Paper presentation at the 11th Biennial Conference of the European Personal Construct Association (EPCA), Dublin, Irland, July 2012.

1. Standardizing inter-element distances in grids
A revision of Hartmann distances
EPCA Conference, Dublin, July 1, 2012
Mark Heckmann
University of Bremen, Germany

2. 1. Inter-element distances
2. Slater‘s standardization
3. Hartmann‘s standardization
4. A new approach to standardization
5. Discussion

3. Types of (inter-element) distances
• Euclidean distance
• City-Block distance
• Minkowski metrics

4. Inter-element distances (Euclidean)
Self
Element X
Ideal Self

5. Self
Ideal
self
Conrad, R., Schilling, G., & Liedtke, R. (2005). Parental Coping
with Sudden Infant Death After Donor Insemination: Case
Report. Human Reproduction, 20(4), 1053–1056.
Norris and Makhlouf-Norris‘ self-identity-plot
Parental coping after sudden death of DI in

6. Issue
(Euclidean) distance
depends on grid size and
rating scale graduation

7. point scale. In the example, the elements are rated to maximum dissimilarity,
the extremes of the scales are used. Though the rating pattern is consistent over
(Euclidean) distance depends on grid size
s, the Euclidean distance changes considerably.
Table 6.1 Dependency of Euclidean distance on grid size and rating scale.
a b c d
self ideal self ideal self ideal self ideal
self self self self
C1 1 3 1 5 1 3 1 5
C2 1 3 1 5 1 3 1 5
C3 1 3 1 5 1 3 1 5
C4 1 3 1 5
C5 1 3 1 5
ED 3.46 6.92 4.47 8.94
Note: ED = Euclidean distance.
Heckmann M. (2011). OpenRepGrid - An R package for the analysis of repertory grids (Unpublished diploma
s property inherent in the definition of the Euclidean distance hinders the compar
thesis). University of Bremen, Germany, p. 84.

8. Challenge
Standardization is
needed to compare
distances across grids

9. z-Transform

10. First approach
Slater 1977

11. Euclidean distance matrix can be rewritten as Ejk = (Sj + Sk + 2Pjk )1/2 .
value for Sj and Sk is the average of S, i.e., Savg = S/m where m is the number
the grid. The average of the off-line diagonals of P is −S/m(m − 1). Inserted
Divide Euclidean distances by
e formula, this yields the following expected average Euclidean distance U =
ch is outputted as “Unit of Expected Distance” in Slater’s INGRID program
tandardized Euclidean distances expected distance
the unit of ES are then calculated as ES = E/U.
Euclidean distance
E matrix (1)
ES = E/U Divide by unit of (2)
expected distance
G

12. Norris & Makhlouf-Norris‘ simulation
92% of distances inside (0.8, 1.2) interval
Cut-offs to determine „significant“
deviation from randomness
Slater‘s Simulation: 78% of
values inside (0.8, 1.2)
and skewed distribution

13. Hartmanns‘s
Simulation
1992

14. Hartmann‘s extended simulation design
Element Comparisons 47
~ Slater‘s
simulation
~ Norris & I I
loo loo loo
Markhlouf-Norris‘
21 loo loo loo loo loo
When probability theory is taken into account, this result is no
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences
of a monte carlo study, JCP, 5(1), p. computation of a zero distance between
longer surprising. For the 47
two vectors of random numbers, these vectors (i.e., elements) must

15. Issue
Slater distances still
depend on the size of
the grid

16. rating poles (producing a maximum distance) will also become more
unlikely. Because the cause of the effect occurs before the computa-
tion ofdepends kindsthe number of constructs
SD distances, all on of distances (euclidean, city-block, etc.)
will be affected.
1.6 Not
1.5 symmetrical
E1.4
1.3
= 1.2
distance
Slater‘s
; 1.1
' 1.0
6
: 0.9
.
I 0.8
8 0.7
0.6
0.5
0.4
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Number
off
constructs
number
o constructs
Percentiles: 1s 5% 10s 9Or 95x 99s
Range (Uin.Max) represented by T-bars
Figure 2 Means of percentiles: QUASIS.
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences
of a monte carlo study, JCP, 5(1), p. 47

17. Skewness of a distribution

18. with the same number of constructs. The first sample contained 64
grids of the size 8E x 1OC. They were produced by students of med-
Skewness courses dealing on the number of
icine participating in
depends with doctor-patient interaction.
These grids were provided constructs
for didactic purposes to explore the stu-
Skewness
skewness
0 03
-c) 25
-0 30
7 E 9 10 11 12 !9 13 15 16 17 15 19 20 2i
numper of constructs
Number
of
constructs
Figure 3 Skewness of distance distributions including linear regression.
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences
of a monte carlo study, JCP, 5(1), p. 47

19. Hartmann‘s results for Slater distances
• SD of distribution depends on number of
constructs
• Distributions are skewed
• Symmetrical cutoffs overrepresent similarities
• Skewness depends on the number of constructs
• No effect of rating scales (5-, 7-, 10-point)

20. Second
approach
Hartmann 1992

21. -
ard deviations of the distance distributio
.217 to SD,, .123.
om SD, Hartmann‘s standardizationThese m
ted by the following formula:
-
the distancesdof a grid are computed acc
Dslater
=
Slater
istances
corresponding mean (or the expected
Mc
=
mean
of
simulated
Slater
distribu;on
d, divided byevia;on
ostandard istribu;on
sdc
=
standard
d the f
simulated
d deviation
nce distribution of quasis and multiplie
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences
of a monte carlo study, JCP, 5(1), p. 49

22. Suggested assymetrical
cutoff values
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences
of a monte carlo study, JCP, 5(1), p. 52

23. Why replicate
Hartmann’s study?

24. • Simulation uses few scale ranges
• Variation in results
• No removal of skewness
 Symmetrical cutoffs more favorable
• Equally skewed after transform (p. 52) X
• Contradicts relation between skewness and
the number of constructs
Replication with bigger
sample size

25. Study design
Scale range
1 - 13
~ Hartmann‘s
.
.
.
simulation
Scale range Elements (by 2)
1-2 6 8 . . . 28 30
4 n = 1000 n = 1000 . . . n = 1000 n = 1000
6 n = 1000 n = 1000 . . . n = 1000 n = 1000
Constructs (by 2)
. . . . . .
. . . . . .
. . . . . .
28 n = 1000 n = 1000 . . . n = 1000 n = 1000
30 n = 1000 n = 1000 . . . n = 1000 n = 1000

26. with the same number of constructs. The first sample contained 64
grids of the size 8E x 1OC. They were produced by students of med-
Skewness courses dealing on the number of
icine participating in
depends with doctor-patient interaction.
These grids were provided constructs
for didactic purposes to explore the stu-
Skewness
skewness
0 03
No
breakdown
by
number
of
elements
-c) 25
-0 30
7 E 9 10 11 12 !9 13 15 16 17 15 19 20 2i
numper of constructs
Number
of
constructs
Figure 3 Skewness of distance distributions including linear regression.
Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences
of a monte carlo study, JCP, 5(1), p. 47

27. Skewness by number constructs and
Number
of
number of elements
elements
6 8 10 12 14 16 18 20
0.00
−0.05
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Pronounced
joint
effect
●
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−0.25
−0.30
on
skewness
8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20 8 12 16 20
Number of constructs
Number
of
constructs

28. 6 10 20 30
Number
of
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elements
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1−2
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Triple
interac;on
effect
●
Scale
●
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−0.15
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1−4
●
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range
● ● ●
−0.20 ●
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on
skewness
●
−0.25 ●
●
−0.30 ●
●
−0.08 ●
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1−5
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−0.14 ●
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−0.16 ●
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−0.18 ● ●
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−0.20 ●
−0.22 ●
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−0.10
1 − 13
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−0.20 ●
5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
Number
of
constructs
Number of constructs

29. Why is varying
skewness an issue?

30. 0.
P1 P99
Effect of skewness on quantiles
0.0
−3 −2 −1 0 1 2 3
Figure 6.6 Effect of distribution form on percentiles. The figure shows a normal
0.4
A
(A: solid line) and skewed normal distribution (B: dashed line). Both have a mean
P5A
1.
of 0 and standard deviation ofP5B
The vertical bars denote the percentiles P1 to P99
0.3
for each distribution. For the quantile values, refer to Table 6.6.
Density
P10
B
P90
0.2
As a consequence, one and the P same cutoff value may correspond to different proportion
P95
5
f the distribution, as shown in Table 6.7. In distribution A (Figure 6.6, solid line), 5%
0.1
P1 P99
f the values were smaller than or equal to -1.64. For the skewed distribution B (dashed
0.0
ne), this is the case for only 2.7%. Hence, when one single value is used as a cutoff to
etermine the 5% lowest values, the results may be flawed.
−3 −2 −1 0 1 2 3
Table 6.6 Effect of distribution form on percentiles.
P1 P5 P10 P90 P95 P99 mean sd skew kurtosis
A -2.31 -1.64 -1.28 1.28 1.65 2.31 0.00 1.00 0.00 -0.02
B -1.91 -1.44 -1.18 1.35 1.82 2.75 -0.00 1.00 0.58 0.42
Note: The table shows the percentiles and the moments of the distributions A (solid)

31. 0.4 Effect of skewness on proportions
A
Δ
0.3
Density
P10 B
P90
0.2
distances revisited P5 P95
0.1
P1 P99
0.0
Table 6.7 Effect of distribution form on proportions.
−3 −2 −1 0 1 2 3
-2.31 -1.64 -1.28 1.28 1.65 2.31
A 0.010 0.050 0.100 0.900 0.950 0.990
B 0.002 0.027 0.078 0.890 0.935 0.978

32. Hartmann Suggested
approach
Normalized
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Quantile value
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Number of constructs