disp_DA: Calculate the dispersion measure D_{A}

In tlda: Tools for Language Data Analysis

Calculate the dispersion measure D_{A}

Description

This function calculates the dispersion measure D_{A}. It offers two computational procedures, the basic version as well as a computational shortcut. It allows the user to choose the directionality of scaling, i.e. whether higher values denote a more even or a less even distribution. It also provides the option of calculating frequency-adjusted dispersion scores.

Usage

disp_DA(
  subfreq,
  partsize,
  procedure = "basic",
  directionality = "conventional",
  freq_adjust = FALSE,
  freq_adjust_method = "even",
  unit_interval = TRUE,
  digits = NULL,
  verbose = TRUE,
  print_score = TRUE,
  suppress_warning = FALSE
)

Arguments

subfreq	A numeric vector of subfrequencies, i.e. the number of occurrences of the item in each corpus part
partsize	A numeric vector specifying the size of the corpus parts
procedure	Character string indicating which procedure to use for the calculation of D_{A}. See details below. Possible values are 'basic' (default), 'shortcut'.
directionality	Character string indicating the directionality of scaling. See details below. Possible values are "conventional" (default) and "gries"
freq_adjust	Logical. Whether dispersion score should be adjusted for frequency (i.e. whether frequency should be 'partialed out'); default is FALSE
freq_adjust_method	Character string indicating which method to use for devising dispersion extremes. See details below. Possible values are "even" (default) and "pervasive"
unit_interval	Logical. Whether frequency-adjusted scores that exceed the limits of the unit interval should be replaced by 0 and 1; default is TRUE
digits	Rounding: Integer value specifying the number of decimal places to retain (default: no rounding)
verbose	Logical. Whether additional information (on directionality, formulas, frequency adjustment) should be printed; default is TRUE
print_score	Logical. Whether the dispersion score should be printed to the console; default is TRUE
suppress_warning	Logical. Whether warning messages should be suppressed; default is FALSE

Details

The function calculates the dispersion measure D_{A} based on a set of subfrequencies (number of occurrences of the item in each corpus part) and a matching set of part sizes (the size of the corpus parts, i.e. number of word tokens).

• Directionality: D_{A} ranges from 0 to 1. The conventional scaling of dispersion measures (see Juilland & Chang-Rodriguez 1964; Carroll 1970; Rosengren 1971) assigns higher values to more even/dispersed/balanced distributions of subfrequencies across corpus parts. This is the default. Gries (2008) uses the reverse scaling, with higher values denoting a more uneven/bursty/concentrated distribution; use directionality = "gries" to choose this option.

• Procedure: Irrespective of the directionality of scaling, two computational procedures for D_{A} exist (see below for details). Both appear in Wilcox (1973), where the measure is referred to as MDA. The basic version (represented by the value "basic") carries out the full set of computations required by the composition of the formula. As the number of corpus parts grows, this can become computationally very expensive. Wilcox (1973) also gives a "computational" procedure, which is a shortcut that is much quicker and closely approximates the scores produced by the basic formula. This version is represented by the value "shortcut".

• Frequency adjustment: Dispersion scores can be adjusted for frequency using the min-max transformation proposed by Gries (2022: 184-191; 2024: 196-208). The frequency-adjusted score for an item considers the lowest and highest possible level of dispersion it can obtain given its overall corpus frequency as well as the number (and size) of corpus parts. The unadjusted score is then expressed relative to these endpoints, where the dispersion minimum is set to 0, and the dispersion maximum to 1 (expressed in terms of conventional scaling). The frequency-adjusted score falls between these bounds and expresses how close the observed distribution is to the theoretical maximum and minimum. This adjustment therefore requires a maximally and a minimally dispersed distribution of the item across the parts. These hypothetical extremes can be built in different ways. The method used by Gries (2022, 2024) uses a computationally expensive procedure that finds the distribution that produces the highest value on the dispersion measure of interest. The current function constructs extreme distributions in a different way, based on the distributional features pervasiveness ("pervasive") or evenness ("even"). You can choose between these with the argument freq_adjust_method; the default is even. For details and explanations, see vignette("frequency-adjustment").

  • To obtain the lowest possible level of dispersion, the occurrences are either allocated to as few corpus parts as possible ("pervasive"), or they are assigned to the smallest corpus part(s) ("even").

  • To obtain the highest possible level of dispersion, the occurrences are either spread as broadly across corpus parts as possible ("pervasive"), or they are allocated to corpus parts in proportion to their size ("even"). The choice between these methods is particularly relevant if corpus parts differ considerably in size. See documentation for find_max_disp() and vignette("frequency-adjustment").

In the formulas given below, the following notation is used:

• k the number of corpus parts

• R_i the normalized subfrequency in part i, i.e. the number of occurrences of the item divided by the size of the part

• r_i a proportional quantity; the normalized subfrequency in part i (R_i) divided by the sum of all normalized subfrequencies

The value "basic" implements the basic computational procedure (see Wilcox 1973: 329, 343; Burch et al. 2017: 194; Egbert et al. 2020: 98). The basic version can be applied to absolute frequencies and normalized frequencies. For dispersion analysis, absolute frequencies only make sense if the corpus parts are identical in size. Wilcox (1973: 343, 'MDA', column 1 and 2) gives both variants of the basic version. The first use of D_{A} for corpus-linguistic dispersion analysis appears in Burch et al. (2017: 194), a paper that deals with equal-sized parts and therefore uses the variant for absolute frequencies. Egbert et al. (2020: 98) rely on the variant using normalized frequencies. Since this variant of the basic version of D_{A} works irrespective of the length of the corpus parts (equal or variable), we will only give this version of the formula. Note that while the formula represents conventional scaling (0 = uneven, 1 = even), in the current function the directionality is controlled separately using the argument directionality.

1 - \frac{\sum_{i = 1}^{k-1} \sum_{j = i+1}^{k} |R_i - R_j|}{\frac{k(k-1)}{2}} \times \frac{1}{2\frac{\sum_i^k R_i}{k}} (Egbert et al. 2020: 98)

The function uses a different version of the same formula, which relies on the proportional r_i values instead of the normalized subfrequencies R_i. This version yields the identical result; the r_i quantities are also the key to using the computational shortcut given in Wilcox (1973: 343). This is the basic formula for D_{A} using r_i instead of R_i values:

1 - \frac{\sum_{i = 1}^{k-1} \sum_{j = i+1}^{k} |r_i - r_j|}{k-1} (Wilcox 1973: 343; see also Soenning 2022)

The value "shortcut" implements the computational shortcut given in Wilcox (1973: 343). Critically, the proportional quantities r_i must first be sorted in decreasing order. Only after this rearrangement can the shortcut version be applied. We will refer to this rearranged version of r_i as r_i^{sorted}:

\frac{2\left(\sum_{i = 1}^{k} (i \times r_i^{sorted}) - 1\right)}{k-1} (Wilcox 1973: 343)

The value "shortcut_mod" adds a minor modification to the computational shortcut to ensure D_{A} does not exceed 1 (on the conventional dispersion scale):

\frac{2\left(\sum_{i = 1}^{k} (i \times r_i^{sorted}) - 1\right)}{k-1} \times \frac{k - 1}{k}

Value

A numeric value

Author(s)

Lukas Soenning

References

Burch, Brent, Jesse Egbert & Douglas Biber. 2017. Measuring and interpreting lexical dispersion in corpus linguistics. Journal of Research Design and Statistics in Linguistics and Communication Science 3(2). 189–216. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1558/jrds.33066")}

Carroll, John B. 1970. An alternative to Juilland’s usage coefficient for lexical frequencies and a proposal for a standard frequency index. Computer Studies in the Humanities and Verbal Behaviour 3(2). 61–65. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1002/j.2333-8504.1970.tb00778.x")}

Egbert, Jesse, Brent Burch & Douglas Biber. 2020. Lexical dispersion and corpus design. International Journal of Corpus Linguistics 25(1). 89–115. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/ijcl.18010.egb")}

Gries, Stefan Th. 2008. Dispersions and adjusted frequencies in corpora. International Journal of Corpus Linguistics 13(4). 403–437. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/ijcl.13.4.02gri")}

Gries, Stefan Th. 2022. What do (most of) our dispersion measures measure (most)? Dispersion? Journal of Second Language Studies 5(2). 171–205. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/jsls.21029.gri")}

Gries, Stefan Th. 2024. Frequency, dispersion, association, and keyness: Revising and tupleizing corpus-linguistic measures. Amsterdam: Benjamins. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1075/scl.115")}

Juilland, Alphonse G. & Eugenio Chang-Rodríguez. 1964. Frequency dictionary of Spanish words. The Hague: Mouton de Gruyter. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1515/9783112415467")}

Rosengren, Inger. 1971. The quantitative concept of language and its relation to the structure of frequency dictionaries. Études de linguistique appliquée (Nouvelle Série) 1. 103–127.

Soenning, Lukas. 2022. Evaluation of text-level measures of lexical dispersion: Robustness and consistency. PsyArXiv preprint. https://osf.io/preprints/psyarxiv/h9mvs/

Wilcox, Allen R. 1973. Indices of qualitative variation and political measurement. The Western Political Quarterly 26 (2). 325–343. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.2307/446831")}

Examples

disp_DA(
  subfreq = c(0,0,1,2,5), 
  partsize = rep(1000, 5),
  procedure = "basic",
  directionality = "conventional",
  freq_adjust = FALSE)

tlda documentation built on June 8, 2025, 11:41 a.m.