# Biological and technical replicates for statistical analysis in cellular biology

These are questions regarding basic statistics/reporting in biology. I have already read a couple of articles on this subject, but couldn't find a clear answer applying to my research.

I have the following scenario:

I have 3 independent cell cultures from which I have analyzed 10 cells each with a confocal microscope and obtained a value X (protein interaction) for each measurement.

I also have 2 independent cell cultures from which I have analyzed 20 cells each obtaining value X again. These samples are my negative control.

Now I would like to compare these groups with a simple student's t-test and there are two thinkable scenarios:

(1) Average each independent sample's measurements as input for my statistical analysis: M1 (n=10), M2 (n=10), M3 (n=10) vs C1 (n=20), C2 (n=20) (n=3 for samples vs n=2 for controls).

In this scenario I would group all biological/experimental replicates together and compare them against all controls.

or

(2) Compare M1 (n=10) vs C1+C2 (n=40), M2 (n=10) vs C1+C2 (n=40), M3 (n=10) vs C1+C2 (n=40).

In general, this appears to me like comparing independent Day 1, Day 2 and Day 3 of an experiment to a control.

Q1: Would it be better to compare to C1 only instead of C1+C2?

In this scenario I would compare each biological replicate with the controls. This seems more appropriate to me, as an overall/grouped result could be heavily affected by errors.

Q2: But which solution is appropriate? Q3: Are these measurements for each independent cell culture technical replicates even though they are not performed on the same cell?

EDIT: As an alternative, I could report just the first experiment as add "Experiment was repeated with similar results" as seen in various publications. However, this appears to be the worst solution to me.

• I don't think you should have gotten this far without a statistical analysis plan that would answer your question. Could you describe your analysis plan briefly? If not, could you state your questions or hypotheses? Finally, many biological measures are highly skewed, eg, concentration values, with quite different skewness in control samples and experimental or abnormal samples. Looking at each of your cell cultures, are the measurements reasonably symmetric? Commented Jun 9, 2017 at 16:19
• In forming an analysis plan, it helps to ask some simple questions. One of them is why do you have three experimental cell cultures and why do you have two control cultures? Why do you need more than one of each? Commented Jun 9, 2017 at 16:25
• Research question in short: Is there protein interaction between two constructed proteins. By assaying and comparing value (with value X from a negative control) I can get information whether both proteins might be interacting. Commented Jun 9, 2017 at 16:26
• 3 independent cell cultures for replication = experimental replicate. 2 control cell cultures were only for the purpose of further assurance. Usually, only 1 control culture is necessary/measured for these kind of experiments. Commented Jun 9, 2017 at 16:29
• What does '=experimental replicate' mean? Commented Jun 9, 2017 at 16:50

The mixed model is suitable for your situation. Put all of 70 Xs into one model. Treat M (experiment) and C (control) as fixed effect. Add the random intercept of cell culture as random effect. Based on the properties of X (protein interaction), you need to select a distribution. If X follows normal distribution conditional on treatment and cell culture, you can fit linear mixed model. It can be written as: $$Y_{ij}=\beta_0 + \beta_1 X_{ij} + \lambda_i + \epsilon_{ij}$$ where $$i$$ index culture, $$j$$ index measurement, $$Y_{ij}$$ is X (protein interaction), $$X_{ij} = 0$$ for control and $$=1$$ for experiment. $$\lambda_i$$ is random intercept, $$\epsilon_{ij}$$ is error term, and both of them follow normal distributions with mean 0 and unknown variance, and they are independent.

If $$\beta_1$$ is significantly differ from 0, it means M and C have different means of X.

• Why if $\beta_0$ is differ from 0, they have different mean of X? It doesn't make sense. Commented Jul 19, 2017 at 16:49
• Typo. Should be $\beta_1$. corrected. Commented Oct 6, 2018 at 4:47

Effectively, you have five observations to compare the experimental treatment with the control treatment: 3 treatment and 2 control.

You can do either of two things.

First, you can compute the average of the measurements for each culture. Then do a t-test or a nonparametric test to compare the means of the two treatments.

Your control culture averages might be more reliable since they are based on more measurements. If you want to evaluate this you will need a hierarchical model.

You have two treatments. Nested within treatments is a random factor, culture number (numbered 1 to 5). Nested within each level of culture number you have another random factor, call it measure, numbered from 1 to 70. You can use an analysis of variance with these three factors and with this nesting. The denominator for the test to compare the two treatments still comes from variation among cultures.

I recommend the first method, at least to start.

You should evaluate whether the measurements are skewed.

• Thanks for the answer! (1) By "compare the means of the two treatments." you meant to say compare treatment to negative control? Because a comparison of treatment with treatment (it's basically the "same treatment") wouldn't allow me to make any interesting inferences. (2) Would I have to "combine" the three t-tests (from comparison of each independent culture with negative controls) afterwards? I am asking because I couldn't find any paper in the field that chose to present the data of three independent experiments/replicates. Commented Jun 9, 2017 at 19:21
• So would you recommend pooling both independent controls C1+C2 (n=40) or rather compare against either C1 (n=20) or C2 (n=20)? Commented Jun 9, 2017 at 19:22
• Neither one. You have effectively two observations, one for each control. I can't imagine why you would pool them, which isn't justifiable, or using only one of them, which gives you a weak test. I must remind you that you have effectively five observations, three treatments and two controls. Measurements made on the same culture are dependent on each other. Commented Jun 9, 2017 at 19:46
• The idea was that it might be better to compare everything to the same negative control. So if I pair C1 with M1 and C2 with M2, with what would you pair M3? Or am I mistaking here? Commented Jun 9, 2017 at 20:01
• I see no reason to do any of this. If you want answers that only agree with you then I can't offer any help. Commented Jun 9, 2017 at 20:50