1. GIS and Spatial Analysis1 Summary  Parametric Test  Interval/ratio data  Based on normal distribution  Difference in Variances  Differences are just sampling error  F test with two variances  Difference in Means  t test with small sample with same variance  t test with small sample with different variance  z test with large sample  Difference in a Mean and a Fixed Value  t test with small sample  z test with large sample

2. GIS and Spatial Analysis2 HYPOTHESIS TESTERS III 1.Significance of Pearson’s C (1)Significance of Pearson’s C (1) 2.Significance of Regression Parameters(1)Significance of Regression Parameters(1) Nonparametric Test 3.Chi-Square Statistic (3)Chi-Square Statistic (3) 4.Spearman’s Rank Correlation Coefficient (3)Spearman’s Rank Correlation Coefficient (3) 5.Kolmogorov-Smirnov Test (1)Kolmogorov-Smirnov Test (1) Summary and Next… Geography KHU Jinmu Choi

3. GIS and Spatial Analysis 3 Significance of Pearson’s C To evaluate the correlation between two sets H 0 : Correlation coefficient is not different from 0 t-Statistics with n-2 degree of freedom Ex: the correlation between medium household income and median housing value for the 50 states and D.C: r = 0.79, n=51 t calculated = 9.0196 Test at 0.05 significance level t critical(49, 0.05) = 1.677, t critical(49, 0.025) = 2.011 Reject H 0

4. GIS and Spatial Analysis4 Significance of Reg. Par. Regression parameters Y = a + bx To test the Validity of the model If the estimated slope parameter,, is significantly different from zero; H 0 : = 0 t-Statistics with (n-1) d.f. To test the meaning of the model f-Test for the coefficient of determination: r 2 (1, n-2) degrees of freedom

5. GIS and Spatial Analysis5 Nonparametric Statistics The distribution of the data is not known or the data are categorical variables Nominal or ordinal scale Test methods Chi-square test for nominal data Spearman’s Rank Correlation Coefficient t test for strong ordinal data Kolmogorov-Smirnov D Test for weak ordinal data

6. GIS and Spatial Analysis6 Chi-Square Test Two Nominal data Correlation has meaning or not H0: the observed statistic is not significantly different from zero Chi-Square Statistics calculated

7. AB32317 Chi-Square Test II critical value Degree of freedom: (c-1)(r-1) = 4 Significance level: 0.05 critical value = 9.488 Test result Reject H 0 is different from 0 Correlation has meaning GIS and Spatial Analysis7

8. AB32318 Spearman’s Rank Correlation Correlation with two strong ordered data Rank for crime rate vs. safety of each city Spearman’s rank correlation coefficient (r s ) d i = x i –y i :difference in rank, not high or low value Just how much they differ r s =1: two ranks are the same or very similar r s =-1: two ranks are exactly opposite GIS and Spatial Analysis8

9. AB32319 Examples of r s r s = Meaning? r s =0.8662 Meaning: rankings are strongly and positively correlated GIS and Spatial Analysis9

10. AB323110 Correlation Test H0: the correlation coefficient is not significantly different from 0 t statistics with d.f. = n-1 t calculated with d.f. = 50, r s = 0.8662 t critical value with d.f = 50 Significance level: 0.05 Test result Reject H 0 Correlation has meaning GIS and Spatial Analysis10

11. AB323111 Kolmogorov-Smirnov Test Weak ordered data with group If the distributions of two samples of a weak- ordered variable are significantly different or not H0: two distributions are not significantly different K-S D statistic cp a : cumulative probability of each category in group a cp b : cumulative probability of each category in group b D critical value with n1, n2, significance level If D calculated > D critical value, reject H 0 GIS and Spatial Analysis11

12. AB323112 K-S D Test example GIS and Spatial Analysis12 D calculated with 20, 24 = 0.2333 D critical (20,24, 0.05) = 0.4118 D calculated < D critical value, fail to reject H 0

13. GIS and Spatial Analysis13 Summary Parametric Test Continue Significance of Pearson’s C using t-Statistic Correlation is not different from 0 Significance of Regression Parameters Validity: t-statistic Meaning: F-test Nonparametric Test Chi-Square Statistic Meaning of the correlation between two nominal or categorical data H0: the observed statistic is not significantly different from zero Spearman’s Rank Correlation Coefficient using t-statistic Correlation between two strong ordered data Kolmogorov-Smirnov Test using K-S D-statistic Correlation between two weak ordered data

14. GIS and Spatial Analysis14 Next Lab8: Significance of Correlation Coefficient and Nonparametric Test Statistics Lec9: Point Pattern Descriptor: mean center, standard distance