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Qualitative data – tests of association

Qualitative data – tests of association. The Chi-Square Distribution and Test for Independence Hypothesis testing between two or more categorical variables. Sporiš Goran, PhD. http://kif.hr/predmet/mki http://www.science4performance.com/. Chi-Square Distribution.

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Qualitative data – tests of association

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  1. Qualitative data – tests of association The Chi-Square Distribution and Test for Independence Hypothesis testing between two or more categorical variables Sporiš Goran, PhD. http://kif.hr/predmet/mki http://www.science4performance.com/

  2. Chi-Square Distribution • The chi-square distribution results when independent variables with standard normal distributions are squared and summed.

  3. Chi-square Degrees of freedom • df = (r-1) (c-1) • Where r = # of rows, c = # of columns • Thus, in any 2x2 contingency table, the degrees of freedom = 1. • As the degrees of freedom increase, the distribution shifts to the right and the critical values of chi-square become larger.

  4. Chi-Square Test of Independence

  5. Using the Chi-Square Test • Often used with contingency tables (i.e., crosstabulations) • E.g., gender x student • The chi-square test of independence tests whether the columns are contingent on the rows in the table. • In this case, the null hypothesis is that there is no relationship between row and column frequencies. • H0: The 2 variables are independent.

  6. Requirements for Chi-Square test • Must be a random sample from population • Data must be in raw frequencies • Variables must be independent • Categories for each I.V. must be mutually exclusive and exhaustive

  7. Example Crosstab: Gender x Student Observed Expected

  8. Special Cases • Fisher’s Exact Test • When you have a 2 x 2 table with expected frequencies less than 5. • Strength of Association • Some use Cramer’s V (for any two nominal variables) or Phi (for 2 x 2 tables) to give a value of association between the variables.

  9. Two chi square tests • Goodness of fit • One variable • Determines how well the sample proportions match a pre-specified distribution • Independence • Two variables • Determines whether there is a relationship between two variables

  10. Steps in hypothesis testing • State the hypotheses • null • research • Select an alpha level and determine the critical value • Compute the test statistic • Make a decision

  11. Test for goodness of fit Forms of the null hypothesis • No preference • There is no difference in proportions among the categories • Participants do not prefer one category over another • Example: Pepsi: 50%, Coke 50% • No difference from a comparison population • There is no difference between the sample distribution and a known (population) distribution • Example: ND: 20% Bl, 75% Br, 5% R US: 20% Bl, 75% Br, 5% R

  12. Test for goodness of fit • Null hypothesis • Specifies a distribution of proportions • Research hypothesis • Specifies that the distribution will be different than that indicated in the null hypothesis

  13. Calculating the test statistic • Observed frequencies • the number of individuals from the sample who are classified in a particular category • fo • Expected frequencies • the number of individuals from the sample who are expected to be classified in a particular category • fe

  14. Calculating the test statistic Coin flip: What percentage of people will predict heads? tails?

  15. Calculating the test statistic • Expected frequency = fe = pn • n = 50 (sample size) • fe = .5 x 50 = 25

  16. n = 50 Question: The last five flips were tails. What do you predict for the next flip? Calculating the test statistic

  17. Calculating the test statistic x2 = ∑(fo - fe)2 fe Steps find the difference between fo and fe for each category square the difference divide the squared difference by fe sum the values from all categories

  18. x2 = ∑(fo - fe)2 = 4 + 4 = 8 fe

  19. Chi square distribution Critical range 0 x2 Low chi square Hi chi square

  20. Critical values for chi square distribution Table B.8

  21. Test for goodness of fit • The greater the number of categories, the greater the likelihood of a large observed chi square value • Degrees of freedom (df) • The number of values that are free to vary • df = C – 1 • C = the number of categories

  22. Chi square distribution

  23. Critical values for chi square distribution

  24. Chi square distribution 5% 0 3.84 x2 Critical value (df = 1,  = .05) = 3.84

  25. Goodness of fit • Make a decision • Critical value = 3.84 with df = 1 and  = .05. • Observed chi square = 8.0 • 8.0 > 3.84 • Observed chi square is greater than critical value • We reject the null hypothesis • Conclude that category frequencies are different • People were more likely to predict heads than tails

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