Gridstat
A Program for Analyzing the Data of A Repertory Grid
Version 3.0
Richard C Bell
Department of Psychology
University of Melbourne
August 2002
Version 3.0
In version three of this program a number of changes have been made:
- A new module for evaluation of asymmetric relationships among constructs and elements has been added
- The singular-value decomposition (principal components) module
- has changed the default scaling
- now prints residual information and a fit measure
- Correspondence analysis has been modified to allow an interactive decision as to the number of dimensions
- Printer plots have been corrected and improved for correspondence analysis and singular value decomposition. Users can now choose solutions centred on constructs, elements, or a symmetric solution.
- In the statistics module the re-scaled correlation matrix to allow invertibility (when necessary) is no longer printed (since the changes cannot be detected at the level of accuracy printed
Introduction
GRIDSTAT is a program oriented towards the flexible analysis of data in a repertory grid. It is not designed for the elicitation of grid data, nor to be client-friendly. It is hoped that it is, however, moderately user friendly.
GRIDSTAT allows the user to carry out a series of analyses on a repertory grid, varying the options for analysis at each step. Methods for the analysis of grid data have often been developed intrinsically and have no counterpart to indices developed within the mainstream of data analysis. In GRIDSTAT, where there is a traditional index that corresponds to a grid index, the former has been preferred. Where there is a view (by the author) of a preferred option, this is indicated in the menu.
GRIDSTAT is principally oriented to carry out the following analyses:
1. Basic statistics
2. Factor analysis - rotated principal components of construct correlations with associated element factor scores, a (joint) singular value decomposition analysis, as in Slater’s INGRID, or correspondence analysis
3. Cluster analysis
4. Predictive analyses
GRIDSTAT also allows for some modification of the grid during its operation, and provides some indices for a particular variety of grid, the dependency grid.
GRIDSTAT detects some problems that can be inherent in grid data and makes appropriate adjustments, indicating to the user that it has done so. Some of these changes will not be discernible to the human eye or the accuracy of printed output. Unfortunately the central processor of a computer has a keener eye for arithmetic misdemeanors and will otherwise crash. This is not to say that GRIDSTAT will not crash. Hopefully it will not crash as often as its predecessor G-PACK.
Data Input
GRIDSTAT reads grid data from a file in MS-DOS text (or ASCII) format. The file for a grid should be arranged in the following sequence:
1. A title line (80 characters or less)
2. A line with two numbers on it, the number of constructs (rows)[NC] and the number of elements (columns) [NE]. These two numbers must be separated by at least one space, but otherwise can be placed anywhere on the line.
3. NC lines, one for each construct, containing left & right labels (use a slash{/} to separate them). Only the first 40 characters are read, and in some output this may be truncated further. [These lines can be omitted (along with element labels} if the first label says NO LABELS]
4. NE lines, one for each element. Only the first 40 characters are read, and in some output this may be truncated further. [These lines can be omitted {along with construct labels} if the first label for constructs says NO LABELS].
5. NC lines of grid data, each containing NE numbers representing element ratings on the construct, each number being separated by at least one space.
The NO LABELS indicator - only the first 4 characters are read, and upper & lower case may be interchanged as in the following combinations - ','NO L', 'NoLa', 'No L', 'Nola',
'No l', 'nola', 'no l'. This causes dummy labels to be created, since for grids where the number of elements and constructs are equal, it may not be always clear which is which.
The file may contain any number of grids in this format. The main menu of the program contains an option ‘Analyse another grid’ which may be read from the same file (or another).
Output
Output is written to a disk file (specified by the user) using the MS-DOS naming convention, and in MS-DOS (ASCII) format. Output is not as pretty as other programs since the focus has been on the answers rather than the way they are presented. Users hankering for pretty pictures should cut & paste output into their favorite spreadsheet and indulge in grandiosity thereafter. All analyses for one grid are written to a single file. When subsequent grids are analyzed, the user has the option of continuing with this file or specifying a new one.
Starting GRIDSTAT
GRIDSTAT is an MS-DOS program and must be run under MS-DOS. However it seems to be able to run by double-clicking on the icon/file-name in File-Manager under Windows 3.1 or Explorer in Windows 95(+), as well as in an MS-DOS window. The file is a compiled fortran file and can be started by typing GRIDSTAT (). Earlier versions were beginning to have problems with current versions of windows, this version has been compiled under the Lahey lf95 compiler which is compatible with windows.
The Opening Screen
The opening screen looks as follows:
GRIDSTAT - A Package for Analysing Repertory Grid Data.
written by Richard C. Bell
Department of Psycholgy
University of Melbourne, Parkville, Victoria 3052 Australia.
phone : 61 (0)3 8344 6364
fax : 61 (0)3 9347 6618
e-mail:
Version 3.0 August 2002
What is the name of the input GRID file ?
Addresses & the reason for these (errors detected):
Please let me know of problems. There will always be some for odd data sets. I can assure users that the test data sets with known solutions have returned the same. And I have tried to eliminate as many as possible problems as I can, but I cannot dream up every possible problem. Nor can those who have helped by testing preliminary versions. This program is destined to belong to the ‘community of scholars’. Unfortunately us scholars don’t have as much time to schol as we did in the past. Consequently errors that are so far undetected and subsequently reported will be dealt with as soon as possible. With an emphasis on the possible.
Input & Output Files
GRIDSTAT asks for the input (data) and output file names on the opening screen. These must be fully specified as name.ext in MS-DOS format where name does not exceed 8 characters and contains no spaces, and ext is of 3 characters (though it may be omitted).
Main Menu
The Main Menu appears as follows. User must type a number and return () as with all menus, to indicate their choice.
GRIDSTAT Analysis Options:
1. Basic Statistics & Correlations.
2. Component Analysis.
3. Cluster Analysis.
4. Ordinal Relationship Analysis.
5. Dependency Grid Analysis.
6. Analyse another Grid.
7. Edit Grid (Reflect Constructs,
remove Rows or Cols).
8. End GRIDSTAT.
Option :
This menu appears initially and subsequently after each option is exercised. To quit GRIDSTAT users must enter 8 and return () .
The first four options are analysis options and are detailed in the following sections. As indicated above, this menu also contains options for the analysis of another grid and for exiting GRIDSTAT. The remaining option (?) has been included to facilitate processing of a grid that contains problems. Of course, users may exit GRIDSTAT, enter some editor and change the grid file. However, GRIDSTAT caters for the impatient and allows for some editing of problem grids, either by deleting constructs or elements (something a user may wish to do if such exhibit zero variance) or reflecting constructs. This only affects the version of the grid that GRIDSTAT has internally stored - not the original grid.
Some analysis options are routine and do not display sub-menus. [A caution - on any reasonably fast machine, some options may operate almost imperceptibly. You enter an option, the screen blinks, and you are back with the option menu. However at the top of the screen you should see the phrase Analysis Completed.]
Other options involve choice-points, and for each of these the user is presented with a menu, and recommendations, if appropriate. Hopefully the following details on analysis options will make clear the reasons for the recommendations.
Option 1. Basic Statistics
This option has no choices and consequently the first the user knows that it is complete is that it has returned to the main menu. Meantime (in the twinkling of an eye on all but ancient computers) it has carried out the following:
Analysis of Variance
The data of a repertory grid can be considered in traditional terms as a two-way within-subject analysis of variance (with no replications) framework. This approach was first suggested by Vannoy (1965). Such an analysis should be of interest to the grid user insofar as it says to what extent there are significant differences between constructs and/or elements. Not all grid users are aware that there are links between analysis of variance and a form of correlation, the intra-class correlation. This index shows the average correlation (in traditional terms) among the constructs (or the elements). Thus it provide an index of the differentiation among constructs (or elements), and as noted by Bell & Keen (1981), may be taken as an indicator of cognitive complexity in the repertory grid.
Analysis of Variance
~~~~~~~~~~~~~~~~~~~~
Source of Sum of Mean F- Intraclass
Variation Squares Square df Ratio Prob. Correlation
Constructs 11.4 1.27 9 3.18 0.543 0.703
Elements 88.8 9.87 9 24.71 0.015 0.179
Remainder 32.4 0.40 81
Total 132.6
Distribution Statistics
Kelly only formulated one corollary relating to the distribution of elements along a construct. This was the Range Corollary, which in statistical terms simply indicates the number of elements for which the construct is relevant. In statistical terms, this would simply be the number of cases. Other statistics can also be invoked to say something about the way elements are allocated along a construct. One notion involved the lopsidedness of the allocation of elements along a construct. For simple bi-polar allocation (i.e., to one pole or the other) the mean contains all information for this. Where ratings are used other distribution statistics become important. As well as the range of convenience, the discriminability of a construct says something about how well a construct enables discriminations to be made between elements. A conventional statistic that says something about this, is the familiar standard deviation. For lopsidedness, the statistic employed is the skewness index, where a negative value indicates ratings are skewed towards high values, and a positive value indicates ratings are skewed towards low values. Users need to keep in mind the relationship between high/low ratings and left/right construct pole descriptors. From a viewpoint of efficiency it would be sensible to maintain a standard relationship (such as low scores are always associated with the left-hand pole, and high scores with the right-hand pole).
Output for constructs is as follows:
Construct Statistics
~~~~~~~~~~~~~~~~~~~~
Mean Std.Dev. Skewness
4.30 1.19 -2.82 kind/unkind
4.90 1.14 -2.00 friendly/unfriendly
4.40 0.66 -0.91 good-looking/ugly
4.80 1.17 -1.56 competent/incompetent
4.90 1.04 -2.65 caring/uncaring
4.80 1.08 -2.11 empathic/unempathic
4.60 1.36 -2.30 nonjudgemental/judgemental
3.80 1.25 -0.75 empowers others/dominates others
4.80 1.08 -2.11 genuine/false
4.30 0.90 -2.01 like me/unlike me
4.56 1.09 1.92 Average of Statistic
0.34 0.18 0.64 St. Dev. of Statistic
The output for statistics also shows the average and standard deviation of the statistics. This enables the statistics themselves to be evaluated. Thus in the above table, empowers others / dominates others has a much lower mean (3.80) thanaverage (4.56), while good-looking / ugly has a much lower standard deviation (0.66) than average (1.09). The average for the skewness index is based on the absolute values of that statistic (i.e., the sign is ignored) in order that the discrepancy in either positive or negative directions can be evaluated.
Correlations
Bannister pioneered the use of correlations as a way of considering the structure among constructs. The coefficient calculated in GRIDSTAT is the standard Pearson product-moment correlation, appropriate for rated or binary data. No concessions are made here for ranked data, since in practice there is little difference between coefficients specifically devised for ranked data and ordinary correlation coefficients.
Construct Correlations
~~~~~~~~~~~~~~~~~~~~~~
1 2 3 4 5 6 7 8 9 10
1 1.00 0.84 0.61 0.84 0.91 0.83 0.88 0.58 0.91 0.76
2 0.84 1.00 0.58 0.66 0.83 0.80 0.82 0.55 0.88 0.81
3 0.61 0.58 1.00 0.75 0.78 0.81 0.73 0.10 0.67 0.64
4 0.84 0.66 0.75 1.00 0.89 0.92 0.90 0.38 0.76 0.53
5 0.91 0.83 0.78 0.89 1.00 0.96 0.96 0.37 0.87 0.78
6 0.83 0.80 0.81 0.92 0.96 1.00 0.97 0.34 0.83 0.68
7 0.88 0.82 0.73 0.90 0.96 0.97 1.00 0.48 0.90 0.67
8 0.58 0.55 0.10 0.38 0.37 0.34 0.48 1.00 0.71 0.14
9 0.91 0.88 0.67 0.76 0.87 0.83 0.90 0.71 1.00 0.68
10 0.76 0.81 0.64 0.53 0.78 0.68 0.67 0.14 0.68 1.00
One of Bannister’s innovations was the intensity score. This was the average correlation with other constructs (times 100). Constructs with high values were seen as more superordinate.
A standard way of averaging correlations, used in GRIDSTAT, is to calculate the root-mean-square [RMS] correlation (i.e. the square root of the average of the squared correlations). This process is necessary because correlations are non-linear and should not be summed directly. The RMS values in GRIDSTAT can be interpreted however in Bannister’s fashion. However, a better index of the relationship between one construct and the others is often the squared-multiple correlation [SMC] which expresses the relationship between one construct and the set of other constructs (taking into account overlaps between constructs and what each adds to the others). This index is often used in factor analysis as an initial estimate of the communality of a variable. Unfortunately it cannot always be calculated for the correlations of grid. Because of the small amount of data in a grid, numerical problems can be encountered in calculating this index. (Technically, the matrix of correlations cannot be inverted.) GRIDSTAT uses an innovation from factor analysis to re-scale the correlations so that SMC’s can be calculated. If this occurs a notes is written to the output.
Correlation Matrix cannot be inverted,
SMC cannot be calculated in above matrix. Re-scaling...
Since it is very rare that these changes can be detected, the current version of GRIDSTAT does not print out the re-scaled matrix.
Root-Mean-Squared (RMS) and Squared-Multiple (SMC) Correlations
RMS SMC
0.80 0.90 kind/unkind
0.76 0.96 friendly/unfriendly
0.66 0.95 good-looking/ugly
0.76 0.81 competent/incompetent
0.83 0.88 caring/uncaring
0.81 0.98 empathic/unempathic
0.83 0.98 nonjudgemental/judgemental
0.45 0.82 empowers others/dominates others
0.81 0.98 genuine/false
0.66 0.90 like me/unlike me
0.74 0.92 Average of Statistic
0.11 0.06 St. Dev. of Statistic
Again, averages and standard deviations of these statistics are provided so that unusual values for the RMS or SMC indices can be readily detected.
Element Statistics and Correlations
The same information is provided for elements.
Element Statistics
~~~~~~~~~~~~~~~~~~
Mean Std.Dev. Skewness
4.50 0.50 5.26 Self now
4.60 0.49 4.88 Nurse who influenced me the most
4.40 0.92 -1.70 Ideal student
5.10 1.14 -2.09 Student with excellent opportunities
4.80 0.40 8.65 Ideal teacher
4.70 0.64 2.99 Me as a clinical/practical nurse
4.90 0.54 4.46 Ideal clinical/practical nurse
1.90 0.54 1.53 Person I dont admire
5.50 0.67 1.32 Person I work with
5.20 0.40 11.12 Rotating teacher
4.56 0.62 4.40 Average of Statistic
0.94 0.22 3.11 St. Dev. of Statistic
Element Correlations
~~~~~~~~~~~~~~~~~~~~
1 2 3 4 5 6 7 8 9 10
1 1.00 -0.41 0.22 0.09 0.00 0.47 -0.19 -0.19 -0.15 0.50
2 -0.41 1.00 0.36 0.61 0.10 -0.06 0.23 0.23 0.61 -0.10
3 0.22 0.36 1.00 0.83 -0.33 0.37 -0.12 0.28 0.16 0.05
4 0.09 0.61 0.83 1.00 0.04 0.32 0.18 -0.15 0.46 -0.04
5 0.00 0.10 -0.33 0.04 1.00 0.16 0.84 -0.56 0.75 0.25
6 0.47 -0.06 0.37 0.32 0.16 1.00 -0.09 -0.09 0.35 0.62
7 -0.19 0.23 -0.12 0.18 0.84 -0.09 1.00 -0.38 0.69 0.09
8 -0.19 0.23 0.28 -0.15 -0.56 -0.09 -0.38 1.00 -0.14 0.09
9 -0.15 0.61 0.16 0.46 0.75 0.35 0.69 -0.14 1.00 0.37
10 0.50 -0.10 0.05 -0.04 0.25 0.62 0.09 0.09 0.37 1.00
Root-Mean-Squared (RMS) and Squared-Multiple (SMC) Correlations
RMS SMC
0.29 0.55 Self now
0.36 1.00 Nurse who influenced me the most
0.37 1.00 Ideal student
0.40 1.00 Student with excellent opportunities
0.44 0.84 Ideal teacher
0.34 0.52 Me as a clinical/practical nurse
0.40 0.64 Ideal clinical/practical nurse
0.27 1.00 Person I dont admire
0.46 0.85 Person I work with
0.31 0.58 Rotating teacher
0.37 0.80 Average of Statistic
0.06 0.19 St. Dev. of Statistic
Correlations among elements are not invariant over reflection of construct poles (i.e., when the construct is reversed, the correlations between elements change). Since the construct orientation is often arbitrary, this can be a problem, as was noted by Mackay (1992).
The problem had been similarly noted (in another context) by Cohen (1969) who showed that invariant correlations could be obtained by centering scores on a constant. GRIDSTAT uses the midpoint of the range of ratings used in the grid as this constant, and computes the invariant correlations in addition to the raw ones, as shown below.
Element Correlations are affected by the orientation
of the constructs. Coefficients which are independent of this
require a construct midpoint to be defined. The Midpoint of the
smallest and largest ratings overall is used here. ( 3.500)
Construct-Invariant Element Correlations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1 2 3 4 5 6 7 8 9 10
1 1.00 0.74 0.70 0.75 0.85 0.89 0.80 -0.87 0.83 0.92
2 0.74 1.00 0.74 0.89 0.89 0.79 0.89 -0.84 0.94 0.88
3 0.70 0.74 1.00 0.91 0.60 0.74 0.62 -0.60 0.70 0.69
4 0.75 0.89 0.91 1.00 0.79 0.81 0.80 -0.80 0.86 0.79
5 0.85 0.89 0.60 0.79 1.00 0.86 0.98 -0.96 0.98 0.95
6 0.89 0.79 0.74 0.81 0.86 1.00 0.81 -0.85 0.89 0.93
7 0.80 0.89 0.62 0.80 0.98 0.81 1.00 -0.93 0.96 0.92
8 -0.87 -0.84 -0.60 -0.80 -0.96 -0.85 -0.93 1.00 -0.91 -0.92
9 0.83 0.94 0.70 0.86 0.98 0.89 0.96 -0.91 1.00 0.95
10 0.92 0.88 0.69 0.79 0.95 0.93 0.92 -0.92 0.95 1.00
Root-Mean-Squared (RMS) Construct-Invariant Correlations
0.82 Self now
0.85 Nurse who influenced me the most
0.71 Ideal student
0.82 Student with excellent opportunities
0.88 Ideal teacher
0.84 Me as a clinical/practical nurse
0.86 Ideal clinical/practical nurse
0.86 Person I dont admire
0.90 Person I work with
0.89 Rotating teacher
0.84 Average of Statistic
0.05 St. Dev. of Statistic
Option 2. Component Analysis
GRIDSTAT currently offers three forms of component analysis: rotated principal components of construct correlations, singular-value-decomposition of constructs and elements jointly (as in Slater’s INGRID) and correspondence analysis.
Principal Components of Construct Correlations
In this solution the constructs are represented by principal component loadings and the elements by factor scores. In the computation of the element factor scores used here the construct correlation matrix must be inverted, consequently the correlation matrix will be re-scaled if necessary (as in the calculation of the squared-multiple-correlations in the basic statistics routine).
The choice of the number of factors is made by the user when the following information is presented on the screen:
Eigenvalues.
Root Value % Cum %
1 7.638 76.4 76.4 **************************************************
2 1.143 11.4 87.8 ********
3 0.634 6.3 94.1 *****
4 0.276 2.8 96.9 **
5 0.164 1.6 98.6 **
6 0.087 0.9 99.4 *
7 0.032 0.3 99.7 *
8 0.018 0.2 99.9 *
9 0.008 0.1 100.0 *
10 0.000 0.0 100.0 *
Enter Number of Components :
If the user wishes to use the rule of eigenvalues-greater-than-one, then that information is available in the column headed ‘Value’. The next two columns show the percentage of variance accounted for; by component and cumulatively. Finally, to the right, a scree-plot representation is presented for those who wish to make a decision on this basis.
The next point of decision is the choice of rotation. GRIDSTAT offers a range of options here, as shown in the following menu:
Factor Rotation Options:
1. Varimax Orthogonal Rotation
2. Equamax Othogonal Rotation