Lecture Notes H: More Hypothesis Testing - ANOVA, the Chi Square Test and the Kolmogorov-Smirnov Test for Normality

Transcription

1 ECON 497 Lecture H Page 1 of 1 Metropolitan State University ECON 497: Research and Forecasting Lecture Notes H: More Hypothesis Testing - ANOVA, the Chi Square Test and the Kolmogorov-Smirnov Test for Normality These are two additional topics in statistics. ANOVA deals with analysis of data from designed (rather than observational) experiments. The Chi-square test is used in the analysis of proportions of qualitative variables. ANOVA Some Definitions The response variable or the dependent variable is the variable of interest that you are looking at in the experiment. The factors or independent factors, which may be qualitative or quantitative, are variables that affect that value of the response or dependent variable. The factor levels are the values or the levels of the factors to be used or applied in the experiment. The treatments are the combinations of factor levels applied in an experiment. The experimental unit is the object, thing or body on which the treatments are imposed and responses are measured. A completely randomized design is a design for an experiment in which experimental units are randomly selected for the different treatments. If there are p different treatments, one hypothesis about the response or dependent variable is that the mean value is the same for all treatment groups, or that H0 : µ 1 = µ 2 = µ 3 =... = µ H A : At least two of the treatement means differ p The test statistic is MST F = (The explanation of this is in the book.) MSE MSE is the sum of squared errors within groups divided by n-p. Put another way, it is basically a weighted the sum of the variances of each group.

2 ECON 497 Lecture H Page 2 of 2 MST is a version of the variance in which each group is treated as an observation, divided by p-1. The underlying assumptions are 1. Samples are random and independent 2. The p populations are normally distributed 3. The p populations have the same variance. The null is rejected if F>F α where F α denominator degrees of freedom. has p-1 numerator degrees of freedom and n-p Happily, as always, you need only look at the p-value to see if the null is rejected or not. Here s an example in SPSS looking at mean age across races. Explore RACE (EDITED/IMPUTED) Descriptives AGE - 12/31/96 (EDITED/IMPUTED) RACE Statistic Std. Error

3 ECON 497 Lecture H Page 3 of 3

4 ECON 497 Lecture H Page 4 of 4 Oneway ANOVA AGE - 12/31/96 (EDITED/IMPUTED) Sum of Squares df Square F Sig. Between Groups Within Groups Total The output lists the sum of squared deviations (basically the variances) and shows that there are 4 degrees of freedom between groups (five races minus one) and 5412 degrees of freedom within groups (5417 observations minus five races). The F-stat is and the all important p-value is 0.000, meaning that we reject the null hypothesis that mean age is the same across races.

5 ECON 497 Lecture H Page 5 of 5 Chi-square Test A Chi-square test allows you to examine the relationship between two qualitative variables to see whether or not they are independent. For example, you might have information on whether or not people smoke and their level of educational achievement (a qualitative variable that can take several different values). A Chi-square test allows you to test the null hypothesis that there is no relationship between education level and smoking versus the null that the two have some relationship to each other. To do a Chi-square test in SPSS, you choose Analyze/Descriptives/Crosstabs and under the Statistics button, choose Chi-square.

6 ECON 497 Lecture H Page 6 of 6 The output is Crosstabs of Told and Save for the Dolphin Question Case Processing Summary TOLD * SAVE Cases Valid Missing Total N Percent N Percent N Percent % 0.0% % Count TOLD Total 0 1 TOLD * SAVE Crosstabulation SAVE 0 1 Total Chi-Square Tests Pearson Chi-Square Continuity Correction a Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases Asymp. Sig. Value df (2-sided) b a. Computed only for a 2x2 table Exact Sig. (2-sided) Exact Sig. (1-sided) b. 0 cells (.0%) have expected count less than 5. The minimum expected count is The p-value of under the Pearson Chi-Square test is a p-value for the hypothesis test where the null hypothesis is that there is no relationship between whether someone was told that the particular type of dolphin is never seen and whether they would support saving them.

7 ECON 497 Lecture H Page 7 of 7 The Kolmogorov-Smirnov Test for Normality Some of you are now totally addicted to hypothesis tests. I will now drag you further down that road by introducing the Kolmogorov-Smirnov test for normality of distribution. This is of no real value to you for purposes of this course, but it is yet another example of a type of hypothesis test you may encounter at some point. As usual, the important issues are the null hypothesis and the p-value. The null hypothesis is that the variable in question is normally distributed. Let s look at hours worked per week at main job Histogram of Hours Worked Per Week at Main Job Std. Dev = = 38.9 N = Hours/week work at main job Now let s test to see whether or not this is normally distributed according to the Kolmogorov-Smirnov test. To do this, select Analyze/NonParametricTests/1-Sample KS As you see below, you can actually test to see whether the data follow any of four distributions, but let s try normal since that looks like the most likely candidate here.

8 ECON 497 Lecture H Page 8 of 8 : NPar Tests: KS test for Normality of Hours Worked at Main Job One-Sample Kolmogorov-Smirnov Test N Normal Parameters a,b Most Extreme Differences Kolmogorov-Smirnov Z Asymp. Sig. (2-tailed) Std. Deviation Absolute Positive Negative a. Test distribution is Normal. b. Calculated from data. Hours/week work at main job The p-value (here it s an asymptotic p-value because the test statistic is actually a Z value, so this is only technically correct for very large sample sizes) is 0.000, meaning that the data follow some distribution that is significantly different from the normal distribution. We don t know what distribution it follows, but it s not normal.

9 ECON 497 Lecture H Page 9 of 9 Histogram of Age Std. Dev = = 30.1 N = Age NPar Tests: K-S tests for Normal and Uniform Distributions on Age For People between the ages of 6 and 55 One-Sample Kolmogorov-Smirnov Test N Normal Parameters a,b Most Extreme Differences Kolmogorov-Smirnov Z Asymp. Sig. (2-tailed) Std. Deviation Absolute Positive Negative a. Test distribution is Normal. b. Calculated from data. Age One-Sample Kolmogorov-Smirnov Test 2 N Uniform Parameters a,b Most Extreme Differences Kolmogorov-Smirnov Z Asymp. Sig. (2-tailed) Minimum Maximum Absolute Positive Negative a. Test distribution is Uniform. b. Calculated from data. Age