Longquiz

• Contingency table


  A table that is used to determine whether the distribution of one variable is conditionally dependent (contingent) on the other variable
• 2 x 2 Contingency Table


  • Used to first arrange data during hypothesis testing when the variables are independent
  • Commonly used in, Chi-Square test, Fisher Exact Probability test, McNemar Chi-Square test
  • It has two cells (2x2) in each direction: Columns - outcome or dependent variable, Rows - exposures/groups compared/interventions
• R x C Table

  • Used if there are more than two cells in each direction of a contingency table
  • R stands for ROWS; C stands for COLUMN
• Chi-Square test


  • A non-parametric testing, used for asymmetric distributions (not a normal distribution) of strictly categorical (nominal/dichotomous) data
  • Chi-square test statistic is denoted by: (x2)
• Null hypothesis


  Statement of no association of variables/statement of independence of variables
• Alternative hypothesis


  Statement of association of variables/statement of dependence of variables
• Chi-Square Distribution


  • Distribution is not symmetric; always skewed to the right/positively skewed
  • Values of the chi-square can be zero or positive but they cannot be negative
• Degrees of Freedom (df)

  • Number of sample values that can vary after certain restrictions have been imposed on all data values
  • As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution
  • Critical region for Chi-Square is always right-tailed
• Computing the Test Statistic (x2)

  1. The Chi Square (x2) statistic compares the observed frequencies (O) in each table cell to the expected frequencies (E) under the null hypothesis

  2. x2 = chi-square value

  3. ∑ = summation sign

  4. Observed frequency (O) = actual count values in each category (value from the data gathering)

  5. Expected frequency (E) = predicted counts in each category if the null hypothesis were true (obtained by calculation)
• Making a Statistical Decision

  1. Compare the test statistic (x2) with the critical region

  2. Compare the p value with the level of significance

  3. Compare the hypothesized (test statistic) with the confidence interval
• In the journal article, "Surgery Unfounded for Tarsal Navicular Stress Fracture," by Bruce Jancin (Internal Medicine News, Vol. 42, No. 14), data on treatments to manage stress fracture of the navicular bone using a randomized sample and assignment was obtained.
• Compute for the Expected Frequency

  1. Write table of observed values (O) from the study. Get the total of each columns and rows and the grand total.

  2. Compute for the expected values (E) using observed values. Get the total of each columns and rows and the grand total.

  3. E = TotalRow × TotalColumn/Totalobservation
• All assumptions for the Chi-Square test were fulfilled: Data randomly selected, Frequency counts, Expected values (E)
• d values (E)


  Expected Frequency
• Computing the Expected Value

  1. Write table of observed values (O) from the study

  2. Get the total of each columns and rows and the grand total

  3. Compute for the expected values (E) using observed values

  4. Get the total of each columns and rows and the grand total
• Observed Frequency (O)


  • Surgery
  • WBC
  • NWBC 6 weeks
  • NWBC < 6 weeks
• Expected Frequency (E)


  𝐸 = 𝑇𝑜𝑡��𝑙��𝑜𝑤 × 𝑇𝑜𝑡𝑎𝑙𝐶𝑜𝑙𝑢𝑚𝑛

  𝑇𝑜𝑡𝑎𝑙𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
• Check the test requirement if all assumptions were fulfilled:
• Hypothesis Testing


  1. State the null and alternative Hypothesis

  2. State the level of significance, (α level)

  3. Select the appropriate test statistic

  4. Determine the critical region

  5. Compute the test statistic

  6. Make a statistical decision

  7. Conclusion
• Null Hypothesis (Ho)


  Success is independent of the treatment
• Alternative Hypothesis (Ha)


  Success is dependent on the treatment
• Level of significance (α)

  0.05
• Test Statistic


  Chi-Square Test of Independence: 𝑥2= ∑ (𝑂 −��)2

  𝐸
• Determine the critical region

  1. Obtain the degrees of freedom (df)

  2. Use the Chi-Square distribution table to determine the critical region
• Critical Region
• Compute the test statistic

  1. Get the difference of O & E

  2. Square the difference

  3. Divide the difference by E

  4. Get the total
• Statistical Decision
• Hypothesis Testing using Chi-Square in SPSS

  1. Click Analyze → Descriptive Statistics→ Cross tabs

  2. Put the grouping variable in rows while the variables you wish to compare in columns

  3. Click Statistics. Check the Chi-square and Phi & Cramer's V. Click Continue

  4. Click Cells. Check the OBSERVED & EXPECTED under Counts

  5. Click Continue then OK.
• Crosstabulation


  Shows a contingency table of observed counts & expected count
• Chi-Square Test Results


  Show the values of the pearson chi-square (aka x2 statistic) under asymptomatic significance (2-sided)
• Phi and Cramer's V

  Shows the size of the effect
• Sample Problem on Chi-Square Test of Independence

  1. State the null and alternative Hypothesis

  2. State the level of significance, (α level)

  3. Select the appropriate test statistic

  4. Determine the critical region

  5. Compute the test statistic

  6. Make a statistical decision

  7. Conclusion
• Null Hypothesis (Ho)


  Gender is not associated with the occurrence of heart attack
• Alternative Hypothesis (Ha)


  Gender is associated with the occurrence of heart attack
• Test Statistic


  Chi-Square Test of Independence: 𝑥2 =

  (𝑂−��)2

  𝐸
• Determine the critical region

  1. Perform CrossTabs to determine the number of rows and columns of the table

  2. Obtain the degrees of freedom (df)

  3. Use the Chi-Square distribution to determine the critical region