# Comparing regression coefficients of same model across different data sets

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:

1. Determine the model to use: $Y = b_0 + b_1S + b_2D + b_3SD + b_4S^2 + b_5D^2 + b_6S^2D + b_7D^2S + b_8D^3 + b_9S^3$
2. Derive coefficients ($b_i$) from the baseline data ($R_1$).
3. Using those coefficients, for each $S$ & $D$ in the $R_2$ data set, calculate each expected amp draw ($\hat{Y}$) and then average.
4. Compare the $\hat{Y}$ average to the actual average amp draw ($Y_2$) of the $R_2$ data.
5. $\text{percent (%) change} = (Y_2 - \hat{Y}) / \hat{Y}$

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.

• 1. A polynomial model is a linear model, because it is linear in the coefficient. 2. I am trying to understand your question. If the refrigeration system has been modified between the time R1 and R2 were used, then they are really not the 'same refrigeration system' (line 1), right? 3. Why is it in your second approach, you started comparing the coefficients of S? 4. Have you consider introducing a covariate 'refrigerants' with levels R1 and R2 into the polynomial fit (maybe with interaction)? Its coefficient might answer the question. – qoheleth Mar 5 '14 at 6:22
• @qoheleth 1. Not sure I follow your line of thinking... The coefficient is always linear - it's a number. When would the coefficient not be linear then? 2. Correct, the refrigeration system has been SLIGHTLY changed, but only to ensure the same output temperature for both refrigerants - "apples to apples". 3. 'S' is the only variable of interest for this specific comparison. 4. I have read about the covariate/interacting variable method, but fail to understand the meaning of the coefficients using such a method. Can you elaborate on interpreting the output? Thank you. – gth826a Mar 5 '14 at 17:40
• 1. from the statistical point of view, linearity in the things that you are estimating is what count, so a polynomial model is linear. An example of a non-linear model would be the mitscherlich function y=alpha(1-exp(beta-lambda*X)), where alpha/beta/lambda are what we are estimating. 3. What are you actually trying to test? is it the coefficient of S? or Y? If it is S, why is your 1st attempt a comparison in \hat{Y}? – qoheleth Mar 5 '14 at 23:29
• Y-hat would be: the actual S & D from the 2nd data set used with the coeffs derived from the 1st data set. This method is common for 'Performance Contracting' energy analyses when comparing previous equipment's energy consumption to the energy consumption after a retrofit/remodel/renovation/etc. The equation would be: energy consumption = y-hat = baseload + energy/degree-day * degree-days... where energy/degree-day is the coeff derived from the baseline regression analysis, and degree-days is from post renovation. The "what would you have consumed" if you didn't do this project scenario... – gth826a Mar 6 '14 at 3:29
• So it seems that ultimately you want to compare Y. I would say forget about calculating % change in the coefficients, in the presence of the higher order terms (S^2, S^3 etc.), the coefficients are not what you think they are. Focus on Y. The question remaining unclear to me is, are you saying the S & D in R2 means different things to the S & D in R1? If not, then you can simply fit one model to the combined dataset, with an extra covariate (X variable) called refrigerant (r1 or r2), and look at the its coefficient to make the inference, assuming your model is adequate. – qoheleth Mar 6 '14 at 9:33

From the ideal gas law here, $PV=nRT$, suggesting a proportional model. Make sure your units are in absolute temperature. Asking for a proportional result would imply a proportional error model. Consider, perhaps $Y=a D^b S^c$, then for multiple linear regression one can use $\ln (Y)=\ln (a)+b \ln (D)+c \ln (S)$ by taking the logarithms of the Y, D, and S values, so that this then looks like $Y_l=a_l+b D_l+c S_l$, where the $l$ subscripts mean "logarithm of." Now, this may work better than the linear model you are using, and, the answers are then relative error type.

To verify what type of model to use try one and check if the residuals are homoscedastic. If they are not then you have a biased model, then do something else like model the logarithms, as above, one or more reciprocals of x or y data, square roots, squaring, exponentiation and so forth until the residuals are homoscedastic. If the model cannot yield homoscedastic residuals then use multiple linear Theil regression, with censoring if needed.

How normally the data is distributed on the y axis is not required, but, outliers can and often do distort the regression parameter results markedly. If homoscedasticity cannot be found then ordinary least squares should not be used and some other type of regression needs to be performed, e.g. weighted regression, Theil regression, least squares in x, Deming regression and so forth. Also, the errors should not be serially correlated.

The meaning of the output: $z = (a_{1} - b_{1}) / \sqrt{SE_{a_{1}}^2 + SE_{b_{1}}^2 )}$, may or may not be relevant. This assumes that the total variance is the sum of two independent variances. To put this another way, independence is orthogonality (perpendicularity) on an $x,y$ plot. That is, the total variability (variance) then follows Pythagorean theorem, $H=+\sqrt{A^2+O^2}$, which may or may not be the case for your data. If that is the case, then the $z$-statistic is a relative distance, i.e., a difference of means (a distance), divided by Pythagorean, A.K.A. vector, addition of standard error (SE), which are standard deviations (SDs) divided by $\sqrt{N}$, where SEs are themselves distances. Dividing one distance by the other then normalizes them, i.e., the difference in means divided by the total (standard) error, which is then in a form so that one can apply ND(0,1) to find a probability.

Now, what happens if the measures are not independent, and how can one test for it? You may remember from geometry that triangles that are not right angled add their sides as $C^2=A^2+B^2-2 A B \cos (\theta ),\theta =\angle(A,B)$, if not refresh your memory here. That is, when there is something other than a 90-degree angle between the axes, we have to include what that angle is in the calculation of total distance. First recall what correlation is, standardized covariance. This for total distance $\sigma _T$ and correlation $\rho_{A,B}$ becomes $\sigma _T^2=\sigma _A^2+\sigma _B^2-2 \sigma _A \sigma _B \rho_{A,B}$. In other words, if your standard deviations are correlated (e.g., pairwise), they are not independent.

• "To verify what type of model to use try one and check if the residuals are homoscedastic", yeah sure... except you do not to make this assumption at all, and even if it is valid - it does in no way ensure that you have a "good" model. – Repmat Jun 26 '16 at 18:25
• If one uses OLS and the residuals are heteroscedastic, then for sure one has a biased model. Homoscedasticity is an OLS requirement, shown here. To have a good model requires other conditions, like avoiding omitted variable bias, but having serial uncorrelated errors, and linearity of the model versus dependent variable. – Carl Jun 28 '16 at 2:54
• You can have an unbiased and/or consistent model (estimates) where the residuals are heteroskedlastic. That would only imply that the usual inference procedures does not work – Repmat Jun 28 '16 at 4:21
• Heteroscedasticity flattens the slope, even if an outlier corrected this, the penalty would be large confidence intervals and a lousy model. Would not use such a model, but, yes, one can make lousy models. The medical literature is full of them. – Carl Jun 28 '16 at 12:00
• The first part of your comment is just plain wrong. I am not even sure what it means. – Repmat Jun 28 '16 at 12:02