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  "Date": "2026-04-06",
  "Title": "Psychometric Functions from the Waller Lab",
  "Authors@R": "c(\nperson(\"Niels\", \"Waller\", role = c(\"aut\", \"cre\"), email=\"nwaller@umn.edu\" ),\nperson(\"Justin\", \"Kracht\", role = \"ctb\"),\nperson(\"Jeff\",\"Jones\", role = \"ctb\"),\nperson(\"Casey\",\"Giordano\", role = \"ctb\"),\nperson(\"Hoang V. Nguyen\", role=\"ctb\"))",
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  "Description": "Computes fungible coefficients and Monte Carlo data.\nUnderlying theory for these functions is described in the\nfollowing publications: Waller, N. (2008). Fungible Weights in\nMultiple Regression. Psychometrika, 73(4), 691-703,\n<DOI:10.1007/s11336-008-9066-z>. Waller, N. & Jones, J. (2009).\nLocating the Extrema of Fungible Regression Weights.\nPsychometrika, 74(4), 589-602, <DOI:10.1007/s11336-008-9087-7>.\nWaller, N. G. (2016). Fungible Correlation Matrices: A Method\nfor Generating Nonsingular, Singular, and Improper Correlation\nMatrices for Monte Carlo Research. Multivariate Behavioral\nResearch, 51(4), 554-568. Jones, J. A. & Waller, N. G. (2015).\nThe normal-theory and asymptotic distribution-free (ADF)\ncovariance matrix of standardized regression coefficients:\ntheoretical extensions and finite sample behavior.\nPsychometrika, 80, 365-378, <DOI:10.1007/s11336-013-9380-y>.\nWaller, N. G.  (2018).  Direct Schmid-Leiman transformations\nand rank-deficient loadings matrices.  Psychometrika, 83,\n858-870. <DOI:10.1007/s11336-017-9599-0>.",
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      "name": "BadRRM",
      "title": "Improper R matrix reported by Rousseeuw and Molenberghs",
      "object": "BadRRM",
      "class": [
        "matrix",
        "array"
      ],
      "fields": {},
      "rows": 3,
      "table": true,
      "tojson": true
    },
    {
      "name": "Boruch70",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Boruch, Larkin, Wolins, and MacKinney (1970)",
      "object": "Boruch70",
      "class": [
        "matrix",
        "array"
      ],
      "fields": [
        "X.Consideration",
        "X.Structure",
        "X.Sup.Satisfaction",
        "X.Job.Satisfaction",
        "X.Gen.Effectiveness",
        "X.Hum.Relations",
        "X.Leadership",
        "Y.Consideration",
        "Y.Structure",
        "Y.Sup.Satisfaction",
        "Y.Job.Satisfaction",
        "Y.Gen.Effectiveness",
        "Y.Hum.Relations",
        "Y.Leadership"
      ],
      "rows": 14,
      "table": true,
      "tojson": true
    },
    {
      "name": "Box20",
      "title": "Length, width, and height measurements for Thurstone's 20 boxes",
      "object": "Box20",
      "class": [
        "data.frame"
      ],
      "fields": [
        "x",
        "y",
        "z"
      ],
      "rows": 20,
      "table": true,
      "tojson": true
    },
    {
      "name": "Box26",
      "title": "R matrix for Thurstone's 26 hypothetical box attributes.",
      "object": "Box26",
      "class": [
        "matrix",
        "array"
      ],
      "fields": [
        "x",
        "y",
        "z",
        "xy",
        "xz",
        "yz",
        "xsqy",
        "xysq",
        "xsqz",
        "xzsq",
        "ysqz",
        "yzsq",
        "x.over.y",
        "y.over.x",
        "x.over.z",
        "z.over.x",
        "y.over.z",
        "z.over.y",
        "twox.plus.twoy",
        "twox.plus.twoz",
        "twoy.plus.twoz",
        "root.xsq.plus.ysq",
        "root.xsq.plus.zsq",
        "root.ysq.plus.zsq",
        "xyz",
        "root.xsq.plus.ysq.plus.zsq"
      ],
      "rows": 26,
      "table": true,
      "tojson": true
    },
    {
      "name": "HS9Var",
      "title": "9 Variables from the Holzinger and Swineford (1939) Dataset",
      "object": "HS9Var",
      "class": [
        "data.frame"
      ],
      "fields": [
        "id",
        "sex",
        "ageyr",
        "agemo",
        "school",
        "grade",
        "x1",
        "x2",
        "x3",
        "x4",
        "x5",
        "x6",
        "x7",
        "x8",
        "x9"
      ],
      "rows": 301,
      "table": true,
      "tojson": true
    },
    {
      "name": "HW",
      "title": "Six data sets that yield a Heywood case",
      "object": "HW",
      "class": [
        "list"
      ],
      "fields": [],
      "table": false,
      "tojson": true
    },
    {
      "name": "Jackson67",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Jackson and Singer (1967)",
      "object": "Jackson67",
      "class": [
        "matrix",
        "array"
      ],
      "fields": [
        "DiS.Fem",
        "DiS.Anx",
        "DiS.SomatComplaint",
        "DiS.SDAttitude",
        "DiO.Fem",
        "DiO.Anx",
        "DiO.SomatComplaint",
        "DiO.SDAttitude",
        "WOFD.Fem",
        "WOFD.Anx",
        "WOFD.SomatComplaint",
        "WOFD.SDAttitude",
        "Freq.Fem",
        "Freq.Anx",
        "Freq.SomatComplaint",
        "Freq.SDAttitude",
        "Harm.Fem",
        "Harm.Anx",
        "Harm.SomatComplaint",
        "Harm.SDAttitude"
      ],
      "rows": 20,
      "table": true,
      "tojson": true
    },
    {
      "name": "Malmi79",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Malmi, Underwood, and Carroll (1979).",
      "object": "Malmi79",
      "class": [
        "matrix",
        "array"
      ],
      "fields": [
        "FR.Words",
        "FR.Triads",
        "FR.Sentences",
        "SL.Words",
        "SL.Triads",
        "SL.Sentences",
        "PA.Words",
        "PA.Triads",
        "PA.Sentences",
        "PA.12s.2r",
        "PA.4s.6r",
        "PA.2s.12r"
      ],
      "rows": 12,
      "table": true,
      "tojson": true
    },
    {
      "name": "Thurstone41",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Thurstone and Thurstone (1941).",
      "object": "Thurstone41",
      "class": [
        "matrix",
        "array"
      ],
      "fields": [
        "X.Prefix",
        "X.Suffix",
        "X.Vocab",
        "X.sentence",
        "Y.FLLetters",
        "Y.Letters",
        "Y.Words",
        "Y.Completion",
        "Y.SameOpposite"
      ],
      "rows": 9,
      "table": true,
      "tojson": true
    },
    {
      "name": "ThurstoneBox20",
      "title": "Factor Pattern and Factor Correlations for Thurstone's 20 hypothetical box attributes.",
      "object": "ThurstoneBox20",
      "class": [
        "list"
      ],
      "fields": [],
      "table": false,
      "tojson": true
    },
    {
      "name": "ThurstoneBox26",
      "title": "Factor Pattern Matrix for Thurstone's 26 box attributes.",
      "object": "ThurstoneBox26",
      "class": [
        "matrix",
        "array"
      ],
      "fields": {},
      "rows": 26,
      "table": true,
      "tojson": true
    }
  ],
  "_help": [
    {
      "page": "ACL",
      "title": "Adjective Checklist Data.",
      "topics": [
        "ACL"
      ]
    },
    {
      "page": "adfCor",
      "title": "Asymptotic Distribution-Free Covariance Matrix of Correlations",
      "topics": [
        "adfCor"
      ]
    },
    {
      "page": "adfCov",
      "title": "Asymptotic Distribution-Free Covariance Matrix of Covariances",
      "topics": [
        "adfCov"
      ]
    },
    {
      "page": "alphaR",
      "title": "Generate random R matrices with a known coefficient alpha",
      "topics": [
        "alphaR"
      ]
    },
    {
      "page": "AmzBoxes",
      "title": "Length, width, and height measurements for 98 Amazon shipping boxes",
      "topics": [
        "AmzBoxes"
      ]
    },
    {
      "page": "BadRBY",
      "title": "Improper correlation matrix reported by Bentler and Yuan",
      "topics": [
        "BadRBY"
      ]
    },
    {
      "page": "BadRJN",
      "title": "Improper R matrix reported by Joseph and Newman",
      "topics": [
        "BadRJN"
      ]
    },
    {
      "page": "BadRKtB",
      "title": "Improper R matrix reported by Knol and ten Berge",
      "topics": [
        "BadRKtB"
      ]
    },
    {
      "page": "BadRLG",
      "title": "Improper R matrix reported by Lurie and Goldberg",
      "topics": [
        "BadRLG"
      ]
    },
    {
      "page": "BadRRM",
      "title": "Improper R matrix reported by Rousseeuw and Molenberghs",
      "topics": [
        "BadRRM"
      ]
    },
    {
      "page": "BiFAD",
      "title": "Bifactor Analysis via Direct Schmid-Leiman (DSL) Transformations",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "BiFAD"
      ]
    },
    {
      "page": "bigen",
      "title": "Generate Correlated Binary Data",
      "topics": [
        "bigen"
      ]
    },
    {
      "page": "Boruch70",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Boruch, Larkin, Wolins, and MacKinney (1970)",
      "topics": [
        "Boruch70"
      ]
    },
    {
      "page": "Box20",
      "title": "Length, width, and height measurements for Thurstone's 20 boxes",
      "topics": [
        "Box20"
      ]
    },
    {
      "page": "Box26",
      "title": "R matrix for Thurstone's 26 hypothetical box attributes.",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "Box26"
      ]
    },
    {
      "page": "cb",
      "title": "Cudeck & Browne (1992) model error method",
      "topics": [
        "cb"
      ]
    },
    {
      "page": "cfi",
      "title": "Calculate CFI for two correlation matrices",
      "topics": [
        "cfi"
      ]
    },
    {
      "page": "CompleteRcvx",
      "title": "Complete a Partially Specified Correlation Matrix by Convex Optimization",
      "topics": [
        "CompleteRcvx"
      ]
    },
    {
      "page": "CompleteRdev",
      "title": "Complete a Partially Specified Correlation Matrix by the Method of Differential Evolution",
      "topics": [
        "CompleteRdev"
      ]
    },
    {
      "page": "CompleteRmap",
      "title": "Complete a Partially Specified Correlation Matrix by the Method of Alternating Projections",
      "topics": [
        "CompleteRmap"
      ]
    },
    {
      "page": "corDensity",
      "title": "Generate the marginal density of a correlation from a uniformly sampled R matrix.",
      "topics": [
        "corDensity"
      ]
    },
    {
      "page": "corSample",
      "title": "Sample Correlation Matrices from a Population Correlation Matrix",
      "topics": [
        "corSample"
      ]
    },
    {
      "page": "corSmooth",
      "title": "Smooth a Non PD Correlation Matrix",
      "topics": [
        "corSmooth"
      ]
    },
    {
      "page": "cosMat",
      "title": "Compute the cosine(s) between either 2 matrices or 2 vectors.",
      "topics": [
        "cosMat"
      ]
    },
    {
      "page": "d2r",
      "title": "Convert Degrees to Radians",
      "topics": [
        "d2r"
      ]
    },
    {
      "page": "eap",
      "title": "Compute eap trait estimates for FMP and FUP models",
      "topics": [
        "eap"
      ]
    },
    {
      "page": "eigGen",
      "title": "Generate eigenvalues for R matrices with underlying component structure",
      "topics": [
        "eigGen"
      ]
    },
    {
      "page": "enhancement",
      "title": "Find OLS Regression Coefficients that Exhibit Enhancement",
      "topics": [
        "enhancement"
      ]
    },
    {
      "page": "erf",
      "title": "Utility fnc to compute the components for an empirical response function",
      "topics": [
        "erf"
      ]
    },
    {
      "page": "faAlign",
      "title": "Align the columns of two factor loading matrices",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faAlign"
      ]
    },
    {
      "page": "faBounds",
      "title": "Bounds on the Correlation Between an External Variable and a Common Factor",
      "topics": [
        "faBounds"
      ]
    },
    {
      "page": "faEKC",
      "title": "Calculate Reference Eigenvalues for the Empirical Kaiser Criterion",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faEKC"
      ]
    },
    {
      "page": "faIB",
      "title": "Inter-Battery Factor Analysis by the Method of Maximum Likelihood",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faIB"
      ]
    },
    {
      "page": "faLocalMin",
      "title": "Investigate local minima in faMain objects",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faLocalMin"
      ]
    },
    {
      "page": "fals",
      "title": "Unweighted least squares factor analysis",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "fals"
      ]
    },
    {
      "page": "faMain",
      "title": "Automatic Factor Rotation from Random Configurations with Bootstrap Standard Errors",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faMain"
      ]
    },
    {
      "page": "faMAP",
      "title": "Velicer's minimum partial correlation method for determining the number of major components for a principal components analysis or a factor analysis",
      "topics": [
        "faMAP"
      ]
    },
    {
      "page": "faMB",
      "title": "Multiple Battery Factor Analysis by Maximum Likelihood Methods",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faMB"
      ]
    },
    {
      "page": "fapa",
      "title": "Iterated Principal Axis Factor Analysis (fapa)",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "fapa"
      ]
    },
    {
      "page": "fareg",
      "title": "Regularized Factor Analysis",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "fareg"
      ]
    },
    {
      "page": "faScores",
      "title": "Factor Scores",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faScores"
      ]
    },
    {
      "page": "faSort",
      "title": "Sort a factor loadings matrix",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faSort"
      ]
    },
    {
      "page": "faStandardize",
      "title": "Standardize the Unrotated Factor Loadings",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faStandardize"
      ]
    },
    {
      "page": "faX",
      "title": "Factor Extraction (faX) Routines",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "faX"
      ]
    },
    {
      "page": "FMP",
      "title": "Estimate the coefficients of a filtered monotonic polynomial IRT model",
      "topics": [
        "FMP"
      ]
    },
    {
      "page": "FMPMonotonicityCheck",
      "title": "Utility function for checking FMP monotonicity",
      "topics": [
        "FMPMonotonicityCheck"
      ]
    },
    {
      "page": "fsIndeterminacy",
      "title": "Understanding Factor Score Indeterminacy with Finite Dimensional Vector Spaces",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "fsIndeterminacy"
      ]
    },
    {
      "page": "fungible",
      "title": "Generate Fungible Regression Weights",
      "topics": [
        "fungible"
      ]
    },
    {
      "page": "fungibleExtrema",
      "title": "Locate Extrema of Fungible Regression Weights",
      "topics": [
        "fungibleExtrema"
      ]
    },
    {
      "page": "fungibleL",
      "title": "Generate Fungible Logistic Regression Weights",
      "topics": [
        "fungibleL"
      ]
    },
    {
      "page": "fungibleR",
      "title": "Generate Fungible Correlation Matrices",
      "topics": [
        "fungibleR"
      ]
    },
    {
      "page": "FUP",
      "title": "Estimate the coefficients of a filtered unconstrained polynomial IRT model",
      "topics": [
        "FUP"
      ]
    },
    {
      "page": "gen4PMData",
      "title": "Generate item response data for 1, 2, 3, or 4-parameter IRT models",
      "topics": [
        "gen4PMData"
      ]
    },
    {
      "page": "genCorr",
      "title": "Generate Correlation Matrices with User-Defined Eigenvalues",
      "topics": [
        "genCorr"
      ]
    },
    {
      "page": "GenerateBoxData",
      "title": "Generate Thurstone's Box Data From length, width, and height box measurements",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "GenerateBoxData"
      ]
    },
    {
      "page": "genFMPData",
      "title": "Generate item response data for a filtered monotonic polynomial IRT model",
      "topics": [
        "genFMPData"
      ]
    },
    {
      "page": "genPhi",
      "title": "Create a random Phi matrix with maximum factor correlation",
      "topics": [
        "genPhi"
      ]
    },
    {
      "page": "get_wb_mod",
      "title": "Find an `lm` model to use with the Wu & Browne (2015) model error method",
      "topics": [
        "get_wb_mod"
      ]
    },
    {
      "page": "HS9Var",
      "title": "9 Variables from the Holzinger and Swineford (1939) Dataset",
      "topics": [
        "HS9Var"
      ]
    },
    {
      "page": "HW",
      "title": "Six data sets that yield a Heywood case",
      "topics": [
        "HW"
      ]
    },
    {
      "page": "irf",
      "title": "Plot item response functions for polynomial IRT models.",
      "topics": [
        "irf"
      ]
    },
    {
      "page": "itemDescriptives",
      "title": "Compute basic descriptives for binary-item analysis",
      "topics": [
        "itemDescriptives"
      ]
    },
    {
      "page": "Jackson67",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Jackson and Singer (1967)",
      "topics": [
        "Jackson67"
      ]
    },
    {
      "page": "kurt",
      "title": "Calculate Univariate Kurtosis for a Vector or Matrix",
      "topics": [
        "kurt"
      ]
    },
    {
      "page": "Ledermann",
      "title": "Ledermann's inequality for factor solution identification",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "Ledermann"
      ]
    },
    {
      "page": "Malmi79",
      "title": "Multi-Trait Multi-Method correlation matrix reported by Malmi, Underwood, and Carroll (1979).",
      "topics": [
        "Malmi79"
      ]
    },
    {
      "page": "monte",
      "title": "Simulate Clustered Data with User-Defined Properties",
      "topics": [
        "monte"
      ]
    },
    {
      "page": "monte1",
      "title": "Simulate Multivariate Non-normal Data by Vale & Maurelli (1983) Method",
      "topics": [
        "monte1"
      ]
    },
    {
      "page": "noisemaker",
      "title": "Simulate a population correlation matrix with model error",
      "topics": [
        "noisemaker"
      ]
    },
    {
      "page": "normalCor",
      "title": "Compute Normal-Theory Covariances for Correlations",
      "topics": [
        "normalCor"
      ]
    },
    {
      "page": "normF",
      "title": "Compute the Frobenius norm of a matrix",
      "topics": [
        "normF"
      ]
    },
    {
      "page": "obj_func",
      "title": "Objective function for optimizing RMSEA and CFI",
      "topics": [
        "obj_func"
      ]
    },
    {
      "page": "Omega",
      "title": "Compute Omega hierarchical",
      "topics": [
        "Omega"
      ]
    },
    {
      "page": "orderFactors",
      "title": "Order factor-loadings matrix by the sum of squared factor loadings",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "orderFactors"
      ]
    },
    {
      "page": "plot.monte",
      "title": "Plot Method for Class Monte",
      "topics": [
        "plot.monte"
      ]
    },
    {
      "page": "print.faMain",
      "title": "Print Method for an Object of Class faMain",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "print.faMain"
      ]
    },
    {
      "page": "print.faMB",
      "title": "Print Method for an Object of Class faMB",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "print.faMB"
      ]
    },
    {
      "page": "promaxQ",
      "title": "Conduct an Oblique Promax Rotation",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "promaxQ"
      ]
    },
    {
      "page": "r2d",
      "title": "Convert Radians to Degrees",
      "topics": [
        "r2d"
      ]
    },
    {
      "page": "rarc",
      "title": "Rotate Points on the Surface on an N-Dimensional Ellipsoid",
      "topics": [
        "rarc"
      ]
    },
    {
      "page": "Ravgr",
      "title": "Generate a random R matrix with an average rij",
      "topics": [
        "Ravgr"
      ]
    },
    {
      "page": "Rbounds",
      "title": "Generate random R matrices with user-defined bounds on the correlation coefficients via differential evolution (DE).",
      "topics": [
        "Rbounds"
      ]
    },
    {
      "page": "rcone",
      "title": "Generate a Cone of Regression Coefficient Vectors",
      "topics": [
        "rcone"
      ]
    },
    {
      "page": "rcor",
      "title": "Generate Random PSD Correlation Matrices",
      "topics": [
        "rcor"
      ]
    },
    {
      "page": "rellipsoid",
      "title": "Generate Uniformly Spaced OLS Regression Coefficients that Yield a User-Supplied R-Squared Value",
      "topics": [
        "rellipsoid"
      ]
    },
    {
      "page": "restScore",
      "title": "Plot an ERF using rest scores",
      "topics": [
        "restScore"
      ]
    },
    {
      "page": "RGen",
      "title": "Generate random R matrices with various user-defined properties via differential evolution (DE).",
      "topics": [
        "RGen"
      ]
    },
    {
      "page": "rGivens",
      "title": "Generate Correlation Matrices with Specified Eigenvalues",
      "topics": [
        "rGivens"
      ]
    },
    {
      "page": "Rmad",
      "title": "Generate a random R matrix with given MAD from a user-defined, population R",
      "topics": [
        "Rmad"
      ]
    },
    {
      "page": "rMAP",
      "title": "Generate Correlation Matrices with Specified Eigenvalues",
      "topics": [
        "rMAP"
      ]
    },
    {
      "page": "rmsd",
      "title": "Root Mean Squared Deviation of (A - B)",
      "topics": [
        "rmsd"
      ]
    },
    {
      "page": "rmsea",
      "title": "Calculate RMSEA between two correlation matrices",
      "topics": [
        "rmsea"
      ]
    },
    {
      "page": "RnpdMAP",
      "title": "Generate Random NPD R matrices from a user-supplied population R",
      "topics": [
        "RnpdMAP"
      ]
    },
    {
      "page": "rPCA",
      "title": "Generate a Correlation Matrix from a Truncated PCA Loadings Matrix.",
      "topics": [
        "rPCA"
      ]
    },
    {
      "page": "Rplus",
      "title": "Generate a random R matrix with all rij > 0",
      "topics": [
        "Rplus"
      ]
    },
    {
      "page": "SchmidLeiman",
      "title": "Schmid-Leiman Orthogonalization to a (Rank-Deficient) Bifactor Structure",
      "concept": [
        "Factor Analysis Routines"
      ],
      "topics": [
        "SchmidLeiman"
      ]
    },
    {
      "page": "seBeta",
      "title": "Standard Errors and CIs for Standardized Regression Coefficients",
      "topics": [
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