Standardizing inter-element distances in gridsA revision of Hartmann distancesEPCA Conference, Dublin, July 1, 2012Mark He...
1.  Inter-element distances2.  Slater‘s standardization3.  Hartmann‘s standardization4.  A new approach to standardization...
Types of (inter-element) distances   •  Euclidean distance   •  City-Block distance   •  Minkowski metrics
Inter-element distances (Euclidean)      Self                           Element X             Ideal Self
Self	  Ideal	  self	                      Conrad, R., Schilling, G., & Liedtke, R. (2005). Parental Coping                ...
Issue   (Euclidean) distancedepends on grid size and rating scale graduation
point scale. In the example, the elements are rated to maximum dissimilarity, the extremes of the scales are used. Though ...
Challenge  Standardization isneeded to comparedistances across grids
z-Transform
First approach    Slater 1977
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., S...
Norris & Makhlouf-Norris‘ simulation 92% of distances inside (0.8, 1.2) interval   Cut-offs to determine „significant“    ...
Hartmanns‘s Simulation   1992
Hartmann‘s extended simulation design                                Element Comparisons                                  ...
Issue Slater distances stilldepend on the size of       the grid
rating poles (producing a maximum distance) will also become moreunlikely. Because the cause of the effect occurs before t...
Skewness of a distribution
with the same number of constructs. The first sample contained 64grids of the size 8E x 1OC. They were produced by student...
Hartmann‘s results for Slater distances•  SD of distribution depends on number of   constructs•  Distributions are skewed•...
SecondapproachHartmann 1992
-ard deviations of the distance distributio            .217 to SD,, .123.om SD, Hartmann‘s standardizationThese m ted by t...
Suggested assymetrical                     cutoff valuesHartmann, A. (1992). Element comparisons in repertory grid techniq...
Why replicateHartmann’s study?
•  Simulation uses few scale ranges•  Variation in results•  No removal of skewness    Symmetrical cutoffs more favorable...
Study design                                                    Scale range                                               ...
with the same number of constructs. The first sample contained 64grids of the size 8E x 1OC. They were produced by student...
Skewness by number constructs and  Number	  of	      number of elements     elements	                                     ...
6                                                       10                                                      20        ...
Why is varyingskewness an issue?
0.                                         P1                                                     P99                     ...
0.4   Effect of skewness on proportions                                                                    A	             ...
Hartmann                                              Suggested	  approach	                                               ...
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
Standardizing inter-element distances in repertory grids
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Standardizing inter-element distances in repertory grids

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

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Standardizing inter-element distances in repertory grids

  1. 1. Standardizing inter-element distances in gridsA revision of Hartmann distancesEPCA Conference, Dublin, July 1, 2012Mark HeckmannUniversity of Bremen, Germany
  2. 2. 1.  Inter-element distances2.  Slater‘s standardization3.  Hartmann‘s standardization4.  A new approach to standardization5.  Discussion
  3. 3. Types of (inter-element) distances •  Euclidean distance •  City-Block distance •  Minkowski metrics
  4. 4. Inter-element distances (Euclidean) Self Element X Ideal Self
  5. 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. 6. Issue (Euclidean) distancedepends on grid size and rating scale graduation
  7. 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 sizes, 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 diplomas property inherent in the definition of the Euclidean distance hinders the compar thesis). University of Bremen, Germany, p. 84.
  8. 8. Challenge Standardization isneeded to comparedistances across grids
  9. 9. z-Transform
  10. 10. First approach Slater 1977
  11. 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 numberthe grid. The average of the off-line diagonals of P is −S/m(m − 1). Inserted Divide Euclidean distances bye formula, this yields the following expected average Euclidean distance U =ch is outputted as “Unit of Expected Distance” in Slater’s INGRID programtandardized 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. 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. 13. Hartmanns‘s Simulation 1992
  14. 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 noHartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequencesof 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. 15. Issue Slater distances stilldepend on the size of the grid
  16. 16. rating poles (producing a maximum distance) will also become moreunlikely. 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. 17. Skewness of a distribution
  18. 18. with the same number of constructs. The first sample contained 64grids of the size 8E x 1OC. They were produced by students of med- Skewness courses dealing on the number oficine 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. 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. 20. SecondapproachHartmann 1992
  21. 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. 22. Suggested assymetrical cutoff valuesHartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequencesof a monte carlo study, JCP, 5(1), p. 52
  23. 23. Why replicateHartmann’s study?
  24. 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. 25. Study design Scale range 1 - 13 ~ Hartmann‘s .    .    .   simulationScale 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. 26. with the same number of constructs. The first sample contained 64grids of the size 8E x 1OC. They were produced by students of med- Skewness courses dealing on the number oficine 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. 27. Skewness by number constructs and Number  of   number of elements elements   6 8 10 12 14 16 18 20 0.00 −0.05 ● ● ● ● ● ●● −0.10 ● ● ● ●● ●●●● ●Skewness   ● ● ● ●● ● ●● ● ● ● ●● ● ●● ●● ● ● ● ● ●● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● Skewness ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● −0.15 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● −0.20 ● Pronounced  joint  effect   ● ● ● −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. 28. 6 10 20 30Number  of   −0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●elements   ● ● ● ● ● ● ● ● ● ● ● −0.6 ● ● ● ● ● 1−2 −0.8 ● ● ● ● −1.0 ● −1.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1−3 −0.25 ● ● ● ● ● ● −0.30 ● ● −0.35 ● ● ● ● −0.40 ● ● −0.10 ● ● ● ● Skewness   ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Triple  interac;on  effect   ● Scale     ● Skewness ● ● ● ● ● ● ● ● −0.15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1−4 ● ● range   ● ● ● −0.20 ● ● on  skewness   ● −0.25 ● ● −0.30 ● ● −0.08 ● ● ● ● ● ● ● ● ● ● ● −0.10 ● ● ● ● ● ● ● ● ● ● ● ● ● −0.12 ● ● ● ● ● ● ● ● ● ● ● ● 1−5 ● ● −0.14 ● ● ● ● ● ● −0.16 ● ● ● ● −0.18 ● ● ● ● −0.20 ● −0.22 ● ● ● −0.05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.10 1 − 13 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.15 ● ● ● ● ● −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. 29. Why is varyingskewness an issue?
  30. 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 0.2As 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. 31. 0.4 Effect of skewness on proportions A   Δ   0.3 Density P10 B   P90 0.2distances 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. 32. Hartmann Suggested  approach   Normalized −2.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2.1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.01 −2.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2.3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.50 ● ● ● ● ● ● ● ● ● ScaleQuantile value ● ● ● ● ● −1.55 1−3 ● ● ● ● ● ● ● ● ● ● ● ● 0.05 ● ● ● ● ● ● 1−4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.60 ● ● ● ● ● 1−5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +   1−13 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.65 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.24 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.26 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.28 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1.30 ● ● ● ● ● ● ● ● 5 10 15 20 25 30 5 10 15 20 25 30 Number of constructs

×