What is my sample size?

Question

I'm a cell biologist. I have a seemingly easy question for which I've received opposing answers.

I'm trying to compare the number of peroxisomes (a cellular organelle) in control cells and those expressing a protein X. For this experiment I've cultured cells from the same batch in two separate dishes and used a transfection reagent to induce the expression of protein X in one of them. After 24 hours these cells were fixed and prepared for fluorescence microscopy. From each condition I then took 7-9 images of cells, and calculated the number of peroxisomes for each individual cells using imaging software. For each condition I've repeated the experiment 2 more times.

My question is: is N=3 or N=21-27 (3 x 7-9)? Is each individual cell a replicate within the experiment or an independent data point?

Of note, despite the seemingly homogeneous nature of cells in culture, the number of organelles between two cells in the same conditions can vary greatly. Additionally, cell transfection results in different levels of protein expression, creating variability between cells.

From looking at statistical information I would point to N=3, but yet the majority of the articles that I've checked that use this sort of analysis go for N=21-27, or don't mention it. I would like to compare this data using a mean +/- SEM and calculate a two-tailed unpaired t-test.

Thank you!

Answer 1

( This should be a comment, but I lack the reputation! I've tried to distil the essence of the question for everyone's benefit. )

So,

Let $N_c$ be the number of peroxisomes in the control population.

Let $N_x$ be the number of peroxisomes in the X-protein population.

Assume

$N_c \sim Distribution(\theta_c, \sigma_c^2)$

$N_x \sim Distribution(\theta_x, \sigma_x^2)$

We are testing

$H_0:\theta_x = \theta_c$

$H_1:\theta_x \neq \theta_c$

With everything unknown. To perform this test we have taken three samples of size $n (\in {7, 8, 9})$ from each population. However, for various reasons, the homogeneity of the samples is questionable. The question is whether to subsume the three samples from each population into one sample of size $3n$, or whether to make further distributional considerations and attempt to model the error.

Your first comment seems to be equivalent to the statement that $\sigma^2$ is large (which will be taken into account in the calculation of the MLE for $\sigma_c^2, \sigma_x^2$). Your second comment points to a source of measurement error, and the question is whether it is significant enough to model.

(I would suggest that you have to choose this yourself, depending on how significant you consider the measurement error to be. At any rate, (assuming by $N$ you mean the sample size) I don't see how $N$ could be 3 - I believe it should either be $3n$ if you think the measurement error is insignificant, or the question becomes more complex if you incorporate the measurement error.)