## 2 PPP  
The vagueness of the original PPP statement is one reason there are so many interpretations of PPP [5]. PPP can occur if there is a large positive covariance between \(y_{1}\) and \(y_{2}\) and the variances of \(y_{1}\) and \(y_{2}\) are very different. Let \(\Sigma\) be the 2-by-2 symmetric covariance matrix for \(y_{1}\) and \(y_{2}\) with diagonal entries \(\sigma_{1}^{2}\), \(\sigma_{2}^{2}\), and off-diagonal entry \(\sigma_{12}\), which denote the variance of \(y_{1}\), the variance of \(y_{2}\), and the covariance of \(y_{1}\) and \(y_{2}\), respectively. [PERSON] and [PERSON] [4] approximated \(\Sigma\) for their definition of PPP (using \(y_{1}=cm_{1}\) and \(y_{2}=cm_{2}\) as described in the Introduction) as  
\[\Sigma = \mathrm{Cov}(y_{1},y_{2}) \tag{1}\] \[= \left( \begin{array}{cc}\sigma_{1}^{2}&\sigma_{12}\\ \sigma_{12}&\sigma_{2}^{2}\end{array} \right)\] \[\approx \mu_{c}^{2}\left( \begin{array}{cc}\sigma_{m1}^{2}&\\ &\sigma_{m2}^{2}\end{array} \right)+\sigma_{c}^{2}\left( \begin{array}{cc}\mu _{m}^{2}&\mu_{m}^{2}\\ \mu_{m}^{2}&\mu_{m}^{2}\end{array} \right)\] \[\approx \left( \begin{array}{cc}0.1125&0.06\\ 0.06&0.05\end{array} \right)\]  
where \(\sigma_{m1}\), \(\sigma_{m2}\), and \(\sigma_{c}\) are the standard deviations in the quantities \(m_{1}\), \(m_{2}\), and \(c\), respectively. The first approximation arises because the relatively small term \(\sigma_{c}^{2}\sigma_{m}^{2}\) was omitted (see below). The second approximation follows by approximating \(\mu_{m}^{2}\) by \(m_{1}m_{2}\). Equation 1 assumes that \(m\) and \(c\) are independent. And, the relative standard deviations of 10%, 10%, and 20% are assumed to be the fractions relative to the measured values of 1.5, 1.0, and 1.0, respectively by definition, so that \(\sigma_{m1}=0.15\), \(\sigma_{m2}=0.10\), and \(\sigma_{c}=0.2\). See Section 4 for further discussion. Readers might find it informative that those with traditional statistical education among the authors were the most willing to accept GLS estimates despite the apparent flaw of lying outside the range of the data. Statisticians will often consider alternatives to GLS, but recognize that GLS estimation is difficult to beat, at least in terms of typical performance measures such as being close on average to the true parameter value over hypothetical repeats of the pair of experiments [6]. Also, note that because \(\frac{y_{1}-y_{2}}{\sigma_{y_{1}-y_{2}}}=0.5/0.21=2.39\), one might consider the original PPP to be an unusual data realization. However, we show using elementary algebra and error modeling in Theorem 2 that when PPP occurs, it occurs for _all_ data realizations. Although [PERSON] [1] originally constructed the covariance matrix \(\Sigma\) as in Equation 1 and other authors followed, this result is not exact if the common error is multiplicative because of the omitted term \(\sigma_{c}^{2}\sigma_{m}^{2}\) and because of the need to estimate \(\mu_{m}\). If we include the \(\sigma_{c}^{2}\sigma_{m}^{2}\) term, then  
\[\mathrm{Var}(cm)=\mu_{c}^{2}\sigma_{m}^{2}+\mu_{m}^{2}\sigma_{c}^{2}+\sigma_{c} ^{2}\sigma_{m}^{2} \tag{2}\]where \(\sigma_{m}\) is \(\sigma_{m1}\) or \(\sigma_{m2}\) in Equation 1. The covariance matrix is therefore estimated as  
\[\Sigma = \mu_{c}^{2}\left( \begin{array}{cc}\sigma_{m1}^{2}&\\ &\sigma_{m2}^{2}\end{array} \right)+\sigma_{c}^{2}\left( \begin{array}{cc}\mu_ {m}^{2}+\sigma_{m1}^{2}&\mu_{m}^{2}\\ \mu_{m}^{2}&\mu_{m}^{2}+\sigma_{m2}^{2}\end{array} \right) \tag{3}\] \[\approx \left( \begin{array}{cc}0.1134&0.06\\ 0.06&0.0504\end{array} \right)\]  
where again as in Equation 1, \(\mu_{m}\) is estimated by \(m_{1}\) or \(m_{2}\). As shown below, prior to substituting the approximation for \(\mu_{m}^{2}\), Equations 1 and 3 satisfy conditions under which PPP cannot occur (Theorem 2 in Section 3).