Statistics for Experimental Biologists

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Is the information below useful? All of Chapter 3 is devoted to the question of "What is N?" in cell culture and animal experiments.




What is "n" in cell culture experiments?

One of the difficulties in analysing cell culture experiments is determining what the experimental unit is, or what counts as a replicate, or "n". This is easy when cells are derived from different individuals, for example if a blood sample is taken from 20 individuals, and ten serve as a control group while the other ten are the treated group. It is clear that each person is a biological replicate and the blood samples are independent of each other, so the sample size is 20. However, when cell lines are used, there isn't any biological replication, only technical replication, and it is important to have this replication at the right level in order to have valid inferences. The examples below will mainly discuss the use of cell lines. In the figures, the tubes represent a vial of frozen cells, the dishes could be separate flasks, separate culture dishes, or different wells in a plate, and represent cells in culture and the point at which the treatment is applied. The flat rectangular objects could represent glass slides, microarrays, lanes in a gel, or wells in a plate, etc. and are the point at which something gets measured. The control groups are blue and the treated groups are red.


Design 1: As bad as it can get

In this experiment a single vial is thawed, cells are divided into two culture dishes and the treatment (red) is randomly applied to one of the two dishes. The cells are allowed to grow for a period of time, and then three samples are pipetted from each dish onto glass slides, and the number of cells are counted (yes there are better ways to count cells, the main point is that from each glass slide we get just one value, in this case the total number of cells). So after the quantification, there are six values--the number of cells on the three control and three treated slides. So what is the sample size--there was one vial, two culture dishes, and six glass slides?


The answer, which will surprise some people, is one, and most certainly not six. The reason for this has to do with the lack of independence between the three glass slides within each condition. A non-laboratory example will clarify why. Suppose I want to know if people gain weight over the Christmas holidays, so I find one volunteer and measure their weight three times on the morning of Dec 20th (within a few minutes of each other). Then, on the morning of Jan 3rd I measure this same person's weight three times. So I have six data points in total, and I can calculate means, SEMs, 95%CIs, and can even do a t-test. But with these six values, can I address the research question? No, because the research question was do people gain weight over the holidays, but I have observations on only one person, and taking more and more observations on this single person will not enable me to make better estimates of weight changes in people. This is discussed in greater detail in Lazic (2010) and Cumming et al., (2007), and the key point is that the variability from slide-to-slide within a condition is only pipetting error (just like measuring someone's weight three times within a few minutes of each other), and therefore those values do not constitute a sample size of three in each condition.


Design 2: Marginally better, but still not good enough

In this modified experiment, the vial of cells is divided into six separate culture dishes, and then cells from each culture dish are pipetted onto a single glass slide. Similar to the previous experiment, there are six values after quantifying the number of cells on each slide. So now is the sample size six?


Unfortunately not, because even though the cells were grown in separate dishes, they are not really independent because they were all processed on the same day, they were all sitting in the same medium, they were all kept in the same incubator at the same time, etc. Cells in two culture dishes from the same stock and processed identically do not become fully independent just because a bit of plastic has been placed between them. However, one might expect some more variability within the groups compared to the first design because the samples were split higher up in the hierarchy, but this is not enough to ensure the validity of the statistical test. To keep with the weight gain analogy, you can think of this as measuring a person's weight in the morning, afternoon, and evening on the same day, rather than taking measurements a few minutes apart. The three measurements are likely to be a bit more variable, but still highly correlated.


Design 3: Often, as good as it can get

In this design, a vial of cells is thawed, divided in two culture dishes, and then eventually one sample from each dish is pipetted onto a glass slide. The main (and key) difference is that the whole procedure is repeated three separate times. Here, they are listed as Day 1, 2, and 3, but they need not be consecutive days and could be weeks or even months apart. This is where independence gets introduced, even though the same starting material is used (i.e. same cell line), the whole procedure is done at one time, and then repeated at another time, and then a third time. There are still six numbers that we get out of the experiment, but the variability now includes the variability of doing the experiment more than once. Note that this is still technical variability, but it is done at the highest level in the hierarchy, and the results of one day are (mostly) independent of the results of another day. And what is the sample size now?


The "independent" aspect of the experiment are the days, and so n = 3. Note, that the two glass slides from the same day can (and should) be treated as paired observations, and so it is the difference between treated and control within each day that is of interest (a paired-samples t-test could be used). An important technical point is that these three replications should be made as independent as possible. This means that it is better to complete the first experiment before starting the second. For example, if the cells will be grown in culture for a week, it is better to do everything over three weeks rather than starting the first experiment on a Monday, the next on Tuesday, and the third on Wednesday. If the three experiments are mostly done in parallel, they will not be as independent as when done back-to-back. Ideally, different media should be made up for each experiment, but this is where reality often places constraints on what is statistically optimal.

Continuing with the weight-gain example, this design is similar to measuring a person's weight before and after the holidays over three consecutive years. This is still not ideal for answering the research question (which was determining whether people gain weight over the holidays), but if we have only one volunteer at our disposal then this is the best we can do. But now at least we can see whether the phenomenon is reproducible over multiple years, which will give us a bit more confidence that the phenomenon is real. We still don't know about other people, and the best we could do was repeated experiments on this one person.


Design 4: The ideal design

Like many ideals, the ideal experiment is often impossible to attain. With cell lines, there are no biological replicates, and so Design 3 is the best that can be done. The ideal design would have biological replicates (i.e. cells from multiple people or animals), and in this case the experiment need only be done once. I hope it is now clear (and after reading the two references) why Design 1 and Design 2 do not provide any reason to believe that the results will be reproducible. Some people may object that it is a weak analogy, and say that they are only interested in whether compound X increases phosphorylation of protein Y, and are not interested in other proteins, other compounds, other cell lines, etc., and so Design 1 or 2 are sufficient. Unfortunately, this is not the case and it has to due with lack of independence, which is a fundamental assumption of the statistical analysis (see Lazic, 2010 and references therein). But even if you don't appreciate the statistical arguments, this analogy might help: if you claim to be a superstar basketball player and sink a 3-pointer to prove it, this is certainly evidence that you have some skill, but let's see if you can do it three times in a row hot-shot.


Replication at multiple levels

The analysis of such cell culture experiments in many published studies is inappropriate, even if there were replicate experiments. You will probably have noticed the hierarchical nature of the data: the experiment can be conducted on multiple days, there can be replications of cell cultures within days, there can be replications of more than one glass slide per culture dish, and often multiple measurements within each glass slide can be taken (in this example the total number of cells was measured, but the soma size of 20 randomly selected cells on each glass slide could have been measured, which would give many more data points). This hierarchical structure needs to be respected during the analysis, either by using a hierarchical model (also known as a mixed-effects or multi-level model) or by averaging the lower level values (see Lazic, 2010). Note that it is NOT appropriate to simply enter all of the numbers into a statistics program and run a simple t-test or ANOVA.

Two more things to note. First, it is possible to have replication at multiple levels, in the previous examples replication was only introduced at one level at a time to illustrate the concepts. However, it is often of interest to know at which level most of the variation comes from, as this will aid in designing future experiments. Cost considerations are also important, if samples are difficult to obtain (e.g. rare clinical samples) then technical replication can give more precise estimates for those precious few samples. However, if the samples are easy to get and/or inexpensive, and you want to do a microarray study (substituting expensive arrays for the glass slides in the previous examples), then there is little point in having technical replicates and it is better to increase the number of biological replicates. Second, if you want to increase the power of the analysis, you need to replicate the "days", not the number of culture dishes within days, or the number of glass slides within a culture dish, or the number of cells on a slide. Alternatively, if biological replicates are available, increasing these will increase power, but not more technical replicates.



References


Cumming G, Fidler F, Vaux DL (2007). Error bars in experimental biology. J Cell Biol 177:7–11. [Pubmed]

Lazic SE (2010). The problem of pseudoreplication in neuroscientific studies: is it affecting your analysis? BMC Neurosci 11:5. [Pubmed]