An example from the study of the repression of austerity protests is used in the following test to illustrate the use of "Truth Tables". Table 1 presents hypothetical data on sixteen countries that experienced austerity protests in the early 1980s. Eight of these countries had governments that because violently repressive in response to austerity protests; the governments of the other eight did not. The table shows differences and similarities among these sixteen countries with respect to conditions believed to be relevant to repression derived from an analytic frame for government repression. The conditions include:

First, whether the country was politically aligned with the Soviet Union or with the United States and Western Europe in the 1980s. (x1)

Second, whether or not the country had undergone substantial industrialization prior to 1980 (x2).

Third, whether or not the country had a democratic government prior to the emergence of austerity protests (x3)

Fourth, whether or not the country had a strong military establishment prior to the emergence of austerity protests (x4)

The goal of comparative analysis is to determine the combinations of causal conditions that differentiate sets of cases. In this analysis, the goal is to find combinations of casual conditions that distinguish the eight countries with governments that became repressive (Y=1) from the other eight countries (Y=0). Careful examination of the similarities among the countries with violently repressive governments shows that they do not share any single causal condition or any single combination of conditions. However, there are two combinations of conditions that are present in the set of countries that had repressive governments that are both absent from the set that did not. The sixteen cases are sorted in table 1 to highlight these two combinations.

The first four cases share an absence of democratic government prior to the emergence of austerity protests combined with a strong military establishment. None of the cases in the lower half of the table (the eight countries lacking violent repression has this combination). The second four countries with violent repression share two different conditions: a presence of democratic government prior to austerity protests combined with an absence of significant industrialization prior to the protests. Again, none of the eight countries lacking violent repression has this combination of conditions

Case	Aligned with USSR (x1)	Industrialized (x2)	Democratic Government (x3)	Strong Military (x4)	Violent Repression (Y )**
1	0	0	0	1	1
2	0	1	0	1	1
3	1	0	0	1	1
4	1	1	0	1	1
5	0	0	1	0	1
6	0	0	1	1	1
7	1	0	1	0	1
8	1	0	1	1	1
9	0	0	0	0	0
10	0	1	0	0	0
11	0	1	1	0	0
12	0	1	1	1	0
13	1	0	0	0	0
14	1	1	0	0	0
15	1	1	1	0	0
16	1	1	1	1	0

___________________________________________________________________________________

The results of the examination of similarities and differences thus leads to the conclusion that there are two different combinations of conditions (or causal configurations) that explain the emergence of violent repression in these cases. The first configuration (non-democratic rule combined with a strong military) suggests a situation where the military establishment has gained the upper hand in part because of the absence of checks (democratic government) on its power. The second configuration (absence of significant industrialization combined with the presence of a democratic government prior to the emergence of violent repression suggests a situation where a breakdown of democratic rule occurred in countries that lacked many of the social structures associated with industrialization (for example, urbanization, literacy, and so on). These social structures are believed to facilitate stable democratic rule. Further research might show important differences between these two sets of cases with respect to the kind of repression that was inflicted on the protesters.

The cases are arranged in Table 1 so that the main patterns of similarity among the countries with violent repression are easy to detect, and the comparison of these cases with countries lacking violent repression is simplified. The findings in Table 1 are easy to see. Usually, however, the patterns are not so simple, and researchers must use more systematic comparative methods to help them analyze similarities and differences. These techniques, explained in the next sections, make it possible for researchers to find patterns that they would probably miss if they tried to unravel differences simply by "eyeballing" their cases.

In the comparative approach each case is understood as a combination of causal conditions linked to a particular outcome. Thus, the selection of the outcome to be studied and the specification of causal conditions relevant to that outcome are crucially important parts of a comparative investigation. Generally, in order to specify causes, the investigator must be familiar with the research literature on the outcome (for example, "government repression") and with the cases included in the study. In this early phase of the research, the investigator explores connections between social scientific thinking (for example, about government repression) and the evidence. These early explorations lead to a clarification of the nature of the outcome to be studied and a specification of the relevant causes.

The comparative methods described in this chapter use what social scientists call presence-absence dichotomies. This means that causal conditions and outcomes are either present or absent in each case and can be coded "yes" or "no," as in Table 1. Thus, instead of using a precise measurement of industrialization (for example, the percentage of the work force employed in manufacturing) in the data analysis, an assessment might be made of whether or not substantial industrialization occurred before a specific year (again, as in Table 1). The use of presence-absence dichotomies simplifies the representation of cases as configurations of causes. Research methods that focus explicitly on conditions that vary by degree or level are variable-oriented.

In comparative analysis the number of causal conditions determines the number of combinations of causal conditions that are possible. For example, the specification of four causal conditions (as in Table 1) provides for 16 logically possible combinations of causal conditions. Specification of five causal conditions provides for 32 combinations, six causal conditions provides for 64 combinations, and so on.

Causal conditions are not examined separately, as in studies focusing on covariation but in combinations. Once causal conditions have been selected, cases conforming to each combination of causal conditions are examined to see if they agree on the outcome. In Table 1, there is only one case for each combination of causal conditions, so there is no possibility of disagreement. But what if there were two cases in the first row (that is, two countries that combined absence of alignment with the Soviet Union, absence of substantial industrialization before 1980, absence of democratic government, and presence of a strong military), but in one country protesters suffered violent repression while in the other they did not? The researcher would have to determine what additional factor (present in one country but absent in the other) caused repression. This new causal condition would then be added to the table for all cases.

If there are many causal combinations with cases thatdisagree on the outcom, investigator should take this as a sign that the specification of causal conditions is incorrect or incomplete. The close examination of cases that have the same presence-absence values on all the causal conditions yet have different outcomes is used as a basis for selecting additional causal variables. Investigators move back and forth between specification of causal conditions (using social science theory and their general substantive knowledge as guides) and examination of evidence to resolve these differences.

Once a satisfactory set of causal conditions for a particular outcome has been identified, evidence on cases can be represented in truth tables. The use of truth tables facilitates the analysis of patterns of similarities and differences.

The first step in constructing a truth table is simply to list the evidence on the cases in the form of a data table. Consider for example, the data presented in Table 2. This table shows hypothetical evidence on thirty suburban school districts surrounding a major metropolitan area. The outcome of interest here is whether or not the elementary schools in each district track students according to ability (Y). When students are tracked, they are grouped together into relatively homogeneous classes (Y=1). Students who learn things quickly are assigned to one class, while students who learn things at an average speed are assigned to another, and so on.

Having students of uniform ability together in the same room is thought to simplify teaching, making it more efficient. After all, it clearly would be a mistake to put first graders and sixth graders in the same room. Why not apply this same principle to students within grade levels? The usual objection is that students who are assigned to the "slow" group become branded low achievers and are rarely given the opportunity to prove otherwise. Plus, being surrounded by "faster" students can motivate a "slow" student to learn faster. Assigning students to the slow group may seal their academic fate.

School District (N)	Racial Diversity (X1)	Class Diversity (X2)	Competitive Elections (X3)	Unionized Teachers (X4)	Ability Tracking (Y)
1	0	0	0	0	0
2	0	0	0	0	0
3	0	0	0	0	0
4	0	0	0	1	1
5	0	0	0	1	1
6	0	0	1	0	0
7	0	0	1	1	1
8	0	1	0	0	0
9	0	1	0	0	0
10	0	1	0	0	0
11	0	1	0	0	0
12	0	1	0	1	1
13	0	1	1	0	0
14	0	1	1	1	1
15	1	0	0	0	1
16	1	0	0	0	1
17	1	0	0	1	1
18	1	0	0	1	1
19	1	0	0	1	1
20	1	0	0	1	1
21	1	0	1	0	0
22	1	0	1	0	0
23	1	0	1	0	0
24	1	0	1	1	0
25	1	1	0	0	1
26	1	1	0	1	1
27	1	1	0	1	1
28	1	1	1	0	0
29	1	1	1	1	0
30	1	1	1	1	0

* In the columns with causal (x) or outcome (y) conditions, the number 1 indicates the presence of a condition or "yes"; 0 indicates its absence or "no".

The researcher in this example wanted to understand why some school districts track elementary school students and others don't. The table lists the causal conditions that the researcher, on the basis of an examination of the relevant research literatures, thought might be important:

First, whether the school district is racially diverse or predominantly white (x1)

Second, whether or not the school district has a broad representation of income groups (poor, working class, middle class, and upper middle class( (x2)

Third, whether or not the school board elections in the district are open and competitive, with good voter turnout (x3)

Fourth, whether or not the teachers in the district are unionized (x4).

The first two factors (racial and class diversity) show the social composition of school districts. These factors are important because where there is more diversity, members of dominant groups (for example, whites in racially diverse districts) generally believe that tracking will benefit their children most. The competitiveness of school board elections is important because the majority of voters usually disapprove of tracking in elementary schools. They believe this practice benefits only a minority of students. In districts where school board elections are routine matters that attract little voter interest, however, the minority of families that benefit from tracking might have more influence. Unionization of teachers is included because the researcher believes that teacher unions prefer tracking because it simplifies teaching.

The school districts are sorted in Table 2 according to the four causal conditions, so that districts that are identical on these factors are next to each other. Inspection of the data shows that there are no districts that have the same combination of scores on the causal conditions but different outcomes. Districts 8-11, for example, all show the same pattern on the four causal conditions; they also are identical on the outcome none of these districts tracks students according to ability. If the cases were not consistent on the outcome, it would be necessary to examine them closely to determine which other causal factors should be added to the table.

Listing the data on the cases, as shown in Table 2, is a necessary preliminary to the construction of the truth table. The idea behind a truth table is simple. The focus is on causal combinations. Each logical combination of values on the causal conditions is represented as one row of the truth table. Thus, truth tables have as many rows as there are logically possible combinations of values on the causal conditions. If there are four dichotomous causal conditions, as in Table 2, the truth table will contain 16 rows. Each row of the truth table is assigned an outcome score (1 or 0, for presence-absence of the outcome) based on the cases in that row. The first three cases in Table 2, for example, have the same combination of scores on the causal conditions (absent on each of the four conditions) and the same outcome (absence of tracking). They are combined to form the first row of the truth table presented in Table 3. The number of districts that make up each row of the truth table is also reported in Table 3, so that the translation of Table 2 to Table 3 is clear.

The truth table (Table 3) summarizes the causal configurations that exist in a data table (Table 2). Listing configurations is however not the same as identifying patterns. Usually, comparative researchers want to examine configurations to see if they can be simplified. When investigators simplify configurations, they identify patterns.

A quick example of simplification: Look at rows 13 and 14 of the truth table reported in Table 3. Row 13 reports that school districts that combine the following four characteristics track students: (1) racial diversity, (2) class diversity, (3) an absence of competitive school board elections, and (4) an absence of teachers' unions. Row 14 reports that school districts that differed on only one of these four conditions - teachers' unions also tracked students. The comparison of these two rows shows that when the first two causal conditions are present (race and class diversity) and the third is absent (competitive school board elections), it does not matter whether teachers are unionized; tracking by ability still takes place.

An easy way to represent this simplification is to use uppercase letters to indicate presence of a condition and lowercase letters to indicate its absence. In this example, the word "RACE" indicates the presence of racial diversity; "race" is used to indicate its absence. "CLASS" is used to indicate the presence of class diversity, "class" to indicate its absence. "ELECTIONS" is used to indicate the presence of open, competitive school board elections, "elections" to indicate the absence of this condition. "UNIONS" indicates the presence of teachers' unions, "unions" the absence of this condition. Finally, "TRACKING" indicates the presence of tracking, and "tracking" its absence

In the columns with causal or outcome conditions, the number I indicates the presence a condition or "yes" 0 indicates its absence or "no". The number of districts is reported simply to remind the reader that each row of a truth table may represent more than one case.

Thus, row 13 can be represented as

TRACKING = RACE*CLASS*elections*unions

and row 14 as:

TRACKING = RACE*CLASS*elections*UNIONS

where multiplication (*) is used to indicate the combination of conditions. These two rows can be simplified through combination because they have the same outcome and differ on only one causal condition -- presence-absence of teachers' unions. This simplification strategy follows the logic of an experiment. Only one condition at a time is allowed to vary (the "experimental" condition). If varying this condition has no discernible impact on the outcome, it can be eliminated as a factor. Thus, the comparison of rows 13 and 14 results in a simpler combination:

TRACKING = RACE*CLASS*elections

This rule for combining rows of the truth table as a way of simplifying them can be stated formally: If two rows of a truth table differ on only one causal condition yet result in the same outcome, then the causal condition that distinguishes the two rows can be considered irrelevant and can be removed to create a simpler combination of casual conditions (a simpler term).

The process of combining rows to create simpler terms can be carried on until no more simplification is possible. Table 4 shows all the simplifications that are possible for the truth table in Table 3, using presence of ability tracking as the outcome of interest. In Table 4 the truth table rows from Table 3 with outcomes of "1" (presence of tracking) have been translated into the uppercase and lowercase names in the manner just described. Panel A of this table simply lists the eight kinds of districts that track students according to ability. Panel B shows the first round of simplification. Each of the terms from panel A can be combined with one or more other terms to create simpler terms. Whenever two terms with four conditions are combined, the new term has three conditions because one condition has been eliminated.

Panel C shows the second round of simplification. In this round, terms with three conditions (from panel B) are combined to form terms with two conditions. For example, the term labeled #17 in panel B (race-class-UNIONS) can be combined with the term #21 (race*CLASS*UNIONS) to form a two-condition term (race*UNIONS). All the terms from panel B combine with one or more terms from the same panel to produce the three two-condition terms listed in panel C.

The three terms in panel C can be represented in a single statement describing the conditions under which tracking in these suburban school districts occurs:

TRACKING = race*UNIONS + RACE*elections * elections*UNIONS

Tracking occurs when:

1. racial diversity is absent and teachers' unions are present
2. racial diversity is present and competitive school board elections are

absent, or

3. competitive school board elections are absent and teachers' unions are present.

Panel A. District That Track Student
Row	Causal Configurations
2 4 6 8 9 10 13 14	race*class*elections*UNIONS race*class*ELECTIONS*UNIONS race*CLASS*elections*UNIONS race*CLASS*ELECTIONS*UNIONS RACE*class*elections*unions RACE*class*elections *UNIONS RACE*CLASS*elections*unions RACE*CLASS*elections*UNIONS

Before accepting these tentative results, it is important to determine if further simplification is possible, as is often the case. Sometimes the process of combining rows to produce simpler terms (presented in Table 4) generates "surplus" terms. A surplus term is redundant with other terms and is not needed in the statement describing the combinations of conditions linked to an outcome. In short, some of the terms that are left after the process of combining rows may be superfluous. Recall that the goal of comparative analysis is to describe diversity in a simple way. If the results can be further simplified by eliminating surplus terms, as is the case here, it is important to do so. The idea of a surplus term is best understood by examining the methods used to detect them.

The best way to check if there are surplus terms is to construct a chart showing which of the original terms in panel A are covered by which simplified terms in panel C. A simplified term covers a truth table row if the row is a subset of the simplified term. For example, RACE*CLASS*elections*UNIONS (row 14 of the truth table) is a subset of the simplified term elections*UNIONS.

The chart showing the coverage of the simplified terms is presented in Table 5. The simplified term race-UNIONS covers the first four terms from panel A of Table 4, while the term RACE-elections covers the other four. The third simplified term (elections-UNIONS) does not cover any of the rows uniquely; it covers two that are covered by the first simplified term and two that are covered by the second. Thus, the third simplified term is surplus; it is redundant with the other terms. By eliminating the third simplified term the results of the analysis of

configurations can be reduced to

TRACKING = race*UNIONS + RACE*elections

This completes the procedure. The final statement says that tracking occurs (1) when racial diversity is absent and teachers' unions are present, or (2) when racial diversity is present and competitive school board elections are absent. The first term indicates that in school districts that are predominantly white, tracking is implemented if there are teachers' unions. This finding supports the researcher's belief that teachers' unions

prefer tracking and specifies the conditions under which their interests are realized in districts where there is an absence of racial diversity. It does not matter whether school board elections are open and competitive or whether the district contains a broad range of income groups. The second term indicates that in school districts where there is racial diversity, tracking occurs when school board elections are not competitive. They are routine matters that do not attract a lot of voter interest. In these districts, it does not matter whether teachers' unions are present or whether the districts contain a broad range of income groups. The second term suggests that if voters become involved in school board elections, tracking would be eliminated in racially diverse districts.

* * From paneC of Table 4

The analysis of school districts presented here shows the major steps in using comparative techniques to unravel causal patterns.

1. Select causal and outcome conditions, using existing social science literature and substantive knowledge to guide the selection.

2. Construct a sorted data table showing the scores of cases on these causal and outcome conditions (Table 2).

3. Construct a truth table from the data table, making sure that cases with the same causal conditions actually have the same score on the outcome (Table 3).

4. Compare rows of the truth table and simplify them, eliminating one condition at a time from pairs of rows (Table 4).

5. Examine the coverage of the simplified terms to see if there are any surplus terms that can be eliminated (Table 5).

The terms that remain after step 5 show the simplest way to represent the patterns of diversity in the data. In the comparative analysis presented in Tables 2 through 5, the goal is to explain why some school districts track elementary students. The results show which types of school districts track elementary students and distinguishes them from those that do not.

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