Chi-Square Test – Two-Way Table Data: Advanced Tests and Measures of Association – Nominal Categories

Checking the Nominal Categories option provides additional chi-square statistics and measures of association, including:

• Adjusted Residual
  • Equivalent to normal z score
  • Red font highlight denotes significant cell residual value
  • Bold red highlight denotes significant cell residual value with Bonferroni adjustment
  • Note: red highlight is only active if Chi-Square P-Value is significant
    
• Cell’s Contribution to Chi-Square
• Additional Chi-Square Tests
• Likelihood Ratio
• McNemar-Bowker Symmetry (Square Table)
  
• For a 2x2 table, McNemar’s test is equivalent to a paired two-proportions test, for example applicable to studying before versus after change in proportion on the same subject. The returned P-Value is exact, based on the binomial distribution.
• Bowker extended McNemar’s test for square tables larger than 2x2. The null hypothesis is that the table is symmetrical (i.e., symmetry of disagreement). This uses Chi-Square as the test statistic.
  
  
• Measures of Association for Nominal Categories
• Pearson's Phi
  
• Pearson’s Phi is equivalent to Pearson’s correlation coefficient for a 2x2 table. It is the most popular measure of association for 2x2 tables.
  
• We recommend the following rules-of-thumb, adapted from Cohen (1988):
• < 0.1 = Very Weak
• 0.1 to < 0.3: Weak (“Small” Effect)
• 0.3 to < 0.5: Moderate (“Medium” Effect)
• > 0.5: Strong (“Large” Effect)
  
• Although Phi is equivalent to Pearson’s correlation for a 2x2 table, we recommend these rules-of-thumb for use in typical contingency tables, rather than those commonly used for correlation (i.e., > 0.9 = Strong).
  
  
• Cramer’s V
• Cramer's V is an extension of Phi for larger tables. It is the most popular measure of association for tables of any size.
• It varies from 0 to 1, with 0 = no association and 1 = perfect association.
• Use Cohen’s rules-of-thumb given above for Phi.
  
• Contingency Coefficient
• An alternative to Phi, varies from 0 to < 1.
• Use Cohen’s rules-of-thumb given above for Phi.
  
• Cohen's Kappa (Agreement - Square Table)
• Kappa is used to measure agreement between two assessors evaluating the same parts or items.
• For an extended Attribute Measurement Systems Analysis use SigmaXL > Measurement Systems Analysis > Attribute MSA.
  
  For Attribute MSA used in Six Sigma quality, the recommendation is Kappa > 0.9 is strong agreement and < 0.7 is weak agreement, but for general use, the less stringent guidelines by Fleiss are recommended:
  
  Kappa: >= 0.75 or so signifies excellent agreement, for most purposes, and <= 0.40 or so signifies poor agreement.
  
  
• Goodman-Kruskal Lambda & Tau and Theil's Uncertainty
• Measures of Proportional Reduction in Predictive Error. The basic concept is a measure that indicates how much knowing the value of the independent variable improves our ability to estimate the value of the dependent variable.
• They are Directional Measures. If the Y dependent variable is in the Rows Category, then use the Rows Dependent measure. If the Y dependent variable is in the Columns Category, then use the Cols Dependent measure. If there is no clear X-Y dependent-independent relationship, then use the Symmetric measures (not available for Tau).
• Use Cohen’s rules-of-thumb for these measures.
  
  
1. Press F3 or click Recall SigmaXL Dialog to recall last dialog. Check Nominal Categories as shown:
   
   
   
2. Click Next. The resulting output is:
   
   
   
   The adjusted residuals are equivalent to normal z values, so for a specified 95% confidence level, any value greater than 1.96 (or less than -1.96) is highlighted in red. This results in a slight difference in interpretation from that of the standardized residuals, but the 3 largest magnitude residuals are consistent.
   
   
   
   As noted above, the Chi-Square P-Value tells us that there is a significant difference across suppliers, in other words, there is association between Supplier and Pass/Fail/Marginal, but it does not tell us the degree or strength of that association. Cramer’s V is used for tables larger than 2x2 and from the rules-of-thumb, the 0.1 value is considered weak (or small effect).