# Chapter 4 Data analysis

## 4.1 The absence of control variables

Under construction

## 4.2 The bottleneck table

After having evaluated *theoretical support*, the *effect size* and the *p value*, the researcher may conclude that empirical evidence exists for the hypothesis that ‘\(X\) is necessary for \(Y\).’ This hypothesis is formulated in a qualitative way and describes the necessary condition *in kind*. For additional insights the hypothesis can be formulated and evaluated in a quantitative way: the necessary condition *in degree*: *level* of \(X\) is necessary for *level* of \(Y\).
The bottleneck table is a helpful tool for evaluating necessary conditions in degree.

The bottleneck table is the tabular representation of the ceiling line. The first column is the outcome \(Y\) and the next columns are the conditions. The values in the table are levels of \(X\) and \(Y\). By reading the bottleneck table row by row from left to right it can be evaluated for which particular level of \(Y\) which particular levels of the conditions \(X\) are necessary.

The bottleneck table includes only conditions that are supposed to be a necessary. Conditions that are not supposed to be necessary (e.g. because the p values is too large) are usually excluded from the table.

### 4.2.1 Levels expressed as percentage of range

The default bottleneck table has 10 rows and the levels of \(X\) and \(Y\) are expressed as ‘percentage.range.’ In the first row the \(Y\) level is 0% and in the eleventh row it is 100%. This corresponds to the percentage of the *range* of \(Y\)(maximum level minus minimal level): 0% means the minimum level of \(Y\), 100% the maximum value of \(Y\) and 50% means middle level between these extremes. The same holds for the percentage.range levels of the \(X\)’s.

### 4.2.2 Levels expressed as percentages

under construction

### 4.2.3 Levels expressed as actual values

under construction

### 4.2.4 Levels expressed as percentiles

under construction

### 4.2.5 NN and NA the bottleneck table

A NN (Not Necessary) in the bottleneck table means that \(X\) is not necessary for \(Y\) for the particular level of \(Y\). With any value of \(X\) it is possible to achieve the particular level of \(Y\).

An NA (Not Applicable) in the bottleneck table is a warning that it is not possible to compute a value for the \(X\). There are two possible reasons for it, the first more often than the second:
1. The maximum possible value of the condition for the particular level of \(Y\) according to the ceiling line is lower than the actually observed maximum value. This can happen for example when the CR ceiling (which is a trend line) runs at \(X\) = \(X_{max}\) (which is the right vertical line of the scope in the scatter plot) under the line \(Y\) = \(Y_{max}\) (which is the upper horizontal line of the scope). If this happens the researcher can either explain why this NA appears in the bottleneck table, or can change NA into the highest observed level of \(X\). The latter can be done with the argument `cutoff`

= 1 in the `nca_analysis`

function.
2. In a bottleneck table with multiple conditions, one case determines the \(Y_{max}\) value and that case has a missing value for the condition with the NA (but not for another condition). When all cases are complete (have no missing values) or when at least one case exist that has a complete observation (\(X\), \(Y_{max}\)) for the given condition, the NA will not appear. The action to be taken is either to explain why this NA appears, or to delete the incomplete case from the bottleneck table analysis (and accept that the \(Y_{max}\) in the bottleneck table does not correspond to the actually observed \(Y_{max}\)).

## 4.3 Combining NCA with regression

under construction see: NCA website.

## 4.4 Combining NCA with QCA

under construction see: NCA website.