Pith. sign in

REVIEW 4 major objections 5 minor 49 references

This paper claims that the correct correlation method for a variable pair can be chosen automatically by variable type, and that doing so changes substantive results on real data.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 05:13 UTC pith:CROQGK2D

load-bearing objection Useful, well-engineered tools paper with a real empirical mini-study, but the simulations validate estimators, not the auto-pipeline, and low-cardinality counts get mislabeled as ordinal. the 4 major comments →

arxiv 2607.22285 v1 pith:CROQGK2D submitted 2026-07-24 stat.ME stat.CO

smartcor: Intelligent Correlation Method Selection for Mixed Variable Types

M Harshvardhan , Pritam Ranjan This is my paper
classification stat.ME stat.CO MSC 62H2062H1762-04
keywords correlation coefficientassociation measuresmixed variable typespolychoric correlationKendall's tauCramér's Vvariable type detectionstatistical software
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Most software defaults every correlation to Pearson, but Pearson is mathematically appropriate for only three of the ten possible type-pairings among continuous, binary, ordinal, count, and categorical variables. This paper presents smartcor, a package that detects each variable's type, selects from 14 correlation or association methods so every pair gets a principled estimator, and explains the choice. The paper proves that Pearson coincides with point-biserial and phi for binary pairs, shows via simulation that ordinal data attenuate Pearson while latent-variable methods recover the association when their normality assumption holds, and demonstrates on survey data that type-aware analysis yields different coefficients and changes which associations reach significance.

Core claim

The central claim is that the 'one correlation fits all' default of statistical software is wrong for most real variable pairs, and that a small, transparent decision rule can fix it. The paper establishes two algebraic identities—Pearson's r equals the point-biserial coefficient for continuous-binary pairs and the phi coefficient for binary-binary pairs under 0/1 coding—and then shows that for the remaining seven pairings, the choice of estimator matters: Pearson attenuates ordinal associations by roughly 2–21% (up to ~37% for dichotomized data), and Cramér's V or related measures are required for nominal variables because Pearson is undefined there. The implemented tool routes each pair to

What carries the argument

The load-bearing object is a decision table: the five variable types (continuous, binary, ordinal, count, categorical) generate ten unordered pairs, and each pair maps to a recommended estimator among 14 supported methods. Two formal equivalences carry the 'Pearson is fine' part: Theorem 2.1 shows Pearson equals point-biserial for continuous-binary, Theorem 2.2 shows Pearson equals phi for binary-binary. For ordinal-involving pairs, a likelihood-ratio goodness-of-fit test of the bivariate-normal threshold model chooses between polychoric/polyserial (if normality is not rejected) and Kendall/Spearman (if it is). The package also separates signed correlation from unsigned association, applying

Load-bearing premise

The default 'auto' choice between latent-variable and rank-based correlation relies on a likelihood-ratio test whose failure to reject normality is treated as evidence for normality, even though at small samples (n around 30) that test has low power, so non-normal latent distributions will often be mistakenly accepted and a biased estimator selected.

What would settle it

Simulate, say, 1,000 datasets of n=32 from a bivariate distribution with a strongly skewed latent (e.g., exponential margins transformed to a normal copula with rho=0.5) and ordinal variables with 5 categories each; if smart_cor's 'auto' option selects polychoric in the majority of cases and the polychoric estimate has bias >0.1 while Kendall's tau is unbiased, the automatic selection logic is not doing what it claims for small samples.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the paper is right, default Pearson correlation tables in software are misleading for the majority of mixed-type pairs; correcting the default would change reported effect sizes and significance in a substantial minority of published correlations.
  • The algebraic equivalences mean practitioners gain nothing by switching methods for continuous-binary and binary-binary pairs with 0/1 coding; the gain is concentrated in ordinal and categorical pairs.
  • Type-aware selection makes it practical to compute a single mixed-type correlation matrix with nine or more distinct methods without manual specification.
  • The data-driven latent-normality test means the choice between latent-variable and rank-based estimators can be audited and overridden, reducing reliance on untested assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The auto latent-normality rule inherits the LR test's low power at small n; a natural extension is a sample-size-aware fallback (e.g., require a minimum n or a confidence margin before selecting polychoric). This is an editorial inference, not a paper claim.
  • Because type detection is heuristic (e.g., integer-valued 'count' vs 'continuous'), the package's results can depend on the ordinal_threshold; a testable extension is a sensitivity analysis that varies the threshold and reports how many cells change method.
  • The correlation-versus-association distinction suggests that presenting signed and unsigned values in one matrix could mislead readers who rank them; a stronger design would separate the two blocks in output. This is a design suggestion beyond the paper's stated scope.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces smartcor (R) and pysmartcor (Python), a package that automatically detects whether each variable is continuous, binary, ordinal, count, or categorical, and then selects one of 14 correlation or association methods for each pair. The main contributions claimed are: (i) algebraic proofs that Pearson correlation equals the point-biserial coefficient for continuous–binary pairs and the phi coefficient for binary–binary pairs; (ii) a simulation study comparing estimator bias across variable-type pairs and latent distributions; (iii) a data-driven latent-normality test that chooses between polychoric/polyserial and rank-based methods; (iv) a mini-study of economics papers showing that 92.3% of correlation calls are Pearson; and (v) case studies on mtcars and General Social Survey data illustrating substantive differences between naive and type-aware analysis.

Significance. If the central claim is fully supported, the package would fill a real practical gap: applied researchers routinely compute Pearson correlations on mixed-type data without checking measurement levels. The paper's strengths include clean algebraic proofs (Appendix A), a transparent and explainable decision architecture with user override, and simulations that separate skew direction from symmetric heavy-tailed departures. The R and Python implementations are distributed through standard repositories, and the tidy API is a genuine usability contribution. However, the paper currently claims more than it validates: the Monte Carlo simulations fix variable types and do not exercise the automatic type-detection and selection pipeline end-to-end, and one concrete detection-rule bug (low-cardinality counts) affects the paper's own flagship example. These issues are fixable, but they currently limit the strength of the paper's main claim.

major comments (4)
  1. [§3.2 (rules 5–6), §1, §3.5] The type-detection order is internally inconsistent. Rule 5 labels any numeric vector with ≤ ordinal_threshold unique values as ordinal before Rule 6 can label non-negative integer-valued vectors as count. Thus a count variable with ≤10 distinct values — e.g., number of children, previous jobs, or carburetors — is labeled ordinal. This contradicts §1's 'all count variables are treated as continuous variables' and §3.2's claim that the count label 'never changes the selected method or the estimate.' It does change the method: mtcars$carb has 6 unique values and is labeled ordinal in §3.5; the reported carb–gear value is 0.098 (Kendall) whereas treating carb as continuous would give Pearson ≈ 0.27. The paper attributes this gap to ordinal structure, but part of it is an artifact of the detection heuristic. Since the Section 2 simulations never call detect_type(), the auto-detection step it
  2. [§2.5, §3.4, §3.6] The default assume_latent_normal='auto' uses the Jöreskog LR goodness-of-fit test and selects polychoric/polyserial when the test fails to reject latent normality. At small n, this test has low power, so non-normal latent distributions will often fail to reject and the auto mode will choose the latent-variable estimator despite the mismatch. The paper explicitly acknowledges this in §2.5 ('Failure to reject is of course not equivalent to accepting normality, particularly at small sample sizes'), but the default behavior does not reflect the caveat. The mtcars example (§3.6) illustrates the risk: at n=32, compare_methods() recommends polychoric for gear–carb on the basis of the LR test. Either the default should be more conservative at small n, the output should warn about low power, or the paper should provide a power analysis justifying the current threshold.
  3. [Abstract; §2.3–§2.4] The abstract states that 'Monte Carlo simulations validate the selection logic,' but the simulations in Section 2 fix the variable types and compare estimators; they do not feed raw data through detect_type() and the full selection rules. The end-to-end pipeline has three components — type detection, method selection, and the latent-normality decision — and none of these is evaluated as a system. For example, there is no measurement of detect_type() misclassification rates, no sensitivity analysis for ordinal_threshold, and no evaluation of the auto latent-normality decision's operating characteristics. The claim should be softened or an end-to-end simulation should be added.
  4. [§2 (paragraph beginning 'Some binary variables may be categorical')] The text states: 'when a scalar variable with two (and only two) unique values exists, it is a binary categorical variable and must use an association measure instead of correlation measures.' Taken literally, this contradicts Table 1 and the package's own behavior: Table 1 lists point-biserial, phi, tetrachoric, and Yule's Q for binary variables, and the GSS case study (§4.3) computes point-biserial correlation for sex. If the sentence were implemented, no binary variable could ever receive a correlation coefficient. The intended distinction is presumably between binary variables with a meaningful 0/1 ordering and binary variables whose two labels are nominal categories, but the sentence as written is internally inconsistent and should be rewritten.
minor comments (5)
  1. [Table 1 vs. Table 2] Table 1 lists Spearman and Kendall as available methods for continuous–binary pairs, but Table 2 gives no alternative for continuous–binary. Clarify whether rank methods are appropriate for this pairing and, if so, list them in Table 2.
  2. [§2.3 (Figures 2 and 4)] Spearman's deviation from the latent ρ is labeled 'attenuation bias,' but Spearman estimates a rank-based monotonic association, not the linear correlation. The text acknowledges this scale issue for Kendall but not for Spearman. Consider relabeling the figures or adding an explicit caveat.
  3. [§3.2] The ordinal_threshold default of 10 is presented without any sensitivity analysis. Because the threshold directly controls the low-cardinality-count misclassification discussed above, a short sensitivity check (e.g., threshold 5, 10, 20) would help users understand the heuristic's robustness.
  4. [§4.2] The text says the surviving Pearson correlations 'underestimate associations involving binary variables (Section 2.2),' but Section 2.2 proves Pearson equals the point-biserial coefficient for continuous–binary data. The underestimation issue arises specifically when binary variables are treated as dichotomized continuous variables (tetrachoric case), not generally. The wording should be corrected.
  5. [§3.5] In the mtcars example, hp is detected as count and then 'counts are treated as numeric continuous variables' for method selection. This is fine, but the paper's repeated statement that the count label 'never changes the selected method' is confusing given the low-cardinality count problem. Consider restricting that claim to count variables that pass the ordinal_threshold.

Circularity Check

0 steps flagged

No circularity found: core equivalences are proved algebraically and simulations use external generative benchmarks; the count/ordinal heuristic and LR-test power are correctness/validation limitations, not circular reductions.

full rationale

The paper's central derivation is self-contained. Theorem 2.1 and Theorem 2.2 are algebraic identities proved in Appendix A: Pearson correlation on 0/1-coded data equals point-biserial and phi coefficients, with no statistical assumptions or fitted parameters. The simulations in Sections 2.3–2.4 evaluate estimator bias under explicit generative models (e.g., bivariate normal latents, skewed and heavy-tailed latents) and are externally checkable; they do not use smartcor's own outputs as inputs. The latent-normality decision relies on the Jöreskog (2005) likelihood-ratio test, an external method, and the paper explicitly acknowledges the small-sample limitation: 'Failure to reject is of course not equivalent to accepting normality, particularly at small sample sizes.' That is a validation/power caveat, not a circular step. The type-detection heuristic can label low-cardinality count variables as ordinal (e.g., mtcars$carb), which creates an internal inconsistency with the statement that all count variables are treated as continuous; however, this is a correctness/accuracy limitation rather than a circularity, because the method-selection rules are evaluated on independent real datasets and the special-purpose estimators are standard, externally validated methods rather than quantities fitted to the paper's own target result. There is no load-bearing self-citation, no imported uniqueness theorem, and no fitted parameter renamed as a prediction. The paper's overbroad claim that Monte Carlo simulations 'validate the selection logic' is incomplete since detect_type and the auto LR-test decision are not simulated end-to-end, but this under-specification is a validation gap, not a circular derivation. Score 0 reflects the absence of circularity in the core derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard psychometric assumptions and a small set of hand-chosen defaults (ordinal_threshold=10, quintile binning, alpha=0.05). No new theoretical entities are postulated.

free parameters (3)
  • ordinal_threshold = 10
    Any numeric variable with <=10 unique values is classified ordinal (Section 3.2 rule 5). This default is chosen by the package authors, not learned; it determines whether columns like mtcars$gear (3 unique) are treated as ordinal (changing the estimator) vs continuous.
  • continuous-to-categorical bin count for Cramér's V = 5 (quintiles)
    For continuous–categorical pairs, the continuous variable is binned into quintiles before computing Cramér's V (Table 1 footnote). The number of bins is fixed at 5 by the package design and affects the resulting V.
  • alpha for latent normality test = 0.05
    Default significance level in assume_latent_normal='auto' (Section 2.5); a user-specified but package-default parameter that controls switching between polychoric/polyserial and Kendall/Spearman.
axioms (3)
  • domain assumption Latent bivariate normality is required for polychoric/polyserial/tetrachoric to estimate the true latent correlation.
    Section 2.1(e) and 2.4. Standard psychometric assumption, not proved in this paper.
  • domain assumption The LR chi-square test of bivariate normality (Jöreskog, 2005) has correct size/power for the package's auto mode.
    Section 3.4. The paper relies on this test to decide between polychoric and Kendall; no Monte Carlo study of the test's size/power is reported in the paper.
  • domain assumption Pearson's r, Spearman, Kendall, etc. are the correct measures for their respective type pairs.
    Table 2 encodes a normative decision table; while standard, it is a modeling choice (e.g., treating count as continuous).

pith-pipeline@v1.3.0-alltime-deepseek · 30428 in / 12696 out tokens · 100919 ms · 2026-08-01T05:13:03.801996+00:00 · methodology

0 comments
read the original abstract

Pearson correlation is the default measure of association in most statistical software, yet it is only appropriate for pairs of continuous variables with a linear relationship. When variables are binary, ordinal, or categorical, specialized methods (e.g., point-biserial, polychoric, tetrachoric, and Cram\'{e}r's~$V$) may be more appropriate, but practitioners rarely know which to select. The \textbf{smartcor} package for \textbf{R} (and its companion \textbf{pysmartcor} package for \textbf{Python}) automatically detects variable types, selects the statistically appropriate correlation method for each pair, and explains its reasoning. The package supports 14~correlation and association methods covering all 10~variable-type pair combinations, and distinguishes true correlation (for ordinal and continuous pairs) from statistical association (for nominal categorical pairs). Monte~Carlo simulations validate the selection logic, and a case study with General Social Survey data demonstrates substantive differences between naive and type-aware correlation analysis.

Figures

Figures reproduced from arXiv: 2607.22285 by M Harshvardhan, Pritam Ranjan.

Figure 1
Figure 1. Figure 1: Boxplots of estimated correlation across [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bias of four estimators on ordinal–ordinal data ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bias of the polychoric estimator grouped by true latent [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bias of three estimators on continuous–ordinal data ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bias by latent distribution type (n = 500, k = 5, ρ ∈ {0.1, 0.2, . . . , 0.9}, 500 replications). Rows: methods (Pearson, Spearman, Kendall, polychoric). Columns: latent distribution; all panels share a common y-axis clipped at ±0.25 for readability. Skewed distributions are split into right- and left-skewed; heavy-tailed distributions into symmetric (t3-like) and asymmetric (mixture). Polychoric is unbias… view at source ↗
Figure 6
Figure 6. Figure 6: Correlation heatmap for the GSS subset using [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Method-selection heatmap showing which correlation method was selected for each variable pair. [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 1 canonical work pages

  1. [1]

    Benesty, J

    J. Benesty, J. Chen, Y. Huang, and I. Cohen. Pearson correlation coefficient. In Noise Reduction in Speech Processing, pages 1--4. Springer, 2009

    2009

  2. [2]

    D. G. Bonett and T. A. Wright. Sample size requirements for estimating pearson, kendall and spearman correlations. Psychometrika, 65 0 (1): 0 23--28, 2000

    2000

  3. [3]

    M. B. Brown and J. K. Benedetti. Sampling behavior of tests for correlation in two-way contingency tables. Journal of the American Statistical Association, 72 0 (358): 0 309--315, 1977

    1977

  4. [4]

    Casella and R

    G. Casella and R. L. Berger. Statistical Inference. Duxbury, Pacific Grove, CA, 2nd edition, 2002

    2002

  5. [5]

    N. Cliff. Ordinal Methods for Behavioral Data Analysis. Lawrence Erlbaum Associates, Mahwah, NJ, 1996

    1996

  6. [6]

    Cram \'e r

    H. Cram \'e r. Mathematical Methods of Statistics. Princeton University Press, Princeton, 1946

    1946

  7. [7]

    E. E. Cureton. Rank-biserial correlation. Psychometrika, 21 0 (3): 0 287--290, 1956. doi:10.1007/BF02289138

    doi:10.1007/bf02289138 1956

  8. [8]

    E. C. Davenport and N. A. El-Sanhurry. Phi/phimax: Review and synthesis. Educational and Psychological Measurement, 51 0 (4): 0 821--828, 1991. doi:10.1177/001316449105100403

    doi:10.1177/001316449105100403 1991

  9. [9]

    J. C. De Winter, S. D. Gosling, and J. Potter. Comparing the P earson and S pearman correlation coefficients across distributions and sample sizes: A tutorial using simulations and empirical data. Psychological Methods, 21 0 (3): 0 273--290, 2016

    2016

  10. [10]

    F. Drasgow. Polychoric and polyserial correlations. The Encyclopedia of Statistics, 7: 0 68--74, 1986

    1986

  11. [11]

    E. C. Fieller, H. O. Hartley, and E. S. Pearson. Tests for rank correlation coefficients. I . Biometrika, 44 0 (3/4): 0 470--481, 1957

    1957

  12. [12]

    D. B. Flora and P. J. Curran. An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9 0 (4): 0 466--491, 2004

    2004

  13. [13]

    J. Fox. polycor : Polychoric and Polyserial Correlations , 2022. URL https://CRAN.R-project.org/package=polycor. R package version 0.8-1

    2022

  14. [14]

    L. A. Goodman and W. H. Kruskal. Measures of association for cross classifications. Journal of the American Statistical Association, 49 0 (268): 0 732--764, 1954

    1954

  15. [15]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al. Array programming with NumPy . Nature, 585 0 (7825): 0 357--362, 2020. doi:10.1038/s41586-020-2649-2

    doi:10.1038/s41586-020-2649-2 2020

  16. [16]

    K. Healy. gssr : General Social Survey Data for R , 2023. URL https://kjhealy.github.io/gssr/. R package version 0.4.0

    2023

  17. [17]

    S. L. Hershberger. Tetrachoric correlation. Encyclopedia of statistics in behavioral science, 2005

    2005

  18. [18]

    J. D. Hunter. Matplotlib : A 2D graphics environment. Computing in Science & Engineering, 9 0 (3): 0 90--95, 2007. doi:10.1109/MCSE.2007.55

    doi:10.1109/mcse.2007.55 2007

  19. [19]

    Iannone, J

    R. Iannone, J. Cheng, B. Schloerke, E. Hughes, A. Lauer, and J. Seo. gt: Easily Create Presentation-Ready Display Tables, 2024. URL https://CRAN.R-project.org/package=gt. R package version 0.10

    2024

  20. [20]

    K. G. J \"o reskog. On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59 0 (3): 0 381--389, 1994

    1994

  21. [21]

    K. G. J \"o reskog. Structural equation modeling with ordinal variables using LISREL . Technical Report, 2005. URL https://www.ssicentral.com/lisrel/techdocs/ordinal.pdf. Scientific Software International, Lincolnwood, IL

    2005

  22. [22]

    M. G. Kendall. A new measure of rank correlation. Biometrika, 30 0 (1-2): 0 81--93, 1938

    1938

  23. [23]

    M. G. Kendall. The treatment of ties in ranking problems. Biometrika, 33 0 (3): 0 239--251, 1945

    1945

  24. [24]

    Kiwanuka, J

    F. Kiwanuka, J. Kopra, N. Sak-Dankosky, R. C. Nanyonga, and T. Kvist. Polychoric correlation with ordinal data in nursing research. Nursing Research, 71 0 (6): 0 469--476, 2022

    2022

  25. [25]

    S. Kranz. Find economic articles with data. Database and search application, Ulm University, 2023. URL https://ejd.econ.mathematik.uni-ulm.de. Snapshot of 2024-05-30

    2023

  26. [26]

    Makowski, M

    D. Makowski, M. S. Ben-Shachar, I. Patil, and D. L \"u decke. Methods and algorithms for correlation analysis in R . Journal of Open Source Software, 5 0 (51): 0 2306, 2020. doi:10.21105/joss.02306

    work page doi:10.21105/joss.02306 2020

  27. [27]

    McKinney

    W. McKinney. Data structures for statistical computing in Python . In Proceedings of the 9th Python in Science Conference, pages 56--61, 2010. doi:10.25080/Majora-92bf1922-00a

    doi:10.25080/majora-92bf1922-00a 2010

  28. [28]

    Muth \'e n and D

    B. Muth \'e n and D. Kaplan. A comparison of some methodologies for the factor analysis of non-normal L ikert variables. British Journal of Mathematical and Statistical Psychology, 38 0 (2): 0 171--189, 1985

    1985

  29. [29]

    Muth \'e n and D

    B. Muth \'e n and D. Kaplan. A comparison of some methodologies for the factor analysis of non-normal L ikert variables: A note on the size of the model. British Journal of Mathematical and Statistical Psychology, 45 0 (1): 0 19--30, 1992

    1992

  30. [30]

    U. Olsson. Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44 0 (4): 0 443--460, 1979

    1979

  31. [31]

    Olsson, F

    U. Olsson, F. Drasgow, and N. J. Dorans. The polyserial correlation coefficient. Psychometrika, 47 0 (3): 0 337--347, 1982. doi:10.1007/BF02294164

    doi:10.1007/bf02294164 1982

  32. [32]

    K. Pearson. Notes on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London, 58: 0 240--242, 1895

  33. [33]

    K. Pearson. Mathematical Contributions to the Theory of Evolution. XIII . On the Theory of Contingency and Its Relation to Association and Normal Correlation . Number 1 in Drapers' Company Research Memoirs, Biometric Series. Dulau and Co., London, 1904

    1904

  34. [34]

    Poon and S.-Y

    W.-Y. Poon and S.-Y. Lee. Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients. Psychometrika, 52 0 (3): 0 409--430, 1987

    1987

  35. [35]

    R : A Language and Environment for Statistical Computing

    R Core Team . R : A Language and Environment for Statistical Computing . R Foundation for Statistical Computing, Vienna, Austria, 2025. URL https://www.R-project.org/

    2025

  36. [36]

    W. Revelle. psych : Procedures for Psychological, Psychometric, and Personality Research . Northwestern University, Evanston, IL, 2024. URL https://CRAN.R-project.org/package=psych. R package version 2.4.3

    2024

  37. [37]

    Robinson, A

    D. Robinson, A. Hayes, and S. Couch. broom : Convert Statistical Analysis Objects into Tidy Tibbles , 2024. URL https://CRAN.R-project.org/package=broom. R package version 1.0.7

    2024

  38. [38]

    T. W. Smith, M. Davern, J. Freese, and S. Morgan. General social surveys, 1972--2024. NORC at the University of Chicago, 2024. URL https://gss.norc.org/. Machine-readable data file

    1972

  39. [39]

    Spearman

    C. Spearman. The proof and measurement of association between two things. The American Journal of Psychology, 15 0 (1): 0 72--101, 1904

    1904

  40. [40]

    H. Theil. Economics and information theory. 1967

    1967

  41. [41]

    A. A. Tschuprow. Principles of the Mathematical Theory of Correlation. W. Hodge & Co., London, 1939. Translated by M. Kantorowitsch

    1939

  42. [42]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, et al. SciPy 1.0: Fundamental algorithms for scientific computing in Python . Nature Methods, 17: 0 261--272, 2020. doi:10.1038/s41592-019-0686-2

    doi:10.1038/s41592-019-0686-2 2020

  43. [43]

    M. L. Waskom. seaborn : Statistical data visualization. Journal of Open Source Software, 6 0 (60): 0 3021, 2021. doi:10.21105/joss.03021

    doi:10.21105/joss.03021 2021

  44. [44]

    H. Wickham. ggplot2 : Elegant Graphics for Data Analysis . Springer-Verlag, New York, 2016. URL https://ggplot2.tidyverse.org

    2016

  45. [45]

    Wickham and L

    H. Wickham and L. Henry. purrr: Functional Programming Tools, 2023. URL https://CRAN.R-project.org/package=purrr. R package version 1.0.2

    2023

  46. [46]

    Wickham, M

    H. Wickham, M. Averick, J. Bryan, W. Chang, L. D. McGowan, R. Fran c ois, G. Grolemund, A. Hayes, L. Henry, J. Hester, M. Kuhn, T. L. Pedersen, E. Miller, S. M. Bache, K. M \"u ller, J. Ooms, D. Robinson, D. P. Seidel, V. Spinu, K. Takahashi, D. Vaughan, C. Wilke, K. Woo, and H. Yutani. Welcome to the tidyverse . Journal of Open Source Software, 4 0 (43):...

    doi:10.21105/joss.01686 2019

  47. [47]

    Wickham, R

    H. Wickham, R. Fran c ois, L. Henry, K. M \"u ller, and D. Vaughan. dplyr: A Grammar of Data Manipulation, 2023. URL https://CRAN.R-project.org/package=dplyr. R package version 1.1.4

    2023

  48. [48]

    G. U. Yule. On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75 0 (6): 0 579--652, 1912

    1912

  49. [49]

    H. Zhu. kableExtra: Construct Complex Table with kable and Pipe Syntax, 2024. URL https://CRAN.R-project.org/package=kableExtra. R package version 1.4.0

    2024