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Thread: Evaluating data via a three-way (2 x 3 x 4) ANCOVA

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    Evaluating data via a three-way (2 x 3 x 4) ANCOVA




    Biomedical researcher here, trying to figure out how to analyze the results of a study on cell culture conditions. Three IVs were chosen: compound A (2 levels), compound B (3 levels), and compound C (4 levels). Cells were grown in microplate wells, 4 wells for each of the 24 permutations of the IVs. 3 microplates were used, each containing the groups for 1 of the 3 levels of compound B. In addition, 8 wells on each microplate were dedicated to a "gold standard" control, cells grown in a different cocktail considered optimal. Cells were treated with the test 24 cocktails (+1 control) and cell activity was measured after treatment.

    Cells were added to all wells in what should be equal numbers, but because of potential imprecision, pre-treatment activity was measured (activity is proportional to cell count). Activity was measured again post-treatment.

    I was advised to perform an ANCOVA on my data (3 IVs, post-treatment measurement as the DV, and pre-treatment measurement as the covariate). Pre-treatment activity was a significant covariate and a three-way interaction between the IVs was identified (as expected). Now I'm stuck. A few questions:
    1. I learned that a 3-way interaction in an ANOVA requires 2-way ANOVAs for each of the levels of the third IV to tease out main effects, but what do I do with a 3-way ANCOVA interaction? Part of my dilemma: I want to report the "adjusted" post-treatment measurements as if all groups had started off equivalently, but the adjusted means are different for the 3-way ANCOVA vs. the 2-way ANCOVAs, since a subset of the data is used in 2-way ANCOVAs. Which adjusted means are the proper ones to report, and in general how is a 3-way ANCOVA interaction decomposed?
    2. How do I handle my "gold standard?" Is it appropriate to include it in the ANCOVA (so that its mean also gets adjusted), or does that invalidate the analysis? I don't want the focal point to be comparing every other cocktail to the standard; I only want the standard included as a point of reference.
    3. Three post-treatment measurements were taken to observe early-, mid-, and late-onset effects. In terms of a timecourse, is it appropriate to simply run an ANCOVA for each time point and observe the introduction/disappearance of significant effects?

    This is a lot, so any help is appreciated!

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    Re: Evaluating data via a three-way (2 x 3 x 4) ANCOVA

    Quote Originally Posted by lnhstats View Post
    [*]I learned that a 3-way interaction in an ANOVA requires 2-way ANOVAs for each of the levels of the third IV to tease out main effects, but what do I do with a 3-way ANCOVA interaction? Part of my dilemma: I want to report the "adjusted" post-treatment measurements as if all groups had started off equivalently, but the adjusted means are different for the 3-way ANCOVA vs. the 2-way ANCOVAs, since a subset of the data is used in 2-way ANCOVAs. Which adjusted means are the proper ones to report, and in general how is a 3-way ANCOVA interaction decomposed?
    Maybe there is a confusion about nomenclature here. I would say that a model could have main effects, two-factor interactions and three-factor interactions. If you include a three-factor interaction it it is thought as a rule to also include two-factor interaction and of course the main effects.

    Here is an example:

    y = constant + A + B + C + A*B + A*C + B*C + A*B*C + error

    where A etc is the main effect, A*B is one of the two-factor interactions and A*B*C is the three-factor interaction. It is customary to include all terms in the model and to delete those who are not significant, while retaining the lower level terms, like if A*B is included then A and B should be in the model even if they are not significant.

    Of course you need to include the pre-variable as well and maybe a few interaction effects with the A, B and C variables.

    Quote Originally Posted by lnhstats View Post
    [*]How do I handle my "gold standard?" Is it appropriate to include it in the ANCOVA (so that its mean also gets adjusted), or does that invalidate the analysis? I don't want the focal point to be comparing every other cocktail to the standard; I only want the standard included as a point of reference.
    Just estimate it separately. Then you will know the mean and standard error and you can compare that with the estimated values in the factorial model (where you also know the estimated standard error).

    Quote Originally Posted by lnhstats View Post
    [*]Three post-treatment measurements were taken to observe early-, mid-, and late-onset effects. In terms of a timecourse, is it appropriate to simply run an ANCOVA for each time point and observe the introduction/disappearance of significant effects?
    You could run the each separately, that's maybe the easiest, and you could estimate it with a repeated measures layout.



    Quote Originally Posted by lnhstats View Post
    3 microplates were used, each containing the groups for 1 of the 3 levels of compound B.
    I did not understand this part. What is the difference between the wells and the microplates?

    Is this study about bacterial growth?

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    Re: Evaluating data via a three-way (2 x 3 x 4) ANCOVA

    Thank you for your reply!

    I'm running a factorial 3-way ANCOVA (I expect synergy between the 3 IVs), so I've included all main effects and interactions. The 3-way interaction between IVs is significant. For ANOVA, I was taught that the 2-way interactions or main effects cannot be interpreted if the 3-way interaction is significant; the data must be split by 1of the 3 variables, and 2-way ANOVAs run for each level of the splitting variable. My question is: is this true also of ANCOVA? In addition, the 3-way ANCOVA reports different estimated marginal means than the split-data 2-way ANCOVAs (since only data subsets are used for the 2-way ANCOVAs). Which marginal means can I report as the "data adjusted for pre-treatment differences?"

    I want to make sure I understand your statement about my "gold standard." Are you saying I should leave it out of the ANCOVA(s) and simply compare its unadjusted value to the estimated marginal means I determine from the ANCOVA?

    Sorry about the confusion with my terminology. I'm studying mammalian cell growth. A single microplate contains 96 wells on it. There are 120 total wells being used (24 cocktails x 4 replicates per cocktail, + 8 control wells per plate * 3 plates). Each plate contains 8 of the 24 cocktails, + 8 control wells. The plates are split by compound B: each plate has every combination of compounds A and C, but only one of the three levels of compound B. In theory, an equal number of wells were added to every well, but in reality this is nearly impossible, hence my pre-treatment measurement to establish a covariate.

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    Re: Evaluating data via a three-way (2 x 3 x 4) ANCOVA

    Quote Originally Posted by lnhstats View Post
    For ANOVA, I was taught that the 2-way interactions or main effects cannot be interpreted if the 3-way interaction is significant; the data must be split by 1of the 3 variables, and 2-way ANOVAs run for each level of the splitting variable. My question is: is this true also of ANCOVA?
    It is true that it is difficult to interpret a two-factor interaction when there is a three-factor interaction. That means that when looking at a result you need to consider all three variables at the same time. Do an interaction plot (with all three factors included.)

    But it is not true that if there is a three-factor interaction, that you then need to re-estimate it as a two-way anova. If you have a three-way anova (or anacova) that fit to the data then you are done. Use that model and look at the estimated results.

    Skip the two-way anova (or anacova)!

    The most usual thing is to report the results by adjusting by the covariate at its mean value.

    Quote Originally Posted by lnhstats View Post
    I want to make sure I understand your statement about my "gold standard." Are you saying I should leave it out of the ANCOVA(s) and simply compare its unadjusted value to the estimated marginal means I determine from the ANCOVA?
    I suggest that you do just the same adjustment to the gold standard data as to the factorial experiment.


    Quote Originally Posted by lnhstats View Post
    In theory, an equal number of wells were added to every well,
    Is there a typing error here?


    Quote Originally Posted by lnhstats View Post
    .... but in reality this is nearly impossible,
    What does this mean?

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    Re: Evaluating data via a three-way (2 x 3 x 4) ANCOVA

    Thanks for your follow up!

    So, you're saying that I don't need to run 2-way ANCOVAs to tease out the simple effects, even though there is a 3-way interactions in the 3-way ANCOVA? This would be nice, but goes against everything I'd been taught to this point. Can you provide some reassurance on how I can tease out main effects when there is a 3-way interaction occurring?

    Maybe a silly question, but how do I adjust the gold standard's mean when it was not included in the factorial ANCOVA? The ANCOVA adjusts it (in SPSS) as part of the data reporting. Is it possible for me to do a calculation on my own to adjust the gold standard's data? What is that calculation?

    No typo. Each "sample" is an individual well with cells in it. When preparing the experiment, I should have added equal numbers of cells to every well, but this is technically very difficult-- the microscopic cells are suspended in a liquid at a certain concentration, and a given volume of that liquid is transferred into every well. This should result in equal numbers of cells in every well, but any inconsistencies in the volume transferred, or if the suspension was not completely uniform (i.e., there were "hot spots" in the suspension where lots of cells were clustered), then slightly different numbers of cells might have been added to the wells. In reality it's almost guaranteed that the wells contain different numbers-- the precision required for ensure equality is almost impossible. This is the main source of variation between samples before treatment. I therefore did a pre-treatment measurement to be able to pick up on any of these preparation differences, to see if the different groups were already different coming out of the gate.

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    Re: Evaluating data via a three-way (2 x 3 x 4) ANCOVA

    I guess that you have already the log (logarithm) of the cell counts (as is customary when using cell counts) as the dependent variable. It you have not, the it is a standard "trick" that can change a multiplicative model to a additive model which makes many of the interaction to "go away". Anyway, try to do the logarithm again and reestimate with 3-way-anacova.

    Quote Originally Posted by lnhstats View Post
    So, you're saying that I don't need to run 2-way ANCOVAs to tease out the simple effects, even though there is a 3-way interactions in the 3-way ANCOVA? This would be nice, but goes against everything I'd been taught to this point. Can you provide some reassurance on how I can tease out main effects when there is a 3-way interaction occurring?
    No, you run a three way anova (or anacova) and use the model that fits to the data. Then you don't need to run any 2-way-anova (or anacova). Aso notice the difference in words/nomenclature: in a 3-way-anova there are main effects, two factor interactions and one three-factor interaction. To understand the meaning of the parameter estimates you need to evaluate all the three factors. E.g. do an interaction plot by plotting the four levels on the x-axis and the 6 means for the other factors, resulting in the 24 points.

    It would be bizarre to first estimate a 3-way-anova and then estimate a 2-way-anova on a subset of the data. Look in any textbook about anova if you don't trust this.


    Quote Originally Posted by lnhstats View Post
    Maybe a silly question, but how do I adjust the gold standard's mean when it was not included in the factorial ANCOVA? The ANCOVA adjusts it (in SPSS) as part of the data reporting. Is it possible for me to do a calculation on my own to adjust the gold standard's data? What is that calculation?
    You did measure the pre -value for both the factorial part and the gold standard, didn't you? and wasn't the slope for the covariate about the same in the factorial part and in the gold part?

    Quote Originally Posted by lnhstats View Post
    No typo. Each "sample" is an individual well with cells in it. When preparing the experiment, I should have added equal numbers of cells to every well, but this is technically very difficult-- the microscopic cells are suspended in a liquid at a certain concentration, and a given volume of that liquid is transferred into every well. This should result in equal numbers of cells in every well, but any inconsistencies in the volume transferred, or if the suspension was not completely uniform (i.e., there were "hot spots" in the suspension where lots of cells were clustered), then slightly different numbers of cells might have been added to the wells. In reality it's almost guaranteed that the wells contain different numbers-- the precision required for ensure equality is almost impossible. This is the main source of variation between samples before treatment. I therefore did a pre-treatment measurement to be able to pick up on any of these preparation differences, to see if the different groups were already different coming out of the gate.
    Have you seen in your area if it is customary to think of the cell count as having a Poisson distribution?

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    Re: Evaluating data via a three-way (2 x 3 x 4) ANCOVA


    I should clarify: cell count is not the dependent variable of interest, so I am not performing any sort of transformation, log or otherwise. The cells I'm interested in express a sort of activity that can be quantitatively measured, and my intent is to investigate how different substances modify that activity's expression level. The measure of activity, though, is dependent on the number of cells: more cells = higher activity measurement. I wanted the number of cells in each well to be constant, but unfortunately this is almost impossible to accomplish. I therefore took a measurement of the activity prior to treating the cells with these different substances to answer the question, "What pre-existing differences have I inadvertently introduced?" In essence, I'm attempting to remove cell count as a confounding variable. I've read that ANCOVA is a great method for prettest-posttest designs to remove the influence of pre-existing differences, so here I am. The activity measurement prior to treating the cells is my covariate.

    In my case, the 3-way ANCOVA identified that the covariate was significant, as expected. It also said that everything was significant: the 3-way interaction between IVs, the three 2-way interactions between IVs, and the individual IVs. How do I interpret that? Slide 10 from this lecture says that if a 3-way interaction is significant in ANOVA, do not interpret the 2-way interactions or main effects-- move to follow-up tests. Wouldn't the same be true of ANCOVA?

    I see what you're saying now about adjusting the gold standard's value. The gold standard's relationship between covariate and DV should be the same as for the other groups (homogeneity of slopes assumption of ANCOVA), so I can simply take the model for the groups in the factorial ANCOVA and apply it to the gold standard myself, but do not include it in the factorial ANCOVA's calculations. Is that correct?

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