3 This weekWhat is chi-squareCHIDISTNon-parameteric statistics
4 Parametric statistics A main branch of statisticsAssuming data with a type of probability distribution (e.g. normal distribution)Making inferences about the parameters of the distribution (e.g. sample size, factors in the test)Assumption: the sample is large enough to represent the population (e.g. sample size around 30).They are not distribution-free (they require a probability distribution)
5 Nonparametric statistics Nonparametric statistics (distribution-free statistics)Do not rely on assumptions that the data are drawn from a given probability distribution (data model is not specified).It was widely used for studying populations that take on a ranked order (e.g. movie reviews from one to four stars, opinions about hotel ranking). Fits for ordinal data.It makes less assumption. Therefore it can be applied in situations where less is known about the application.It might require to draw conclusion on a larger sample size with the same degree of confidence comparing with parametric statistics.
6 Nonparametric statistics Nonparametric statistics (distribution-free statistics)Data with frequencies or percentageNumber of kids in difference gradesThe percentage of people receiving social security
7 One-sample chi-square One-sample chi-square includes only one dimensionWhether the number of respondents is equally distributed across all levels of education.Whether the voting for the school voucher has a pattern of preference.Two-sample chi-square includes two dimensionsWhether preference for the school voucher is independent of political party affiliation and gender
8 Compute chi-square One-sample chi-square test O: the observed frequencyE: the expected frequency
9 ExampleQuestion: Whether the number of respondents is equally distributed across all opinionsOne-sample chi-squarePreference for School Voucher formaybeagainsttotal23175090
10 Chi-square steps Step1: a statement of null and research hypothesis There is no difference in the frequency or proportion in each categoryThere is difference in the frequency or proportion in each category
11 Chi-square stepsStep2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis0.05
12 Chi-square steps Step3: selection of proper test statistic Frequencynonparametric procedureschi-square
13 Chi-square stepsStep4. Computation of the test statistic value (called the obtained value)categoryobserved frequency (O)expected frequency (E)D(difference)(O-E)2(O-E)2/Efor23307491.63maybe17131695.63against502040013.33Total9020.60
14 Chi-square stepsStep5: Determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statisticTable B5df=r-1 (r= number of categories)If the obtained value > the critical value reject the null hypothesisIf the obtained value < the critical value accept the null hypothesis
15 Chi-square stepsStep6: a comparison of the obtained value and the critical value is made20.6 and 5.99
16 Chi-square steps Step 7 and 8: decision time What is your conclusion, why and how to interpret?
17 Another exampleWe’ll settle the age-old debate of whether people can actually detect their favorite cola based solely on taste. For 30 coke-lovers, I blindfold them, and have them sample 3 colas…is there a true difference, or are these preference differences explainable by chance?
18 HypothesisNull: There are no preferences: The population is divided evenly among the brandsAlternate: There are preferences: The population is not divided evenly among the brands
19 Chance Model df = C -1 = 3 -1 = 2, set α = .05 For df = 2, X2-crit = 5.99
20 Calculate Chi-Square category observed frequency (O) expected frequency (E)D(difference)(O-E)2(O-E)2/ECoke1310390.9Pepsi10.1RC Cola8240.4Total301.4
21 Decision and Conclusion Conclude that the preferences are evenly divided among the colas when the logos are removed.
22 Excel functions CHIDIST (x, degree of freedom) CHIDIST(20.6,2) >0.05
23 More non parametric statistics Table 15.1 (P297)