Past exam of the mathematics course of the University of Cambridge 2001 iii Paper 29 5 d Solution Created 2026-10-03 Updated 2026-10-14
A case chromosome is transmitted and a pseudo-control chromosome is untransmitted. Summing the paired table by columns for cases and by rows for controls givesFor the ordinary unpaired chi-squared test of independence, the expected cells are 30,30,70,70. Its uncorrected Pearson statistic isFor the paired McNemar's test, only the discordant transmissions matter, with counts 24 and 18. ThusBoth uncorrected statistics happen to coincide and have the same approximate one-degree-of-freedom chi-squared distribution reference, giving . The data do not show persuasive transmission imbalance at the conventional 5% level.
The equality is a numerical coincidence, not a justification for ignoring pairing. To see the equality criterion for paired and unpaired allele tests, write the original matched cells as and . The unpaired marginal-table statistic iswhereas McNemar's is . Here , , and , giving equality. Changing concordant counts can change the unpaired statistic without changing McNemar's. The transmission disequilibrium test should retain its conditional paired interpretation.
If a continuity correction is used, state it explicitly: McNemar's corrected statistic is ; the corresponding Yates-corrected marginal statistic also coincides here. An exact conditional test uses and gives the two-sided probability .