Statistics for Experimental Biologists

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Is the information below useful? Chapter 2 covers spatial effects (where they might occur and how to deal with them) in more detail.




Randomisation I: spatial arrangement of samples


Most scientists are aware of the benefits of randomisation, at least in regard to allocating subjects to treatments (or treatments to subjects, however you want to think about it). What is less widely appreciated are the many other areas where an experiment can benefit from randomisation, and where lack of randomisation or appropriate blocking can make the experiment a complete waste of time and effort. There are two key areas where randomisation and blocking can benefit an experiment. The first is the spatial arrangement of samples, and the second is order of sample collection and processing, which is discussed in the second article on randomisation.

Many in vitro and molecular biology studies use multi-well microtitre plates. The arrangement of samples on these plates is important, as technical artifacts can make interpretation of the results difficult, and sometimes not even possible. One type of artifact is an edge-effect, where wells near the edge tend to be different than wells in the middle of the plate, often because wells near the edge tend to dry out faster, which can change the concentration of substances remaining in the well. This is particularly true for plates with small wells (e.g. 384-well plates and higher). Another type of artifact is a gradient across a plate, with higher values being obtained at one side compared to the other. A third effect is a "plate effect", which occurs when there are systematic differences between plates. The layout of samples within and across plates should be arranged so that valid inferences can be obtained even if these artifacts are present.

The figure below shows a layout with 48 control (light blue) and 48 treated (red) samples. Each well could represent a blood sample from an individual (n = 96), and the concentration of a cytokine will be measured. This arrangement of samples is problematic because any gradient in cytokine concentration in the left-to-right direction will be confounded with the treatment effect. For example, if the right side of the plate has higher values for technical reasons, it will appear that this is due to an effect of the treatment. (Note that in this particular example, it would likely be possible to detect such a gradient within each treatment group and adjust for it, partly because of the large sample size.) A positive aspect of this design is that a gradient from top-to-bottom would not be confounded with the treatment effect, and there are an equal number of samples on the edges for each treatment.


96 well plate

A better (from an experimental design and statistical perspective) is the randomised layout in the figure below. Here, gradients in any direction will not be confounded with treatment effects, and (on average) the number of samples on the edges for each treatment will be similar. It would be possible to detect a gradient in cytokine levels by examining the residuals of the statistical model, as a function of well position (this will be the topic of another article), and then this could be taken into account.


96 well plate

One drawback of the above layout is that if the treatment is being applied one well at a time by a researcher, it might be more likely that the treatment is accidentally applied to the wrong well (they all look the same!). This would be less of a concern for an automated system that could be programmed. Therefore a compromise between what is statistically optimal and the constraints of what is experimentally feasible is shown in the figure below. Here, the treatment and controls are alternated in each column. This still provides protection against gradients and edge-effects, while being less error-prone.


96 well plate

Below is a more complex (and realistic) example, with four conditions (different colours), samples from eight individuals (each sample receives each treatment), and three technical replicates per sample for each treatment. An unsuitable arrangement would have the three technical replicates beside each other, and all of the experimental conditions clustered together; for example, columns 1-3 are the controls, columns 4-6 are treatment one, etc. This layout is prone to gradients and edge-effects being confounded with experimental condition. One reasonable layout is shown below, where treatments alternate by columns, and each technical replicate for an individual is in the same row. So for example, A1, A5, and A8 are three technical replicates from one individual in the control condition; A2, A6, and A9 are the technical replicates from the same individual for treatment one, etc. A key thing to note is that the technical replicates are not beside each other (e.g. A1, B1 and C1). One drawback of this layout is that all of the wells for two subjects are all on the edges (rows A and H), so these subjects might be different than the rest. However, this may not be a major concern because the appropriate analysis would compare differences within individuals; for example, if all of the values for the individual located in row A are 20% higher than the other samples due to an edge-effect, the difference between treated and control values for this sample would likely not be affected. However, even if this were not true (maybe due to a ceiling effect) this would not systematically bias the results, it is just one individual that is a bit different, not a whole experimental condition.


96 well plate

Note that in the above design, if there were another eight individuals, these could be placed on another plate with the same layout. One additional useful feature would be to have the first column on the second plate different from the first plate, so that the control condition is not always in the first column. More control (light blue) wells are on the edge on the first plate, so this could be balanced out across plates, for example by ordering the conditions red, light blue, green, and purple on the second plate. The drawback is that different plates have a different arrangement of treatments, which is one more thing to keep track of and therefore one more place where human error can be introduced. There are other suitable layouts and there is nothing special about this one; the aim is to find a layout that is easy to implement while also ensuring that any technical artifacts are not confounded with treatment effects.

Another important point is that all of the control samples should not be on one plate while all of the treated samples are on another, because it is not possible to separate plate effects (where the average value between plates differs due to technical reasons) from treatment effects. A good rule of thumb is to put whatever you are interested in comparing on the same plate. Design layouts can quickly become complex, and it might be worth consulting a statistician. Zimmermann et al., (2009) give an example of how they dealt with potential spatial artifacts and sources of variation for a particular type of assay.

96-well plates were used in these examples, but the general ideas extend to plates of other sizes, lanes on a Western blot gel, the arrangement of samples under a photographic film for autoradiography, some types of microarrays (e.g. Illumina chips can have up to twelve arrays per chip), and in many other areas. If randomisation or blocking are not done, then technical artifacts can masquerade as treatment effects, and it may not be possible to determine whether differences between groups are due to the treatment or are an experimental artifact. Technical artifacts are often present to various degrees, and they can often be large. It is important therefore to appropriately arrange the samples so that valid inferences can be obtained.



References


Zimmermann H, Gerhard D, Dingermann T, Hothorn LA (2009). Statistical aspects of design and validation of microtitre-plate-based linear and non-linear parallel in vitro bioassays. Biotechnology Journal 5(1):62–74. [Pubmed]