assoc_scores: Association scores used in collocation analysis and keyword...

In mclm: Mastering Corpus Linguistics Methods

Association scores used in collocation analysis and keyword analysis

Description

assoc_scores and assoc_abcd take as their arguments co-occurrence frequencies of a number of items and return a range of association scores used in collocation analysis, collostruction analysis and keyword analysis.

Usage

assoc_scores(
  x,
  y = NULL,
  min_freq = 3,
  measures = NULL,
  with_variants = FALSE,
  show_dots = FALSE,
  p_fisher_2 = FALSE,
  haldane = TRUE,
  small_pos = 1e-05
)

assoc_abcd(
  a,
  b,
  c,
  d,
  types = NULL,
  measures = NULL,
  with_variants = FALSE,
  show_dots = FALSE,
  p_fisher_2 = FALSE,
  haldane = TRUE,
  small_pos = 1e-05
)

Arguments

x	Either an object of class freqlist or an object of class cooc_info. If x is a freqlist, it is interpreted as the target frequency list (i.e. the list with the frequency of items in the target context) and y must be a freqlist with the frequency of items in the reference context. If x is an object of class cooc_info instead, it is interpreted as containing target frequency information, reference frequency information and corpus size information.
y	An object of class freqlist with the frequencies of the reference context if x is also a freqlist. If x is an object of class cooc_info, this argument is ignored.
min_freq	Minimum value for a[[i]] (or for the frequency of an item in the target frequency list) needed for its corresponding item to be included in the output.
measures	Character vector containing the association measures (or related quantities) for which scores are requested. Supported measure names (and related quantities) are described in Value below. If measures is NULL, it is interpreted as short for the default selection, i.e. c("exp_a", "DP_rows", "RR_rows", "OR", "MS", "Dice", "PMI", "chi2_signed", "G_signed", "t", "fisher"). If measures is "ALL", all supported measures are calculated (but not necessarily all the variants; see with_variants).
with_variants	Logical. Whether, for the requested measures, all variants should be included in the output (TRUE) or only the main version (FALSE). See also p_fisher_2.
show_dots	Logical. Whether a dot should be shown in console each time calculations for a measure are finished.
p_fisher_2	Logical. only relevant if "fisher" is included in measures. If TRUE, the p-value for a two-sided test (testing for either attraction or repulsion) is also calculated. By default, only the (computationally less demanding) p-value for a one-sided test is calculated. See Value for more details.
haldane	Logical. Should the Haldane-Anscombe correction be used? (See the Details section.) If haldane is TRUE, and there is at least one zero frequency in a contingency table, the correction is used for all measures calculated for that table, not just for measures that need this to be done.
small_pos	Alternative (but sometimes inferior) approach to dealing with zero frequencies, compared to haldane. The argument small_pos only applies when haldane is set to FALSE. (See the Details section.) If haldane is FALSE, and there is at least one zero frequency in a contingency table, adding small positive values to the zero frequency cells is done systematically for all measures calculated for that table, not just for measures that need this to be done.
a	Numeric vector expressing how many times some tested item occurs in the target context. More specifically, a[[i]], with i an integer, expresses how many times the i-th tested item occurs in the target context.
b	Numeric vector expressing how many times other items than the tested item occur in the target context. More specifically, b[[i]], with i an integer, expresses how many times other items than the i-th tested item occur in the target context.
c	Numeric vector expressing how many times some tested item occurs in the reference context. More specifically, c[[i]], with i an integer, expresses how many times the i-th tested item occurs in the reference context.
d	Numeric vector expressing how many times items other than the tested item occur in the reference context. More specifically, d[[i]], with i an integer, expresses how many times other items than the i-th tested item occur in the reference context.
types	A character vector containing the names of the linguistic items of which the association scores are to be calculated, or NULL. If NULL, assoc_abcd() creates dummy types such as "t001", "t002", etc.

Details

Input and output

assoc_scores() takes as its arguments a target frequency list and a reference frequency lists (either as two freqlist objects or as a cooc_info object) and returns a number of popular measures expressing, for (almost) every item in either one of these lists, the extent to which the item is attracted to the target context, when compared to the reference context. The "almost" is added between parentheses because, with the default settings, some items are automatically excluded from the output (see min_freq).

assoc_abcd() takes as its arguments four vectors a, b, c, and d, of equal length. Each tuple of values (a[i], b[i], c[i], d[i]), with i some integer number between 1 and the length of the vectors, is assumed to represent the four numbers a, b, c, d in a contingency table of the type:

	tested item	any other item	total
target context	a	b	m
reference context	c	d	n
total	k	l	N

In the above table m, n, k, l and N are marginal frequencies. More specifically, m = a + b, n = c + d, k = a + c, l = b + d and N = m + n.

Dealing with zeros

Several of the association measures break down when one or more of the values a, b, c, and d are zero (for instance, because this would lead to division by zero or taking the log of zero). This can be dealt with in different ways, such as the Haldane-Anscombe correction.

Strictly speaking, Haldane-Anscombe correction specifically applies to the context of (log) odds ratios for two-by-two tables and boils down to adding 0.5 to each of the four values a, b, c, and d in every two-by-two contingency table for which the original values a, b, c, and d would not allow us to calculate the (log) odds ratio, which happens when one (or more than one) of the four cells is zero. Using the Haldane-Anscombe correction, the (log) odds ratio is then calculated on the bases of these 'corrected' values for a, b, c, and d.

However, because other measures that do not compute (log) odds ratios might also break down when some value is zero, all measures will be computed on the 'corrected' contingency matrix.

If the haldane argument is set to FALSE, division by zero or taking the log of zero is avoided by systematically adding a small positive value to all zero values for a, b, c, and d. The argument small_pos determines which small positive value is added in such cases. Its default value is 0.00001.

Value

An object of class assoc_scores. This is a kind of data frame with as its rows all items from either the target frequency list or the reference frequency list with a frequency larger than min_freq in the target list, and as its columns a range of measures that express the extent to which the items are attracted to the target context (when compared to the reference context). Some columns don't contain actual measures but rather additional information that is useful for interpreting other measures.

Possible columns

The following sections describe the (possible) columns in the output. All of these measures are reported if measures is set to "ALL". Alternatively, each measure can be requested by specifying its name in a character vector given to the measures argument. Exceptions are described in the sections below.

Observed and expected frequencies

• a, b, c, d: The frequencies in cells a, b, c and d, respectively. If one of them is 0, they will be augmented by 0.5 or small_pos (see Details). These output columns are always present.

• dir: The direction of the association: 1 in case of relative attraction between the tested item and the target context (if \frac{a}{m} ≥ \frac{c}{n}) and -1 in case of relative repulsion between the target item and the target context (if \frac{a}{m} < {c}{n}).

• exp_a, exp_b, exp_c, exp_d: The expected values for cells a, b, c and d, respectively. All these columns will be included if "expected" is in measures. exp_a is also one of the default measures and is therefore included if measures is NULL. The values of these columns are computed as follows:

  • exp_a = \frac{m \times k}{N}

  • exp_b = \frac{m \times l}{N}

  • exp_c = \frac{n \times k}{N}

  • exp_d = \frac{n \times l}{N}

Effect size measures

Some of these measures are based on proportions and can therefore be computed either on the rows or on the columns of the contingency table. Each measure can be requested on its own, but pairs of measures can also be requested with the first part of their name, as indicated in their corresponding descriptions.

• DP_rows and DP_cols: The difference of proportions, sometimes also called Delta-p (Δ p), between rows and columns respectively. Both columns are present if "DP" is included in measures. DP_rows is also included if measures is NULL. They are calculated as follows:

  • DP_rows = \frac{a}{m} - \frac{c}{n}

  • DP_cols = \frac{a}{k} - \frac{b}{l}

• perc_DIFF_rows and perc_DIFF_cols: These measures can be seen as normalized versions of Delta-p, i.e. essentially the same measures divided by the denominator and multiplied by 100. They therefore express how large the difference of proportions is, relative to the reference proportion. The multiplication by 100 turns the resulting 'relative difference of proportion' into a percentage. Both columns are present if "perc_DIFF" is included in measures. They are calculated as follows:

  • perc_DIFF_rows = 100 * \frac{(a / m) - (c / n)}{c / n}

  • perc_DIFF_cols = 100 * \frac{(a / k) - (b / l)}{c / n}

• DC_rows and DC_cols: The difference coefficient can be seen as a normalized version of Delta-p, i.e. essentially dividing the difference of proportions by the sum of proportions. Both columns are present if "DC" is included in measures. They are calculated as follows:

  • DC_rows = \frac{(a / m) - (c / n)}{(a / m) + (c / n)}

  • DC_cols = \frac{(a / k) - (b / l)}{(a / k) + (b / l)}

• RR_rows and RR_cols: Relative risk for the rows and columns respectively. RR_rows represents then how large the proportion in the target context is, relative to the proportion in the reference context. Both columns are present if "RR" is included in measures. RR_rows is also included if measures is NULL. They are calculated as follows:

  • RR_rows = \frac{a / m}{c / n}

  • RR_cols = \frac{a / k}{b / l}

• LR_rows and LR_cols: The so-called 'log ratio' of the rows and columns, respectively. It can be seen as a transformed version of the relative risk, viz. its binary log. Both columns are present if "LR" is included in measures. They are calculated as follows:

  • LR_rows = \log_2≤ft(\frac{a / m}{c / n}\right)

  • LR_cols = \log_2≤ft(\frac{a / k}{b / l}\right)

Other measures use the contingency table in a different way and therefore don't have a complementary row/column pair. In order to retrieve these columns, if measures is not "ALL", their name must be in the measures vector. Some of them are included by default, i.e. if measures is NULL.

• OR: The odds ratio, which can be calculated either as \frac{a/b}{c/d} or as \frac{a/c}{b/d}. This column is present measures is NULL.

• log_OR: The log odds ratio, which can be calculated either as \log≤ft(\frac{a/b}{c/d}\right) or as \log≤ft(\frac{a/c}{b/d}\right). In other words, it is the natural log of the odds ratio.

• MS: The minimum sensitivity, which is calculated as \min(\frac{a}{m}, \frac{a}{k}). In other words, it is either \frac{a}{m} or \frac{a}{k}, whichever is lowest. This column is present measures is NULL.

• Jaccard: The Jaccard index, which is calculated as \frac{a}{a + b + c}. It expresses a, which is the frequency of the test item in the target context, relative to b + c + d, i.e. the frequency of all other contexts.

• Dice: The Dice coefficient, which is calculated as \frac{2a}{m + k}. It expresses the harmonic mean of \frac{a}{m} and \frac{a}{k} This column is present measures is NULL.

• logDice: An adapted version of the Dice coefficient. It is calculated as 14 + \log_2≤ft(\frac{2a}{m + k}\right). In other words, it is 14 plus the binary log of the Dice coefficient.

• phi: The phi coefficient (φ), which is calculated as \frac{(a \times d) - (b \times c)}{ √{m \times n \times k \times l}}.

• Q: Yule's Q, which is calculated as \frac{(a \times d) - (b \times c)}{(a \times d)(b \times c)}.

• mu: The measure mu (μ), which is calculated as \frac{a}{\mathrm{exp\_a}} (see exp_a).

• PMI and pos_PMI: (Positive) pointwise mutual information, which can be seen as a modification of the mu measure and is calculated as \log_2≤ft(\frac{a}{\mathrm{exp\_a}}\right). In pos_PMI, negative values are set to 0. The PMI column is present measures is NULL.

• PMI2 and PMI3: Modified versions of PMI that aim to give relatively more weight to cases with relatively higher a. However, because of this modification, they are not pure effect size measures any more.

  • PMI2 = \log_2≤ft(\frac{a^2}{\mathrm{exp\_a}}\right)

  • PMI3 = \log_2≤ft(\frac{a^3}{\mathrm{exp\_a}}\right)

Strength of evidence measures

The first measures in this section tend to come in triples: a test statistic, its p-value (preceded by p_) and its signed version (followed by _signed). The test statistics indicate evidence of either attraction or repulsion. Thus, in order to indicate the direction of the relationship, a negative sign is added in the "signed" version when \frac{a}{k} < \frac{c}{l}.

In each of these cases, the name of the main measure (e.g. "chi2") and/or its signed counterpart (e.g. "chi2_signed") must be in the measures argument, or measures must be "ALL", for the columns to be included in the output. If the main function is requested, the signed counterpart will also be included, but if only the signed counterpart is requested, the non-signed version will be excluded. For the p-value to be retrieved, either the main measure or its signed version must be requested and, additionally, the with_variants argument must be set to TRUE.

• chi2, p_chi2 and chi2_signed: The chi-squared test statistic (χ^2) as used in a chi-squared test of independence or in a chi-squared test of homogeneity for a two-by-two contingency table. Scores of this measure are high when there is strong evidence for attraction, but also when there is strong evidence for repulsion. The chi2_signed column is present if measures is NULL. chi2 is calculated as follows:

  \frac{(a-\mathrm{exp\_a})^2}{\mathrm{exp\_a}} + \frac{(b-\mathrm{exp\_b})^2}{\mathrm{exp\_b}} + \frac{(c-\mathrm{exp\_c})^2}{\mathrm{exp\_c}} + \frac{(d-\mathrm{exp\_d})^2}{\mathrm{exp\_d}}

  .

• chi2_Y, p_chi2_Y and chi2_Y_signed: The chi-squared test statistic (χ^2) as used in a chi-squared test with Yates correction for a two-by-two contingency table. chi2_Y is calculated as follows:

  \frac{(|a-\mathrm{exp\_a}| - 0.5)^2}{\mathrm{exp\_a}} + \frac{(|b-\mathrm{exp\_b}| - 0.5)^2}{\mathrm{exp\_b}} + \frac{(|c-\mathrm{exp\_c}| - 0.5)^2}{\mathrm{exp\_c}} + \frac{(|d-\mathrm{exp\_d}| - 0.5)^2}{\mathrm{exp\_d}}

  .

• chi2_2T, p_chi2_2T and chi2_2T_signed: The chi-squared test statistic (χ^2) as used in a chi-squared goodness-of-fit test applied to the first column of the contingency table. The "2T" in the name stands for 'two terms' (as opposed to chi2, which is sometimes the 'four terms' version). chi2_2T is calculated as follows:

  \frac{(a-\mathrm{exp\_a})^2}{\mathrm{exp\_a}} + \frac{(c-\mathrm{exp\_c})^2}{\mathrm{exp\_c}}

  .

• chi2_2T_Y, p_chi2_2T_Y and chi2_2T_Y_signed: The chi-squared test statistic (χ^2) as used in a chi-squared goodness-of-fit test with Yates correction, applied to the first column of the contingency table. chi2_2T_Y is calculated as follows:

  \frac{(|a-\mathrm{exp\_a}| - 0.5)^2}{\mathrm{exp\_a}} + \frac{(|c-\mathrm{exp\_c}| - 0.5)^2}{\mathrm{exp\_c}}

  .

• G, p_G and G_signed: G test statistic, which is also sometimes called log-likelihood ratio (LLR) and, somewhat confusingly, G-squared. This is the test statistic as used in a log-likelihood ratio test for independence or homogeneity in a two-by-two contingency table. Scores are high in case of strong evidence for attraction, but also in case of strong evidence of repulsion. The G_signed column is present if measures is NULL. G is calculated as follows:

  2 ≤ft( a \times \log(\frac{a}{\mathrm{exp\_a}}) + b \times \log(\frac{b}{\mathrm{exp\_b}}) + c \times \log(\frac{c}{\mathrm{exp\_c}}) + d \times \log(\frac{d}{\mathrm{exp\_d}}) \right)

• G_2T, p_G_2T and G_2T_signed: The test statistic used in a log-likelihood ratio test for goodness-of-fit applied to the first column of the contingency table. The "2T" stands for 'two terms'. G_2T is calculated as follows:

  2 ≤ft( a \times \log(\frac{a}{\mathrm{exp\_a}}) + c \times \log(\frac{c}{\mathrm{exp\_c}}) \right)

The final two groups of measures take a different shape. The _as_chisq1 columns compute qchisq(1 - p, 1), with p being the p-values they are transforming, i.e. the p right quantile in a χ^2 distribution with one degree of freedom (see p_to_chisq1()).

• t, p_t_1, t_1_as_chisq1, p_t_2 and t_2_as_chisq1: The t-test statistic, used for a t-test for the proportion \frac{a}{N} in which the null hypothesis is based on \frac{k}{N}\times\frac{m}{N}. Column t is present if "t" is included in measures or if measures is "ALL" or NULL. The other four columns are present if t is requested and if, additionally, with_variants is TRUE.

  • t = \frac{ a/N + k/N + m/N }{ √{((a/N)\times (1-a/N))/N} }

  • p_t_1 is the p-value that corresponds to t when assuming a one-tailed test that only looks at attraction; t_1_as_chisq1 is its transformation.

  • p_t_2 is the p-value that corresponds to t when assuming a two-tailed test, viz. that looks at both attraction and repulsion; t_2_as_chisq1 is its transformation.

• p_fisher_1, fisher_1_as_chisq1, p_fisher_1r, fisher_1r_as_chisq1: The p-value of a one-sided Fisher exact test. The column p_fisher_1 is present if either "fisher" or "p_fisher" are in measures or if measures is "ALL" or NULL. The other columns are present if p_fisher_1 as been requested and if, additionally, with_variants is TRUE.

  • p_fisher_1 and p_fisher_1r are the p-values of the Fisher exact test that look at attraction and repulsion respectively.

  • fisher_1_as_chisq1 and fisher_1r_as_chisq1 are their respective transformations..

• p_fisher_2 and fisher_2_as_chisq1: p-value for a two-sided Fisher exact test, viz. looking at both attraction and repulsion. p_fisher_2 returns the p-value and fisher_2_as_chisq1 is its transformation. The p_fisher_2 column is present if either "fisher" or "p_fisher_1" are in measures or if measures is "ALL" or NULL and if, additionally, p_fisher_2 is TRUE. fisher_2_as_chisq1 is present if p_fisher_2 was requested and, additionally, with_variants is TRUE.

Properties of the class

An object of class assoc_scores has:

• associated as.data.frame(), print(), sort() and tibble::as_tibble() methods,

• an interactive explore() method and useful getters, viz. n_types() and type_names().

An object of this class can be saved to file with write_assoc() and read with read_assoc().

Examples

assoc_abcd(10 , 200, 100,  300, types = "four")
assoc_abcd(30, 1000,  14, 5000, types = "fictitious")
assoc_abcd(15, 5000,  16, 1000, types = "toy")
assoc_abcd( 1,  300,   4, 6000, types = "examples")

a <- c(10,    30,    15,    1)
b <- c(200, 1000,  5000,  300)
c <- c(100,   14,    16,    4)
d <- c(300, 5000, 10000, 6000)
types <- c("four", "fictitious", "toy", "examples")
(scores <- assoc_abcd(a, b, c, d, types = types))

as_data_frame(scores)
as_tibble(scores)

print(scores, sort_order = "PMI")
print(scores, sort_order = "alpha")
print(scores, sort_order = "none")
print(scores, sort_order = "nonsense")

print(scores, sort_order = "PMI",
      keep_cols = c("a", "exp_a", "PMI", "G_signed"))
print(scores, sort_order = "PMI",
      keep_cols = c("a", "b", "c", "d", "exp_a", "G_signed"))
print(scores, sort_order = "PMI",
     drop_cols = c("a", "b", "c", "d", "exp_a", "G_signed",
                    "RR_rows", "chi2_signed", "t"))

mclm documentation built on Oct. 3, 2022, 9:07 a.m.