• A significant chi-square test requires post-hoc analysis to identify specific differences.
• Pairwise Z-tests effectively compare proportions between categories of a categorical variable.
• The Bonferroni correction controls for Type I error during multiple comparisons.
• This method provides clear identification of significant cells within contingency tables.

1. Goodness-of-Fit test: Determines how well the observed data fit a specified theoretical distribution.
2. Test of homogeneity: Assesses whether the distribution of a categorical variable is the same across several populations or groups.
3. Test of independence: Evaluates whether there is a statistically significant association between two qualitative variables.

Based on Table 2, the observed frequencies are 15, 30, 55, and 100. The expected frequency for each cell is calculated using the formula: expected frequency = (row total * column total) / grand total (Preacher, 2001). Applying this to the data in Table 2 yields the following expected frequencies:
(45 × 70)/200 = 15.75
(45 × 130)/200 = 29.25
(155 × 70)/200 = 54.25
(155 × 130)/200 = 100.75
The chi-square test is a non-parametric method that quantifies the discrepancy between these observed and expected frequencies to determine if a significant association exists between the variables. A key advantage of this test, like other non-parametric methods, is that it does not require assumptions about the underlying data distribution, equality of variances, or homogeneity (McHugh, 2013; Sharpe, 2015).
To perform the test, the chi-square statistic is first calculated. The degrees of freedom (df) are then determined as (number of rows - 1) × (number of columns - 1). Finally, the calculated test statistic is compared to the critical value from the chi-square distribution table for the corresponding df. If the test statistic exceeds the critical value, the null hypothesis of independence is rejected (Sharpe, 2015).
The test statistic for the Pearson chi-square test of independence for a contingency table with *r* rows and *c* columns is calculated as follows (Preacher, 2001):

                                                                                     (1)

1. The data must be in the form of frequency counts for each cell.
2. The categories of the variables must be mutually exclusive; that is, each observation can belong to only one category of each variable.
3. Each subject or case must contribute to one and only one cell in the contingency table.
4. The observations must be independent; the value of one observation must not influence another.
5. No more than 20% of the expected frequencies should be less than five, and no expected frequency should be less than one. If these conditions are not met, Fisher's exact test is a more appropriate alternative.

1. Analysis of standardized residuals
2. Partitioning
3. Cell comparison
4. Ransacking

1. Unstandardized residual for ij- the cell =

2. Standardized residual for ij- the cell =                                                                                     (3)
3. Adjusted standardized residual for ij- the cell =                                                                                      (4)

In these formulas, EFij​ denotes the expected frequency for cell (i,j); OFij​ denotes the observed frequency; Ti​ is the marginal total for the i-th row; Tj​ is the marginal total for the j-th column; and T is the grand total of all observations.
As a rule of thumb, standardized residuals are typically interpreted within a range of -2 to +2. A value exceeding an absolute magnitude of 2 suggests that the corresponding cell makes a statistically significant contribution to the overall significance of the chi-square test (MacDonald and Gardner, 2000).
Cell comparison technique
This post-hoc method begins after a significant overall chi-square test is established. It involves selecting pairs of level combinations from the two qualitative variables and performing a statistical test for each pair. The resulting test statistic for a pair is compared to the critical chi-square value for the entire table. A test statistic exceeding this critical value indicates a significant difference in the column proportions between the two levels.
The test statistic in this method is directly influenced by the type of linear contrast used. A contrast is a linear combination of parameters (e.g., column proportions) where the sum of the coefficients is zero (Turner, 2020). Common contrast types include:
-Orthogonal contrasts: A set where the sum of the cross-products of coefficients for any two contrasts is zero (assuming equal sample sizes). A maximum of k-1 orthogonal contrasts are possible for k group means.
-Polynomial contrasts: A specialized subset of orthogonal contrasts used to test for polynomial trends (e.g., linear, quadratic) across ordered means.
-Orthonormal contrasts: Orthogonal contrasts with the additional constraint that the sum of the squared coefficients for each contrast equals one.
For instance, a simple contrast to compare two column proportions would be defined as: Contrast = (1)*P₁ + (-1)*P₂, where the coefficients sum to zero (1+(-1)=0). The subsequent test statistic can be formulated and summarized using this straightforward contrast.
Test statistics
For a pairwise comparison, a simple contrast can be defined as p₁− p₂. The test statistic for this contrast is given by:
          (5)
In this formula, p₁​ and p₂​ represent the estimated proportions for columns 1 and 2, respectively, with qi=1−pi ​. The terms T₁ and T₂​ denote the marginal totals for columns 1 and 2.
The primary limitation of this post-hoc method is the inflation of the Type I error rate due to multiple comparisons. Furthermore, the results can be sensitive to the specific choice of contrast, which may inadvertently influence the interpretation.
Ransacking post-hoc technique
The ransacking technique involves decomposing a larger contingency table into a series of 2x2 subtables for analysis, rather than comparing all cells simultaneously (Goodman, 1969). A significant challenge with this method, particularly for large tables, is the inflation of the Type I error rate. Conducting multiple statistical tests on these subtables increases the probability of falsely rejecting a true null hypothesis.
Partitioning post-hoc technique
The partitioning method systematically reorganizes an *r* x *c* contingency table into a set of independent, orthogonal 2 x 2 subtables, thereby reducing the dimensionality of the original table. Within these subtables, cells that contribute significantly to the overall association are identified. Various techniques for partitioning exist (Goodman, 1971; Read, 1977).
A key advantage of orthogonal partitioning is that it allows for precise control over the Type I error rate. However, the method has two primary limitations: the number of possible orthogonal partitions is limited by the df of the original table, and many of the resulting partitions may not be substantively meaningful. Furthermore, research indicates that unless comparisons are statistically independent, they cannot be treated as separate and unrelated inquiries, a condition that orthogonal partitioning is designed to meet.
Bonferroni correction for post-hoc analysis of contingency tables
Purpose of the test
In post-hoc pairwise comparisons, the family-wise Type I error rate (alpha) increases with each additional test performed. To control this inflation, an adjustment to the significance level is required (Cabin and Mitchell, 2000). The Bonferroni correction is the most common method for this purpose, which involves lowering the per-comparison alpha level to maintain a desired overall error rate (Goodman, 1969).
The total number of pairwise comparisons CC is determined by the number of column levels being compared. For nn column levels, the number of comparisons is given by C=n(n−1)2C=2n(n−1)​. The Bonferroni-adjusted significance level αadjαadj​ is then calculated as:
                                                                                            (6)
The Bonferroni adjustment is applied to the p-values obtained from a series of Z-tests for comparing two independent proportions. The test statistic for each pairwise comparison is calculated as follows:
                                  (7)
Where:
p₁​ and p₂​ are the observed sample proportions for groups 1 and 2,
n₁​ and n₂​ are the sample sizes for the two groups,
p^ is the pooled proportion, calculated as p^=x₁+x₂/n₁+n₂​​, where x₁​ and x₂​ are the number of successes in each group.
The Z-test for pairwise comparisons is used to evaluate differences in column proportions across different row levels. In the results presentation, a common practice is to annotate cell counts with letter codes. Cells that share the same letter indicate that their column proportions are not significantly different from one another within the context of the specific row level being compared (Sharpe, 2015).
Z-test for two independent proportions: assumptions
The validity of the Z-test for comparing two independent proportions relies on the following assumptions (Casella and Berger, 2021):
-Sample size: The sample size should be sufficiently large such that the sampling distribution of the proportion is approximately normal. This condition is typically met when n×p>5 and n×(1−p)>5 for each sample, where n is the sample size and p is the proportion.
-Independence: The data points within each group and between the two groups must be independent.
-Randomization: The data should be obtained through a random process, such as simple random sampling.

-Real data: Isfahan Diabetes Prevention Study (IDPS) Cohort

This study utilizes data from the IDPS, a longitudinal cohort study initiated in 2003. The IDPS cohort originally consisted of 3,483 first-degree relatives of patients diagnosed with type 2 diabetes, who were consecutively selected for participation (Abdoli et al., 2021; Safari et al., 2021).
-Example 2: relationship between education level and patient status
This example investigates the association between education level and patient status, where status is categorized as normal, pre-diabetic (Impaired Glucose Tolerance [IGT] or Impaired Fasting Glucose [IFG]), or diabetic. A chi-square test indicated a significant relationship between these two variables (p=0.027), suggesting that the distribution of patient status differs across education levels.
To identify which specific patient status proportions differed significantly between education levels, a post-hoc analysis was performed using pairwise Z-tests with a Bonferroni correction. The column variable (patient status) has 4 levels, resulting in 6 unique pairwise comparisons
( ). The significance level was set at α = 0.05 for the overall family of tests. The results of this Bonferroni-adjusted post-hoc analysis are presented in Table 3.

1. Navigate to: Analyze > Descriptive Statistics > Crosstabs
2. Specify your Row and Column variables.
3. Click Statistics and select Chi-square.
4. Click Cells. In the "Counts" section, ensure Observed is selected. In the "Z-test" section, check Compare column proportions and select Adjust p-values (Bonferroni method).
5. Click Continue and then OK to run the analysis.