The sensR package
Main purposes of sensR
The sensR package: Difference and similarity testing
Statistical tests of sensory discrimation and similarity data
Per B Brockhoff
Power and sample size computations for discrimination and similarity
tests
DTU Compute
Section for Statistics
Technical University of Denmark
[email protected]
Thurstonian analyses via d0 (d-prime) estimation
Improved confidence intervals via profile likelihood methods
Linking ”one sample at a time” Thurstonian analysis to more generic
statistical analysis (regression and anova)
August 17 2015
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
c Per B Brockhoff (DTU)
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The sensR package
The sensR package
c Per B Brockhoff (DTU)
plot
ROC
AUC
us
on
The sensR package
an
ell
mp
isc
co
M
d0
dprime compare
dprime test
dprime table
posthoc
Official site:
www.cran.r-project.org/packages=sensR
eo
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Illu
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an
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rescale
psyfun
psyinv
psyderiv
pd2pc
pc2pd
s
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mp
Sa
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CI
,t
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discrimPwr
d.primePwr
discrimSS
d.primeSS
twoACpwr
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The sensR package
siz
e
Main functions in sensR
discrim
AnotA
samediff
twoAC
betabin
glm
clm
clmm
The sensR package: Difference and similarity testingDTU Sensometrics 2015
findcr
clm2twoAC
SDT
discrimR
discrimSim
samdiffSim
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Reference manual (51 pages)
News / change-log
Vignettes:
Statistical methodology for sensory discrimination tests and its
implementation in sensR
Examples for papers
Supporting package: ordinal for ratings data, A-not A with sureness, 2-AC
etc.
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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The sensR package
The sensR package
The sensR package
Obtaining sensR
The sensR package
Citation
citation("sensR")
##
## To cite the sensR-package in publications use:
##
##
Christensen, R. H. B. & P. B. Brockhoff (2015). sensR - An
##
R-package for sensory discrimination. R package version 1.4-5.
##
http://www.cran.r-project.org/package=sensR/.
##
## A BibTeX entry for LaTeX users is
##
##
@Misc{,
##
title = {sensR---An R-package for sensory discrimination},
##
author = {R. H. B. Christensen and P. B. Brockhoff},
##
year = {2015},
##
note = {R package version 1.4-5. http://www.cran.r-project.org/package=sensR/},
##
}
1 Download and install R from http://www.cran.r-project.org/
2 Open R and run install.packages("sensR")
3 Tell R that you want to use it with library(sensR)
4 Analyze your data — try for example
discrim(10, 15, method="triangle")
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
The sensR package
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The sensR package
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Regression analysis
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Replicated
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Likelihood CI
Simulation
Similarity test
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io
in
Sample size
Difference test
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(X)
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(X)
Power
d0 estimation
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The sensR package
Publications related to sensR (and ordinal)
Duo-Trio, Triangle, Tetrad
2-AFC, 3-AFC
A-not A
Same-Different
2-AC
A-not A w. Sureness
isc
The sensR package: Difference and similarity testingDTU Sensometrics 2015
The sensR package
Protocols supported in sensR
c Per B Brockhoff (DTU)
c Per B Brockhoff (DTU)
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The sensR package: Difference and similarity testingDTU Sensometrics 2015
Brockhoff , P. B., & Christensen, R. H. B. (2010). Thurstonian models for sensory discrimination tests as generalized
linear models. Food Quality and Preference, 21, 330–338.
Christensen, R. H. B., & Brockhoff, P. B. (2009). Estimation and Inference in the Same Different Test. Food Quality
and Preference, 20, 514–524.
Christensen, R. H. B., Cleaver, G., & Brockhoff, P. B. (2011). Statistical and Thurstonian models for the A-not A
protocol with and without sureness. Food Quality and Preference, 22, 542-549.
Christensen, R. H. B., Lee, H-S. & Brockhoff , P. B. (2012). Estimation of the Thurstonian Model for the 2-AC
Protocol. Food Quality and Preference, 24, 119-128.
Christensen, R. H. B., & Brockhoff, P. B. (2013). Analysis of sensory ratings data with cumulative link models. J. Soc.
Fr. Stat. & Rev. Stat. App., 154(3), 58-79.
Christensen, R.H.B., Ennis, J.M., Ennis, D.M. & Brockhoff, P.B.(2014). Paired preference data with a no-preference
option - Statistical tests for comparison with placebo data. Food Quality and Preference, 32, 48-55.
John M Ennis, Rune HB Christensen (2014). Precision of measurement in Tetrad testing. Food Quality and Preference,
Vol 32, 98-106.
Ennis, J.M. & Christensen, R.H.B.(2015). A Thurstonian comparison of the Tetrad and Degree of Difference tests.
Food Quality and Preference, Vol 40, 263-269.
T. Næs, P.B. Brockhoff and O. Tomic, (2010). Statistics for Sensory and Consumer Science, John Wiley & Sons,
Chapter 7.
7 / 48
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Why discrimination testing?
Outline
Discrimination testing - what can we use it for?
Outline
1
The sensR package
2
Basic sensory difference testing
3
Five basic discrimination protocols
4
Proportion discriminators
5
The Thurstonian model
6
Power and sample size computations
7
Power and the five protocols
Sensory discrimination testing is used to
Show difference or similarity between products
Detect unwanted product changes in product development
Guide ingredient substitution (e.g. health initiatives)
Detect production stability
Measure consumer sensitivity
Substantiate claims (“75% of consumers can tell the difference”)
Dispute claims (“our product is at least as good as yours”)
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
Basic sensory difference testing
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Difference testing with the Triangle protocol
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
Basic sensory difference testing
Example 1: Ingredient substitution
Difference testing with the Triangle protocol
Example 1: Ingredient substitution
In a calorie reduced yoghurt natural sweetener from from the Stevia plant
is used. Unfortunately, consumers complained about a bitter after taste.
The company decided to change to a different supplier of Stevia sweetener
and now wants to know if consumers can actually tell the difference
between the current production and the new yoghurt.
It is decided to use a Triangle test with 100 consumers.
The experiment was performed:
Sensory discrimination protocol: Triangle
Sample size: N = 100, no. correct answers: X = 45
Do consumers perceive the current and new yoghurts as significantly
different?
Each subject is presented with 3 samples, two of which are equal
What are high and low values of X?
Question: which sample is most different from the two others?
What is the alternative hypothesis, HA ?
X
c Per B Brockhoff (DTU)
Y
What is the null hypothesis, H0 ?
X
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Basic sensory difference testing
Difference testing with the Triangle protocol
Basic sensory difference testing
What can we do to test our hypotheses?
Difference testing with the Triangle protocol
What is the expected proportion of correct answers?
Which range of values of pc are plausible given 45 correct answers out of
100?
How many correct answers would we expect if H0 is true and
pc = 1/3?
We don’t know the true pc
What would an extreme observation be if H0 is true?
Could the true pc be 0.40?
How extreme is 45 correct out of 100 if products are not different?
Could the true pc be 0.33?
We can estimate pc : p̂c = X/N = 0.45
Could the true pc be 0.90?
Which values for the true pc are in fact plausible?
We need a confidence interval!
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
Basic sensory difference testing
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Difference testing with the Triangle protocol
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Duo-Trio, Triangle, Tetrad, 2-AFC, and 3-AFC
pc : The expected proportion of correct answers
p̂c : The observed proportion of correct answers
H0 : Products are not different, pc = 1/3
HA : Products are different, pc > 1/3
xc = 45 is the critical value: P r(X ≥ 45) ≤ 0.05 if products are not
different
p-value: The probability of observing 45 correct answers or more if
the products are not different (H0 is true) is p = 0.01
Plausible values of pc are (0.35; 0.55) based on a 95% confidence
interval
The sensR package: Difference and similarity testingDTU Sensometrics 2015
The sensR package: Difference and similarity testingDTU Sensometrics 2015
Five basic discrimination protocols
Conclusions from Example 1
c Per B Brockhoff (DTU)
c Per B Brockhoff (DTU)
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Duo-Trio
1 reference sample XR and 2 test samples X and Y
Which sample is most similar to the reference?
Triangle:
3 samples (X, X 0 , Y )
which sample is most different from the two others?
Tetrad:
4 samples (X, X 0 , Y , Y 0 )
Group the samples based on similarity?
2-AFC:
2 samples X and Y
which sample has the strongest/weakest sensory magnitude?
3-AFC:
3 samples (X, X 0 , Y )
which sample has the strongest/weakest sensory magnitude?
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Five basic discrimination protocols
Five basic discrimination protocols
Duo-Trio, Triangle, Tetrad, 2-AFC, and 3-AFC
Protocol
Duo-Trio
2-AFC
Triangle
Tetrad
3-AFC
Samples
3
2
3
4
3
Alternatives
2
2
3
3
3
p0
1/2
1/2
1/3
1/3
1/3
Which protocol should I choose?
Different criteria:
Type
Unspecified
Specified
Unspecified
Unspecified
Specified
Specified or unspecified problem?
No. samples to taste in each trial
Statistical power
Recommendations:
The Triangle test is often preferred over the Duo-Trio test due to its
higher power.
Table: Four protocols with binomial answers.
The Tetrad test is more powerful than the Triangle test, but requires
that 4 instead of 3 samples be assessed in each trial.
p0 : The probability of a correct answer by guessing
The 2-AFC test is often preferred over the 3-AFC test because it has
similar power, but fewer samples in each trial.
Unspecified: General product differences
Specified: Attribute specific differences (sweetness, saltiness, etc.)
c Per B Brockhoff (DTU)
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c Per B Brockhoff (DTU)
Proportion discriminators
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Proportion discriminators
Which proportion of subjects can actually tell the
difference between the products?
Example 2: Proportion discriminators in a Triangle test
Assume two kinds of subjects: Discriminators and Guessers
Discriminators can tell the difference and always choose the right
sample.
Guessers can only guess and will sometimes choose the right sample.
If N = 100 in a Triangle test and 10% can tell the difference, how many
correct samples can I expect?
We denote the proportion discriminators with pd .
Is this model realistic?
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Proportion discriminators
Proportion discriminators
Proportion discriminators and proportion correct
Example 3: What is the proportion of discriminators?
From pd to pc :
Tetrad, Triangle, 3-AFC: pc =
2
1
+ pd
3
3
Duo-Trio, 2-AFC: pc =
1
1
+ pd
2
2
What is the proportion of discriminators in the sweetener example?
Examples in R
From pc to pd :
Tetrad, Triangle, 3-AFC: pd =
c Per B Brockhoff (DTU)
pc − 1/3
2/3
Duo-Trio, 2-AFC: pd =
What proportion of correct answers should we expect if we had used the
Duo-Trio protocol instead?
pc − 1/2
1/2
The sensR package: Difference and similarity testingDTU Sensometrics 2015
c Per B Brockhoff (DTU)
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The sensR package: Difference and similarity testingDTU Sensometrics 2015
The Thurstonian model
The Thurstonian model
Gridgeman’s paradox
Gridgeman’s paradox
1.0
Masuoka et al. (1995)
Delwiche & O’Mahony (1996)
Rousseau & O’Mahony (1997)
Beer
Choc. pudding
Yoghurt
no. tests
45
108
720
240
240
108
156
180
Triangle
21
42
363
104
99
50
106
105
3-AFC
32
62
539
161
148
75
145
152
Triangle
3−AFC
0.8
Proportion discriminators
Product
Bitter solutions
Onion dip
Salt solutions
1.0
Triangle
3−AFC
Proportion correct
Study
Byer & Abrams (1953)
Stillman (1993)
Tedje et al. (1994)
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0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.0
a
b
c
d
e
Studies
f
g
h
a
b
c
d
e
f
g
h
Studies
Why does 3-AFC consistently give more correct answers than Triangle?
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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The Thurstonian model
The Thurstonian model
The Thurstonian model, 3-alternatives
Variability in Thurstonian models
d': sensory difference
d': sensory difference
A
B
A
a1
a2
b
a1
Normal distributions with equal variance
d0 = (µ2 − µ1 )/σ — a signal-to-noise measurement
Each decision rule (protocol) leads to a specific relation between d0
and pc
Is the answer correct for Triangle? for 3-AFC?
c Per B Brockhoff (DTU)
B
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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a2
b
Variability in sensory intensity comes from
Variability in stimulus (sample-sample variation, temperature, etc.)
Variability within the subject (fatigue, learning, presentation order)
c Per B Brockhoff (DTU)
The Thurstonian model
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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The Thurstonian model
Psychometric functions
Gridgeman’s paradox resolved
2.5
Triangle
3−AFC
1.0
2−AFC
Thurstonian d−prime
2.0
0.9
3−AFC
0.8
Duo−trio
pc
0.7
Triangle
0.6
1.5
1.0
0.5
0.5
0.0
0.4
a
b
0.3
c
d
e
f
g
h
Studies
0
1
2
3
4
5
Triangle and 3-AFC agree for all studies when d0 is used as a measure of
sensory difference.
6
d'
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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The Thurstonian model
The Thurstonian model
Examples of d-prime
Example 4: d-prime estimation
obvious difference (d' = 4)
What is d-prime in the sweetener example?
confusable (d' = 1)
How low or high could we reasonably expect the true d-prime (δ) to be?
Almost indistinguishable (d' = 0.2)
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
Power and sample size computations
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Power and sample size computations
Why plan sensory difference tests?
Type I and Type II error rates
H0 is true
HA is true
Because
Accept H0
X
Type II error (β)
Reject H0
Type I error (α)
X
Five parameters involved when designing a sensory difference test:
Too many tests is expensive and a waste of resources
Sample size
Too few tests is useless and a waste of resources
Power (1 − β)
Significance level (α)
Sensory difference that you want to detect (d0 , pc , or pd )
Test protocol (Triangle, Duo-Trio, Tetrad, 2-AFC, 3-AFC)
Usually we want a power between 80% and 90%.
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Power and sample size computations
Power and sample size computations
What is the power of a sensory difference test really?
Example 5: Power estimation
Power is the probability
of finding a difference (if it is really there)
What is the power of detecting d0 = 2 in a Triangle test with α = 0.05 and
n = 15?
of finding a difference if HA is true
of rejecting H0 if HA is true
of correctly rejecting H0
What are the null and alternative hypotheses?
that the number of correct answers is at least as large as the critical
value
What is the critical value for the test?
that the p-value from a sensory difference test is less than α
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
Power and sample size computations
Power versus pd
Some new technical equipment has been bought to your company to
replace an old machine used in the production of your calorie reduced
yoghurt. The producer of the new machine promises that your yoghurt will
remain exactly as before, but upon tasting the result, you strongly believe
there is a subtle but detectable difference.
You decide to use the Duo-Trio test and have a budget for 100 consumers.
d0
= 1?
1.0
0.8
Power
To make sure you don’t get complaints from your costumers, you decide to
test whether consumers can tell the difference between the yoghurts made
with the old and the new equipment.
0.6
2 alternatives
3 alternatives
0.4
0.2
0.0
0.0
How many consumers would you need to achieve 80% power?
The sensR package: Difference and similarity testingDTU Sensometrics 2015
0.1
0.2
0.3
0.4
0.5
pd
If you are convinced that the difference is primarily a difference in
smoothness, then what can you do?
c Per B Brockhoff (DTU)
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Power and the five protocols
Example 6: Power and sample size estimation
What is the probability that you can detect a difference of
The sensR package: Difference and similarity testingDTU Sensometrics 2015
More alternatives means more power
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Power and the five protocols
Power and the five protocols
Power versus d-prime
Power versus sample size
pd = 0.25
1.0
1.0
Power
0.8
n = 100
Power
3 alternatives
0.8
0.6
0.2
2 alternatives
0.4
0.2
Duo−Trio
Triangle
2−AFC
3−AFC
Tetrad
0.4
0.6
0.0
10
30
50
70
90
Sample size
0.0
0.0
0.5
1.0
1.5
130
150
2.0
Power is a zig-zag function of sample size.
This is due to the discreteness of the binomial distribution
Smoother behavior for 3 alternatives
d−prime
Specified tests are more powerful than unspecified tests
c Per B Brockhoff (DTU)
110
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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c Per B Brockhoff (DTU)
Power and the five protocols
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Power and the five protocols
Reject the next consumer?
First exact and stable exact sample sizes
Power can decrease with increasing sample size!
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Power
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Sample size
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120
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100
105
110
Sample size
115
First exact sample size (conventional):
120
The smallest sample size that has the required power
Stable exact:
The smallest sample size that gives the required power
Sample sizes provide guidelines — not conclusive requirements.
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
The smallest sample size that has the required power such that no
greater sample size has a power smaller than that required.
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c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Power and the five protocols
Power and the five protocols
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
Power and the five protocols
c Per B Brockhoff (DTU)
X
Regression analysis
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
The sensR package: Difference and similarity testingDTU Sensometrics 2015
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Publications related to sensR (and ordinal)
siz
e
plot
ROC
AUC
us
on
dprime compare
dprime test
dprime table
posthoc
M
d0
isc
co
ell
mp
an
eo
ari
s
n
ati
o
str
Illu
Tr
an
sfo
rm
ati
rescale
psyfun
psyinv
psyderiv
pd2pc
pc2pd
Brockhoff , P. B., & Christensen, R. H. B. (2010). Thurstonian models for sensory discrimination tests as generalized
linear models. Food Quality and Preference, 21, 330–338.
s
on
le
mp
Sa
&
Po
we
r
d 0,
CI
,t
est
s
discrimPwr
d.primePwr
discrimSS
d.primeSS
twoACpwr
X
X
Power and the five protocols
Main functions in sensR
discrim
AnotA
samediff
twoAC
betabin
glm
clm
clmm
X
X
Replicated
at
io
rim
c Per B Brockhoff (DTU)
41 / 48
X
X
Likelihood CI
X
X
X
(X)
X
(X)
Simulation
X
X
X
X
X
X
isc
Lets see if we can replicate this number with sensR.
Sample size
Similarity test
X
X
X
X
X
X
in
The ISO standard for the Triangle test describes that for pd = 0.10,
α = 0.05 and β = 0.001, the required (first exact) sample size is 1181.
Power
Difference test
Duo-Trio, Triangle, Tetrad
2-AFC, 3-AFC
A-not A
Same-Different
2-AC
A-not A w. Sureness
n
d0 estimation
Protocols supported in sensR
D
Example 7: First-exact and stable-exact sample sizes
findcr
clm2twoAC
SDT
discrimR
discrimSim
samdiffSim
The sensR package: Difference and similarity testingDTU Sensometrics 2015
43 / 48
Christensen, R. H. B., & Brockhoff, P. B. (2009). Estimation and Inference in the Same Different Test. Food Quality
and Preference, 20, 514–524.
Christensen, R. H. B., Cleaver, G., & Brockhoff, P. B. (2011). Statistical and Thurstonian models for the A-not A
protocol with and without sureness. Food Quality and Preference, 22, 542-549.
Christensen, R. H. B., Lee, H-S. & Brockhoff , P. B. (2012). Estimation of the Thurstonian Model for the 2-AC
Protocol. Food Quality and Preference, 24, 119-128.
Christensen, R. H. B., & Brockhoff, P. B. (2013). Analysis of sensory ratings data with cumulative link models. J. Soc.
Fr. Stat. & Rev. Stat. App., 154(3), 58-79.
Christensen, R.H.B., Ennis, J.M., Ennis, D.M. & Brockhoff, P.B.(2014). Paired preference data with a no-preference
option - Statistical tests for comparison with placebo data. Food Quality and Preference, 32, 48-55.
John M Ennis, Rune HB Christensen (2014). Precision of measurement in Tetrad testing. Food Quality and Preference,
Vol 32, 98-106.
Ennis, J.M. & Christensen, R.H.B.(2015). A Thurstonian comparison of the Tetrad and Degree of Difference tests.
Food Quality and Preference, Vol 40, 263-269.
T. Næs, P.B. Brockhoff and O. Tomic, (2010). Statistics for Sensory and Consumer Science, John Wiley & Sons,
Chapter 7.
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
44 / 48
Details about the four basic protocols
Details about the four basic protocols
The Triangle test
The Duo-Trio test
3 samples (X, X 0 , Y ) OR (X, Y , Y 0 )
1 reference sample XR and 2 test samples X and Y
6 presentation orders XX 0 Y , XY X 0 , Y XX 0 , Y Y 0 X, Y XY 0 , XY Y 0
4 presentation orders XR XY , XR Y X, YR Y X, YR XY
Unspecified test
Unspecified test
Question: Which sample is most similar to the reference?
Question: which sample is most different from the two others?
Constant or variable reference?
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
45 / 48
Details about the four basic protocols
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
46 / 48
Details about the four basic protocols
The 2-AFC test
The 3-AFC test
2 samples X and Y
3 samples (X, X 0 , Y ) OR (X, Y , Y 0 )
2 presentation orders XY and Y X
6 presentation orders XX 0 Y , XY X 0 , Y XX 0 , Y Y 0 X, Y XY 0 , XY Y 0
Specified test
Specified test
Question: which sample has the strongest/weakest sensory
mangnitude?
Question: which sample has the strongest/weakest sensory
mangnitude?
Example: Which sample is the sweetest?
Example: Which sample is the sweetest?
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
47 / 48
c Per B Brockhoff (DTU)
The sensR package: Difference and similarity testingDTU Sensometrics 2015
48 / 48