Upsilon test by example

Promoting dominant function patterns

The Upsilon test promotes non-constant functions that dominate counts in a contingency table, while most other methods promote all mathematical functions.

Here, a contingency table is formed by observed frequencies of two categorical variables: one as the row index and the other as the column index.

Table 1 contains a perfect function pattern tested significant by all methods at \(\alpha=0.05\). The non-constant function pattern is covered by the entire 3 \(\times\) 3 table, thus dominating the table.

Table 2 also contains a perfect function pattern spanning the entire table. However, the table is dominated by the single entry of 16, which can be considered a constant function. The p-value by the Upsilon test is 0.4 demoting the table, while in sharp contrast all other tests call the table significant at much smaller p-values.

In the p-value bar plots hereafter, the \(y\)-axis is \(-\log_{10}\) p-value, with the original p-values printed at the top of bar.

Demoting non-dominant function patterns

Table 3 contains a function pattern dominant in the top-left 2 \(\times\) 2 sub-table. All tests declared this table significant. The function is not perfect but strong, where the entries containing 1 can be considered noise.

Table 4 contains the same 2 \(\times\) 2 sub-table which is no longer dominant. The bottom row of the table dominates the count and presents a constant function pattern. The Upsilon test gave an insignificant p-value of 0.4, while all other tests returned substantially lower p-values calling the pattern significant.

Robust to change in low expected count

Tables 5 and 6 are dominated by the 1 \(\times\) 2 sub-table on the top-left. Both represent constant function patterns and are declared to be insignificant by the Upsilon test at similar p-values close to 1.0. The USP test also declared them insignificant. However, the tables differ in the last column with the entry at the bottom-right corner having low expected counts (given the marginals) of 2/63 and 3/63, respectively. Despite the two tables being highly similar except for the sparse last columns, the remaining tests consider them differ dramatically with Table 5 being insignificant and Table 6 being significant.

This demonstrates that the Upsilon test is not swayed by the instability of small numbers; a tiny shift in count is insufficient to turn a pattern from non-function to function and vice versa.

Lung transplant surgery type and outcome

Table 8 is from a clinical study of lung transplant surgeries (Jung et al., 2003) . Columns represent two surgery options A and B; rows represent four possible outcomes from grade G0 to G3. The p-value by the Upsilon test is 0.12 ( > 0.05), giving an insignificant result, the same as the original study. The USP test gave p-values ranging from 0.05 to 0.10 run-to-run. However, all remaining tests returned p-values smaller than 0.05, contrary to expert intuition in the original study.

A contingency table showing similar testing results

Table 7 presents cross classification of party affiliation by gender from the 2018 General Social Survey (Kim & Eom, 2022). All tests declared a significant association between gender and party affiliation, with the Upsilon test result being the most significant.

How to cite this document

Luo, Xuye, & Song, Mingzhou. (2025). Upsilon test by example. Vignettes, Upsilon: Another Test of Association for Count Data. R package.