Another approach to composite indicators

One of the issues with composite indicators is to ensure that it can be as flexible to changes of risk level as ordinary KRIs. Till now proposed composite indicators fulfil this condition when threshold(s) value(s) is (are) exceeded but are unable to give useful information when individual KRI remains in the acceptable (green) zone. I propose a modification of Taylor approach in order to overcome this difficulty.

Original Taylor Model:

Charles Taylor has proposed[1] a way to derive a simple composite indicator.  His approaches consist of 2 steps:

1)    Transforming any KRI into what he calls a T-value,

2)    Combining as many T-values as we want into a composite.

In order to derive the T-value of an KRI realisation (noted as X) he defines 4 threshold  values (L₁, L₀, H₀, H₁) with L_1< L_0<H₀<H_1. 

     T = 1 when L₀< X < H₀                                                                      (1)

T = (X+L₁-2L₀)/ (L₁-L₀) when X < L₀                                        (2)

T = (X+H₁-2H₀) / (H₁-H₀) when X > H₀                                  (3)

The first case (T equal to 1) corresponds to the “green zone” that means acceptable risk and no need to take corrective action

A value of T below L₁ or above H₁ corresponds to the “red zone”_, unacceptable risk and urgent necessity to take corrective action.

The remaining cases L₁< X < L₀   and  H_o< X < H₁ correspond to the “yellow zone” generally considered as the intermediate case[2].

After establishing T value for each indicator, he proposed the following par of equation for composite indicators:

I_rel = max {1 , α[T(1)^w(1)T(2)^w(2)]^β}                              (4)

I_unrel = max {1 , α max[T(1)^w(1);T(2)^w(2] ^β }                  (5)

With w(i) representing the weight of the „i” indicator, α and β adjustment parameters. Equation (4) is used for related indicators and equation (5)   for unrelated indicators.

Taylor approach seems very attractive as it permits to calculate easily composite indicator. However, it flattens all green values of KRI to a single T-value equal to 1.  In this way unfortunately the T-value is equal for a KRI in the middle of the “green” zone (safe) and for a KRI close to the “yellow “zone”. Moreover this T-value does not convey the information which can be obtained by analysing a trend in the green zone (see figure below).

The proposed solution

My suggestion is to keep the Taylor broad framework but to modify the T-value definition.


Let us:

     1)    consider that realisation of KRI variable are always positive numbers,

2)    define 2 critical and 2 threshold values for a KRI determining the “red”, “yellow” and “green” zone[3],

X_c,low<X_t,low<X_t,high<X_c,high

3)    define by X a realization of KRI,

4)    define the t-value for KRI in the following way:

If X≤ X_c,low

T=2+ X_c,low / X                                                      (6a)

If X_c,low≤X≤X_t,low

T=2+ (X_t,low –X)/(X_t,low –X_c,low )                              (6b)

If X_t,low≤X<X_t,high

T=max{1+(X_t,low /X);   1+(X/ X_t,high)}                      (6c)

If X_t,high≤X≤X_c,high

T=2+(X–X_t,high)/(X_c,high –X_t,high)                              (6d)

If X≥X_c,high

T=2+(X/X_c,high)                                                     (6e)

If there is only one (upper) threshold value and one (upper) critical value then  equations (6d) - (6e) will remain and equation (6c) will be replaced by:

 T=1+(X/ X_t,high)  with X≤X_t,high                                                         (6f)  

Please note that now we can distinguish between KRI belonging to the safe “green zone”. 
 All T-values are above one. As a consequence Taylor equation (4) and (5) may be simplified to:

I’_rel= α’[T(1)^w(1)T(2)^w(2)]^β’                                    (7)

I”_unrel= α” max[T(1)^w(1);T(2)^w(2] ^β”}                       (8)

As the above T-values are different from those obtained by Taylor, parameters (α’, β’) and (α”, β”) should be recalibrated by using boundary conditions. Weights w(i) can remain the same.

To clarify, let us suppose that we have 2 related KRI which may take value from 1 onwards. Then the key question is how should be consider a situation when the first of the KRI reaches its critical value and the second remains at its optimal level.

We will have the first equation:

N1 = α’[3^w(1)1^w(2)]^β’  = α’[3^w(1)]^β’                                             (9)

where N1 means the first number .

To obtain the second equation we will make the opposite hypothesis:

N2= α’[3^w(1)1^w(2)]^β’  = α’[3^w(2)]^β’= α’[3^1-w(1)]^β’                       (10)

where N2 means the second number.

To obtain another equation (if needed) we can consider a combination of one parameter reaching the threshold value and the second remaining at its optimal level (equal to 1).

 N3 = α’[2^w(1)1^w(2)]^β’  = α’[2^w(1)]^β’                                            (11)

where N3 represents the third number.

Having 3 equations and 3 unknown parameter values we can solve the equation system. If we add another condition we may face an optimization problem as probably some of the conditions will be only partially met.

With 3 individual KRIs we will need 4 equations, with n KRIs we will need (n+1) equations obtained in a similar way.

A similar approach can be developed for the case of unrelated KRI.

Conclusion

Amending the framework previously developed by Taylor, we can obtain composite indicators based on T-value sensible to the variation of KRI even in the green zone.

Robert M. Korona, PhD

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[1] Charles Taylor (2006)  Composite Indicators: Reporting KRIs to Senior Management, The RMA Journal , April 2006 , p 16-20

[2] In general in this situation one should consider the necessity to take preventive action before KRI reaches an unacceptable value (reaches the “red zone”).

[3] There is no need to define 2 critical and threshold values. If the KRI is monotonically related to risk one critical and one threshold value are quite enough.