# Create a truth table

## Description

Function to create a truth table from all types of calibrated data (binary crisp, multi-value crisp and fuzzy). For fuzzy data, an improved verson of Ragin's (2008) procedure is applied to assign cases to the vector space corners (the truth table rows, combinations of causal conditions).

## Usage

    truthTable(data, outcome = "", conditions = "", incl.cut = 1, n.cut = 1, pri.cut = 0,
               exclude = NULL, complete = FALSE, use.letters = FALSE, show.cases = FALSE,
               dcc = FALSE, sort.by = "", inf.test = "", ...)

## Arguments

`data` |  |  |  | A data frame containing calibrated causal conditions and an outcome
`outcome` |  |  |  | String, the name of the outcome.
`conditions` |  |  |  | A single string containing the conditions' (columns) names separated by commas, or a character vector of conditions' names.
`incl.cut` |  |  |  | The inclusion cutoff(s): either a single value for the presence of the output, or a vector of length 2, the second for the absence of the output.
`n.cut` |  |  |  | The minimum number of cases under which a truth table row is declared as a remainder.
`pri.cut` |  |  |  | The minimal score for the `PRI` - proportional reduction in inconsistency, under which a truth table row is declared as negative.
`exclude` |  |  |  | A vector of row numbers from the truth table, to code as negative output configurations.
`complete` |  |  |  | Logical, print complete truth table.
`use.letters` |  |  |  | Logical, use letters instead of causal conditions' names.
`show.cases` |  |  |  | Logical, print case names.
`dcc` |  |  |  | Logical, if `show.cases = TRUE`, the cases being displayed are the deviant cases consistency in kind.
`sort.by` |  |  |  | Sort the truth table according to various columns.
`inf.test` |  |  |  | Specifies the statistical inference test to be performed (currently only `"binom"`) and the critical significance level. It can be either a vector of length 2, or a single string containing both, separated by a comma.
`...` |  |  |  | Other arguments (mainly for backward compatibility).

## Details

The `data` should always be provided as a data frame, with calibrated columns.

Calibration can be either crisp, with 2 or more values starting from 0, or fuzzy with continous scores from 0 to 1. Raw data containing relative frequencies can also be continous between 0 and 1, but these are not calibrated, fuzzy data.

Some columns can contain the placeholder `"-"` indicating a “don't care”, which is used to indicate the temporal order between other columns in tQCA. These special columns are not causal conditions, hence no parameters of fit will be calculated for them.

The argument `outcome` specifies the column name to be explained. If the outcome is a multivalue column, it can be specified in curly bracket notation, indicating the value to be explained (the others being automatically converted to zero).

The outcome can be negated using a tilde operator `~X`. The logical argument `neg.out` is now deprecated, but still backwards compatible. Replaced by the tilde in front of the outcome name, it controls whether `outcome` is to be explained or its negation. Note that using both `neg.out = TRUE` and a tilde `~` in the outcome name cancel each other out.

If the outcome column is multi-value, the argument `outcome` should use the standard curly-bracket notation `X{value}`. Multiple values are allowed, separated by a comma (for example `X{1,2}`). Negation of the outcome can also be performed using the tilde `~` operator, for example `~X{1,2}`, which is interpreted as: "all values in X except 1 and 2" and it becomes the new outcome to be explained.

The argument `conditions` specifies the causal conditions' names among the other columns in the data. When this argument is not specified, all other columns except for the outcome are taken as causal conditions.

A good practice advice is to specify both `outcome` and `conditions` as upper case letters. It is possible, in a next version, to negate outcomes using lower case letters, situation in which it really does matter how the outcome and/or conditions are specified.

The argument `incl.cut` replaces the (deprecated, but still backwards compatible) former arguments `incl.cut1` and `incl.cut0`. Most of the analyses use the inclusion cutoff for the presence of the output (code `"1"`). When users need both inclusion cutoffs (see below), `incl.cut` can be specified as a vector of length 2, in the form: `c(ic1, ic0)` where:

`ic1` | is the inclusion cutoff for the presence of the output,
| a minimum sufficiency inclusion score above which the output value is coded with `"1"`.
`ic0` | is the inclusion cutoff for the absence of the output,

If not specifically declared, the argument `ic0` is automatically set equal to `ic1`, but otherwise `ic0` should always be lower than `ic1`.

Using these two cutoffs, the observed combinations are coded with:

`"1"` | if they have an inclusion score above `ic1`
`"C"` | if they have an inclusion score below `ic1` and above `ic0` (contradiction)

The argument `n.cut` specifies the frequency threshold under which a truth table row is coded as a remainder, irrespective of its inclusion score.

When argument `show.cases` is set to `TRUE`, the case names will be printed at their corresponding row in the truth table. The resulting object always contains the cases for each causal combination, even if not printed on the screen (the print function can later be used to print them).

The `sort.by` argument orders all configurations by any of the columns present in the truth table. Typically, sorting occurs by their outcome value, and/or by their inclusion score, and/or by their frequency, in any order.

Sorting decreasingly (the default) or increasingly can be specified adding the signs `-` or `+` next after the column name in argument `sort.by` (see examples). Note that `-` is redundant because it is the default anyways.

The order specified in this vector is the order in which the configurations will be sorted. When sorting based on the OUTput column, remainders will always be sorted last.

The argument `use.letters` controls using the original names of the causal conditions, or replace them by single letters in alphabetical order. If the causal conditions are already named with single letters, the original letters will be used.

The argument `inf.test` combines the inclusion score with a statistical inference test, in order to assign values in the output column OUT. For the moment, it is only the binomial test, which needs crisp data (it doesn't work with fuzzy sets). Following a similar logic as above, for a given (specified) critical significance level, the output for a truth table row will be coded as:

`"1"` | if the true inclusion score is significanly higher than `ic1`,
`"C"` | contradiction, if the true inclusion score is not significantly higher than `ic1`
| but significantly higher than `ic0`,

It should be noted that statistical tests perform well only when the number of cases is large, otherwise they are usually not significant. For a low number of cases, depending on the inclusion cutoff value(s), it will be harder to code a value of `"1"` in the output, and also harder to obtain contradictions if the true inclusion is not signficantly higher than `ic0`.

The argument `complete` controls how to print the table on the screen, either complete (when set to `TRUE`), or just the observed combinations (default). For up to 7 causal conditions, the resulting object will always contain the complete truth table, even if it's not printed on the screen. This is useful for multiple reasons: researchers like to manually change output values in the truth table (sometimes including in this way a remainder, for example), and it is also useful to plot Venn diagrams, each truth table row having a correspondent intersection in the diagram.

## Value

An object of class “tt”, a list containing the following components:

`tt` |  |  |  | The truth table itself.
`indexes` |  |  |  | The line numbers for the observed causal configurations.
`noflevels` |  |  |  | A vector with the number of values for each causal condition.
`initial.data` |  |  |  | The initial data.
`recoded.data` |  |  |  | The crisp version of the `initial.data`, if fuzzy.
`cases` |  |  |  | The cases for each observed causal configuration.
`options` |  |  |  | The command options used.
`rowsorder` |  |  |  | The order of the rows after sorting, if using `sort.by`.
`minmat` |  |  |  | A membership scores matrix of cases in the observed truth table combinations.

## References

Cronqvist, L.; Berg-Schlosser, D. (2009) “Multi-Value QCA (mvQCA)”, in Rihoux, B.; Ragin, C. (eds.) Configurational Comparative Methods. Qualitative Comparative Analysis (QCA) and Related Techniques, SAGE.

Dusa, A. (2019) QCA with R. A Comprehensive Resource. Springer International Publishing.

Lipset, S.M. (1959) “Some Social Requisites of Democracy: Economic Development and Political Legitimacy”, American Political Science Review vol.53, pp.69-105.

Ragin, C.C. (1987) The Comparative Method: Moving beyond Qualitative and Quantitative Strategies. Berkeley: University of California Press.

Ragin, C.C. (2008) Redesigning Social Inquiry: Fuzzy Sets and Beyond. Chicago: University of Chicago Press.

Ragin, C.C.; Strand, S.I. (2008) “Using Qualitative Comparative Analysis to Study Causal Order: Comment on Caren and Panofsky (2005).” Sociological Methods & Research vol.36, no.4, pp.431-441.

Schneider, C.Q.; Wagemann, C. (2012) Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis (QCA). Cambridge: Cambridge University Press.

## Examples

    # -----
    # Lipset binary crisp data
    ttLC <- truthTable(LC, "SURV")

    # inspect the truth table
    ttLC

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
     1   0   0   0   0   0   0  3  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     5   0   0   1   0   0   0  2  0.000 0.000
     6   0   0   1   0   1   0  1  0.000 0.000
    22   1   0   1   0   1   1  2  1.000 1.000
    23   1   0   1   1   0   0  1  0.000 0.000
    24   1   0   1   1   1   1  2  1.000 1.000
    31   1   1   1   1   0   0  1  0.000 0.000
    32   1   1   1   1   1   1  4  1.000 1.000

    # print the cases too, even if not specifically asked for
    print(ttLC, show.cases = TRUE)

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI   cases
     1   0   0   0   0   0   0  3  0.000 0.000 GR,PT,ES
     2   0   0   0   0   1   0  2  0.000 0.000 IT,RO
     5   0   0   1   0   0   0  2  0.000 0.000 HU,PL
     6   0   0   1   0   1   0  1  0.000 0.000 EE
    22   1   0   1   0   1   1  2  1.000 1.000 FI,IE
    23   1   0   1   1   0   0  1  0.000 0.000 AU
    24   1   0   1   1   1   1  2  1.000 1.000 FR,SE
    31   1   1   1   1   0   0  1  0.000 0.000 DE
    32   1   1   1   1   1   1  4  1.000 1.000 BE,CZ,NL,UK

    # the printing function also supports the complete version
    print(ttLC, show.cases = TRUE, complete = TRUE)

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI   cases
     1   0   0   0   0   0   0  3  0.000 0.000 GR,PT,ES
     2   0   0   0   0   1   0  2  0.000 0.000 IT,RO
     3   0   0   0   1   0   ?  0    -     -
     4   0   0   0   1   1   ?  0    -     -
     5   0   0   1   0   0   0  2  0.000 0.000 HU,PL
     6   0   0   1   0   1   0  1  0.000 0.000 EE
     7   0   0   1   1   0   ?  0    -     -
     8   0   0   1   1   1   ?  0    -     -
     9   0   1   0   0   0   ?  0    -     -
    10   0   1   0   0   1   ?  0    -     -
    11   0   1   0   1   0   ?  0    -     -
    12   0   1   0   1   1   ?  0    -     -
    13   0   1   1   0   0   ?  0    -     -
    14   0   1   1   0   1   ?  0    -     -
    15   0   1   1   1   0   ?  0    -     -
    16   0   1   1   1   1   ?  0    -     -
    17   1   0   0   0   0   ?  0    -     -
    18   1   0   0   0   1   ?  0    -     -
    19   1   0   0   1   0   ?  0    -     -
    20   1   0   0   1   1   ?  0    -     -
    21   1   0   1   0   0   ?  0    -     -
    22   1   0   1   0   1   1  2  1.000 1.000 FI,IE
    23   1   0   1   1   0   0  1  0.000 0.000 AU
    24   1   0   1   1   1   1  2  1.000 1.000 FR,SE
    25   1   1   0   0   0   ?  0    -     -
    26   1   1   0   0   1   ?  0    -     -
    27   1   1   0   1   0   ?  0    -     -
    28   1   1   0   1   1   ?  0    -     -
    29   1   1   1   0   0   ?  0    -     -
    30   1   1   1   0   1   ?  0    -     -
    31   1   1   1   1   0   0  1  0.000 0.000 DE
    32   1   1   1   1   1   1  4  1.000 1.000 BE,CZ,NL,UK

    # formally asking the complete version
    truthTable(LC, "SURV", complete = TRUE)

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
     1   0   0   0   0   0   0  3  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     3   0   0   0   1   0   ?  0    -     -
     4   0   0   0   1   1   ?  0    -     -
     5   0   0   1   0   0   0  2  0.000 0.000
     6   0   0   1   0   1   0  1  0.000 0.000
     7   0   0   1   1   0   ?  0    -     -
     8   0   0   1   1   1   ?  0    -     -
     9   0   1   0   0   0   ?  0    -     -
    10   0   1   0   0   1   ?  0    -     -
    11   0   1   0   1   0   ?  0    -     -
    12   0   1   0   1   1   ?  0    -     -
    13   0   1   1   0   0   ?  0    -     -
    14   0   1   1   0   1   ?  0    -     -
    15   0   1   1   1   0   ?  0    -     -
    16   0   1   1   1   1   ?  0    -     -
    17   1   0   0   0   0   ?  0    -     -
    18   1   0   0   0   1   ?  0    -     -
    19   1   0   0   1   0   ?  0    -     -
    20   1   0   0   1   1   ?  0    -     -
    21   1   0   1   0   0   ?  0    -     -
    22   1   0   1   0   1   1  2  1.000 1.000
    23   1   0   1   1   0   0  1  0.000 0.000
    24   1   0   1   1   1   1  2  1.000 1.000
    25   1   1   0   0   0   ?  0    -     -
    26   1   1   0   0   1   ?  0    -     -
    27   1   1   0   1   0   ?  0    -     -
    28   1   1   0   1   1   ?  0    -     -
    29   1   1   1   0   0   ?  0    -     -
    30   1   1   1   0   1   ?  0    -     -
    31   1   1   1   1   0   0  1  0.000 0.000
    32   1   1   1   1   1   1  4  1.000 1.000

    # sorting by multiple columns, decreasing by default
    truthTable(LC, "SURV", complete = TRUE, sort.by = "incl, n")

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
    32   1   1   1   1   1   1  4  1.000 1.000
    22   1   0   1   0   1   1  2  1.000 1.000
    24   1   0   1   1   1   1  2  1.000 1.000
     1   0   0   0   0   0   0  3  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     5   0   0   1   0   0   0  2  0.000 0.000
     6   0   0   1   0   1   0  1  0.000 0.000
    23   1   0   1   1   0   0  1  0.000 0.000
    31   1   1   1   1   0   0  1  0.000 0.000
     3   0   0   0   1   0   ?  0    -     -
     4   0   0   0   1   1   ?  0    -     -
     7   0   0   1   1   0   ?  0    -     -
     8   0   0   1   1   1   ?  0    -     -
     9   0   1   0   0   0   ?  0    -     -
    10   0   1   0   0   1   ?  0    -     -
    11   0   1   0   1   0   ?  0    -     -
    12   0   1   0   1   1   ?  0    -     -
    13   0   1   1   0   0   ?  0    -     -
    14   0   1   1   0   1   ?  0    -     -
    15   0   1   1   1   0   ?  0    -     -
    16   0   1   1   1   1   ?  0    -     -
    17   1   0   0   0   0   ?  0    -     -
    18   1   0   0   0   1   ?  0    -     -
    19   1   0   0   1   0   ?  0    -     -
    20   1   0   0   1   1   ?  0    -     -
    21   1   0   1   0   0   ?  0    -     -
    25   1   1   0   0   0   ?  0    -     -
    26   1   1   0   0   1   ?  0    -     -
    27   1   1   0   1   0   ?  0    -     -
    28   1   1   0   1   1   ?  0    -     -
    29   1   1   1   0   0   ?  0    -     -
    30   1   1   1   0   1   ?  0    -     -

    # sort the truth table decreasing for inclusion, and increasing for n
    # note that "-" is redundant, sorting is decreasing by default
    truthTable(LC, "SURV", complete = TRUE, sort.by = "incl-, n+")

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
    22   1   0   1   0   1   1  2  1.000 1.000
    24   1   0   1   1   1   1  2  1.000 1.000
    32   1   1   1   1   1   1  4  1.000 1.000
     6   0   0   1   0   1   0  1  0.000 0.000
    23   1   0   1   1   0   0  1  0.000 0.000
    31   1   1   1   1   0   0  1  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     5   0   0   1   0   0   0  2  0.000 0.000
     1   0   0   0   0   0   0  3  0.000 0.000
     3   0   0   0   1   0   ?  0    -     -
     4   0   0   0   1   1   ?  0    -     -
     7   0   0   1   1   0   ?  0    -     -
     8   0   0   1   1   1   ?  0    -     -
     9   0   1   0   0   0   ?  0    -     -
    10   0   1   0   0   1   ?  0    -     -
    11   0   1   0   1   0   ?  0    -     -
    12   0   1   0   1   1   ?  0    -     -
    13   0   1   1   0   0   ?  0    -     -
    14   0   1   1   0   1   ?  0    -     -
    15   0   1   1   1   0   ?  0    -     -
    16   0   1   1   1   1   ?  0    -     -
    17   1   0   0   0   0   ?  0    -     -
    18   1   0   0   0   1   ?  0    -     -
    19   1   0   0   1   0   ?  0    -     -
    20   1   0   0   1   1   ?  0    -     -
    21   1   0   1   0   0   ?  0    -     -
    25   1   1   0   0   0   ?  0    -     -
    26   1   1   0   0   1   ?  0    -     -
    27   1   1   0   1   0   ?  0    -     -
    28   1   1   0   1   1   ?  0    -     -
    29   1   1   1   0   0   ?  0    -     -
    30   1   1   1   0   1   ?  0    -     -

    # -----
    # Lipset multi-value crisp data (Cronqvist & Berg-Schlosser 2009, p.80)
    truthTable(LM, "SURV", sort.by = "incl")

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
    22   1   0   1   0   1   1  2  1.000 1.000
    32   1   1   1   1   1   1  1  1.000 1.000
    40   2   0   1   1   1   1  2  1.000 1.000
    48   2   1   1   1   1   1  3  1.000 1.000
     1   0   0   0   0   0   0  3  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     5   0   0   1   0   0   0  2  0.000 0.000
     6   0   0   1   0   1   0  1  0.000 0.000
    23   1   0   1   1   0   0  1  0.000 0.000
    31   1   1   1   1   0   0  1  0.000 0.000

    # using a frequency cutoff equal to 2 cases
    ttLM <- truthTable(LM, "SURV", n.cut = 2, sort.by = "incl")
    ttLM

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEV URB LIT IND STB OUT n  incl  PRI
    22   1   0   1   0   1   1  2  1.000 1.000
    40   2   0   1   1   1   1  2  1.000 1.000
    48   2   1   1   1   1   1  3  1.000 1.000
     1   0   0   0   0   0   0  3  0.000 0.000
     2   0   0   0   0   1   0  2  0.000 0.000
     5   0   0   1   0   0   0  2  0.000 0.000

    # the observed combinations coded as remainders
    ttLM$removed

        DEV URB LIT IND STB OUT n  incl  PRI   cases
     6   0   0   1   0   1   ?  1  0.000 0.000  EE
    23   1   0   1   1   0   ?  1  0.000 0.000  AU
    31   1   1   1   1   0   ?  1  0.000 0.000  DE
    32   1   1   1   1   1   ?  1  1.000 1.000  CZ

    # -----
    # Cebotari & Vink fuzzy data
    ttCVF <- truthTable(CVF, "PROTEST", incl.cut = 0.8, sort.by = "incl")

    # view the Venn diagram for this truth table
    library(venn)
    venn(ttCVF)

    # each intersection transparent by its inclusion score
    venn(ttCVF, transparency = ttCVF$tt$incl)

    # the truth table negating the outcome
    truthTable(CVF, "~PROTEST", incl.cut = 0.8, sort.by = "incl")

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEMOC ETHFRACT GEOCON POLDIS NATPRIDE OUT n  incl  PRI
    10    0      1       0      0       1      1  1  0.976 0.914
     6    0      0       1      0       1      1  1  0.953 0.852
    28    1      1       0      1       1      1  1  0.933 0.775
    14    0      1       1      0       1      1  3  0.929 0.816
     2    0      0       0      0       1      1  4  0.922 0.818
    20    1      0       0      1       1      1  1  0.902 0.714
     8    0      0       1      1       1      1  1  0.902 0.705
    22    1      0       1      0       1      1  4  0.846 0.597
    30    1      1       1      0       1      1  2  0.843 0.333
     5    0      0       1      0       0      1  2  0.823 0.416
    27    1      1       0      1       0      0  1  0.776 0.217
    16    0      1       1      1       1      0  1  0.742 0.409
    29    1      1       1      0       0      0  1  0.738 0.142
    32    1      1       1      1       1      0  1  0.726 0.283
    24    1      0       1      1       1      0  2  0.699 0.229
    31    1      1       1      1       0      0  1  0.662 0.109
    15    0      1       1      1       0      0  2  0.629 0.161

    # allow contradictions
    truthTable(CVF, "PROTEST", incl.cut = c(0.8, 0.75), sort.by = "incl")

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        DEMOC ETHFRACT GEOCON POLDIS NATPRIDE OUT n  incl  PRI
    31    1      1       1      1       0      1  1  0.959 0.891
    29    1      1       1      0       0      1  1  0.957 0.858
    27    1      1       0      1       0      1  1  0.938 0.783
    15    0      1       1      1       0      1  2  0.929 0.839
    30    1      1       1      0       1      1  2  0.922 0.667
    24    1      0       1      1       1      1  2  0.904 0.754
    32    1      1       1      1       1      1  1  0.892 0.717
     5    0      0       1      0       0      1  2  0.874 0.584
    16    0      1       1      1       1      1  1  0.821 0.591
    28    1      1       0      1       1      C  1  0.768 0.225
     8    0      0       1      1       1      0  1  0.749 0.248
    10    0      1       0      0       1      0  1  0.741 0.086
     6    0      0       1      0       1      0  1  0.732 0.148
    22    1      0       1      0       1      0  4  0.725 0.281
    20    1      0       0      1       1      0  1  0.711 0.162
    14    0      1       1      0       1      0  3  0.653 0.098
     2    0      0       0      0       1      0  4  0.648 0.182

    # -----
    # Ragin and Strand data with temporal QCA
    # truth table containing the "-" placeholder as a "don't care"
    truthTable(RS, "REC")

      OUT: outcome value
        n: number of cases in configuration
     incl: sufficiency inclusion score

        P  E  A  S  EBA OUT n  incl  PRI
     3  0  0  0  0   -   0  3  0.000 0.000
    15  0  1  0  0   -   0  1  0.000 0.000
    22  0  1  1  1   0   1  1  1.000 1.000
    27  1  0  0  0   -   0  1  0.000 0.000
    30  1  0  0  1   -   0  3  0.000 0.000
    36  1  0  1  1   -   0  2  0.000 0.000
    42  1  1  0  1   -   1  1  1.000 1.000
    44  1  1  1  0   1   1  2  1.000 1.000
    46  1  1  1  1   0   1  1  1.000 1.000
    47  1  1  1  1   1   1  2  1.000 1.000

## See also

`minimize`

## Author

Adrian Dusa