Which statistical test should be used to compare two groups with biological and technical replicates? I'am conducting a drug screening experiment samples grouped by two conditions (expression or not of a specific molecular receptor). For the group expressing the receptor I have four diferent samples, while for those not expressing that receptor I have five different samples. And I am testing 10 different drugs on all samples. Each drug-sample screen is conducted in three technical replicates.
Now, I'm interested to know if there is statistically different sensitivity to each drug between both groups of samples.
How can I deal with the technical replicates statistically? Do I need to caculate the mean value of sensitivity for the three replicates for each sample? And then employ a non-parametric test, like the Kruskal-Wallis test?
Do I need to consider the multiple testing issue?
Eg:

*

*Drug A tested in samples A-D (group of cell lines expressing the receptor) in three technical replicates and in samples E-I (group of cell lines not expressing the receptor) in three technical replicates.


*Drug B tested in samples A-D (group of cell lines expressing the receptor) in three technical replicates and in samples E-I (group of cell lines not expressing the receptor) in three technical replicates.
...
And I want to compare the sensitivity of both groups, for each drug tested.
 A: A mixed-effects model (a.k.a. hierarchical model) with a random effect to account for the technical replicates would be the standard way to analyse your data. Random effects are not ideal when the number of groups is as small, though, and that may introduce problems.
Other comments:

*

*There's no need for a nonparametric test, and no need for multiple testing corrections.


*"I want to compare the sensitivity of both groups, for each drug tested." This suggests that you want to put the response to both drugs in one model. It's likely possible but I can't recommend a good solution without more details about your experiment.


*FWIW, what you mean by 'sample' is not very clear. Looking at the original version of your question helped me understand it better; I think 'cell line' is more clear, though more detail about the experiment in general would help.
A: Your primary interest is differences between two groups defined by expression of the receptor. Each individual cell line represents the corresponding class of cell lines: those either expressing the receptor, or not. The main variance of interest is thus the variance among cell lines within each group.
The answer differs somewhat depending on the nature of your "sensitivity" measures.
If it's OK to average those measures over your cell lines within each group (the data meet the assumptions of ordinary least squares regression, with constant variance among cell lines within each combination of group and drug), then it's simplest to average the technical replicates for each combination of cell line and drug treatment.* There's no need to use non-parametric tests thereafter, unless you have reason to believe that the linear-model assumptions aren't met.
Such measures, however, often shouldn't be simply averaged. Measures like the fraction of living cells following exposure to a drug are often better modeled with binomial statistics, for example a logistic regression. See this page and its links, among many others on this site.
A mixed model, with drug and group as fixed effects and cellLine as a random effect, would directly use all the technical replicates and accommodate ordinary, logistic, or Poisson (count-based) regressions. The data setup would be one row per technical replicate, with a column representing outcome (2 columns for some binomial-regression setups) and other columns containing the group, drug, and cellLine.
The general form (with the lme4 package in R) could be:
model <- glmer(outcome ~ drug*group + (1 + drug|cellLine),   
    family = yourChoice, data = yourData)

The choice of family represents the type of regression (e.g., gaussian or binomial) appropriate for the type of outcome (e.g., outcome could be a matrix of two columns representing numbers of dead and alive cells in each replicate for a binomial regression).
The drug*group term gets expanded into terms for each of drug and group and a set of drug:group interaction terms that allow differences between the groups with respect to sensitivity to the drugs. The random effects term (1+drug|cellLine) allows for random intercepts (baseline sensitivities) and different sensitivities to drugs among the cell lines.  See the lmer cheat sheet for details.
In the question you seemed to propose multiple non-parametric tests, one for each drug, followed by multiple-comparison correction. A single model evaluating all drugs at once has advantages in combining information from all experiments together and in allowing omnibus tests of whether there are any differences at all. If you want to work more non-parametrically, a proportional-odds model (a generalization of the Kruskal-Wallis test, Mann-Whitney for 2 groups, that you considered) could put all of the data together.
One final thought: it's seldom the case that a cell line has a simply binary yes/no expression of a receptor. There can be some low level of expression in "non-expressing" lines, and "expressing" cell lines can differ substantially in expression levels. You might consider using the actual receptor-expression levels of the cell lines as a predictor instead of binary grouping.

*If you have different numbers of technical replicates within each cell-line/drug combination, you weight each average by the corresponding number of replicates in the regression.
A: I'd use the Friedman test. It's used when the groups are dependent (e.g. repeated measures or randomized block design).
The format is friedman.test(y ~ A | B, data) where y is the numeric outcome variable, A is a grouping variable (Group A or Group B), and B is a blocking variable that identifies matched observations.
Here, y is your sensitivity, A is the group (whether it expresses or does not express the receptor), and B is the subgroup/block, each of the samples. This makes logical sense because consider another example where we test if urban, rural, suburban groups differ across a certain metric. These are the groups. In each group we have a sample (of people). We could take repeated measurements on these people. These are the repeated measurements on each sample in your cells.
