# Tests for consistent measurements and outliers

I have 3 experiments, where some quantity was measured 3 times. Thus, 3 biological replicas, 3 technical replicas in each biological replica, 9 measurements in total. I need to answer the following questions:

1. For each biological replica: are my 3 measurements consistent? To do this, I'm using Dixon's Q test.
2. For average measurements derived for each biological replica: are these 3 measurements consistent?

So, how should I go about the second question? Is it a good idea to use Dixon's Q test here too? What other tests can I use to address the above questions? I guess ANOVA is not suitable because samples are too small, right?

• Do you have any reason to believe your values should be normally distributed? Dixon's Q-test is based on normality, while real data is often non-normal. May 16, 2011 at 20:37
• Yes, I believe they are normally distributed.
– Leo
May 16, 2011 at 20:43

## 1 Answer

It seems highly unlikely that there would be a test that, based upon three observations, decides whether one is an outlier! The fact that 'the other two' are closer could just as well be an anomaly.

At best, you could use Dixon's Q test to find out whether the largest/smallest value of your 9 observations is an outlier. Note that even the almighty Wikipedia advises to only use it once within a dataset.

Either way: the terminology of '3 measurements being consistent' is confusing: in statistics, normally consistency points to big sample sizes...

• Ok, assume I have 20 biological replicas with 20 measurements in each, and I'm asking the same questions. Can I work at the level of 20 averaged measurements corresponding to 20 replicas as if this was my original sample? Or is it better to stay at the level of 400 measurements?
– Leo
May 16, 2011 at 20:38
• From what I quickly gather from The R outliers package description, I tend to say you may be able to say something about 20 observations, yes. But don't quote me on that. May 16, 2011 at 21:05
• The problem with doing a separate outlier detection for each replica is multiple testing, i.e. with enough tests you start getting false positives. May 16, 2011 at 21:08
• @Aniko: I'm not even going there :-) I guess if there are 3 biological replicates with each 20 technical ones, the impact of multiple (3) testing will not be insurmountable (even a FWER correction only implies multiplying p-values by 3, so really strong deviations should get picked up easily). But you're right of course: @Leo did ask for 20 biological replicas :-( May 16, 2011 at 21:56
• Dixon's test is robust in small samples and has historically been used to see if there is one outlier in a set of three. Multiple testing can always be a problem. May 10, 2012 at 22:27