Necessary Conditions
A necessary condition asserts that some outcome Y cannot happen without the occurrence of some causal or prior variable X. The easiest way to think about what this means is to look at a simple 2×2 table. Imagine an X variable and a Y variable, each of which is dichotomous (i.e., has only yes/no, on/off, 1/0 values). The values in the cells show the number of cases in which the X variable was or was not present, and the corresponding value of Y.[1] We hypothesize that X is necessary for Y. If X is a necessary condition for Y, then we should not observe any cases of success without also observing X. Table 3.1 shows a necessary condition.
Table 3.1 Data Showing A Necessary Condition
| X variable à ↓ Y variable | 0 | 1 |
| 0 | 17 | 6 |
| 1 | 0 | 12 |
Let’s think about what the data should look like if our hypothesis is correct. If X is necessary for Y, and we observe Y=1, then we should also observe X=1. Conversely, if we observe X=0, then we should not observe any cases of Y=1 – the cell for (X=0, Y=1) has to be empty if X really is necessary for Y. Looking at the table, we see that for the data shown here, X is indeed necessary for Y. If we look at the bottom row of the table, where Y=1, we observe no cases of Y=1 where X=0. We only observe cases of Y=1 where X=1.[2] The key thing about a necessary hypothesis is that it’s interested in predicting cases of Y=1 – it’s a story about the DV.
One of the best-known examples of necessary conditions in the political science literature is Theda Skocpol’s (1979) work, States and Social Revolutions. She argues that state breakdown and peasant revolt are both necessary conditions for the occurrence of a social revolution, that is, a revolution which changes the entire order of society, not just the government structure. Skocpol tests her argument with process-tracing analyses of three major social revolutions (French, Chinese, and Russian), sprinkling it liberally with other brief cases to present counterfactual analyses and cases of alternate variable values. While later scholars have cast some doubt on Skocpol’s work, at least in part because of her case selection, States and Social Revolutions remains one of the few major works to tackle hypotheses of necessity. As we’ve discussed before, the social world is a messy place, with many factors contributing to any specific outcome. Most social phenomena exhibit equifinality, meaning that more than one route to a particular outcome exists. Arguments of necessity are thus difficult to support, but nonetheless, they do exist and scholars do test them.
Sufficient Conditions
A sufficient condition, shown in Table 3.2, asserts that some cause X always leads to the occurrence of some outcome Y. Again, let’s picture a 2×2 table of dichotomous variables, with outcome counts as the cell entries. We hypothesize that X is a sufficient condition for Y.
Table 3.2. Data Showing a Sufficient Condition
| X variable à ↓ Y variable | 0 | 1 |
| 0 | 6 | 0 |
| 1 | 3 | 4 |
If X is, indeed, a sufficient condition for Y, then we should expect to find no cases of X = 1, Y = 0 – that is, no cases should exist where X was present but Y did not occur. Table X shows this; the cell for X = 1, Y = 0 is empty.
A hypothesis of sufficiency is, like its necessity counterpart, a fairly rare thing. Unlike necessary conditions, though, sufficient conditions allow for multiple routes to the outcome of interest. Observing X will automatically produce Y, but the possibility of observing Y without X exists. A hypothesis of a necessary condition precludes alternate routes to observing Y: Y can only happen if X happens. To think about this distinction a different way, a sufficient condition is a story about an IV, specifically about its presence. We are not interested, in this case, in cases where X = 0. For a hypothesis of sufficiency, Y may occur even without X’s presence; the hypothesis says nothing about other ways that Y may or may not occur. The key thing of interest is that when X = 1, Y always occurs.
Necessary and Sufficient Conditions
Hypotheses of joint necessity and sufficiency are possible. In these hypotheses, X is both necessary and sufficient to produce Y. Where X occurs, Y will always occur, and no other way exists to obtain Y without X. In terms of our 2×2 table, a necessary and sufficient condition has no cases where X = 1, Y = 0, and no cases where X = 0, Y = 1. Events are observed only in the main diagonal of the table, where X = 0, Y = 0, and X = 1, Y = 1.[3] Table 3.3 shows data that supports a hypothesis of a necessary and sufficient condition.
Table 3.3 Data Showing a Necessary and Sufficient Condition
| X variable à ↓ Y variable | 0 | 1 |
| 0 | 11 | 0 |
| 1 | 0 | 9 |
For hypotheses of joint necessity and sufficiency, both the IV and DV typically must be dichotomous. When considering hypotheses of necessary or sufficient conditions, however, we can easily expand this discussion from dichotomous variables to ones of degree – high, medium, low, and no value of some characteristic, for example, or few, some, and many. In theory, we can expand it all the way to interval-ratio variables, where it turns into a threshold effect. In this, we identify through inductive or deductive means some threshold over which we consider X to be ‘present’ and any values below that, X is effectively absent. For example, the international policy community has a widely held belief, supported by quite a bit of research, that a national income of at least $5,000 per capita (in 1990s US$) is a necessary condition for democracy to be stable. This is not, however, a sufficient condition; a wide range of countries have GDP over US$5,000 but lack democracy, such as Saudi Arabia and Singapore.
Example of Necessary and Sufficient condition in anti-DEI
Finally, we can have a hypothesis that calls for both necessity and sufficiency. We should only observe the outcome under certain situations. An example would be, “We will only see unilateral executive action in states with a Republican governor with national political ambitions.” When X takes on any other value (D gov, R without national ambitions), we will not observe unilateral action; in short, “Republican governor with national political ambitions” is a sufficient condition. In this hypothesis, it is also necessary: I expect that whenever we do observe unilateral executive action (Y = 1), it will have been produced by a Republican governor with national ambitions
[1] In accordance with convention, I list the X variable on the horizontal axis. In a table, this means the columns, with the lowest value farthest left. The Y value is on the vertical axis, which means the rows, again with the lowest value at the ‘origin’ of the table. (More on table construction in Chapter 10 and Appendix A.)
[2] We don’t need to look at the top row of data for a ‘necessary’ hypothesis. The distribution of cases where Y=0 is entirely not of interest – we’re only concerned about cases where Y occurred (Y=1) (Braumoeller and Goertz 2000, 846).
[3] The main diagonal of a (square) table always runs from top left to bottom right, where X = Y. If you do not have a square table – if the number of rows is not equal to the number of columns – you cannot use the language of the main diagonal and the off diagonal (the diagonal going from top right to bottom left) to describe your table’s contents.