### Conflict measures bases on correlations

#### Description

The first approach to mathematically derive a conflict measure based
on grid data was presented by (1979). They
based their operationalization on an approach by Lauterbach (1975) who applied Heider’s (1946) balance theory for a quantitative
assessment of psychological conflict. The measure uses inter-construct
correlation as a basis. It assess via the sign of the correlations of a
triad of constructs if the constructs triad is balanced or imbalanced. A
triad is imbalanced if one or three of the correlations are negative, i.
e. when they leading to contrary implications (Slade & Sheehan, 1979). The function
`indexConflict1`

in `OpenRepGrid`

calculates the
proportion of conflictive triads in the grid as devised by Slade and
Sheehan.

Slade and Sheehan’s approach has several drawbacks, as it does not
take into account the magnitude of the inter-construct correlations. As
a consequence significant and insignificant correlations are not
differentiated. Also, correlations of small magnitude, i.e. near zero,
which may have a positive or negative signe due to chance alone distort
the measure (Bassler, Krauthauser, &
Hoffmann, 1992; Winter, 1982). An improved version of the index
that overcomes these shortcomings was proposed by (1992) and incorporated into the program
*CT* (correlation test). The index enhances the identification of
imbalanced triads by using a criterion that incorporates the magnitudes
of correlations. From a psychological viewpoint, it remains unclear
though what is measured. Connections to the concept of cognitive
differentiation have been presumed (Krauthauser,
Bassler, & Potratz, 1994). The function
`indexConflict2`

in `OpenRepGrid`

calculates the
proportion of conflictive triads in the grid as devised by (1992).

#### Calculation

##### Slade & Sheehan (1979) approach

Multiply all correlations from a triad (formed by the constructs A, B and C) $r_{AB} r_{AC} r_{BC}$. If the sign of the result is negative, the triad is imbalanced.

The table below shows when a triad made up of the constructs A, B, and C is balanced and imbalanced.

$r_{AB}$ | $r_{AC}$ | $r_{BC}$ | Triad characteristic |
---|---|---|---|

+ | + | + | balanced |

+ | + | - | imbalanced |

+ | - | + | imbalanced |

+ | - | - | balanced |

- | + | + | imbalanced |

- | + | - | balanced |

- | - | + | balanced |

- | - | - | imbalanced |

##### Bassler et al. (1992) approach

Order correlations of the triad by absolute magnitude, so that $r_{max} > r_{mdn} > r_{min}$.

Apply Fisher’s Z-transformation and devision by 3 to yield values between 1 and -1 Check whether the triad is balanced by assessing if the following relation holds:

- If $Z_{max} Z_{mdn} > 0$, the triad is balanced if $Z_{max} Z_{mdn} - Z_{min} <= crit$
- If $Z_{max} Z_{mdn} < 0$, the triad is balanced if $Z_{min} - Z_{max} Z_{mdn} <= crit$

#### R-Code

To use the Slade and Sheehan approach use

```
indexConflict1(boeker)
#
# ################################
# Conflicts based on correlations
# ################################
#
# As devised by Slade & Sheehan (1979)
#
# Total number of triads: 364
# Number of imbalanced triads: 106
#
# Proportion of balanced triads: 70.9 %
# Proportion of imbalanced triads: 29.1 %
```

For the Bassler et al. approach use

```
indexConflict2(boeker)
#
# ###############################
# Conflicts based on correlations
# ###############################
#
# As devised by Bassler et al. (1992)
#
# Total number of triads: 364
# Number of imbalanced triads: 240
#
# Proportion of balanced triads: 34.1 %
# Proportion of imbalanced triads: 65.9 %
indexConflict2(boeker, crit = .05) # change critical value
#
# ###############################
# Conflicts based on correlations
# ###############################
#
# As devised by Bassler et al. (1992)
#
# Total number of triads: 364
# Number of imbalanced triads: 219
#
# Proportion of balanced triads: 39.8 %
# Proportion of imbalanced triads: 60.2 %
```

### Conflict measures based on distances

#### Description

Measure of conflict or inconsistency as proposed by Bell (2004). The identification of conflict is based on distances rather than correlations as in other measures of conflict indexConflict1 and indexConflict2. It assesses if the distances between all components of a triad, made up of one element and two constructs, satisfies the “triangle inequality” (cf. Bell, 2004). If not, a triad is regarded as conflictive. An advantage of the measure is that it can be interpreted not only as a global measure for a grid but also on an element, construct, and element by construct level making it valuable for detailed feedback. Also, differences in conflict can be submitted to statistical testing procedures.

#### R-Code

```
indexConflict3(leach2001a)
#
# ##########################################################
# CONFLICT OR INCONSISTENCIES BASED ON TRIANGLE INEQUALITIES
# ##########################################################
#
# Potential conflicts in grid: 819
# Actual conflicts in grid: 340
# Overall percentage of conflict in grid: 41.51 %
#
# ELEMENTS
# ########
#
# Percent of conflict attributable to element:
#
# percentage
# 1 Child self 1088.24
# 2 Self now 794.12
# 3 Women in general 176.47
# 4 Men in general 823.53
# 5 Father 1647.06
# 6 Partner 1029.41
# 7 Ideal self 1470.59
# 8 Mother 1058.82
# 9 Abuser in childhood 1911.76
#
# Chi-square test of equal count of conflicts for elements.
#
# Chi-squared test for given probabilities
#
# data: x$e.count
# X-squared = 65, df = 8, p-value = 4.826e-11
#
#
# CONSTRUCTS
# ##########
#
# Percent of conflict attributable to construct:
#
# percentage
# 1 assertive - not assert 8.09
# 2 confident - unconfiden 7.65
# 3 does not f - feels guil 9.12
# 4 abusive - not abusiv 7.35
# 5 frightenin - not fright 6.47
# 6 untrustwor - trustworth 6.76
# 7 powerful - powerless 6.18
# 8 big headed - not big he 6.18
# 9 independen - dependent 6.03
# 10 confusing - not confus 6.91
# 11 guilty - not guilty 6.18
# 12 cold - shows feel 6.03
# 13 masculine - feminine 7.50
# 14 interested - not intere 9.56
#
# Chi-square test of equal count of conflicts for constructs.
#
# Chi-squared test for given probabilities
#
# data: x$c.count
# X-squared = 16.171, df = 13, p-value = 0.24
```

### Literature

*Journal of Constructivist Psychology*,

*5*(1), 95–111.

*Personal Construct Theory & Practice*,

*1*, 53–59.

*Journal of Psychology*,

*21*, 107–112.

*Journal of Constructivist Psychology*,

*7*(4), 283–299.

*The British Journal of Social and Clinical Psychology*,

*14*(1), 43–47. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/1091319

*British Journal of Psychology*,

*70*(4), 519–524.

*The British Journal of Medical Psychology*,

*55 (Pt 3)*, 257–269. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/7126491