Triangle tests
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Introduction
A triangle test is a test beloved by sensory scientists for its simplicity and general use in detecting presence of product differences. The principle is simple. Test subjects get served three samples. One of these contains A, two of these contain B. The test subject has the task to indicate which is the odd sample. If enough odd samples are selected then the samples are declared different. The reverse may also happen. If few enough odd samples are selected the products are declared ‘same’. This is a commercially relevant question as A may contain ingredients which are cheaper or from a competing supplier.Statistical Testing
As follows from the setup, the chance to select the odd sample is 1/3 under H0; products are the same. This allows calculation of the critical values. For 10 to 50 triangles these are plotted below.
This leads to the question: what could be detected with 90% power? This is plotted below. Extreme cases, say 70% or 80% correct can be detected quite easy. Problem is, these are not relevant. In general creation or application staff would reject these before going to the sensory lab. Hence 30 to 40 tests is in general found the minimum to test when searching to declare samples close enough.
R script
triangle <- data.frame(n=10:50)
triangle$crit.val95 <- qbinom(.95,triangle$n,prob=1/3)
triangle$ph0 <- pbinom(triangle$crit.val,triangle$n,1/3)
triangle$pow.6 <- pbinom(triangle$crit.val95,triangle$n,.6,lower.tail=FALSE)
triangle$get.pow.90 <- sapply(1:nrow(triangle), function(row) {
uniroot(function(x)
pbinom(triangle$crit.val95[row],
triangle$n[row],x,lower.tail=FALSE)
-.90,
interval=c(1/3,1))$root
}
)
plot(crit.val95~n,data=triangle,xlab=’Number of triangles’,type=’b’,
ylab=’Critical value’,main=’Triangle test at 95%’)
plot(ph0~n,data=triangle,xlab=’Number of triangles’,type=’b’,
ylab=’1-alpha’,main=’Triangle test at 95%’)
plot(pow.6~n,data=triangle,xlab=’Number of triangles’,type=’b’,
ylab=’Power for H1 (60% correct)’,main=’Triangle test at 95%’)
plot(get.pow.90~n,data=triangle,xlab=’Number of triangles’,type=’b’,
ylab=’Proportion correct for 90% power’,main=’Triangle test at 95%’)
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