Moreover an analysis of the combined levels only - such as Prod:Ass - may in
some cases ... Pic:Ass show significances whereas Prod:Ass itself is very far from
.
Automated and visual mixed model ANOVA for 4-way sensory data Niels Axel Sommer and Per Bruun Brockhoff, I MM, Richard Petersens Plads, DTU - Building 321, DK-2800 Kgs. Lyngby, Denmark
Completely balanced 4-way design for each of 15 attributes/variables. The factors and the number of levels are:
Initially six F-tests are performed; within each of the three strata given in figure 2 the coarser interaction terms are tested directly against the finer interaction term and from this point further testing will depend upon the results of these six tests.
3 Tv sets (Tv) 4 pictures with different motives (Pic)
F = MS ( prod ) ( MS ( Ass ∗ prod ) + MS (Re p ∗ prod ) − MS ( Error ))
Data were kindly provided by Søren Bech, B&O, Denmark.
Or similarly
F = MS ( prod ) ( MS1 + MS 2 − MS 3)
Introduction
Corresponding to the 3 strata in Figure 2. And Satterthwaithe approximation of degrees of freedom are used.
Sensory data may often be viewed as typical 3-way “product-byassessor-by-replicate” sensory profile data justifying a 3-way mixed model analysis (for each attribute) in which all effects apart from Product are considered random (see figure 1).
The 4-way test for e.g. TV similarly becomes :
Decomposing the fixed main effect of Prod into its components Tv and Pic is straightforward and similarly each of the random effects in the 3-way mixed model can be decomposed as shown in the three strata in figure 2. The purpose of this decomposition is twofold:
If TV:Ass is NS and Pic:Ass is NS:
1) To obtain information about the possible interactions determining e.g. if assessors primarily disagree about subfactor 1 or 2 (Tv or Pic).
In e.g. Assessor stratum (1):
SS1 = SS (Tv : Ass ) + SS ( Pic : Ass) + SS (Tv : Pic : Ass ) If TV:Ass is NS and Pic:Ass is Sign.:
SS1 = SS (Tv : Ass) + SS (Tv : Pic : Ass) If TV:Ass is Sign:
SS1 = SS (Tv : Ass)
2) To account properly for the variability in the data when making final conclusions about Prod and its components.
(Effects corresponding “zero/negative” variance components are always pooled.)
Compared to the 3-way model the 4-way model includes 6 additional random components and typically it is not feasible to run this mixed model in R. We give a generally applicable practical strategy for investigating the products and the product components and the strategy is automated in a user friendly R-package.
Graphical results
Comments and discussion
Figure 3. F-statistics for Prod in 3-way mixed effect models
The 4 last figures all show F-test statistics colored by test significance. Figure 3 shows the proper mixed 3-way analysis of Prod and figure 4 elaborates on these results showing in addition the proper mixed model decomposition of Prod into Tv, Pic and TV:Pic. This reveals a new level of information. For example it can be seen that regarding the attribute ”Dim.glass” the significance of Prod is due to differences among Tv. Moreover an analysis of the combined levels only - such as Prod:Ass may in some cases totally disguise effects concerning the underlying factors. Figure 5 displays an example of this when it comes to the attribute ”Flick.stat”. Both the two underlying structures Tv:Ass and Pic:Ass show significances whereas Prod:Ass itself is very far from being significant. This is an example of how effects that have relatively few degrees of freedom associated ”drowns” in comparison to other effects. Figure 6 shows several cases of these disguised effects. It might seem a bit strange at first glance to find effects concerning interactions with Rep (replicate) but one should bear in mind that Rep sometimes is used as Session.
Using as the basic structure a 4-way fixed model corresponding to the mixed model in figure 1, proper tests can be constructed for nicely balanced data like the B&O data.
Figure 1. 3-way mixed model factor diagram. 96
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Figure 4. F-statistics for decomposing Prod into components
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MATFORSK and IMM collaborates in a project producing the free of charge sensometrical software Panelcheck. A course in using the software will be given the 28-30 November 2007.
Figure 2. 4-way mixed model factor diagram. 24
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Read more on http://www.matforsk.no/Panelcheck
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Figure 6. F-statistics for decomposing Prod:Rep
Tv:Rep
where each of the three strata terms depends on the outcome of the initial testing: All additional non-significant interactions are removed/pooled again. Pic:Rep
F = MS (TV ) ( MS1 + MS 2 − MS 3)
However, two-way product structure designs where product levels are combinations of levels of two known sub-factors are not uncommon. In these cases the 3-way analysis will not provide direct information as to how potential effects might be related to the product components (the two sub factors and their interaction).
R-package Sensmixed
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• Performs 3- and 4 way mixed model ANOVA for sensory data • Standard 3-way analysis as cuurently available in PanelCheck • Extended 4-way analysis • Visually based reports of results • Automated analyses • Inbuilt model investigation strategies • Identification of significant product effects • Identification of important noise structures • Will be able to handle missing data • Soon to be available from: www.imm.dtu.dk/~pbb
