# Stage 3: Formulating the prior

The information and data collected during the qualitative and quantitative data collection stages of a SQUEAC assessment reveal a great deal about coverage and about how it can be improved in a programme. However they do not provide an overall estimate of programme coverage.

To do this, the data and information collected during Stages 1 and 2 of a SQUEAC assessment are used by surveyors to come up with a belief of what coverage is in a programme, known as the prior. It is called the prior as it is based on prior information which is already available based on the analysis of existing data and information.

Using a Bayesian technique known as a conjugate analysis, the prior is combined with new survey data collected during Stage 3 known as the likelihood to come up with the posterior, the final coverage estimate, which in Bayesian terms is defined as "the result of modifying the prior belief using new information" (Myatt et al. 2012).

Figure 1: Conjugate Analysis

A conjugate analysis requires that the prior and the likelihood are expressed in similar ways. This means that the prior information about coverage (i.e. the findings from the analysis of routine programme data; the intelligent collection of qualitative data; and the finding of small studies, small surveys, and small-area surveys) needs to be expressed as a probability density.

This page outlines the steps taken to express the prior information as a probability density. The next section,, outlines how to undertake a likelihood survey, finalise the conjugate analysis and calculate the coverage estimation.

# 1. Establishing the prior mode:

The first step in expressing the prior information as a probability density is to make an informed guess about the most likely coverage value (the mode of the probability density) given the prior information. There are a number of methods which can be used to do this:

The results of the above exercises (the totals of the boosters and barriers in each exercise), can be added to the Prior mode estimation template (available in Tools) in order to generate an average prior mode.

These exercises return a first credible value for the mode of the prior and should be reviewed by returning to the prior information and, if necessary, recalculated or adjusted. The mode of the prior may be changed any time before surveyors start collecting information for the likelihood survey.

# 2. Applying levels of uncertainty to the prior mode:

There is always uncertainty about the value of the prior mode. The amount of uncertainty about the prior mode is the same as the probable range of values of coverage that is consistent with the prior information. This is specified using:

• The minimum probable value for coverage that is consistent with the prior information.
• The maximum probable value for coverage that is consistent with the prior information.

A simple way of doing this is to use a fixed quantity. Two credibility intervals can be applied to prior modes for SQUEAC assessments:

• ± 25 percentage points if surveyors are conducting a SQUEAC assessment in an area for the first time, and / or if there is very little certainty about the prior.
• ± 20 percentage points if there is very little uncertainty about the prior mode.

For example, if the prior mode is estimated to be 40%, the minimum and maximum probability values would be as follows:

 Credibility interval ± 25% Credibility interval ± 20% Minimum probable value Maximum probable value Minimum probable value Maximum probable value 15% 65% 20% 60%

There is no requirement that the distribution of the prior be symmetrical about its mode. If, for example, a maximum probable value of 81.5% is calculated and is considered to be extremely unlikely (i.e., it is considered extremely unlikely that coverage could be as high as 81.5%) then it could be replaced with a more credible value (e.g., 75%).

Another situation when a symmetrical prior is likely to be unsuitable is when coverage is expected to be either very low or very high. If, for example, coverage is expected to be about 20% then values for the minimum and maximum probable values of 10% and 40% might be specified. Note that coverage cannot be below 0% or above 100%. This means that the minimum probable value cannot be below 0% and the maximum probable value cannot be above 100%.

# 3. Calculating shape parameters:

The conjugate analysis method used in SQUEAC requires the distribution of the prior to be summarised by two numbers called shape parameters, which are labelled αPrior and βPrior.

Suitable values for αPrior and βPrior may be calculated using the mode and the minimum probable value and maximum probable value of the prior with the following formulas:

It should be noted that these formulas require values to be expressed as proportions, not percentages.

To convert a percentage to a proportion:



Once the αPrior and βPrior have been calculated, they can be entered into the Bayes calculator (in Tools). This will then provide the survey team with a suggested sample size for the likelihood survey (see below). With this, they can proceed with the final part of Stage 3 (see next section).

Figure 2: Screen shot of Bayes calculator

# 4. Tools:

Prior mode estimation template: