### Description

#### Correlation measures

The correlations between constructs are used in a lot of occasions, indices etc, and present a standard statistic. Several types of correlations can be requested via the function constructCor.

• Pearson or Product-moment correlation (PMC)
• Kendall’s tau rank correlation
• Spearman’s rank correlation

#### Root mean square correlation

Also, the Root mean square (RMS) correlation, also known as ‘quadratic mean’ of the inter-construct correlations, can be calculated using the function constructRmsCor. The RMS serves as a simplification of the correlation table. It reflects the average relation of one construct with all other constructs. Note that as the correlations are squared during its calculation, the RMS is not affected by the sign of the correlation (cf. Fransella, Bell & Bannister, 2003, p. 86).

#### Somers’ D

As suggested by Hinkle (1965) the relationships between constructs may take several different forms (reciprocal, hierarchical etc.). Several authors have suggested the use of asymmetric measures of association as an indicator for the type of relationship present (cf. Fransella et al., 2003). Currently only the software Gridstat (Bell, 2009) allows to calculate the asymmetric measure Somers’ D (Somers, 1962). The calculation of Somers’ D in OpenRepGrid is prompted via the function constructD.

### R-Code

#### Correlation measures

constructCor(mackay1992) 

##############################
Correlation between constructs
##############################

Type of correlation:  pearson

1    2    3    4    5    6
Quick - *Slow       1     0.38 0.77 0.13 0.52 0.29
*Satisfied - Bitter 2          0.18 0.82 0.56 0.29
Talkative - *Quiet  3               0.14 0.72 0.58
*Succesful - Loser  4                    0.64 0.47
Emotional - *Calm   5                         0.92
*Caring - Selfish   6                             

you can select between the different types of correlation by setting the argument method to "pearson", "kendall" or "spearman". To request a Spearman rank correlation type

constructCor(mackay1992, method="spearman") 

##############################
Correlation between constructs
##############################

Type of correlation:  spearman

1    2    3    4    5    6
Quick - *Slow       1     0.50 0.83 0.00 0.56 0.19
*Satisfied - Bitter 2          0.09 0.56 0.64 0.13
Talkative - *Quiet  3               0.00 0.39 0.21
*Succesful - Loser  4                    0.69 0.49
Emotional - *Calm   5                         0.81
*Caring - Selfish   6                             

To format the output several arguments are available. See ?print.constructCor for more printing options.

r <- constructCor(mackay1992)
print(r, digits=5, col.index=F)

##############################
Correlation between constructs
##############################

Type of correlation:  pearson

1       2       3       4       5       6
Quick - *Slow              0.37709 0.77226 0.12913 0.51682 0.28911
*Satisfied - Bitter                0.18383 0.81969 0.56239 0.28917
Talkative - *Quiet                         0.13771 0.71984 0.57656
*Succesful - Loser                                 0.63623 0.47133
Emotional - *Calm                                          0.92394
*Caring - Selfish                                                 

#### Root mean square correlation

To calulate the RMS correlation

constructRmsCor(fbb2003)  

##########################################
Root-mean-square correlation of constructs
##########################################

RMS
(1) clever - not bright                0.66
(2) disorganized - organized           0.58
(3) listens - doesn't hear             0.61
(4) no clear view - clear view of life 0.46
(5) understands me - no understanding  0.53
(6) ambitious - no ambition            0.30
(7) respected - not respected          0.62
(8) distant - warm                     0.25
(9) rather aggressive - not aggressive 0.29

Average of statistic 0.48
Standard deviation of statistic 0.15 

#### Somers’ D

Somers’ D is an asymmetric measure. As a default the columns are the dependent the rows the independent variables.

constructD(fbb2003)

############################
Somers' D between constructs
############################

Direction: columns are set as dependent
1    2    3    4    5    6    7
(1) clever - not bright              1  1.0 -0.6  0.9 -0.5  0.6  0.2  0.8
(2) disorganized - organized         2 -0.7  1.0 -0.6  0.4 -0.3 -0.2 -0.7
(3) listens - doesn't hear           3  1.0 -0.5  1.0 -0.4  0.8  0.1  0.7
(4) no clear view - clear view of l  4 -0.6  0.4 -0.5  1.0 -0.2 -0.6 -0.5
(5) understands me - no understandin 5  0.8 -0.3  0.9 -0.2  1.0  0.0  0.6
(6) ambitious - no ambition          6  0.2 -0.2  0.1 -0.5  0.0  1.0  0.3
(7) respected - not respected        7  0.8 -0.7  0.7 -0.4  0.5  0.3  1.0
(8) distant - warm                   8  0.0 -0.1  0.0 -0.2  0.0  0.2 -0.1
(9) rather aggressi - not aggressive 9 -0.1 -0.4 -0.2 -0.2 -0.2  0.1  0.0
8    9
(1) clever - not bright               0.0 -0.1
(2) disorganized - organized         -0.1 -0.4
(3) listens - doesn't hear            0.0 -0.2
(4) no clear view - clear view of l  -0.2 -0.2
(5) understands me - no understandin  0.0 -0.2
(6) ambitious - no ambition           0.2  0.1
(7) respected - not respected        -0.1  0.0
(8) distant - warm                    1.0  0.5
(9) rather aggressi - not aggressive  0.5  1.0

To set the rows as dependent, type

constructD(fbb2003, dep="r")

The returned data is in all cases a matrix or a data.frame, so you can easily extract the data you need for further calculations.

### Literature

• Bell, R. C. (2009). Gridstat version 5 - A Program for Analyzing the Data of A Repertory Grid (manual). University of Melbourne, Australia: Department of Psychology.
• Fransella, F., Bell, R. C., & Bannister, D. (2003). A Manual for Repertory Grid Technique (2. ed.). Chichester: John Wiley & Sons.
• Hinkle, D. (1965). The Change of Personal Constructs from the Viewpoint of a Theory of Construct Implications (Unpublished Ph.D. thesis). Ohio State University, Ohio.
• Somers, R. H. (1962). A New Asymmetric Measure of Association for Ordinal Variables. American Sociological Review, 27(6), 799-811.