That depends. The problem is that you haven't put in all the necessary information.

I am not clear on "sometimes there is variability."

        Is there a pattern?

        Why would you expect the response between days to be invariant? Do you include "day" as a variable in your model?

Why are the two quantitative variables not random factors? Are these quantitative factors that have been coded? Are the two quantitative variables and 1 random factor the independent variables, or is one of these the dependent variable?

How many replicates are there?

What is your goal? If you graph your results can you see your answer?

Who will be reading the paper, and what are their expectations? I would have different expectations if this is a dissertation versus a paper in the journal Science. At least some journals state in the "Instructions to Authors" what they want as far as statistical tests. Sometimes there are Federal guidelines for data reporting to insure some minimum standard of quality.

Dear Mariska

If I understanding your question , you have two fixed variables and one random , but you must tell us about your hypothesis , and the goal of the data analysis , such as you want to determine the significant effect of treatment on the response or estimate the variance components .

You must also discuss the important questions that mentioned by Timothy.

Good Luck

I think the crucial question (in addition to those raised above) is what your dependent variable is. It sounds like you have two fixed independent variables (Treatment and Day) and one random factor (Cell), but as far as I can tell you haven't mentioned what your dependent variable is. This is actually the most important thing to know. A given mixed-effects regression paradigm can handle pretty much any kind of independent variable (for instance, linear mixed effects can handle any mixture of continuous, ordinal, and categorical IVs) but which paradigm to use depends on the dependent variable (for instance, standard linear mixed effects needs a continuous outcome variable, whereas for a binary outcome you would instead use generalized [binomial] linear mixed effects).

@Stephen,

   That is an interpretation that I had not thought of. I had assumed that "this" referred to the entire experiment, but I can now see that "this" could refer to the response. There is not enough information to figure out which interpretation is correct.

Thanks for all your answers. My research question for this experiment is if a PLC antagonist inhibits the G-protein agonist induced activation of bioluminescence in dinoflagellates. I tested the light response of vials with 2000 cells in suspension to either a fixed concentration of agonist, or a control  (so 2 treatments). But the cells were also pretreated with a fixed concentration of antagonist/control (so this gives 2x2 treatments). Every day I had time for 5 replicates. I performed the same experiment on three different days. Two days the antagonist inhibited the response of the agonist but on one of the days the antagonist did not inhibit the reponse, it even amplified the response of the agonist. This is why I thought that Pretreatment, Treatment and Pretreatment * Treatment were my fixed variables and 'Day' was my random variable.

I would guess that there is an error in your data. It may be caused by you, or it may be chance. The possibilities are nearly endless. I would check the data to make sure that all the numbers are correct. You are looking for things like a group of numbers that were copied and pasted into the wrong column. You are looking for an observation with a decimal point in the wrong place, or maybe a 9 was entered when it should have been a 0. Look for unusual observations. Sometimes a sort will turn up the bad entry. If that turns up no problems, then the other option is to try running more replicates. Run for 2 or 3 more days and see if you can find another example of the contrary pattern. It was a case of a bad batch of media, or some glassware not properly cleaned, a pipette not set to the right value, or someone messing with your experiment then the problem should not show up again (we hope). The more complex the experiment, the more likely something will go wrong once in a while.

If you can show that one observation is clearly in error (and you cannot correct that error), then delete that observation (or observations). If you are being completely honest, then make note of the unusual outcome in your report and state that the observations from that day were dropped and discuss what you think the problem was that caused the error.

I think the model you need is

lightresponse ~ Pretreatment*Treatment + (1|Cell) + (Pretreatment*Treatment|Day)

I don't know what kind of variable lightresponse is, but I'm assuming it's continuous. I have the impression that a given cell was tested 5 times in a day, but I could be wrong, since this wasn't clear; if a cell was tested only once then the random effect of Cell is not needed. For an explanation of why I chose the random effects I did (and how to handle convergence failures if you get them), see http://idiom.ucsd.edu/~rlevy/papers/barr-etal-2013-jml.pdf. Also note that I've heard some experts say if there are only a small number of levels (e.g. just 3 days) then th is should be put in as a fixed effect rather than a random effect (I don't have a citation for this, it's just something I've been told by a couple experts), in that case you would have

lightresponse ~ Pretreatment*Treatment*Day + (1|Cell)

or, if cells weren't measured repeatedly, then just a basic OLS regression

lightresponse ~ Pretreatment*Treatment*Day

Dear Stephen, thanks for your answer. A cell was not tested 5 times per day, 5 different vials, each containing 2000 cells, received the same treatment on one day. Light response is indeed continuous since it is the amount of photons emitted per cell per second.

If I would consider date as third fixed factor, couldn't I just use a three-way ANOVA with pretreatment, treatment and date as my independent variables and lightresponse as my dependent variable?

Yes, ANOVA is really just a special case of OLS regression. If I've understood the design properly, then it's appropriate for regression (the last code sample I gave; note those were all based on R syntax), in that there appear to be no repeated measures. So if it's appropriate for regression it would also be appropriate for ANOVA.