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  "Title": "Conditionally Symmetric Multidimensional Gaussian Mixture Model",
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  "Description": "Implements the conditionally symmetric multidimensional\nGaussian mixture model (csmGmm) for large-scale testing of\ncomposite null hypotheses in genetic association applications\nsuch as mediation analysis, pleiotropy analysis, and\nreplication analysis. In such analyses, we typically have J\nsets of K test statistics where K is a small number (e.g. 2 or\n3) and J is large (e.g. 1 million). For each one of the J sets,\nwe want to know if we can reject all K individual nulls. Please\nsee the vignette for a quickstart guide. The paper describing\nthese methods is \"Testing a Large Number of Composite Null\nHypotheses Using Conditionally Symmetric Multidimensional\nGaussian Mixtures in Genome-Wide Studies\" by Sun R, McCaw Z, &\nLin X (Journal of the American Statistical Association 2025,\n<doi:10.1080/01621459.2024.2422124>).",
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    "calc_dens_ind_3d",
    "calc_dens_ind_multiple",
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    "create_plots",
    "find_2d",
    "find_3d",
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    "find_max_means",
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    "prepare_csmgmm_data",
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    "read_gwas_one",
    "read_gwas_set",
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    "symm_fit_cor_EM_fulllik",
    "symm_fit_cor_EM_noAssumption",
    "symm_fit_cor_EM_rho",
    "symm_fit_ind_EM",
    "symm_fit_ind_EM_noAssumption"
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      "topics": [
        "calc_dens_ind_2d"
      ]
    },
    {
      "page": "calc_dens_ind_3d",
      "title": "Calculate J trivariate normal densities (all dimensions are independent) under fitted csmGmm.",
      "topics": [
        "calc_dens_ind_3d"
      ]
    },
    {
      "page": "calc_dens_ind_multiple",
      "title": "Calculate the density of K-dimensional multivariate normal (all dimensions are independent) under fitted acsGmm.",
      "topics": [
        "calc_dens_ind_multiple"
      ]
    },
    {
      "page": "check_incongruous",
      "title": "check_incongruous.R",
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    {
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      "title": "Tells if row x if allTestStats is an incongruous result (has a higher lfdr than a set of test statistics with lower magnitudes). For K=2 case.",
      "topics": [
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      ]
    },
    {
      "page": "find_3d",
      "title": "Tells if row x if allTestStats is an incongruous result (has a higher lfdr than a set of test statistics with lower magnitudes). For K=3 case.",
      "topics": [
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      ]
    },
    {
      "page": "find_4d",
      "title": "Tells if row x if allTestStats is an incongruous result (has a higher lfdr than a set of test statistics with lower magnitudes). For K=4 case.",
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