Solved – Using chi-squared or binomial for triangle test of discrimination

binomial distributionchi-squared-testdiscriminant analysis

Regarding beer taste testing. The wonderful people at Brulosophy do many experiments with beer, changing brew and fermentation methods and then using triangle tests to see whether there's a difference. The literature on sensory discrimination mentions use of both the chi square and binomial methods. E.g.
I can't glean any notion of when it's appropriate to use one or the other.

Can you give me some pointers?


Best Answer

I assume each subject is given three samples of beer to taste, two of them made according to the current recipe and one according to the new recipe. The subject tries to identify the one that tastes different from the other two.

Your null hypothesis is that the new recipe does not taste different. Suppose $n$ subjects taste independently in this way, and $X$ of of them make a correct identification. Then under the null hypothesis $X \sim \mathsf{Binom}(n, 1/3).$

If there are $n = 10$ tasters and $X = 8$ of them make correct identifications, then the P-value of the test is $P(X \ge 8) = 0.0034 < 0.01$ and you have good evidence to reject the null hypothesis at the 1% level, concluding that the new recipe tastes different. (Computation in R statistical software.)

sum(dbinom(8:10, 10, 1/3))
[1] 0.003403953

This is an example of an exact binomial test. For larger $n,$ you might use a normal approximation to the binomial distribution to find the P-value. In a more complex design you might use a chi-squared approximation.

Notes: (1) Experience in food and beverage industries has shown that it is best to use professional tasters for such tests, especially if $n$ has to be small. Randomly chosen 'beer lovers' may vary greatly in their tasting ability, some unable to make distinctions immediately under test conditions that may prove to be important over the long run. (2) If beer samples differ in appearance (color, clarity, etc.) and taste is the important issue, then it may be necessary to blindfold subjects or to serve samples in opaque mugs under dim lighting.

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