How is the Outperformance Indicator Calculated?

The Outperformance Indicator is calculated by comparing the posterior distribution of likely values of a tactic's brand lift to the distribution of likely values of the overall campaign brand lift.  

OI = (post_mean - overall_mean)/ sqrt(post_var)

The cut-level distribution is a posterior distribution that uses the overall distribution as a prior distribution, which thus shrinks the mean and standard error of the cut towards the mean and standard error of the overall campaign.  This enables interim evaluation of Outperformance Indicators at any sample size.  

post_var = 1 / ((1 / cut_se^2) + (1 / overall_se^2))

post_mean = post_var * (cut_mean / (cut_se^2) + overall_mean / (overall_se^2))

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