Problems understanding design matrices and matrix models

Question

I have to analyze gene expression data acquired by DNA microarray experiments. Unfortunately, I never had a good lecture on statistics, which is why I'm often struggling with data analysis.

The biological experiment I'm analyzing was designed as follows:

A cell culture was split into 4 equal batches, three of which were treated with different molecules – let's name them M1, M2, and M3 – and one was not treated (negative control, NC) for one and a half weeks. At different time points (t0, t2, t5, t7, and t10), cells were harvested for DNA microarray analysis.

I now want to compare the effects of the different treatments over time. My idea was to use the model function ~ 0 + treatment + time in order to simulate both effects. This formula generates a model matrix with eight columns: treatmentM1, treatmentM2, treatmentM3, treatmentNC, time2, time5, time7, and time10. I do not understand why there is no time0 column, even though the time level includes all five time points.

treatmentM1	treatmentM2	treatmentM3	treatmentNC	time2	time5	time7	time10
0	0	0	1	0	0	0	0
0	0	1	0	0	0	0	0
0	1	0	0	0	0	0	0
1	0	0	0	0	0	0	0
...	...	...	...	...	...	...	...

Do you think that my formula is well-suited for analyzing this type of experiment? Do you have good resources that I can understand without having deep knowledge of statistics? Most resources I find (1, 2, 3) are way too complex.

Answer 1

I was going to suggest the following paper:

Law CW, Zeglinski K, Dong X, Alhamdoosh M, Smyth GK, Ritchie ME (2020). A guide to creating design matrices for gene expression experiments. F1000Research 9, 1444. https://f1000research.com/articles/9-1444

which would seem to be exactly what you are after. However I've now noticed that you have already rejected this paper, along with all others, as "way too complex". We went to a lot of effort to make that paper accessible to as wide an audience as possible, so I have to give up here. A famous quote attributed to Albert Einstein says that Everything should be made as simple as possible, but not simpler.

The model formula that you have proposed has the flaw that it assumes the time effects to be the same for each treatment, which seems an unwarranted assumption. However the experiment you are analyzing seems itself to have the fundamental flaw that there is no replication of any of the treatment/time combinations, so it cannot be analyzed by the standard simple analysis approach that we usually recommend for replicated microarray experiments. There is an analysis strategy based on splines that I would use myself for an unreplicated time-course experiment like yours. The approach is described in the "Time course" sections of the limma and edgeR User's Guide, but it is a lot more statistically sophisticated than what you seem willing to consider.