Nonparametrics for Sensory Science:
A More Informative Approach

Instructions for Beginners

• First click here to open another webpage for Rweb. Once you have opened the Rweb page you will need to move between this webpage and the one you have just opened. Locate the big box near the top of the Rweb webpage and the "Submit" button underneath it. DON'T PRESS THIS BUTTON until the very end of ALL these instructions!
• Now click here to open another page for the program definitions, select all the text in this page, copy this program definition text, then move to the Rweb webpage you opened earlier, and paste the text into the big box near the top of the Rweb webpage. Once you have done this you can close the webpage you have just now opened (the one containing program definitions text).
• Next choose one of the examples below, and click on the relevant code link for this example (DON'T FORGET you need to press the "Back" button on your web browser to get back to these instructions). Copy all the example code text, then move to the Rweb webpage you opened earlier, and paste the text into the big box near the top of the Rweb webpage (take care to paste it BELOW the text you pasted there earlier).
• Finally, press the "Submit" button on the Rweb webpage (located underneath the big box near the top of the Rweb webpage where you have been pasting text). After a short delay, all the code which you pasted in earlier is printed in a window with your analysis results following it - you will probably need to scroll down to see the end.

Chapter 1

• 1.3 Cigarette Example code and result

Chapter 2

• 2.3 Untied Analysis Example code and result
• 2.4 Japanese Chocolate Example code and result
• 2.6 Bradley et. al. Example code and result

Chapter 3

• 3.2 Peach Example code and result (alternative code and result)
• 3.3 Softdrink Size Example code and result
• 3.5 Tomato Example code and result (alternative code and result)
• 3.6 Geese Meat Example code and result
• 3.8 Catfood Example code and result

Chapter 4

• 4.2.1 Dried Egg Example code and result
• 4.2.2 Cleaning Materials Example code and result
• 4.2.3 Ice Cream Flavour Example code and result
• 4.4 Canned Fruit Example code and result
• 4.6.1 Ice Cream (Off Flavour) Example code and result
• 4.6.2 Ice Cream (Vanilla Flavour) Example code and result

Chapter 5

• 5.3 Plate Count Example code and result
• 5.4 Whiskey Example code and result
• 5.4.1 Lemonade Example code and result
• 5.4.2 Geese Meat Example code and result

Chapter 6

• 6.2 Wine Example code and result
• 6.3 Ordinal Data Example code and result
• 6.5 Hot Chip Example code and result
• 6.6 CMH Analysis of Ordinal Trinary Example code and result
• 6.7 CMH Analysis of Hot Chip Example code and result
• 6.8 Page Analysis of Ordinal Trinary Data code and result

Chapter 7

• 7.3 Examination Mark Example code and result
• 7.5 Cordial Drink Preference Example code and result
• 7.7 Radioactive Counts Example code and result
• 7.8 Milk Bacteria Example code and result
• 7.9 Fat and Protein Content Example code and result

Comments and Corrections

Note that for a few of the examples above the data x is transformed to x*10 or x+1 before analysis (eg examples 3.6, 4.6.2, 6.2 and 6.3). This is because the rank table program requires whole numbers (integers greater than zero) as input.

Similarly, for example 4.2.1 the text explains that the ranks are inverted - that is, the lowest rank of 1 is given to the highest score and the highest rank of 4 is given to the lowest score. In addition, the input data x is fractional. To produce rank equivalent whole number data, the original data x is transformed to 10*(20-x) prior to analysis.

In section 3.5 of the book where the ties in the Tomato example data are randomly assigned, in addition to the ties noted in the text (for consumers 4 and 12) there is also a tie for consumer 22 between the Florade and Momotaro varieties. For this consumer, the tie was eliminated by ranking Momotaro above Florade. Note in the output above for this example both the broken tie data as well as the original data are analysed to produce results that do not materially disagree.

In section 6.2 of the book (table 6.5 dealing with wine example 6.2) the total number of degrees of freedom is mis-stated. As the output above for wine example 6.2 shows, because the relevant U matrix for this data is not of full rank, then 4 degrees of freedom are "lost" in the analysis. Refer to the discussion near the end of section 3.6 and the paper:

Brockhoff, P. B., Best ,D. J. & Rayner, J. C. W. (2004). Partitioning Anderson's Statistic for Tied Data. Journal of Statistical Planning and Inference, No. 121, p93-111.

In section 6.6 the book incorrectly states that "For the hot chips data S takes the value 14.9 on 8 degrees of freedom with a p-value 0.06...". As the example 6.7 output shows, the correct value for the CMH analysis Extended Stuart Test Statistic applied to this data is 12.18, which with 8 degrees of freedom corresponds to a p-value of 0.14. The conclusion in the text is largely unchanged though once again the Anderson statistic is more sensitive.

In section 7.3, the Examination Mark dataset example, the statistic V₃ is given as negative when it should be positive. In addition, the reader should understand that the p-value of 0.045 given for S₄ is in fact correct (it has been obtained via parametric bootstrap) - this is because S₄ only asymptotically has the χ₄ distribution. For the small sample size in this example (n=20) the asymptotics are not yet reliable.

About half way through section 7.5 (just after figure 7.2) two orthogonal polynomials are defined, the first should be g₁(i) rather than g_i(i).

In the third paragraph of section 7.6 the text reads "To get a statistic with an approximate chi-squared distribution with m-q degrees of freedom we should use (X_P²-Sum[V_r²,r=1,...,m-1]). Often Sum[V_r²,r=1,...,m-1] is negligible, but this needs checking for each data set." In these expressions the upper limit of the sums is meant to be q rather than m-1, so the expressions should be (X_P²-Sum[V_r²,r=1,...,q]) and Sum[V_r²,r=1,...,q].

In section 7.7 the correct value residual is R=4.48 (as shown in example 7.7) rather than 3.49. For this case, where the parameter estimate is essentially found by fitting location, we recommend "attributing" the zero df component to location (as in the second GOF analysis shown in example 7.7) which produces a p-value of 0.81141559 though the conclusion is essentially unchanged wherever this is attributed.

In section 7.8 the X_P²=6.859 value has p-value of 0.65 rather than 0.81 as in the text (see output for example 7.8). Also a few expected values in table 7.2 (bacterial cell counts) are incorrect. This table should be:

In section 7.9 the components V_r,s are defined incorrectly - the divisor should be sqrt(n) rather than n to give V_r,s=Sum[g_r(y_i,1)g_s(y_i,2)/sqrt(n),r=1,...,n]

We welcome information about potential misprints etc, please email details of any such information to pbb@imm.dtu.dk

Last updated: 5 April, 2005.