S519: Evaluation of Information Systems Social Statistics Inferential Statistics Chapter 15: Chi-square

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This week What is chi-square CHIDIST CHITEST Nonparameteric statistics

Parametric statistics A main branch of statistics Assuming 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) Most of the well-known elementary statistical methods are parametric. 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)

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 is opposite of parametric statistics It has its own non-parametric statistical models, inference and statistical tests 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.

Nonparametric statistics Nonparametric statistics (distribution-free statistics) Data with frequencies or percentage Number of kids in difference grades The percentage of people receiving social security Chi-square allows you to test whether a sample of data came from a population with a specific distribution.

One-sample chi-square One-sample chi-square or goodness of fit test includes only one dimension Whether 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 dimensions Whether preference for the school voucher is independent of political party affiliation and gender

Example No CollegeSome CollegeCollege DegreeTotal Level of Education: Question: whether the number of respondents is equally distributed across all levels of education? Approach: 1.84/3=28, 2.Calculate the difference among these three categories

Compute chi-square O: the observed frequency E: the expected frequency One-sample chi-square test

Example Preference for School Voucher formaybeagainsttotal Question: Whether the number of respondents is equally distributed across all opinions One-sample chi-square

Chi-square steps Step1: a statement of null and research hypothesis There is no difference in the frequency or proportion in each category There is difference in the frequency or proportion in each category

Chi-square steps Step2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis 0.05

Chi-square steps Step3: selection of proper test statistic Frequency  nonparametric procedures  chi- square

Chi-square steps Step4. Computation of the test statistic value (called the obtained value) category observed frequency (O) expected frequency (E)D(difference)(O-E) 2 (O-E) 2 /E for maybe against Total

Chi-square steps Step5: Determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic Table B5 df=r-1 (r= number of categories) If the obtained value > the critical value  reject the null hypothesis If the obtained value < the critical value  accept the null hypothesis

Chi-square steps Step6: a comparison of the obtained value and the critical value is made 20.6 and 5.99

Chi-square steps Step 7 and 8: decision time What is your conclusion, why and how to interpret?

Excel functions CHIDIST (x, degree of freedom) CHITEST (actual_range, expected_range)

More non parametric statistics Table 15.1 (P297)

Exercises S-p