimated $$et_{50}$$ and $$\gamma$$ parameters highly depends on the initial setup of slope parameter
2. $$p_{max}$$ 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.

## 2. $$p_{max}$$ estimation

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.

# Incidence rate for low titer population
IncidenceRate <- 0.02

# pmax estimation
pmax <- PmaxEstimation(IncidenceRate, params_et50_slope, control)

# combining PoD curve parameters
PoDParams <- unlist(c(params_et50_slope, pmax))

PoDParams
#>       et50      slope       pmax
#> 5.26803128 6.17962043 0.03552708