standardizing inter-element distances in repertory grids

Standardizing inter-element distances in grids A revision of Hartmann distances

EPCA Conference, Dublin, July 1, 2012

Mark Heckmann University of Bremen, Germany

1.  Inter-element distances

2.  Slater‘s standardization

3.  Hartmann‘s standardization

4.  A new approach to standardization

5.  Discussion

Types of (inter-element) distances

•  Euclidean distance

•  City-Block distance

•  Minkowski metrics

Inter-element distances (Euclidean)

Element X

Ideal Self

Self

As can be seen in Figure 1B, there is a discrepancybetween Mrs S.’s actual self and her ideal self (divergentconstellation); the childlessness deeply affects her self-esteem. In her self system, her husband and the dead childare distant from her self and her ideal self. The partner ischaracterized as ill, childless and obstinate.

Finally, the couple filled in the Family AssessmentMeasure (Skinner et al., 1983), a self-rating scale that inves-tigates the individual’s perception of his/her functioning inthe partnership/family. In the Family Assessment Measure,Mrs S. showed no pathological scores, whereas her husbandshowed pathological scores on the subscale ‘task accomplish-ment’ (T ! 66) which can be interpreted as a sign of hisdifficulty in performing important tasks in family life. Fur-thermore, he showed higher scores on the subscale ‘affectiveexpression’ (T ! 65) which can be interpreted as a sign ofhis inadequate affective communication.

After the first examination, the couple were informed ofthe possibilities of psychological counselling. However, itbecame evident that both partners were reluctant to continueconsultations in the context of the fertility clinic. Theyrefused further counselling in order to avoid questions fromfriends and relatives.

In the second interview, 6 years after the incident, thecouple’s relationship appeared to be destabilized. Mrs S.talked about divorce. She complained that there was not suf-ficient communication in the relationship.

In the interview, she appeared to be dominant, while heshowed a submissive attitude. After the SID, they hadmade no further attempt to have a child. We were told thatthere had been no further communication about the loss inthe past years. ‘There is no use talking about things youcannot change.’ Mr S. said. Once again, the coupledeclined any further psychological counselling.

Discussion

In our case, the DI couple is traumatized not only by the experi-ence of SID but also by the preceding illness of their child.

With respect to Mr S., his coping with the SID is overshadowedby his concern about being infertile. In the interview, Mr S. sta-ted that this diagnosis affected his self-esteem and causedambivalent feelings towards the child. However, he deniedthese ambivalent feelings since they were associated withstrong feelings of shame and guilt. Thus, he characterized him-self as healthy and idealized the dead child. Only by means ofstrong defence mechanisms such as denial could he keep theexperience of infertility from endangering his self-esteem.

The taboo surrounding his infertility even made it imposs-ible to obtain a certain relief by talking to friends or relatives.Furthermore, it became apparent that he was aware of certaindifficulties in the functioning of the partnership. However, heinsisted on denying a possible link between these deficienciesand his infertility.

Mrs S. said that she was grief-stricken because of thedeath of their child. However, she attributed it to fate and didnot blame anyone. Being aware of her husband’s ambiva-lence towards the child, she felt guilty about her love towardsthe child. Though she did not verbalize this in the first inter-view, the position of her husband in her self system—who ischaracterized as ill, childless and obstinate—seems to be dueto her being angry with him and she obviously reproacheshim for being responsible for her unfulfilled wish for a child.In her self system, the dead child has nearly the same pos-ition as her husband and is distant to her self and ideal self.One might say that 6 months after the incident the dead childstands for his failure.

In the course of time, the aggression against her husbandbecomes obvious; she blames him for his failures and thinksabout separation. No plans or ideas for a mutual futurebecame visible.

Most couples experiencing SID are (or want to be) preg-nant again within 1 year after the loss of the child (Dyregrov,1990). The complete renouncement of having another childafter the loss supports the view of a pathological griefreaction.

In our case, the secrecy due to feelings of shame and guiltbecause of DI apparently does not only surround the life but

Figure 1. (A) Self-system of the husband (Mr. S.). (B) Self-system of the wife (Mrs. S.).

Parental coping after sudden death of DI infant

1055

Norris and Makhlouf-Norris‘ self-identity-plot

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., Sc

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, G.,

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Ideal self

Self

Issue

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

84 Chapter 6. OpenRepGrid use case: inter-element distances revisited

6.1 Slater distances

One of the most frequently used measures of inter-element similarity is the Euclidean

distance. By observing patterns of element distances, clinically relevant information and

diagnostic hints can be derived by comparing certain elements such as “self,”“ideal self,”

and “social self” (Schoeneich & Klapp, 1998). Changes of distances across the course

of a therapy may also be monitored (e.g. Ben-Tovim & Greenup, 1983; Leach et al.,

2001). Likewise, we may ask whether distances are associated with a particular outcome

or symptoms (Fransella & Crisp, 1970; Schwab, 2006). The use of the Euclidean distance

as a measure of similarity is not unproblematic, however, as its magnitude and range

depends on the graduation of the rating scale and the number of constructs used – i.e.,

on the size of the grid. To demonstrate this point, we will consider the elements self and

ideal self for four different grids (Table 6.1). The grids a and b have three constructs

and the c and d have five constructs. The rating scale alternates between a 3-point and

a 5-point scale. In the example, the elements are rated to maximum dissimilarity, i.e.,

only the extremes of the scales are used. Though the rating pattern is consistent over the

grids, the Euclidean distance changes considerably.

Table 6.1 Dependency of Euclidean distance on grid size and rating scale.

a b c dself ideal

selfself ideal

selfself ideal

selfself ideal

selfC1 1 3 1 5 1 3 1 5C2 1 3 1 5 1 3 1 5C3 1 3 1 5 1 3 1 5C4 1 3 1 5C5 1 3 1 5ED 3.46 6.92 4.47 8.94

Note: ED = Euclidean distance.

This property inherent in the definition of the Euclidean distance hinders the comparison

of element distances across grids of different size and rating scale. Slater (1977, p. 94)

proposed an approach to standardizing the Euclidean distance to make it independent

from size and scale and thus comparable across grids. He suggested dividing the Euclidean

distances by the “Unit of Expected Distance,” i.e., the expected Euclidean distance across

all pairs of elements to standardize the inter-element distances. 1

1 To calculate the “Unit of Expected Distance,” the raw grid matrix G is centered on the constructmeans, yielding D. The Euclidean distance matrix, which is identical for D and G, is denoted E. Thematrix P is defined as P = DTD. The leading diagonal of P is denoted S, i.e., S = diag(P); the sum

(Euclidean) distance depends on grid size

Heckmann M. (2011). OpenRepGrid - An R package for the analysis of repertory grids (Unpublished diploma thesis). University of Bremen, Germany, p. 84.

Challenge

Standardization is needed to compare distances across grids

z-Transform

First approach

Slater 1977

Divide Euclidean distances by the unit of expected distance

U is the“Unit of Expected Distance”, a measure for the average expected euclidean distance.

The raw grid matrix G is centered on the construct means, yielding D. The Euclidean

distance matrix, which is identical forD andG, is denotedE. The matrixP is defined asP =

DTD. The leading diagonal of P is denoted S, i.e., S = diag(P); the sum of S is denoted

S. Hence the Euclidean distance matrix can be rewritten as Ejk = (Sj + Sk + 2Pjk)1/2

.

The expected value for Sj and Sk is the average of S, i.e., Savg = S/m wherem is the number

of elements in the grid. The average of the off-line diagonals of P is −S/m(m− 1). Inserted

into the above formula, this yields the following expected average Euclidean distance U =

(2S

m−1 )1/2

which is outputted as “Unit of Expected Distance” in Slater’s INGRID program

(1977). The standardized Euclidean distances ES are then calculated as ES = E/U.

In short:

E (1)

ES = E/U (2)

G

D

P = DTD

S = tr(P)

U =

�2S

m− 1

�1/2

(3)

6

Euclidean distance matrix

Divide by unit of expected distance

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

Hartmanns‘s Simulation

1992

Hartmann‘s extended simulation design Element Comparisons 47

21

I I

loo loo loo loo loo loo loo loo

When probability theory is taken into account, this result is no longer surprising. For the computation of a zero distance between two vectors of random numbers, these vectors (i.e., elements) must contain the same values. The theory of multinomial distributions makes evident that this event becomes more and more unlikely when we increase the length of the vectors (i.e., number of constructs). In the other extreme, two vectors filled only with opposite rating poles (producing a maximum distance) will also become more unlikely. Because the cause of the effect occurs before the computation of distances, all kinds of distances (euclidean, city-block, etc.) will be affected.

1.6 1.5 1.4 E 1.3

= 1.2 ; 1.1 '6 1.0 .: 0.9 I 0.8 8 0.7

0.6 0.5 0.4

7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 number of constructs

Percentiles: 1s 5% 10s 9Or 95x 99s Range (Uin.Max) represented by T-bars

Figure 2 Means of percentiles: QUASIS.

Downloaded By: [German National Licence 2007] At: 02:00 6 September 2009

Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences of a monte carlo study, JCP, 5(1), p. 47

~ Slater‘s simulation

~ Norris & Markhlouf-Norris‘

Issue

Slater distances still depend on the size of

the grid

SD depends on the number of constructs

Element Comparisons 47

21

I I

loo loo loo loo loo loo loo loo

When probability theory is taken into account, this result is no longer surprising. For the computation of a zero distance between two vectors of random numbers, these vectors (i.e., elements) must contain the same values. The theory of multinomial distributions makes evident that this event becomes more and more unlikely when we increase the length of the vectors (i.e., number of constructs). In the other extreme, two vectors filled only with opposite rating poles (producing a maximum distance) will also become more unlikely. Because the cause of the effect occurs before the computation of distances, all kinds of distances (euclidean, city-block, etc.) will be affected.

1.6 1.5 1.4 E 1.3

= 1.2 ; 1.1 '6 1.0 .: 0.9 I 0.8 8 0.7

0.6 0.5 0.4

7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 number of constructs

Percentiles: 1s 5% 10s 9Or 95x 99s Range (Uin.Max) represented by T-bars

Figure 2 Means of percentiles: QUASIS.



Slater‘s

distance

Number of constructs

Not symmetrical

Skewness of a distribution

Skewness depends on the number of constructs

48 A. Hartmann

Furthermore, the present study replicated Slater's (1976) result: Distance distributions were considerably skewed. If we use symmetrical cutoffs we will always find an artificial overrepresentation of similarities. The skewness was dependent on the number of constructs (see Figure 3).

The two main results of the present Monte Carlo study can be summarized as follows:

1. The distribution of distances was clearly dependent on the number of constructs of a grid and not on the number of elements. This phenomenon was not pointed out by Slater (1976). The statement, then, that Slater's standardized distances can be compared between grids of different size becomes questionable.

2. The distributions were skewed, a fact that has to be taken into consideration when cutoffs are used for the classification of similarity or dissimilarity.

DISCUSSION The question may be raised whether the present results are the very special properties of quasis or whether they can be found in empiri- cally elicited grids as well. To answer this question, I compared the distance distributions of two grid samples, each containing grids 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- icine participating in courses dealing with doctor-patient interaction. These grids were provided for didactic purposes to explore the stu-

skewness 0 03

-c) 25

-0 30 7 E 9 10 1 1 12 ! 9 13 15 16 17 15 19 20 2 i

numper of constructs

Figure 3 Skewness of distance distributions including linear regression.



Skewness


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)

Second approach

Hartmann 1992

Hartmann‘s standardization

Element Comparisons 49

dents constructions of real and ideal doctors. The second sample contains 21 grids of the size 21E x 21C, provided by a research proj- ect on anxiety in the hospital (Schonpflug et al., 1985)’. Both grid samples were connected to one of the extremes of the quasi’s sizes. Figure 4 shows the distance distributions of the grids and their corresponding quasis.

Although the grids expressed a distribution that was considerably more “flat” and more skewed than their corresponding quasis, the real grids also showed a dependency between the number of constructs and the shape of the distributions. We can clearly see that the laws of probability are not limited to random functions but also apply to real persons providing real grids.

A possible solution to the removal of the dependency between the numbers of constructs and distance distributions may be some kind of Z transformation. The distances of any individual grid should be standardized with means and standard deviations of the corresponding distance distribution of quasis. The means and standard deviations of distribution parameters for all quasis of the same construct numbers are listed in Table 3.

The standard deviations of the distance distributions decrease by almost 50% from SD, - .217 to SD,, - .123. These marked effects can be eliminated by the following formula:

In words, the distances of a grid are computed according to Sla- ter (1976). The corresponding mean (or the expected mean of 1) is then subtracted, divided by the standard deviation of the corresponding distance distribution of quasis and multiplied with - 1.

The mean of the new Z-transformed distances will be zero. In an analogy to correlation coefficients, a positive value indicates similarity of elements, and a negative one indicates dissimilarity of elements. A distance of D2 = 1 means that the empirical distance is exactly 1 SD from the indifferent, expected distance in the similarity field of the distribution. The element distances of the two real grid samples were Z-transformed according to the formula. A comparison between the original distributions and the distributions of Z- transformed distances revealed new information (Figure 4c).

Before the transformation, the distribution of the 21C grids showed a higher density in the interval 0.8 to 1.2 and faded out much quicker in the extremes. After the transformation, distributions did not differ substantially in the interval 1.0 to 2.2, but densi-

’1 thank Dr. Arne Raeithel for the transfer of the data.



Dslater = Slater distances Mc = mean of simulated Slater distribu;on sdc = standard devia;on of simulated distribu;on

Suggested assymetrical cutoff values


Why replicate Hartmann’s study?

•  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

Study design

Scale range 1 - 13

. . .

Scale range 1 - 2

Elements (by 2) 6 8 . . . 28 30

Con

stru

cts (

by 2

)

4 n = 1000 n = 1000 . . . n = 1000 n = 1000

6 n = 1000 n = 1000 . . . n = 1000 n = 1000 . . .

.

.

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.

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.

28 n = 1000 n = 1000 . . . n = 1000 n = 1000

30 n = 1000 n = 1000 . . . n = 1000 n = 1000

~ Hartmann‘s simulation

Skewness depends on the number of constructs

48 A. Hartmann

Furthermore, the present study replicated Slater's (1976) result: Distance distributions were considerably skewed. If we use symmetrical cutoffs we will always find an artificial overrepresentation of similarities. The skewness was dependent on the number of constructs (see Figure 3).

The two main results of the present Monte Carlo study can be summarized as follows:

1. The distribution of distances was clearly dependent on the number of constructs of a grid and not on the number of elements. This phenomenon was not pointed out by Slater (1976). The statement, then, that Slater's standardized distances can be compared between grids of different size becomes questionable.

2. The distributions were skewed, a fact that has to be taken into consideration when cutoffs are used for the classification of similarity or dissimilarity.

DISCUSSION The question may be raised whether the present results are the very special properties of quasis or whether they can be found in empiri- cally elicited grids as well. To answer this question, I compared the distance distributions of two grid samples, each containing grids 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- icine participating in courses dealing with doctor-patient interaction. These grids were provided for didactic purposes to explore the stu-

skewness 0 03

-c) 25

-0 30 7 E 9 10 1 1 12 ! 9 13 15 16 17 15 19 20 2 i

numper of constructs

Figure 3 Skewness of distance distributions including linear regression.



No breakdown by number of elements

Skewness


Skewness by number constructs and number of elements


Skew

ness

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Number of elements



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Triple interac;on effect on skewness

Scale range

Skew

ness

Number of elements


Why is varying skewness an issue?

Effect of skewness on quantiles

−3 −2 −1 0 1 2 3

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Density

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94 Chapter 6. OpenRepGrid use case: inter-element distances revisited

In other words, if the form of the distribution of Hartmann distances differs significantly

due to grid size and rating scale, the same may be true for the percentiles and cutoff values.

Yet, if percentiles and cutoff values vary considerably, then inter-element distances have

not been fully standardized.

To demonstrate this point, Figure 6.6 shows a standard normal and a skewed normal

distribution and selected percentiles. Note that both distributions shown in the figure are

standardized, i.e., they have a mean of zero and a standard deviation of one (as in the

case of Hartmann distances). The skewness for the distribution shown by the dotted line

is 0.42. This value is within the range we will discover for Hartmann distances (see, e.g.,

Figure 6.9(b) on page 102). The important fact for this study is that percentiles may

change due to different skewness of a z-transformed distribution, as shown in Figure 6.6

and Table 6.6.

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

Density

P1

P5

P10 P90

P95

P99

Figure 6.6 Effect of distribution form on percentiles. The figure shows a normal(A: solid line) and skewed normal distribution (B: dashed line). Both have a meanof 0 and standard deviation of 1. The vertical bars denote the percentiles P1 to P99

for each distribution. For the quantile values, refer to Table 6.6.

As a consequence, one and the same cutoff value may correspond to different proportions

of the distribution, as shown in Table 6.7. In distribution A (Figure 6.6, solid line), 5%

of the values were smaller than or equal to -1.64. For the skewed distribution B (dashed

line), this is the case for only 2.7%. Hence, when one single value is used as a cutoff to

determine the 5% lowest values, the results may be flawed.

Table 6.6 Effect of distribution form on percentiles.

P1 P5 P10 P90 P95 P99 mean sd skew kurtosisA -2.31 -1.64 -1.28 1.28 1.65 2.31 0.00 1.00 0.00 -0.02B -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)and B (dashed) from the distributions displayed in Figure 6.6.

A

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P5A P5B

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Effect of skewness on proportions

6.3. Hartmann distances revisited 95

Table 6.7 Effect of distribution form on proportions.

-2.31 -1.64 -1.28 1.28 1.65 2.31A 0.010 0.050 0.100 0.900 0.950 0.990B 0.002 0.027 0.078 0.890 0.935 0.978

Note: The table shows the proportion of values smallerthan or equal to the value specified in the column header.The header values correspond to the percentiles P1 toP99. The results refer to the distributions A (solid line)and B (dashed line) from Figure 6.6.

As a consequence, the question arises whether the systematic change in the form of the

Hartmann distribution will have a considerable effect on its quantiles. If the quantiles used

to determine “significance” of distances differ to a large extent, the region of significance

will vary according to the grid parameters “size” and “scale range.” Hence, it may not

be possible to only use one single set of “significance” values independent from the grid

parameters as proposed by Hartmann (1992).

To investigate the effects of grid size and scale range on the percentiles, selected results

are shown in Table 6.8. The 5th

percentile ranges from a maximum of -1.31 to a minimum

of -1.63, depending on the number of constructs and the scale range. The 10th

percentile

also varies between -0.97 and -1.29. It can be noted that the differences between a 5- and

an 11-point scale are small compared to the differences between 2-, 3-, and 5-point scales.

Also, for a 3-, 5-, and 11-point scale with a number of constructs greater than 10, the 5th

and 10th

percentiles are relatively stable.

To further explore how the percentiles are affected, the 5th

percentile, which often serves

as a cutoff value, will be compared across different grid parameters. Table 6.9 shows

that particularly a 2-point scale exhibits more variation than the other scale ranges. An

increasing number of constructs and elements and a scale range bigger than 1-3 seems to

lead to more stable values. The 5th

percentiles of a moderate sized grid, i.e., 10C x 10E,

will be affected primarily by a 2-point rating scale. A higher graduation of the scale leads

to low differences in the percentiles.

The question of whether the variation found in the percentiles is of practical importance

needs to be considered with respect to the particular situation in which standardized dis-

tances are used. In the case of a comparison across grids with identical scale and size (e.g.,

10C x 10E for all participants), the comparison of Hartmann distance is unproblematic,

as Hartmann’s standardization is performed with reference to an identical sampling distri-

bution across all grids. Also, when the number of elements and the number of constructs

between two grids do not differ to a large extent, neither will the sampling distributions,

and hence Hartmann standardization yields similar results. However, if grid size varies

A

B

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0.1

Scale● 1−3

1−41−51−13

Suggested approach

+


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ntile

valu

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1.25

1.30

1.35

1.40

1.60

1.65

1.70

1.75

2.2

2.4

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3.0

3.2

Hartmann

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5 10 15 20 25 30

0.90.95

0.99

Scale● 1−3

1−41−51−13

Suggested approach

+

Issues

Hartmann distances are still affected by the

-  number of constructs -  number of elements -  range of the rating scale

(but much less than Slater distances)

New approach

Diminishing skewness by optimal Box-Cox

transformation

original boxcox transformed

0.0

0.2

0.4

0.6

0.8

−6 −4 −2 0 2 4 −6 −4 −2 0 2 4 −6 −4 −2 0 2 4value

density

Box-Cox Transformation

Box-‐Cox transformed

c = min(Y) λ = .35

Search op;mal λ

1 2


0.0

0.2

0.4

0.6

0.8

−6 −4 −2 0 2 4 −6 −4 −2 0 2 4 −6 −4 −2 0 2 4value

density

Final z-Transformation

Box-‐Cox transformed original boxcox transformed

0.0

0.2

0.4

0.6

0.8

−2 0 2 4 −2 0 2 4 −2 0 2 4value

density

2 3 Suggested approach

1.  Calculate Hartmann distances

2.  Optimal Box-Cox transformation

3.  z-transform

Three steps


0.0

0.2

0.4

0.6

0.8

−6 −4 −2 0 2 4 −6 −4 −2 0 2 4 −6 −4 −2 0 2 4value

density

1 2 3

Suggested approach

Hartmann distances

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

Density

Q1

Q5

Q10 Q90

Q95

Q99

10 constructs

Standard normal

30 constructs

Hartmann distances

−3.0 −2.5 −2.0 −1.5 −1.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Density

Q1

Q5

Q10

10 constructs

Standard normal 30 constructs

Power transformed distances

−3.0 −2.5 −2.0 −1.5 −1.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Density

Q1

Q5

Q10

Hartmann distances

1.0 1.5 2.0 2.5 3.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Density

Q90

Q95

Q99

10 constructs

Standard normal 30 constructs

Power transformed distances

1.0 1.5 2.0 2.5 3.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Density

Q90

Q95

Q99


Qua

ntile

valu

e−2.4

−2.3

−2.2

−2.1

−2.0

−1.65

−1.60

−1.55

−1.50

−1.30

−1.28

−1.26

−1.24

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Wrap-up

Approach Transforms Mean Sd Skew

Slater ini;al

Hartmann z-‐Transform x x

Suggested approach Box-‐Cox + z-‐Transform x x x

Results

+  Less influence of grid size and scale range

+  More stable and symmetrical cutoff values

+  Usage of standard normal distribution percentiles as cut-offs

-  Currently involves simulation

Implications for distance comparisons

•  Do not use unstandardized Euclidean

distances

•  Do not use Slater distances

•  Either use Hartmann distances

•  or use the presented approach for stable and symmetric cutoffs

What‘s next?

•  Make approach available in OpenRepGrid* software package

•  Find approximation to estimate Lambda that avoids simulation

•  Compare with empirical grids

•  Generalize to other types of distances

Interested? Get in touch!

*Heckmann, 2011. Software freely available at www.openrepgrid.org

Literature

•  Ben-Tovim, D. I., & Greenup, J. (1983). The representation of transference through serial grids: A methodological study. British Journal of Medical Psychology, 56, 255–261.

•  Borkenhagen, A., Klapp, B. F., Schoeneich, F., & Brähler, E. (2005). Differences in body image between anorexics and in-vitro-fertilization patients: a study with Body Grid. GMS Psychosoc Med., 2(Doc10). Conrad, R., Schilling, G., & Liedtke, R. (2005). Parental Coping with Sudden Infant Death After Donor Insemination: Case Report. Human Reproduction, 20(4), 1053–1056. doi:10.1093/humrep/deh705

•  Fransella, F., & Crisp, A. H. (1970). Conceptual organization and weight change. Psychotherapy and Psychosomatics, 18(1-6), 176–185.

•  Hartmann, A. (1992). Element comparisons in repertory grid technique: Results and consequences of a monte carlo study. International Journal of Personal Construct Psychology, 5(1), 41–56

•  Heckmann, M. (2011). OpenRepGrid - An R package for the analysis of repertory grids (Unpublished diploma thesis). University of Bremen, Bremen, Germany.

•  Leach, C., Freshwater, K., Aldridge, J., & Sunderland, J. (2001). Analysis of repertory grids in clinical practice. The British Journal of Clinical Psychology, 40, 225–248.

•  Schoeneich, F., & Klapp, B. F. (1998). Standardization of interelement distances in repertory grid technique and its consequences for psychological interpretation of self-identity plots: An empirical study. Journal of Constructivist Psychology, 11(1), 49–58.

•  Schwab, K. (2006). Die Bedeutung standardisierter Maße der Repertory-grid-Technik als ein idiografisches Verfahren in der Beziehungsdiagnostik in der Psychosomatik : eine statistische Analyse der Zusammenhänge zwischen nomothetischen diagnostischen Fragebogenverfahren und standardisierten Repertory-Grid-Maßen (Dissertation). Universität Marburg, Fachbereich Medizin, Marburg.

•  Slater, P. (1977). The measurement of intrapersonal space by grid technique: Dimensions of intrapersonal space (Vol. 2). London: Wiley & Sons.

•  Winter, D. A., Bell, R. C., & Watson, S. (2010). Midpoint Ratings on Personal Constructs: Constriction or the Middle Way? Journal of Constructivist Psychology, 23(4), p. 343

standardizing inter-element distances in repertory grids

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