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

Checking the Ordinal Categories option provides statistics and measures of association appropriate when both row and column category variables are ordinal:

Adjusted Residuals and Cell’s Contribution to Chi-Square
Tests of Association for Ordinal Categories
- Concordant – Discordant
  
  The P-Value for this hypothesis test is from Kendall’s Tau-B, but is the same for all of the Concordant – Discordant ordinal measures: Tau-C, Gamma and Somers’ D. See Agresti (2010). This may differ from other software using an approximation formula.
- Spearman Rank Correlation
Measures of Association for Ordinal Categories with Confidence Intervals

Spearman Rank Correlation

Equivalent to Pearson’s correlation on ranks

Kendall's Tau-B (Square Table), Kendall-Stuart Tau-C (Rectangular Table), Goodman-Kruskal Gamma

Use Tau-B for square tables (no. rows = no. columns) and Tau-C for rectangular tables (no. rows <> no. columns).

These are all Concordant – Discordant measures.

Somers' D (Cols & Rows Dependent, Symmetric)
SigmaXL provides rules-of-thumb for Kendall’s Correlation in Ordinal Attribute MSA (strong association is > 0.8) and Pearson or Spearman Correlation (strong association is > 0.9), however these are in the context of measurement systems analysis, design of experiments or a controlled process study. For typical contingency table applications, we recommend the rules-of-thumb, adapted from Cohen (1988):

Open the file Attribute Data.xlsx, click Example 5 – Salary Sat Sheet tab. This data is in two-way table format and has ordinal categories: Salary in the Rows and Satisfaction Level in the Columns. Note that cells A1:E5 have been pre-selected.

Click SigmaXL > Statistical Tools > Chi-Square Tests > Chi-Square Test & Association – Two-Way Table Data. Note the selection of data includes the Row and Column labels (if we had Row and Column Totals these would NOT be selected). Check Nominal Categories and Ordinal Categories as shown:

Tip: Even if the categories are ordinal, it is sometimes useful to select nominal categories as well for comparison purposes.

Click Next. The resulting output is:

Note that the Chi-Square P-Value is 0.1, indicating that there is no significant association between Salary and Satisfaction when they are treated as nominal categories (although the significant result for McNemar-Bowker does show that there is lack of symmetry in the off diagonals).

Since the Chi-Square P-Value is not significant, the Adjusted Residuals are not highlighted, even though some values are greater than 1.96 (and less than -1.96). This follows the concept used in ANOVA called “Fisher Protected”

Note: 4 out of 16 cells have expected counts less than 5. If more than 20% of the cells have expected counts less than 5 (or if any of the cells have an expected count less than 1), the Chi-Square approximation may be invalid, and Fisher’s Exact should be used. This will be discussed later, but for this example the Fisher’s Monte-Carlo Exact P-Value = 0.095 so does not change the interpretation of the results for the above Chi-Square analysis.

When Salary and Satisfaction are treated as ordinal categories, the more powerful Concordant – Discordant and Spearman Rank Correlation P-Values clearly show that there is a significant association. The Measures of Association for Ordinal Categories show that this is positive, i.e., an increase in Salary is associated with an increase in Satisfaction. However, using the rules-of-thumb given above, we see that the association is weak.

The table row and column cell percentages can be visualized using Excel’s 100% Stacked Column Chart. Select cells A3:E7 of the Chi-Square sheet. Click Excel’s Insert > Insert Column or Bar Chart and select 100% Stacked Column as shown.