I'm evaluating two (2) refrigerants (gases) that were used in the same refrigeration system. I have saturated suction temperature ($S$), condensing temperature ($D$), and amperage ($Y$) data for the evaluation. There are two (2) sets of data; 1st refrigerant ($R_1$) & 2nd refrigerant ($R_2$). I'm using a linear, multivariate ($S$ & $D$), 3rd order polynomial model for the regression analyses. I would like determine how much less / more amperage (or, some similar metric as a performance comparison) on average, as a percentage, is being drawn by the second refrigerant.
My first thought was:
- Determine the model to use: $Y = b_0 + b_1S + b_2D + b_3SD + b_4S^2 + b_5D^2 + b_6S^2D + b_7D^2S + b_8D^3 + b_9S^3$
- Derive coefficients ($b_i$) from the baseline data ($R_1$).
- Using those coefficients, for each $S$ & $D$ in the $R_2$ data set, calculate each expected amp draw ($\hat{Y}$) and then average.
- Compare the $\hat{Y}$ average to the actual average amp draw ($Y_2$) of the $R_2$ data.
- $\text{percent (%) change} = (Y_2 - \hat{Y}) / \hat{Y}$
However, since the 2nd refrigerant has slightly different thermal properties & small changes were made to the refrigeration system (TXV & superheat adjustments) I don't believe this 'baseline comparison method' is accurate.
My next thought was to do two (2) separate regression analyses: \begin{align} Y_1 &= a_{0} + a_{1}S_1 + a_{2}D_1 + a_{3}S_1D_1 + a_{4}S_1^2 + a_{5}D_1^2 + a_{6}S_1^2D_1 + a_{7}D_1^2S_1 + a_{8}D_1^3 + a_{9}S_1^3 \\ Y_2 &= b_{0} + b_{1}S_2 + b_{2}D_2 + b_{3}S_2D_2 + b_{4}S_2^2 + b_{5}D_2^2 + b_{6}S_2^2D_2 + b_{7}D_2^2S_2 + b_{8}D_2^3 + b_{9}S_2^3 \end{align}
and then, for saturated suction temp ($S$), compare coefficients ($a_{1}$ vs $b_{1}$) like so: $$ \text{% change} = \frac{b_{1} - a_{1}}{a_{1}} $$
However, again, these coefficients should be weighted differently. Therefore, the results would be skewed.
I believe I could use a z-test to determine how differently weighted the coefficients are, but I'm not sure I fully understand the meaning of the output: $z = (a_{1} - b_{1}) / \sqrt{SE_{a_{1}}^2 + SE_{b_{1}}^2 )}$. But, that still wouldn't give me a performance metric, which is the overall objective.