Actualidad en Jurisprudencia

5.4.5.1 Pair-wise data

This section refers to the analysis required for Part II of the survey response – the pair-wise matrix. The objective of the data preparation process for pair-wise data is to create a single column of data with 27 rows. The resulting value associated with each row/characteristic represents a consolidation of the effects of every characteristic upon the characteristic in question.

Each respondent was asked to complete ‘half’ a pair-wise matrix. Each cell requires a rating to describe the effect of two characteristics upon each other. Each cell

quantifies the degree of synergy or conflict between a pair of characteristics. By this we mean that either two practices are mutually “helpful” and reinforce each other

(synergetic), or on the contrary, they are mutually exclusive and conflict with each other (conflicting). Values permitted are: +5, +4, +3, +2, +1, 0, -1, -2, -3, -4, -5. If two practices are strongly synergetic (or reinforcing) a response of +5 is required in the cell that joins up these two practices. If the pair of practices have a neutral effect or are independent of each other, then a score of 0 should be supplied. If two

practices conflict or have a severe weakening effect upon each other then a score of -5 would be relevant. An example of a completed pair-wise matrix is shown in Figure 5- 10.

Figure 5-10 Sample Pair-wise matrix

The assumption in requiring only ‘half’ of the matrix to be completed is that each characteristic had the same magnitude of effect regardless of the order in which the characteristic was adopted. That is, the characteristics have a symmetrical effect upon each other. This assumption is a small limitation of the research study.

The effect of this strategy within the context of the current research relating to 27 characteristics is that respondents complete 26 + 25 + 24 …+ 1 cells, a total of 351 cells, compared to a full 27 x 27 matrix, excluding the diagonal (-27) which would be some 702 cells.

5.4.5.2 Method for calculation of pair-wise effects

Matrix multiplication is the method used to calculate the inter-characteristic effects (Allen et al., 2008a). The effect of matrix multiplication is to find the total effect of a characteristic upon every other characteristic. The mathematical process of matrix multiplication takes two input matrices and calculates an output matrix. For this research study the two input matrices are the same and constructed from the pair-wise survey data. The effect of the matrix multiplication and its relevance is discussed below. The calculation of matrix a, which is the product of two input arrays b and c is shown in Equation 5-6. i is the row number, and j is the column number.

Equation 5-6 Matrix Multiplication

Each cell in the output matrix a contains the sum of the products of the pair interactions. This explanation is more easily grasped visually by providing an example. Matrices b and c are the same because the desired outcome is the matrix multiplied by itself. Note that the diagonal is zero in accordance with our assumption in the research study that a characteristic has no effect upon itself. Figure 5-11 shows an example with the diagonal of matrix a containing the results of interest to the study. The result for characteristic 2 (=65) is contained in cell (2,2) of the output matrix a, is the sum of the products of the pair interactions in row 2 of the input matrix b and column 2 of the input matrix c.

The effect of the matrix multiplication is to give more weight to larger scores. For example, if a respondent gives the scores 1, 3 and 5 to three different characteristics, the effect is that the characteristic with a score of 1 is 25 times smaller than the characteristic with a score of 5, and 9 times smaller than a characteristic with a score of 3. This effect is helpful and reflects to some extent the fact that interaction and feedback often creates non-linear and sometimes runaway effects on performance.

A complication arises with matrix multiplication due to the nature of the research problem in that pair-wise ratings can be negative. We require the matrix

multiplication result to reflect a reduction if a cell is negative. This is achieved by using the absolute value (ignoring the sign of the rating) when making the input matrix symmetrical. An example is shown in Figure 5-12 whereby all cells in the source data are negative to show that the result in Figure 5-11 can be delivered. In the actual data collected, there is a mixture of positive and negative data, and the scheme works for either as addition is commutative.

Figure 5-12 Matrix Multiplication Sample 2

Having made the assumption that the effect of any characteristic x is the same upon any characteristic y regardless of the order in which the characteristics are adopted, we need to construct a full matrix by using the transpose of the source matrix.

Microsoft Excel® incorporates a function to do this: Copy, Paste Special, Transpose. A full matrix can then be constructed using the source data, zeros along the diagonal and the absolute values in the transposed half of the matrix, in order to accommodate negative values described above. The newly formed matrix can then be multiplied by itself to create the output matrix. The formula adopted is shown in Equation 5-7.

(M+│MT│) matrix multiplication function (M+│MT│) = output Matrix

The diagonal is moved into a single column, and used as the source data for Table 2.

The detailed steps undertaken to manipulate the input pair-wise data in order to calculate pair-wise effects is shown fully in Appendix K.

5.4.5.3 Normalization

In the same way that normalization of first order data is required, normalization of pair-wise data is needed to allow fair comparison of cases. The matrix multiplication process creates a single score for each characteristic (using the results in the

diagonal). The average of all scores is calculated and divided into each characteristic value.

Some characteristics scored a negative overall value. The standard normalization process above is not sufficient to accommodate this. The absolute value of the matrix multiplication output must be used to calculate the average of all scores. Once

normalized, any negatively rated characteristics can be multiplied by -1. If this process is not completed in this way, it is not a 50:50 comparison to the first order results which was the requirement for Scenario 2 (see Figure 5-1).