# Combining independent variables in linear regression - does it make sense?

I try to model energy consumption for a set of about 50 relatively similar production facilities. I have annual data of energy consumption and 3 independent variables that - from a technical point of view - must have a significant impact on energy consumption:

• Y: gas consumption (gas is used for room heating and to a smaller extent for process heat),
• X1: heated floor area,
• X2: staff working hours as a measure of economic activity and
• X3: heating degree days as a mesaure for weather influence

The general reason for this analysis is to create an energy benchmark and rate sites' energy efficiency. The underlying thesis is that the energy efficiency of a site lies in the differences between modelled and actual energy consumption of the individual sites.

However, when putting all variables into a multiple linear regression, only two of them turn out to be statistically significant: Floor area (X1) and staff working hours (X2), (both have P-Values < 0,0001). While weather (X3) must have an impact on the energy consumption from a technical point of view, the effect is not very significant compared to the standard error of the coefficient (X3 P-value > 0,5).

Yet for the application of the model it would be important to have the weather influence somehow represented. This is because the reference period for the model is only one year, but it will serve as a baseline also for future years. If future years are significantly warmer or colder than the reference year, this would bias the whole ranking...

I see three different paths to proceed:

Path A: Drop X3, use only variables X2 and X3

Path B: Use all three variables, even if X3 has a high SE and P-Value

Path C: Combine X1 and X3 into one variable by multiplying the values (X13=X1*X3)

@A: This is the model with the best R2 (R2-adj=0.89) but we loos information on the impact of weather

@B: The value of the coefficient lies in a plausible range, and r2-adj=0.88, only marginally worse than in option A). After all the high standard error of the coefficient might also be a result of the different thermal insulation standards of the sites - in which case the model would express exactly what I wanted.

@C: I'm not sure if this is an Idiots' or a Genius' Idea: The multiplication of the values seems plausible, as the effect of X3 from a logical point of view must increase linearly with X1 (=more floor area to heat). In fact, option B would be expected to to increase statistical error compared to option C, as it would consider the same impact of cold weather in absolute terms (e.g. 1.000 kWh) for both small and large sites (yet if you have a larger site, keeping the other variables constant you have to put more energy into heating!). R2-adj of the model is slightly lower than in option A but still very good (0,88). However, I cant rationally explain but I feel a bit uneasy about mixing up variables and units in my analysis...

question: Which one of the three models would you choose?

Related questions: Is it enough to have a technically plausible theory available in order to perform a variable transformation such as described in Path C? Which general transformations are there, and is there a guideline when to apply which? Would I need to standardize X1 and X3 before I multiply them?

thanks!

Lukas

Edit: for your reference I added the residual-plots of varialbe x1,x2 and x3 in path B:

• "Heating degree days" doesn't seem to account for the effect of needing to cool, only the need to heat; and even in places that do only need heating it will be poorly (and certainly nonlinearly) related to energy consumption. I wouldn't be at all surprised that it didn't work very well -- I'd have expected exactly that a priori. – Glen_b Jan 17 at 8:07
• As we are looking on the gas consumption, we can ignore cooling energy need (cooling would normally happen with electricity). I am working a lot with HDD and monthly data, and there is usually a very clear linear relationship between heat demand and HDD. – Lukas Eggler Jan 17 at 8:23
• Okay, on the gas vs electricity but it doesn't alter the nonlinearity in the relationship even with heating. You say that it's linear, but wikipedia (and several other sites I found first) are pretty emphatic that it isn't. What do the residuals vs x3 look like? – Glen_b Jan 17 at 8:52
• @Glen_b I added the residual plots for all 3 variables in B in the main post. As you can see the HDD of the sites vary between 2500 and 3700, and I cant see a systematic trend from it... here you can find some information on HDD and energy correlations, and it is also a very good, free data source for localized HDD information! – Lukas Eggler Jan 17 at 11:23
• Given your residual plots I would first check the extreme values of X1 and X2. A point to keep in mind is that in linear regression it is th effect of each variable over and above the others already in the model, not its effect full stop. Having said that in the absence of further information I would choose your option B but perhaps explore nonlinear effects of X3 on Y. – mdewey Jan 17 at 13:19