# Making sense of an actual equation obtained after Box-Cox transformation

I have recently performed and analysed an experiment and I am currently stuck on making sense of the outcome. Any help would be much appreciated.

The experiment consists of a simple DC voltage source in series with two resistors and a small incandescent lightbulb. The idea was to simply vary the voltage and the resistor values, and to measure the illuminance of the lightbulb. So, three inputs and one output. The measuring of the illuminance was done with a luxmeter, which was kept at the same distance throughout the experiment. A certain statistical design (Two Level Full Factorial design) was kept in mind for the experiment. This allowed for an ANOVA to be done. In this case you would expect all inputs to be significant, such a conclusion was indeed drawn during the analysis. The design required a Box-Cox transformation to be done, and the transformation exponent was found to be optimal at roughly $$0.25$$. This is where my problem arises.

It makes virtually no sense physics wise that the illuminance of the lightbulb is proportional to voltage to the fourth or resistance to the fourth, yet this appears to be exactly the case. I can't for the life of me think of an explanation. Does this simply mean my experiment was improperly designed? I am still very new to the idea of using statistics in this manner, so I hope you'll go easy on me for not 100% knowing what I am doing.

Thanks for at least reading. I'm looking forward to any responses.

EDIT 1:

I would expect the illuminance to be proportional to the voltage squared and the resistance to the oneth. But I don't think it is possible for the transformation exponents to be different for each term in the actual equation on Design-Expert 12 (or Minitab 19). I think the only possible transformation in those software packages is one which is the same for all terms..

The reason I speak of an optimal transformation exponent is because otherwise the residuals does not follow a normal distribution, as can be seen in the figure below. Left graph is before the transformation, right graph is after.

The optimal value was obtained not manually but through the software Design-Expert 12. I have added the corresponding graph for the sake of giving complete information.

• You say $0.25$ was "optimal," but the important question is what values are consistent with the data? Would $0$ work? What about $0.5$? Or even $1$? Do any of these have physical interpretations? For a worked example of Box-Cox transformations in a physics experiment see stats.stackexchange.com/a/35717/919 and for a discussion of how to select a reasonable parameter when the software suggests a strange looking one (in the context of the same experiment) see stats.stackexchange.com/a/60455/919.
– whuber
Feb 11, 2021 at 21:58
• I don't understand the edits, because I can't square them with the experiment you describe. The simplest statistical model even worth considering would be of the form $$E[I^\xi]=\beta_0 + \beta_1(R_1)^{\lambda} + \beta_2(R_2)^{\lambda} + \beta_3 (V)^\mu$$ (which forces the Box-Cox parameters of the resistances to be the same) but that is more flexible than the limited models you describe.
– whuber
Feb 11, 2021 at 22:52
• This sounds like purely a software issue of squaring the voltage values before analysis. Am I missing something?
– whuber
Feb 12, 2021 at 14:07
• I am lost at "square the voltages ... outcome does not change," because if that's the case then the data must be unusually degenerate! The problem is that you haven't at any point clearly indicated what your model is. I think that level of rigor will be essential for communicating your question and using mathematical expressions is a must.
– whuber
Feb 12, 2021 at 14:57

Consider that you haven't reported any confidence interval for your estimate of the $$\lambda$$ constant. Similarly, you did not anticipate a priori an adequate sample size to determine an appropriately precise estimate of the Box-Cox constant, rather you expected the prespecified model to be within "oneth" of your estimates. I see perhaps N=50 observations in the QQ-plot.