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Chi square

chi square test
Goodness of fit
Independence

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Chi square

  1. 1. Chi-square
  2. 2. Applications of Chi-Square • Goodness of fit (適合度檢驗) • Independence (獨立性檢驗) • Homogenecity (同質性檢驗)
  3. 3. Chi-Square Test on Numerical Data • The researcher may believe there’s a relationship between X and Y, but doesn’t want to use regression. • There are outliers or anomalies that prevent us from assuming that the data came from a normal population. • The researcher has numerical data for one variable but not the other.
  4. 4. Chi-Square Goodness of Fit Test • The test is applied when you have one categorical variable from a single population. • It is used to determine whether sample data are consistent with a hypothesized distribution. • For example, suppose a company printed baseball cards. It claimed that 30% of its cards were rookies; 60%, veterans; and 10%, All-Stars. We could gather a random sample of baseball cards and use a chi- square goodness of fit test to see whether our sample distribution differed significantly from the distribution claimed by the company.
  5. 5. Chi-Square Test for Independence • The test is applied when you have two categorical variables from a single population. • It is used to determine whether there is a significant association between the two variables. • For example, in an election survey, voters might be classified by gender (male or female) and voting preference (Democrat, Republican, or Independent). We could use a chi-square test for independence to determine whether gender is related to voting preference.
  6. 6. Chi-Square Test of Homogeneity • The test is applied to a single categorical variable from two different populations. • To determine whether frequency counts are distributed identically across different populations. • In a survey of TV viewing preferences, we might ask respondents to identify their favorite program. We might ask the same question of two different populations, such as males and females. We could use a chi-square test for homogeneity to determine whether male viewing preferences differed significantly from female viewing preferences.
  7. 7. Chi-Square Test for Goodness-of-Fit • The goodness-of-fit (GOF) test helps you decide whether your sample resembles a particular kind of population. • The chi-square test will be used because it is versatile and easy to understand. Purpose of the Test
  8. 8. Chi-Square Test for Goodness-of-Fit • A multinomial distribution is defined by any k probabilities p1, p2, …, pk that sum to unity. • For example, consider the following “official” proportions of M&M colors. Multinomial GOF Test calc
  9. 9. Chi-Square Test for Goodness-of-Fit • The hypotheses are H0: p1 = .30, p2 = .20, p3 = .10, p4 = .10, p5 = .10, p6 = .20 H1: At least one of the pj differs from the hypothesized value • No parameters are estimated (m = 0) and there are c = 6 classes, so the degrees of freedom are n = c – m – 1 = 6 – 0 - 1 Multinomial GOF Test
  10. 10. Chi-Square Test for Goodness-of-Fit • The hypotheses are: H0: The population follows a _____ distribution H1: The population does not follow a ______ distribution • The blank may contain the name of any theoretical distribution (e.g., uniform, Poisson, normal). Hypotheses for GOF
  11. 11. Chi-Square Test for Goodness-of-Fit • Assuming n observations, the observations are grouped into c classes and then the chi-square test statistic is found using: Test Statistic and Degrees of Freedom for GOF where fj = the observed frequency of observations in class j ej = the expected frequency in class j if H0 were true calc
  12. 12. Chi-Square Test for Goodness-of-Fit • If the proposed distribution gives a good fit to the sample, the test statistic will be near zero. • The test statistic follows the chi-square distribution with degrees of freedom n = c – m – 1 where c is the no. of classes used in the test m is the no. of parameters estimated Test Statistic and Degrees of Freedom for GOF
  13. 13. Chi-Square Test for Goodness-of-Fit Test Statistic and Degrees of Freedom for GOF 110  ccmcv 211  ccmcv 312  ccmcv
  14. 14. • For example, for n = 6 and a = .05, c2 .05 = 12.59. Chi-Square Test for Goodness-of-Fit
  15. 15. Chi-Square Test for Goodness-of-Fit • Instead of “fishing” for a good-fitting model, visualize a priori the characteristics of the underlying data-generating process. Data-Generating Situations • Mixtures occur when more than one data-generating process is superimposed on top of one another. Mixtures: A Problem
  16. 16. Chi-Square Test for Goodness-of-Fit • A simple “eyeball” inspection of the histogram or dot plot may suffice to rule out a hypothesized population. Eyeball Tests • Goodness-of-fit tests may lack power in small samples. As a guideline, a chi- square goodness-of-fit test should be avoided if n < 25. Small Expected Frequencies
  17. 17. Normal Chi-Square Goodness-of-Fit Test • Two parameters, m and s, fully describe the normal distribution. • Unless m and s are know a priori, they must be estimated from a sample by using x and s. • Using these statistics, the chi-square goodness-of-fit test can be used. Normal Data Generating Situations
  18. 18. Normal Chi-Square Goodness-of-Fit Test • Transform the sample observations x1, x2, …, xn into standardized values. • Count the sample observations fj within intervals of the form and compare them with the known frequencies ej based on the normal distribution. Method 1: Standardizing the Data x + ks
  19. 19. Normal Chi-Square Goodness-of-Fit Test Method 1: Standardizing the Data Advantage is a standardized scale. Disadvantage is that data are no longer in the original units. Figure 15.14
  20. 20. • The chi-square test is unreliable if the expected frequencies are too small. • Rules of thumb: • Cochran’s Rule requires that ejk > 5 for all cells. • Up to 20% of the cells may have ejk < 5 Small Expected Frequencies • Most agree that a chi-square test is infeasible if ejk < 1 in any cell. • If this happens, try combining adjacent rows to enlarge the expected frequencies.
  21. 21. Chi-Square Test for Independence • A contingency table is a cross-tabulation of n paired observations into categories. • Each cell shows the count of observations that fall into the category defined by its row (r) and column (c) heading. Contingency Tables A B
  22. 22. Chi-Square Test for Independence • For example: Contingency Tables Table 15.1
  23. 23. Chi-Square Test for Independence • In a test of independence for an r x c contingency table, the hypotheses are H0: Variable A is independent of variable B H1: Variable A is not independent of variable B • Use the chi-square test for independence to test these hypotheses. • This non-parametric test is based on frequencies. • The n data pairs are classified into c columns and r rows and then the observed frequency fjk is compared with the expected frequency ejk. Chi-Square Test
  24. 24. Chi-Square Test for Independence • In a test of independence for an r x c contingency table, the hypotheses are H0: Variable A is independent of variable B H1: Variable A is not independent of variable B H0: There is no relationship between the variables. H1: There is a relationship between the variables. • If two categorical variables are related, it means the chance that an individual falls into a particular category for one variable depends upon the particular category they fall into for the other variable. Chi-Square Test
  25. 25. Chi-Square Test for Independence • The critical value comes from the chi-square probability distribution with n degrees of freedom. n = degrees of freedom = (r – 1)(c – 1) where r = number of rows in the table c = number of columns in the table • Appendix E contains critical values for right-tail areas of the chi-square distribution. • The mean of a chi-square distribution is n with variance 2n. Chi-Square Distribution
  26. 26. Chi-Square Test for Independence • Assuming that H0 is true, the expected frequency of row j and column k is: ejk = RjCk/n where Rj = total for row j (j = 1, 2, …, r) Ck = total for column k (k = 1, 2, …, c) n = sample size Expected Frequencies
  27. 27. Chi-Square Test for Independence • Step 1: State the Hypotheses H0: Variable A is independent of variable B H1: Variable A is not independent of variable B • Step 2: Specify the Decision Rule Calculate n = (r – 1)(c – 1) For a given a, look up the right-tail critical value (c2 R) from Appendix E or by using Excel. Reject H0 if c2 R > test statistic. Steps in Testing the Hypotheses
  28. 28. Chi-Square Test for Independence • For example, for n = 6 and a = .05, c2 .05 = 12.59. Steps in Testing the Hypotheses
  29. 29. Chi-Square Test for Independence • Here is the rejection region. Steps in Testing the Hypotheses Figure 15.3
  30. 30. Chi-Square Test for Independence • Step 3: Calculate the Expected Frequencies ejk = RjCk/n • For example, Steps in Testing the Hypotheses
  31. 31. Chi-Square Test for Independence • The chi-square test is unreliable if the expected frequencies are too small. • Rules of thumb: • Cochran’s Rule requires that ejk > 5 for all cells. • Up to 20% of the cells may have ejk < 5 Small Expected Frequencies • Most agree that a chi-square test is infeasible if ejk < 1 in any cell. • If this happens, try combining adjacent rows or columns to enlarge the expected frequencies.
  32. 32. Chi-Square Test for Independence • Chi-square tests for independence can also be used to analyze quantitative variables by coding them into categories. Cross-Tabulating Raw Data For example, the variables Infant Deaths per 1,000 and Doctors per 100,000 can each be coded into various categories:

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    Jul. 16, 2021

chi square test Goodness of fit Independence

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