Categorical Association

The problem, from first principles

02-correlation-matrix measures how two numeric columns move together. It cannot say anything about two categorical columns such as device type and subscription plan, because there is no order and no arithmetic to correlate. The tool for them starts from a contingency table: a grid counting how many rows fall in each combination of the two categories. If the two columns are unrelated, the counts in the grid follow a predictable pattern, and how far the real counts depart from that pattern is a measure of association. This question builds the table, the departure measure (the chi-square statistic) and a normalized strength score between 0 and 1.

From theory to code

Implement contingency_table(a, b), which counts the rows in every combination of categories, then expected_counts(table), the counts that independence would predict, then chi_square_statistic(table), the total departure from them, then cramers_v(table), the departure rescaled to a 0-to-1 strength. The signatures and docstrings are already in the editor.

Constraints

• a and b are 1D arrays of hashable labels of the same length. contingency_table returns an integer 2D array whose rows are the sorted unique labels of a and whose columns are the sorted unique labels of b, with entry (i, j) the number of rows where a is the i-th label and b is the j-th.
• expected_counts(table) returns a float array of the same shape where entry (i, j) is row_total_i * column_total_j / grand_total.
• chi_square_statistic(table) returns the sum over all cells of (observed - expected) ** 2 / expected as a float. Cells with expected count 0 are skipped.
• cramers_v(table) returns sqrt(chi2 / (n * min(r - 1, c - 1))) for a table with r rows, c columns and grand total n. It returns 0.0 when min(r - 1, c - 1) is 0 or n is 0.
• Inputs must not be modified.

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