Analysis of Frequency Data - Contingency Tables

Uses

• test for Independence between (nominal or ordinal) variables containing frequencies
• test whether the occurrence of one variable (e.g. eye color {blue, green}) is independent of another (e.g., sex {male, female}) using a Contingency table
• test for Goodness of fit
• identify those cells that are the main contributors for any overall effect using Cell-wise examination

How is this done

• calculate expected frequencies under the null hypothesis (Note: use Goodman's Test for quasi-independence if the diagonal or other cells are structurally zero)
• calculate overall significance of a contingency table with
  • X² Statistic: Σ(Observed frequency_i - Expected frequency_i)²
  • Likelihood-Ratio Statistic (also known as Likelihood Ratio Chi-square, LRX², G², G-statistic, negative log likelihood, nLL): - 2Σf_iln (f_i/f_{i exp})
• compare these values to a Χ² distribution
• These analyses can be performed using Java DataGrinder applets
• If sample sizes are small you can use a Randomization Test (i.e., Monte Carlo Simulation)

Note

• make sure that no expected frequencies are < 3. lump cells with adjacent classes
• X² and G are sample statistics; Χ² is a theoretical frequency distribution.
• G statistic is preferrable to X²:
  • The distribution of G fits a Χ² distribution more closely than does the distribution of X²
  • G-values are additive while X²- values are not. Thus, total G can be partitioned into two components, the G for heterogeneity and the pooled G
  • in more complex designs G is easier to compute than X² as it is calculated from the observed frequencies only
• perform William's correction for estimationg actual type 1 error. Yate's correction is too conservative
• perform a cell-wise examination of the matrix using Freeman-Tukey deviates (o - observed cell frequency; e - expected cell frequency) as:

F-T = Math.sqrt(o)+Math.sqrt(o+1)-Math.sqrt((4*e)+1));
compare to: Math.sqrt(df*c2a[1]/nCells);