## Aggregation: Within and between levels

Aggregation has to be “in synch” with the structure of the concept and its levels. In line with the basic conceptual framework and the structure of components we derive methods of aggregation, between and within levels.

StIx differentiates between three levels of aggregation: Within dimensions (first level indicators; second level indicators), between dimensions (total composite score) and between formal and informal institutions (first and second level indicators).

 Aggregation of Levels Method of Aggregation First Level Indicators – Within Dimensions Weighted multiplicative function, root Composite Index aggregation – Total score Multiplicative with equal weights, root Second Level Indicators – Informality Arithmetic mean with equal weights First and Second level Indicators – Formal and Context Multiplicative function without weights, Malus

The total composite Index score is calculated across the three monopolies by multiplicative aggregation with equal weights to reflect the logic of conceptualization.

The dimensional indices (within monopoly of law, monopoly of violence, monopoly of administration) are the multiplicative products of the indicators resp. mean values of remuneration und recruitment criteria within the public services and armed forces, each raised to their assigned weights.

The composite value of informality for each dimension is based on the logic of substitutability, i.e., the indicators are “of the same thing”: Informality. Assuming that informal institutions are partial substitutes, we average them with the mean value.

The third level of aggregation allocates formal and informal institutions for each monopoly: We focus on informal institutions that undermine and distort the formal institutions of the state. To calculate the impact of informality on the formal state we use a multiplication procedure without weights (malus system).

## Typology: Regular and diminished subtypes

Categories of quality of stateness reduce the complexity of dimensionality by grouping similar cases together in accordance with their performance to reflect the gradual nature of stateness.

State fragility describes a situation in which the state is not capable to uphold and enforce its monopolies. Fragility as a continuous phenomenon encompasses states with small defects (defective states), severe defects (highly defective states) and situations of institutional collapse (collapsed states), where (formal) stateness is de facto absent.

The first threshold at 0.7 separates the “full” presence of components and attributes in line with the root concept of a functioning state from empirical patterns that, although not completely corresponding to the concept of stateness, still fulfil criteria with some minor deficits, i.e., moderate functioning states.

The second threshold is located between moderate functioning and defective states at 0.5. The third threshold at 0.3 is located within diminished subtypes to further differentiate between defective and profound defective states that differ regarding the severity of defects. The fourth threshold specifies at which point attributes and components are no longer sufficiently given, i.e., when the minimal standards are no longer existent (collapsed state; 0.1).

 Typology Thresholds and Rules of Classification High functioning State 1 – 0.7, all dimensional scores above 0.7 Moderate functioning State < 0.7 – 0.5, one or more dimensional scores below 0.7 and all above 0.5 Defective State < 0.5 – 0.3, one or more dimensional scores below 0.5 and all above 0.3 Profound defective State < 0.3 – 0.1, one or more dimensional scores below 0.3 and all above 0.1 Collapsed State < 0.1, one or more dimensional scores below 0.1

To classify differing types of stateness in accordance to varying degrees of stateness we use a weakest link approach: the classification is determined by the lowest value among dimensional values (i.e., monopolies of law, violence, and administration). This approach prevents compensation and accommodates for the logical structure of necessary components of each dimension of a state.

The Typology of StIx is primarily designed to display discrepancies between groups.