Two-Way Tables Math Example 2

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Example 2

hard
From a 2ร—2 table: Group/Outcome frequencies: A-Success=40, A-Fail=10, B-Success=25, B-Fail=25. Test independence using the chi-square approach and calculate the relative risk.

Solution

  1. 1
    Totals: A=50, B=50; Success=65, Fail=35; Grand=100
  2. 2
    Expected: EA,S=50(65)/100=32.5E_{A,S}=50(65)/100=32.5; EA,F=17.5E_{A,F}=17.5; EB,S=32.5E_{B,S}=32.5; EB,F=17.5E_{B,F}=17.5
  3. 3
    ฯ‡2=(40โˆ’32.5)232.5+(10โˆ’17.5)217.5+(25โˆ’32.5)232.5+(25โˆ’17.5)217.5=1.73+3.21+1.73+3.21=9.88\chi^2 = \frac{(40-32.5)^2}{32.5}+\frac{(10-17.5)^2}{17.5}+\frac{(25-32.5)^2}{32.5}+\frac{(25-17.5)^2}{17.5} = 1.73+3.21+1.73+3.21 = 9.88
  4. 4
    Relative risk: RR=P(SโˆฃA)P(SโˆฃB)=40/5025/50=0.800.50=1.6RR = \frac{P(S|A)}{P(S|B)} = \frac{40/50}{25/50} = \frac{0.80}{0.50} = 1.6; Group A is 60% more likely to succeed

Answer

ฯ‡2=9.88>3.841\chi^2 = 9.88 > 3.841 (df=1). Groups differ significantly. Relative risk = 1.6.
Two-way tables support both chi-square tests of independence and relative risk calculations. Relative risk (A's success rate / B's success rate) = 1.6 means Group A has 60% higher probability of success than Group B โ€” a directly interpretable effect measure.

About Two-Way Tables

A table that displays frequencies for two categorical variables simultaneously, organized with one variable in rows and the other in columns. It shows joint frequencies (individual cells), marginal frequencies (row/column totals), and enables calculation of conditional frequencies.

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More Two-Way Tables Examples

Example 1 medium

A two-way table shows: Smoker/Cancer=30, Smoker/No-Cancer=70, Non-smoker/Cancer=20, Non-smoker/No-Ca

Example 3 easy

A two-way table: Left-handed/Male=12, Left-handed/Female=8, Right-handed/Male=88, Right-handed/Femal

Example 4 hard

Construct a two-way table from this information: 200 students surveyed; 120 prefer online learning;

โ† All Two-Way Tables Examples Two-Way Tables Concept