{
  "alt": "A figure with three tabs. The first is a bar chart of nine eigenvalues with a reference line at one, the first far above it and the second just above it. The second tab is a bar chart of loadings on the first component, positive for measures that rise with deprivation. The third is a bar chart of loadings on the second component, with homeownership on one side and labor-force participation and the college share on the other.",
  "caption": "The first tab is the scree, the eigenvalue of each component. The second gives the weights of the nine measures in the first component, which is the index, and the third gives the weights in the second component, which the index drops.",
  "data": [],
  "id": "nine_measure_pca",
  "source": "ACS 2019-2023, census tracts; principal components of nine standardized measures, unweighted (each tract counted once)",
  "spec": {
    "panels": [
      {
        "figure": {
          "alt": "Bar chart of nine eigenvalues in decreasing order. The first bar is far taller than the rest, the second is above the horizontal reference line at one, and the other seven are below it. Whiskers are short.",
          "caption": "The first bar is the tallest by a wide margin, and the second is the only other bar above the line. Each bar is a component's eigenvalue, which is its variance in units of one standardized measure, with a 95% interval from a bootstrap over counties.",
          "data": [
            {
              "component": "PC1",
              "conf_high": 4.693,
              "conf_low": 4.539,
              "eigenvalue": 4.613
            },
            {
              "component": "PC2",
              "conf_high": 1.478,
              "conf_low": 1.397,
              "eigenvalue": 1.439
            },
            {
              "component": "PC3",
              "conf_high": 0.907,
              "conf_low": 0.854,
              "eigenvalue": 0.88
            },
            {
              "component": "PC4",
              "conf_high": 0.743,
              "conf_low": 0.699,
              "eigenvalue": 0.72
            },
            {
              "component": "PC5",
              "conf_high": 0.543,
              "conf_low": 0.52,
              "eigenvalue": 0.532
            },
            {
              "component": "PC6",
              "conf_high": 0.303,
              "conf_low": 0.274,
              "eigenvalue": 0.286
            },
            {
              "component": "PC7",
              "conf_high": 0.283,
              "conf_low": 0.263,
              "eigenvalue": 0.276
            },
            {
              "component": "PC8",
              "conf_high": 0.186,
              "conf_low": 0.167,
              "eigenvalue": 0.177
            },
            {
              "component": "PC9",
              "conf_high": 0.082,
              "conf_low": 0.073,
              "eigenvalue": 0.077
            }
          ],
          "id": "pca_scree",
          "notes": [
            "The nine eigenvalues sum to nine, so an eigenvalue divided by nine is the share of variance."
          ],
          "spec": {
            "category": "component",
            "categoryLabel": "Principal component",
            "domain": [
              0,
              5.5
            ],
            "format": {
              "digits": 2
            },
            "hi": "conf_high",
            "lo": "conf_low",
            "orientation": "v",
            "reference": 1,
            "referenceLabel": "Kaiser line (eigenvalue 1)",
            "sort": "data",
            "value": "eigenvalue",
            "valueLabel": "Eigenvalue (variance in units of one standardized measure)"
          },
          "subtitle": "Eigenvalue of each principal component of the correlation matrix of nine standardized tract measures, with the Kaiser line at one",
          "title": "Two of nine components have an eigenvalue above one, and the first holds {fact:pca_evr1} of the variance",
          "type": "bar"
        },
        "label": "Scree"
      },
      {
        "figure": {
          "alt": "Horizontal bars for nine loadings on the first component. Five bars extend right of zero and four extend left. No bar is much longer than the others, and whiskers are short.",
          "caption": "Poverty, cash assistance, unemployment, low education and single-parent households load positively, and income, college, labor-force participation and homeownership load negatively. Bars are the loadings and whiskers are 95% intervals from a bootstrap over counties.",
          "data": [
            {
              "conf_high": 0.391,
              "conf_low": 0.386,
              "loading": 0.388,
              "measure": "Poverty rate"
            },
            {
              "conf_high": 0.404,
              "conf_low": 0.4,
              "loading": 0.402,
              "measure": "Households on cash assistance or SNAP"
            },
            {
              "conf_high": 0.262,
              "conf_low": 0.242,
              "loading": 0.252,
              "measure": "Unemployment rate"
            },
            {
              "conf_high": 0.384,
              "conf_low": 0.372,
              "loading": 0.378,
              "measure": "Adults with a high school diploma or less"
            },
            {
              "conf_high": 0.31,
              "conf_low": 0.299,
              "loading": 0.304,
              "measure": "Single-parent households"
            },
            {
              "conf_high": -0.41,
              "conf_low": -0.416,
              "loading": -0.412,
              "measure": "Median household income (log)"
            },
            {
              "conf_high": -0.359,
              "conf_low": -0.37,
              "loading": -0.364,
              "measure": "Adults with a bachelor's degree or more"
            },
            {
              "conf_high": -0.161,
              "conf_low": -0.192,
              "loading": -0.177,
              "measure": "Labor-force participation"
            },
            {
              "conf_high": -0.218,
              "conf_low": -0.247,
              "loading": -0.233,
              "measure": "Homeownership"
            }
          ],
          "id": "pca_pc1",
          "notes": [
            "Loadings are the entries of the first eigenvector of the correlation matrix. Because every measure is standardized, a larger absolute loading means a larger weight on that measure in the score."
          ],
          "spec": {
            "category": "measure",
            "diverging": true,
            "domain": [
              -0.6,
              0.6
            ],
            "format": {
              "digits": 2,
              "signed": true
            },
            "hi": "conf_high",
            "lo": "conf_low",
            "orientation": "h",
            "sort": "desc",
            "value": "loading",
            "valueLabel": "Loading on the first component"
          },
          "subtitle": "Loading of each measure on the first principal component, signed so that a positive loading means the measure rises with deprivation, with 95% intervals",
          "title": "The first component gives weight to all nine measures, with a largest absolute loading of {fact:pca1_max}",
          "type": "bar"
        },
        "label": "First component"
      },
      {
        "figure": {
          "alt": "Horizontal bars for nine loadings on the second component. Homeownership and the share with a high school diploma or less extend left of zero, and labor-force participation, the college share and the other measures extend right.",
          "caption": "Homeownership and the share of adults with a high school diploma or less sit on one side, and labor-force participation and the college share sit on the other. Bars are the loadings and whiskers are 95% intervals from a bootstrap over counties.",
          "data": [
            {
              "conf_high": 0.169,
              "conf_low": 0.132,
              "loading": 0.152,
              "measure": "Poverty rate"
            },
            {
              "conf_high": 0.134,
              "conf_low": 0.098,
              "loading": 0.116,
              "measure": "Households on cash assistance or SNAP"
            },
            {
              "conf_high": 0.211,
              "conf_low": 0.148,
              "loading": 0.181,
              "measure": "Unemployment rate"
            },
            {
              "conf_high": -0.285,
              "conf_low": -0.323,
              "loading": -0.304,
              "measure": "Adults with a high school diploma or less"
            },
            {
              "conf_high": 0.264,
              "conf_low": 0.195,
              "loading": 0.228,
              "measure": "Single-parent households"
            },
            {
              "conf_high": 0.059,
              "conf_low": 0.021,
              "loading": 0.04,
              "measure": "Median household income (log)"
            },
            {
              "conf_high": 0.372,
              "conf_low": 0.334,
              "loading": 0.354,
              "measure": "Adults with a bachelor's degree or more"
            },
            {
              "conf_high": 0.584,
              "conf_low": 0.524,
              "loading": 0.555,
              "measure": "Labor-force participation"
            },
            {
              "conf_high": -0.573,
              "conf_low": -0.61,
              "loading": -0.593,
              "measure": "Homeownership"
            }
          ],
          "id": "pca_pc2",
          "notes": [
            "This component is orthogonal to the first by construction, so the index built from the first component discards it."
          ],
          "spec": {
            "category": "measure",
            "diverging": true,
            "domain": [
              -0.8,
              0.8
            ],
            "format": {
              "digits": 2,
              "signed": true
            },
            "hi": "conf_high",
            "lo": "conf_low",
            "orientation": "h",
            "sort": "desc",
            "value": "loading",
            "valueLabel": "Loading on the second component"
          },
          "subtitle": "Loading of each measure on the second principal component, signed so that the college share loads positively, with 95% intervals",
          "title": "The second component separates homeownership ({fact:pca2_own}) from labor-force participation ({fact:pca2_lfp}) and the college share ({fact:pca2_col})",
          "type": "bar"
        },
        "label": "Second component"
      }
    ]
  },
  "subtitle": "Principal components of poverty, cash assistance, unemployment, education, single-parent households, income, labor-force participation and homeownership across tracts, 2019-2023",
  "title": "The first of nine components holds {fact:pca_evr1} of the variance in nine census measures and the second holds {fact:pca_evr2}",
  "type": "tabs",
  "width": "wide"
}