Two-way Chi-square

Internet
Demonstrations

Up
2-way Chi-square Ans.

You can link to a two-way chi-square applet that simulates outcomes that reflect varying degrees of dependency between two qualitative variables and varying sample sizes.

Throughout this demonstration, focus only on the chi-square test described in the text, and ignore references to the chi-square test with Yate’s correction, which is rarely appropriate. The "corrected" chi-square test is designed for use with small samples only when the marginal (column and row) totals are fixed in advance. Almost always in practice, including the current demonstration, the marginal totals for at least one of the two qualitative variables are not fixed in advance by the investigator, but instead they are outcomes (dependent variables) produced by the study.

The default values specify that the probability of success equals .60 for Condition 1 and .60 for Condition 2, with a sample size (N) of 10 for each condition. Therefore, since the success/failure ratios (.60/.40) are identical for both conditions, knowledge about Condition 1 or Condition 2 doesn’t improve the predictability of success, and the null hypothesis is true; the two qualitative variables are independent.

  • Read Instructions (printing a hard copy, if possible, for easy reference.)
  • Using the default values, click on "simulate 1" and note how the value of chi-square originates from the observed and expected frequencies to the right. Also notice the representation of each outcome, as significant or non-significant, in the lower right. Repeat until you are satisfied with the process.
  • Still using the default values, click on the "Simulate 1000" or ‘Simulate 5000" and note the proportions of significant outcomes. Repeat after changing the sample size (N) to 100 for each condition.
  • Click on "Begin" to start demonstration, and when finished, return to this page.

Question 1: Does the proportion of statistically significant outcomes (type I errors) correspond approximately to the specified (.05) level of significance, regardless of sample size? Reminder: ignore the results for the "corrected" chi-square (under "E <5").  Answer.

  • Change the probabilities of success (for example, to .70 and .50) and click on "Simulate 5000," noting the proportion of significant outcomes (now correct decisions because the null hypothesis is false) when N = 10 and then when N = 100.
  • Repeat this process by increasing the dependency between variables, that is, increase the difference between the probabilities of success (for example, to .80 and .40).
  • Click on "Begin" to return to demonstration, and when finished, answer the question below.

Question 2: What happens to the proportion of significant outcomes -- that is, the power of the test – with increases in sample size and with increases in the degree of dependency between variables? Answer.