BIOSTATS 3.6 - CH. 22: THE CHI SQUARE TEST FOR TWO-WAY TABLES

Two-Way Tables

• Experiments employ a two-way or block design when examining two categorical factors across multiple levels.
• Two-way tables arrange data involving two categorical variables, each categorized by levels or treatments derived from a two-way or block design.

Two-Way Table Example

• High schoolers were surveyed about their smoking habits and their parents' smoking habits.
• Parent smoking status is the first factor.
• Student smoking status is the second factor.

Marginal Distribution

• Marginal distributions summarize each factor independently using the "margins" of the table.
• The marginal distribution for parental smoking calculates probabilities for 'both parents', 'one parent', or 'neither parent' smoking.
• With two factors, there are two marginal distributions.
• The marginal distribution for student smoking calculates probabilities for students who 'smoke' or 'don't smoke'.

Conditional Distribution

• Cells in a two-way table represent the intersection of levels.
• These cells can be used for conditional distributions.
• The conditional distribution of student smoking with respect to parental smoking habits determines the probability of a student smoking given their parents' smoking status.
  • P(student smokes | both parents) = 22.5%
  • P(student smokes | one parent) = 18.6%
  • P(student smokes | neither parent) = 13.9%

Hypotheses for Two-Way Tables

• A two-way table has 'r' rows and 'c' columns.
• H0: There is no association between row and column variables.
• Ha: There is an association or relationship between the two variables.
• Comparison involves actual counts from sample data versus expected counts under the null hypothesis of no relationship.

Expected Counts in a Two-Way Table

• The null hypothesis (H0) states that there is no association between row and column variables.
• The expected count in any cell of a two-way table with a true null hypothesis is calculated as: expected count = (row total x column total) / table total

Expected Counts Example

• Cocaine induces short-term physical and mental well-being, potentially leading to increased frequency and dosage.
• A study assesses rehabilitation success rates for cocaine addicts using three treatments: the antidepressant desipramine, standard treatment (lithium), and a placebo.

Chi-Square Test Conditions

• The chi-square test for two-way tables assesses the association between two categorical variables, and sample data can be drawn in two ways:
  • Randomly selecting SRSs from different populations or from a population subjected to different treatments.
    • Examples: girls vaccinated for HPV among 8th and 12th graders or remission status for different treatments
  • By taking one SRS and classifying the individuals according to two categorical variables
    • Example: obesity and ethnicity among high school students
• A Chi-square test can be used when:
  • Very few (no more than 20%) expected counts are less than 5.0
  • All expected counts are ≥ 1.0
  • If a factor has many levels and too many expected counts are too low, levels can be "collapsed" (regrouped) to achieve adequate expected counts.

Chi-Square Test for Two-Way Tables

• H0 states that no association exists between row and column variables.
• Ha states that H0 is not true.
• The X2 statistic is summed over all r x c cells in the table:
  • X2 = Σ ((observed count - expected count)2 / expected count)

Chi-Square Distribution

• When H0 is true, the X2 statistic follows an approximate chi-square distribution with (r – 1)(с – 1) degrees of freedom.
• The P-value is expressed as P(chi-square variable ≥ calculated X2).

Using Table D to Determine P-Value

• Reference Table D to find the p-value based on the calculated X2 and degrees of freedom (df). Degrees of freedom is calculated as (r-1)(c-1)
• If for example X2 = 15.9 and df = 6, then the P-value falls between 0.01 and 0.02.

Chi-Square Statistic Example

• Given a table of "actual/expected counts" with three rows and two columns, determining the degrees of freedom.

Two-Sample Z Test

• A chi-square test in a 2x2 table can be calculated using the two-sample z test.
• The Chi-square test can be calculated using 2-PropZTest; these tests will always give identical P-values.

Interpreting Chi-Square Output

• When the X2 test is statistically significant:
  • The largest components indicate conditions most different from H0.
  • The observed vs. expected counts, or computed proportions, can be compared.

Parental Smoking Example: Chi-Square Test Results

• The sample size is 5375.
• H0: no association between parental and student smoking habits. Ha: H0 is not true.
• Data is okay for a chi-square test.
• Strong evidence reveals an association between parental and student smoking habits (P < 0.001).

Physician-Assisted Suicide Example

• Study: 2013 Gallup study on phrasing affecting physician-assisted suicide opinions.
  • 1,535 adults participated in telephone interviews.
  • 719 participants heard the question in Form A (“end the patient’s life by some painless means”).
  • 816 participants heard the question in Form B (“assist the patient to commit suicide”).
• The Chi-square test statistic for these data is X2 = 57.88.
• Phrasing significantly influences opinions about physician-assisted suicide (P < 0.0005).
• “Painless means” resulted in a substantially higher approval (70% in favor) than “commit suicide” (51% in favor).

Caution with Categorical Data: Simpson's Paradox

• Beware of lurking variables (aka confounders)!
• Associations that hold for groups can reverse when data is combined into a single group.
• This reversal is Simpson's Paradox.

Kidney Stones Example of Simpson's Paradox

• A study compared kidney stone removal success rates between open surgery and percutaneous nephrolithotomy (PCNL).
• Procedures are not chosen randomly; PCNL is for smaller stones with a good chance of success.
  • Open surgery is for more problematic conditions.
• Open surgery had a lower failure rate for both small and large stones.
• More challenging cases with large stones tend to be treated more often with open surgery, making it appear as if the procedure was less reliable overall.
• For small stones, open surgery failure rate was 7% and for percutaneous nephrolithotomy (PCNL) it was 13%.
• For large stones, open surgery failure rate was 27% and for percutaneous nephrolithotomy (PCNL) it was 31%.