## Abstract

### Background

Immunological correlates of protection are biological markers such as disease-specific antibodies which correlate with protection against disease and which are measurable with immunological assays. It is common in vaccine research and in setting immunization policy to rely on threshold values for the correlate where the accepted threshold differentiates between individuals who are considered to be protected against disease and those who are susceptible. Examples where thresholds are used include development of a new generation 13-valent pneumococcal conjugate vaccine which was required in clinical trials to meet accepted thresholds for the older 7-valent vaccine, and public health decision making on vaccination policy based on long-term maintenance of protective thresholds for Hepatitis A, rubella, measles, Japanese encephalitis and others. Despite widespread use of such thresholds in vaccine policy and research, few statistical approaches have been formally developed which specifically incorporate a threshold parameter in order to estimate the value of the protective threshold from data.

### Methods

We propose a 3-parameter statistical model called the a:b model which incorporates parameters for a threshold and constant but different infection probabilities below and above the threshold estimated using profile likelihood or least squares methods. Evaluation of the estimated threshold can be performed by a significance test for the existence of a threshold using a modified likelihood ratio test which follows a chi-squared distribution with 3 degrees of freedom, and confidence intervals for the threshold can be obtained by bootstrapping. The model also permits assessment of relative risk of infection in patients achieving the threshold or not. Goodness-of-fit of the a:b model may be assessed using the Hosmer-Lemeshow approach. The model is applied to 15 datasets from published clinical trials on pertussis, respiratory syncytial virus and varicella.

### Results

Highly significant thresholds with p-values less than 0.01 were found for 13 of the 15 datasets. Considerable variability was seen in the widths of confidence intervals. Relative risks indicated around 70% or better protection in 11 datasets and relevance of the estimated threshold to imply strong protection. Goodness-of-fit was generally acceptable.

### Conclusions

The a:b model offers a formal statistical method of estimation of thresholds differentiating susceptible from protected individuals which has previously depended on putative statements based on visual inspection of data.