1. Pearson linear correlation calculating testing multiple correlation. 2. Spearman rank correlation calculating testing 3. Other correlation measures

Transcription

1 STATISTICAL METHODS 1. Introductory lecture 2. Random variables and probability theory 3. Populations and samples 4. Hypotheses testing and parameter estimation 5. Most widely used statistical tests I. Most widely used statistical tests II 7. Linear regression 8. Nonlinear regression 9. Regression model fit 10. Correlation 11. Elements of statistical data modelling 12. Model comparison 13. Variance analysis 14. Covariance analysis 15. Summary of the material, analysis of eamples, discussion

2 INTRODUCTION 1. Pearson linear correlation calculating testing multiple correlation 2. Spearman rank correlation calculating testing 3. Other correlation measures

3 PEARSON LINEAR CORRELATION COEFFICIENT

4 PEARSON CORRELATION COEFFICIENT - definition r y n n i1 y y i n 2 y y i i1 i1 original deviations weight fat weight fat deviations from the mean: n i yi y i mean: i i 2

5 PEARSON CORRELATION COEFFICIENT - definition r y n n i1 y y i n 2 y y i i1 i1 i i 2 1. Measures relationship between 2 variables (,y) 2. Assumptions: Continuous values Values normally distributed Measures a linear relationship 3. Values [ -1, 1 ]

6 PEARSON CORRELATION COEFFICIENT - eamples y y r y = -1 r y = -0.9 y r y = r y = 1 15 y

7 PEARSON CORRELATION COEFFICIENT - eamples y11 1 r y = y11 1 r y = y11 1 r y = y11 1 r y =

8 PEARSON CORRELATION COEFFICIENT - eamples SAMPLE weight fat r =

9 PEARSON CORRELATION COEFFICIENT - testing 1. Hypotheses H 0 : no correlation between weight and fat H 1 : correlation between weight and fat H 0 : r = 0 H 1 : r 0 2. Maimum type I error a MAX = Test: t = r n 2 1 r 2 ~t n 2 4. Type I error probability for t=7.47 is a T = a MAX > a T. H 1 7. There eists a positive correlation between weight and fat

10 WSPÓŁCZYNNIK KORELACJI WIELOKROTNEJ - definicja 1. Measures how one variable can be predicted by a combination of other variables 2. Measures the strength, but not the direction of correlation 3. Values [ 0, 1 ]

11 SPEARMAN RANK CORRELATION COEFFICIENT

12 SPEARMAN CORRELATION COEFFICIENT - definition y 1. Measures relationship between 2 variables (,y) 2. No assumptions regarding variable distribution 3. No assumptions on the linear relationship 4. Measures a monotonic relationship 5. Is based on ranks. Values [ -1, 1 ] i1 1 2 N n N d 2 i 1 Difference in ranks between and y

13 SPEARMAN CORRELATION COEFFICIENT - eamples y11 1 y = y11 1 y = y11 1 y = y11 1 y =

14 SPEARMAN CORRELATION COEFFICIENT - eamples SAMPLE volume [cm 3 ] frequency [Hz] males of Fregata magnificens 2. Correlation between pouch volume and sound frequency = - 0.7

15 SPEARMAN CORRELATION COEFFICIENT - testowanie 1. Hypotheses H 0 : no correlation between volume and frequency H 1 : correlation between volume and frequency H 0 : = 0 H 1 : 0 2. Maimum type I error a MAX = Test: 4. Type I error probability for t=-4.8 is a T = a MAX > a T. H 1 t N ~ t N 7. There eists a negative correlation between volume and frequency 2

16 y y y PEARSON vs. SPEARMAN CORRELATION COEFFICIENTS y y y y y y P S P S P S

17 OTHER CORRELATION MEASURES

18 TAU COEFFICIENT (Kendall) τ = n c n d n(n 1) 2 volum e [cm 3 ] frequency [Hz] Rank 1 Rank 2 # concorda nt # discordan t Sum of rank Rank based correlation measure τ [ 1,1] τ = (18 1) 2 τ = 0.57

19 TAU COEFFICIENT (Kendall) 1. Hypotheses H 0 : no correlation between volume and frequency H 1 : correlation between volume and frequency H 0 : t = 0 H 1 : t 0 2. Maimum type I error a MAX = Test: z = 4. Type I error probability for z=-3.30 is a T = a MAX > a T τ 0 2(2n + 5) 9n(n 1) = = 3.30 ~N(0,1). H 1 7. There eists a negative correlation between volume and frequency

20 GAMMA COEFFICIENT (Goodman i Kruskal) γ = n c n d Rank based correlation measure n c + n d Similar to t γ [ 1,1] 3. Test: z = 3γ n(n 1) 2(2n + 5) = = 3.17 ~N(0,1) 4. Type I error probability for z=-3.17 is a T = a MAX > a T. H 1 7. There eists a negative correlation between volume and frequency

21 LITERATURE

22 1. Pearson linear correlation calculating testing multiple correlation 2. Spearman rank correlation calculating testing 3. Other correlation measures