Correlation and Regression. Tudor Călinici 2017

Transcription

1 Correlation and Regression Tudor Călinici

2 Objectives To verify the existence of a relation between two quantitative continuous variables using the coefficient of correlation If the correlation exists, to estimate one variable using the other one If the correlation exists, to decide if it has statistical significance To use Microsoft Excel for that 2

3 Goal of studies To describe To explain To predict 3

4 Bi-dimensional statistic series Col. X: X 1, X 2,..., X n S.B.P. Y: Y 1, Y 2,..., Y n. 1. Is there a relation between X and Y? If it is, how strong it is? Correlation coefficient 2. Can I predict the value of Y knowing only the value of X? Linear regression If the relation exist, is it statistical significant? 4

5 The variables are quantitative and continuous Two observations A reason to study the existence of the relation 5

6 Describe For a sample of n persons the researcher collect the two attributes Each person will be associate with a pair (x i, y i ) Each pair will represent a point on a XY chart (x i, y i ) 6

7 Chart example I 7

8 Chart example II 8

9 Chart example III 9

10 Adding the means on X and Y (I) Y II III I IV X 10

11 Adding the means on X and Y (II) II I Y III IV X 11

12 Adding the means on X and Y (III) II I Y III IV X 12

13 Explain n i 1 Sum of the products SPE ( X X)( Y Y) Covariance COV(X,Y) n COV ( X, Y) 1 n ( X X )( Y Y) i 1 i i i i Pearson s coefficient r COV ( X, Y) S S X Y Coeficient of determination d = r 2. 13

14 Correlations Colton (1974) i) r between and 0,25 = no correlation ii) r between 0.25 and 0.50 (or ) = weak going to acceptable correlation iii) r between 0.5 şi 0.75 (or si -0.5) = acceptable going to good correlation iv) r between 0.75 şi 1 (or 1 si -0.75) = strong correlation 14

15 Interpretation of R Pearson s coefficient of correlation The interval [-1, 1] The sign the direction The value the intensity 15

16 Spearman s Coefficient of correlation It describes the relationship between two ordinal variables or between one ordinal and one quantitative variable. The rules for interpretation are the same as for the Pearson s coefficient of correlation 16

17 Regresions to predict Trend line Y(X): y = a + b x 17

18 Situation 100 students had a test. To have an ideea about the results of the test, the professor correct and show the results for 20 students (anonimous) 8, 4, 6, 6, 9, 7, 9, 10, 8, 10, 10, 7, 9, 6, 7, 9, 8, 10, 5, 5 Mean= 7.7; Standard deviation=

19 Situation At the half of the semester, the students had a verification test. For those students whose grades were displayed, the teacher also displayed the verification test grade. Nota test Nota finala

20 Correlation chart 20

21 Regression line 21

22 Compute the coefficients Regresion Line Y(X) min ( a bxi Y i ) a, b R b n i 1 COV ( X, Y). S X a Y b X 2 22

23 Correlation chart 23

24 Statistics 24

25 Cranial perimeter Graphical representation Correlation between lenght and cranial perimeter Lenght 25

26 Cranial perimeter Trend line Correlation between lenght and cranial perimeter Lenght 26

27 Cranial perimeter Correlation chart Correlation between lenght and cranial perimeter y = 0,546x + 3,7441 R² = 0, Lenght 27

28 Remember Scatter Chart On X axis will be represented the variable in the left It has NO legend To be complete you must add the trend line It must contain the equation of the trend line and the determination coefficient 28

29 The statistical significance P between 0 and 1 If p <= 0,05 then the correlation has statistical significance 29

30 Algorithm Check the type of the variable The idea of relation between variables must make sense Verify the existence of the relation If it exists check the statistical significance 30

31 The direction Sign of R or Sign of x from the trend line equation Must be the same! 31

32 Warning! Relation cause 32

33 33

34 Q1 Variables: SBP, DBP R = 0,73 p=0,02 34

35 Q2 Variables: Value of Blood Sugar and Date of Birth R 2 =0,84, p = 0,12 35

36 Q3 Variables: age and height R =0,98 Trend line y=-3x

37 Q4 Variables: age and height R =0,9 Trend line y=3x+110 p=0,1 37

38 Thank you! Happy Holiday and a significantly better New Year! (one-tail test, p<0,05)