Describe the chi-square test and its applications.

The Chi-square Test

• χ² = the sum of (observed - expected)² / expected, summed over every cell of the table
• Its logic is straightforward — the larger the discrepancy between what was observed and what would be expected by chance, the larger the value of chi-square, and the less likely the null hypothesis becomes
• The expected frequency for any cell = (row total × column total) / grand total
• Degrees of freedom = (number of rows - 1) × (number of columns - 1), so a 2 × 2 table has one degree of freedom
• The three applications:
• 1. Test of association (independence) — whether two categorical variables are associated: is disease related to exposure, is treatment related to outcome. This is much the commonest use
• 2. Test of goodness of fit — whether an observed distribution matches a theoretical one, as in testing Mendelian ratios
• 3. Test of homogeneity — whether several populations are similar with respect to a characteristic
• The assumptions must be stated, because they determine when the test may be used at all:
• The data must be frequencies (actual counts), not percentages, proportions or means. Applying chi-square to percentages is a common and serious error, since the test depends on the actual numbers
• Observations must be independent — each subject contributes to one cell only. Paired data therefore require McNEMAR’S test instead
• The sample must be reasonably large, with NO expected frequency below 5 in more than about 20% of cells, and none below 1
• Categories must be mutually exclusive and exhaustive
• When the expected numbers are too small, there are three options — combine adjacent categories; apply YATES’S continuity correction in a 2 × 2 table, which subtracts 0.5 from the absolute difference and makes the test more conservative; or use FISHER’S exact test, which calculates the exact probability and is the preferred solution for small numbers
• The interpretation and its limits:
• Chi-square tells you whether an association exists; it does not tell you its strength, its direction, or whether it is causal
• For strength one must calculate the relative risk or odds ratio with its confidence interval, and a good answer says so
• A significant chi-square in a table larger than 2 × 2 tells you only that the groups differ somewhere, not which ones differ, which requires further examination of the cells
• Like every test, its significance depends on sample size, so a trivial association becomes significant in a very large table
• The chi-square test for trend is used where the exposure is ordered (dose categories, social class, age bands); it is more powerful than an ordinary chi-square because it uses the ordering, and it directly tests for a dose-response relationship, which is a Bradford Hill criterion
• Its uses in community medicine — comparing attack rates in an outbreak, immunisation coverage between districts, outcomes between treatment groups, and the association between an exposure and a disease in a case-control study
• Chi-square needs counts, not percentages, since the test depends entirely on the actual numbers
• It tells you whether, not how strongly; the odds ratio or relative risk with an interval gives the strength
• Expected equals row times column over grand total, which is the calculation to show rather than assert
• A two by two table has one degree of freedom, from rows minus one times columns minus one
• Use Fisher exact test for small numbers, which is preferable to Yates correction where expected counts are low
• Yates correction makes the test more conservative by subtracting a half from the absolute difference
• Paired categorical data need McNemar, since chi-square assumes each subject contributes to one cell only
• Chi-square for trend uses the ordering and directly tests a dose-response relationship
• A trend test is more powerful than an ordinary chi-square when the exposure categories are genuinely ordered
• A significant result in a large table says the groups differ somewhere, not which ones differ
• Combine sparse categories where sensible, which is often preferable to abandoning the analysis
• It is the commonest test in community medicine, because so much public health data is categorical
• Goodness of fit compares with a theoretical distribution, which is the second of its three applications