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There are two types of study objectives. One has to do with the technology and one has to do with the biology. 

The biological samples are often an organism or a cell culture. The most basic biological sample is an independently grown organism or culture. When we take replicates from the sample, they should differ only from the technical effects of being handled and are technical replicates.

organism to data process

Many of you have likely read papers where people did a high throughput gene expression study which was followed up with PCR. They took some of the genes that appeared to differentially express and some that didn't and then did PCR on just a few genes. To me, this seems to focus on inference about the technology. If we have well tested technologies this type of technical replication seems to be a waste of resources unless you found something unusual (and therefore need to verify that the technology is working).

When we are introducing a new technology or new measurement protocol, it is necessary to make some inferences about its reliability.  Often this is done in conjunction with an experiment that also targets biological inference.  For example, if you want to know whether your RNA-Seq samples are good, then you do might do quantitative PCR on the same samples to see if you get the same results.  Since the technology is rapidly evolving, it is not unusual to follow-up some measurements from newer technologies with some type of "gold standard".  However, once we are comfortable with our protocols, we can focus on biological inference.

Technical replication and remeasurement of the same biological samples is typically not cost-effective when making biological inferences.  If you want to know whether there are differences between normal tissues and tumor tissues you take different tumor samples and different tissue samples not technical replicates, even if the data are noisy.  Technical replication reduces measurement error (by averagings) but biological replication reduces the standard error of estimates of parameters of the biological population (which includes both biological variation and measurement error) and is therefore more effective. [1]

[1] Krzywinski, Martin, Naomi Altman, and Paul Blainey. "Points of Significance: Nested designs." Nature methods 11.10 (2014): 977-978.