Testing whether replicate number has an effect on outcome I have four biological (independent) replicates in each of my experimental conditions. There is a potential issue with our equipment - it's possible that the first replicate for each condition is contaminated and needs to be excluded from the analysis. What's the right way to test whether the first replicates are significantly different from the others? I was thinking of multivariate regression using replicate numbers and conditions as independent variables. Is there a better choice?

If you have serious reason to question the equipment used in the first set of biological replicates (other than you just didn't like the way the results looked), you are probably safest if you discard those data.
You could proceed the way that you propose, using additive fixed-effect terms to distinguish the replicates. But that would only account for differences in baseline values; it wouldn't account for different responses to treatment conditions (e.g., from improper slope calibration of the equipment). For that you would need to include a replicate:treatment interaction, too. If you suspect systematic differences between biological replicates, one might argue that you should do that anyway.
A potential advantage is that you can not only estimate the systematic differences among biological replicates but also get estimates of treatment effects that are corrected for those differences. The potential disadvantage is that you now have more regression coefficients to estimate, potentially lowering your power to find true treatment effects.
Your situation, however, poses an additional problem: if there was external evidence of equipment malfunction, the results from the replicates examined with that equipment might not represent the underlying biological phenomena faithfully. As a rule of thumb, the precision of an estimate increases with the square root of the number of replicates. On that basis, working with 3 replicates instead of 4 only decreases your precision by about 13% $(1-\sqrt{3/4})$. If, for reasons beyond your observations of the outcomes, you think that 1 of the replicates had technical difficulties, I think that's a reasonable tradeoff to make.

A couple of extra thoughts. First, I recall a relevant mantra from early in my training: "When in doubt, throw it out." Second, you used the word "multivariate" in your question. In this context, that word is best reserved for multiple outcomes rather than multiple predictors. The analysis here with multiple predictors should be called "multiple regression." This is a common confusion, and I confess to having published at least one paper where I consistently wrote "multivariate" instead of "multiple."
