Random block mixed ANOVA

We tested proliferation of 4 types of cells (between-subject factor) during 3 consequent time points (within-subject factor). All test were done in 4 blocks with 1 plate of each type of cells in each block (so 4 independent experiments with n = 1 of each group in each block). Blocks were separated by each other by several weeks. Of course blocks differ between each other a little, but general differences between proliferation rate of different cell lines are evident. I use STATISTICA (Statsoft). How should I enter a program for correct analysis?

Without addition of blocks as a factor there is no differences between groups due to differences between blocks. I tried to add block as an independent factor add got it as a factorial design. I used interaction of block x group as the between-subject MS error term for MS group (between-subject factor) and interaction of block x group x time point as the within-subject MS error term for MS time and MS time x group (within-subject tests). Was I correct? (I'm confused by very high F-ratios in F time point x group due to very low MS block x group x time. Generally at each block differences are evident, but it is confusing...).

• If I need to nest some factor within other, then can you help me to specify right interactions. – Дима Неназванный Dec 15 '15 at 16:14
• As a note, this site is not for assistance with the use of Statsoft, but it seems your question concerns appropriateness of design of the blocking factors, which is okay for the site. – AdamO Dec 15 '15 at 16:25
• Yes. Ill specify design by myself, I just do not understand how should I designate factors (fixed or random) and whom of them should I nest (if I should) and within what factor. – Дима Неназванный Dec 15 '15 at 16:37
• For me now it looks like: Between subject effects: -Type of cells - MS type of cells - F = MS type / MS type x block -Block - MS block - F = MS block / MS type x block -type of cells x block - between error term Within subject effects: -Time - MS time - F = MS time / MS time x type x block -Time x Type of cells - MS time x type of cells - F = MS time x type of cells / MS time x type x block -Time x Block - MS time x block - F = MS time x block / MS time x type x block -time x type of cells x block- within error term (Of note, very low). – Дима Неназванный Dec 15 '15 at 16:38

I assume your scientific question concerns whether the rate of growth for each of the cell types is different. I furthermore assume you have used some appropriate method of titration so that the cell counts may be assumed to be equal at time $t=0$ for each of the $n=4$ cell types.

You have used overlapping notation to indicate the cell types and the blocks. Were all 4 cell types propogated on a single plate 4 different times (for a total of 16 cell cultures that were measured over time) or did each plate get 1 cell type (so that 4 cell cultures were measured over time). I will call the 16 cultures case a factorial design and the 4 cultures case a block design.

If you have the block design with 4 plates having one cell count measured over 3 time periods for a total of 12 observations, then the time x group x block model is overspecified. That would explain why you have such a large F value, since it's an approximation of $\infty$ obtained by the numerical solver in your statistics package. This is the wrong modeling choice. You should drop the effect due to block and only test the interaction between group and time.

If you have a factorial design, you can fit a factorial model and test for a group effect only if you use a GLM. This is because, techincally, there are no replicates in this study. A block x group x time model will have a total of 4 x 4 x 3 = 48 effects for which you only have 48 observations. Fitting a saturated model is okay if you make some probabilistic assumptions about the rate of growth for your cell counts, you would have to resolve to using a Poisson model which could be okay. This is also known as a loglinear modeling approach.

Otherwise, I would drop the block effect entirely. You have simply replicated your experiment 4 times. If this is the case, as before, testing the time x group effect will address the hypothesis of whether cell counts grow at different rates for the 4 cell types or groups.

Factors in this study are absolutely not nested.

• So it looked like: - Block 1: - Cells from plate 1 (group 1), Cells from plate 2 (group 2), Cells from plate 3 (group 3), Cells from plate 4 (group 4) - time point 1 - Cells from plate 1 (group 1), Cells from plate 2 (group 2), Cells from plate 3 (group 3), Cells from plate 4 (group 4) - time point 2 - Cells from plate 1 (group 1), Cells from plate 2 (group 2), Cells from plate 3 (group 3), Cells from plate 4 (group 4) - time point 3 - Block n......... the same as above except of course plate numbers within each block were other. – Дима Неназванный Dec 15 '15 at 16:45
• Well, blocks differed indeed. And I had 48 observations = total 16 plated (in 4 blocks - as 4 types of cells in each) in 3 repeated measures. – Дима Неназванный Dec 15 '15 at 16:49
• Blocks were separated by time of several weeks from one another. I assume factorial design as Ok, right? – Дима Неназванный Dec 15 '15 at 16:55
• @ДимаНеназванный no, it is not factorial because there are no replicates of group within your block (plates). That's okay, but--and I stress--drop the notion that you can model plate effects. You have a 1-1 plate-cellgroup, so you can't model plates. There is no blocking by plates. – AdamO Dec 15 '15 at 17:58
• I meant factorial-like design (as you have stated in your answer). Ive used GLM module with blocks as a random factor. It has retrieved meaningful results) – Дима Неназванный Dec 16 '15 at 6:02