Hi everyone,
P-value (or fdr), calculated by the t-test, is the common criteria of statistical significance of the measurements between conditions. It is enough to have mean, SD/SEM and N to calculate a p-value between, say, 2 conditions of particular measurement. But there are also cases, when the measurement for the condition is continuous. In my case, I measure the sizes of muscle fibres on the crossections between control and treatment conditions. The common way to interpret the results, and how I also did so far, is to show the distribution of fibres into size bins, say 50-100 um2, 200-300 um2, etc. To claim, for example, that treatment induces increase of proportion of fibres of particular size, we compare the means for particular size-bin and calculate the significance based on the parameters for particular size-bin. Anyway, rather artificial binning of sizes may lead to the incorrect evaluation of the results. For example, the mean and p-value of comparison bin_(100-200 um2) between conditions may be different from those for bin_(100-210 um2), which is not quite biologically meaningful.
To make this more honest, I used the density, which gives the interpolated plot of the continuous frequency-function from the fibre size (see example pic). The red bump is an example of an enrichment of particular fibre-sizes in the treatment group. (This is the summarised plot for couple of replicates per condition) Anyway, now I wonder how to calculate the statistical significance of such enrichment without selecting particular point on the plot (which will be the same story as with binning), but rather using the diapason of values.
Do you have any suggestions how to calculate that? Is there a way to calculate the p-value as a differential equation and then integrate by the size in the particular diapason?
Thanks!
Best,
Michail
You can generally test the equality of distributions with the Kolmogorow-Smirnow test ("KS test"). However, this is not specific to any particular feature of the distribution. In the diagrams above it seems that there are at least 2 possible modes (at 50 and at 120), and the blue density seems to show a shift from the left mode at 50 into the mode at 120. If you have some theoretical idea about such modes (and what they represent) you can construct a model of the distribution as a compound or mixture distribution (e.g. of some Gaussians for the modes and a "background" distribution). You could then specifically test the shift in the mode heights.
Dear Jochen,
Thank you for the nice answer!
PS. Maybe you can also suggest me how to extract the function used for plotting the density curve in R? Or how to detect the extremal points of the plotted curve?
# plot(density(vector_of_num))
Best,
Michail
The code line you provided will plot the density. The function density() returns a list with the items "x" and "y". The plot is simply a curve through these coordinates.
Finding an absolute maximum is simple:
d
Is your sample size large enough that the pink area is real? The distribution will change from sample to sample. So these shapes have variability. I could measure 5000 fibers from one slide. I could make several slides from different muscles in the same patient. I could make several slides from the same muscle at different times for the same patient. I could take multiple samples from multiple patients all subjected to the same treatment.
Dear Timothy,
Thank you for your answer. The attached picture is more an example of an observation, significance of whose I want to calculate.
Overall, for an experiment I have 4 mice per treatment group, from each mice I have 1 same muscle, at 1 time-point, with 5 images each. Every image contain 500-1000 fibres. So I have roughly 10k-20k fibres per condition. Would you suggest to use only 4 as sample size (biological replicates), or use 20 (bio + technical replicates) for statistics?
Best,
Michail
The top-level of replication is the mouse: any lab/group that is going to replicate your experiment has to use different mice. When you report the uncertainty of your estimates, this should refer to what you think may be found in a replication study. Therefor, the uncertainty should be calculated based on 4 mice.
PS: you can use hierarchical models to estimate the contribution of the different layers of replication (fiber/picture/mouse) to the total variance. This cann allow you to optimize the design of your experiment (you could, for instance, possibly find that taking more pictures may considerably improve the results, more than increasing the numebr of mice; that would be good news for future experiments. Or you may find that the pictures of the same mouse are so similar that you don't gain much by taking several pictures - so you can save making pictures, but you would have to use more mice to get better results).
I agree with Jochen's answer. From a statistical perspective my guess is that 4 mice are too few. With only 4 replicates there is a low probability that another lab running 4 replicates will arrive at the same conclusions as you do. However your ethical review board will have the deciding vote.
My guess is that one needs more than one slide per mouse. The shape of the distribution given 500 to 1000 fibers on one slide will be too irregular and the underlying distribution hard to see. Five slides/mouse may not be enough especially if you are looking at the distribution of rare fiber sizes (>250 um).
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