# Accounting for unknown error in multiple regression?

I have a very naive question about multiple regression and errors...one that isn't addressed here already (Choosing a robust estimator to account for measurement error in dependent variable)

I would like to untangle the effect of three independent variables - P,T,S on a dependent variable CA. The problem is that I'm using pooled data across multiple experiments (not conducted by me). The experiments involve different methodologies and have introduced an un-quantified error into the measurements of CA. So, for example, even if the response to P,T,S in CA is similar across studies the values of CA might be offset between studies because of different amounts of contamination.

How do I account for this error (which is actually lab-based contamination) in the multiple regression model? Would adding the study/experiment as a categorical predictor variable help? Or is there another way?

(Note, I'm not a statistician and am not familiar with the symbology that is often used on this forum...)

Update: After some searching I'm starting to think a Random Effects Model (i.e. Proc TSCSREG in SAS) might be appropriate? Any advice would be great...

Many thanks.

Update 2: Actually, Proc Mixed seems to do the trick...using the study as a random intercept but not as one of the effects...that is:

proc mixed data ...;
class study;
model ca = p t s /solution;
random study;
run;

• Do you believe that these errors will bias the means, or just add noise (w/ mean 0) to the observations. Also, for the sake of clarity, you believe that the effect of these errors will not change the relationship b/t P,T,&S & CA, is that correct? Jan 19, 2015 at 4:02
• Thank you for the comment. I suppose that will bias the means; for example, under certain conditions (that is, certain values of P,T and S) the dependent variable should be, say, from zero to thirty. But the contamination means that CA in some experiments might start at much higher values and then (for example) increase as P increases or decrease as S increases. It's not known whether the errors will change the relationships between the variables but if it does it's not as critical as the original displacement of the dependent variables (and beyond the scope of my analyses). Jan 19, 2015 at 4:18
• The errors are at least somewhat random, right? Ie, they aren't just adding a constant to every value from a particular experiment, right? Jan 19, 2015 at 4:27
• Well, sort of...it's not really known how the contamination affects the overall relationship (this type of contamination has only very recently been 'uncovered'). I'm willing to assume that it is functionally like adding an unknown constant to every value in the experiment. Jan 19, 2015 at 4:31
• If you have a number of experiments and each experiment has a different additive error then you will not be able to identify the intercept in the model. If each experiment adds one observation to the pooled sample, you will not be able to tell anything about the additive error in experiment $i$ as compared with experiment $j$. If, on the other hand, each experiment adds more than one observation to the pooled sample, you will be able to estimate the mean difference between the additive error in experiment $i$ as compared with experiment $j$. Does that make sense? Jan 19, 2015 at 20:17

## 1 Answer

Taking your update to the OP one step further, your situation can be set up as a mixed-effects model, where P, T, and S are fixed effects and the sets of experiments represent random effects. In principle, a mixed-effects model allows you to investigate not only whether there are differences in CA values among the sets of experiments but also, if you wish, whether the slopes of the relations between CA and each of P, T, and S differs among experiments. The trade-off for looking at differences in slopes, of course, is that as you estimate more parameters from the data you use up degrees of freedom and lose statistical power.

I don't have experience with this in SAS, but the way to specify these models in R is nicely described in this lmer cheat-sheet. A close read through that cheat-sheet will help clarify the issues in what you are trying to accomplish even if you don't end up using R for the analysis.