1. Multivariate Data Summary

2. Linear Regression and Correlation

3. Pearson’s correlation coefficient r.

4. Slope and Intercept of the Least Squares line

5. Scatter Plot Patterns
r = 0.0
r = +0.7
r = +0.9
r = +1.0

6. r = -0.7
r = -0.9
r = -1.0

7. Non-Linear Patterns
r can take on arbitrary values between -1 and +1 if the pattern is non-linear depending or how well your can fit a straight line to the pattern

8. The Coefficient of Determination

9. An important Identity in Statistics
(Total variability in Y) = (variability in Y explained by X) + (variability in Y unexplained by X)

10. It can also be shown:
= proportion variability in Y explained by X.
= the coefficient of determination

11. Techniques for summarizing, displaying and graphing
Categorical Data
Techniques for summarizing, displaying and graphing

12. The frequency table The bar graph
Suppose we have collected data on a categorical variable X having k categories – 1, 2, … , k.
To construct the frequency table we simply count for each category (i) of X, the number of cases falling in that category (fi)
To plot the bar graph we simply draw a bar of height fi above each category (i) of X.

13. Example
In this example data has been collected for n = 34,188 subjects.
The purpose of the study was to determine the relationship between the use of Antidepressants, Mood medication, Anxiety medication, Stimulants and Sleeping pills.
In addition the study interested in examining the effects of the independent variables (gender, age, income, education and role) on both individual use of the medications and the multiple use of the medications.

14. Anxiety medication use, Stimulant use and Sleeping pills use. gender,
The variables were:
Antidepressant use,
Mood medication use,
Anxiety medication use,
Stimulant use and
Sleeping pills use.
gender,
age,
income,
education and
Role –
Parent, worker, partner
Parent, partner
Parent, worker
worker, partner
worker only
Parent only
Partner only
No roles

15. Frequency Table for Age

16. Bar Graph for Age

17. Frequency Table for Role

18. Bar Graph for Role

19. The pie chart An alternative to the bar chart Draw a circle (a pie)
Divide the circle into segments with area of each segment proportional to fi or pi = fi /n

20. Example
In this study the population are individuals who received a head injury. (n = 22540)
The variable is the mechanism that caused the head injury (InjMech) with categories:
MVA (Motor vehicle accident)
Falls
Violence
Other VA (Other vehicle accidents)
Accidents (industrial accident)
Other (all other mechanisms for head injury)

21. Graphical and Tabular Display of Categorical Data.
The frequency table
The bar graph
The pie chart

22. The frequency table

23. The bar graph

24. The pie chart

25. Multivariate Categorical Data

26. The two way frequency table
The c2 statistic
Techniques for examining dependence amongst two categorical variables

27. Situation We have two categorical variables R and C.
The number of categories of R is r.
The number of categories of C is c.
We observe n subjects from the population and count
xij = the number of subjects for which R = i and
C = j.
R = rows, C = columns

28. Example
Both Systolic Blood pressure (C) and Serum Chlosterol (R) were meansured for a sample of n = 1237 subjects.
The categories for Blood Pressure are:
<
The categories for Chlosterol are:
<

29. Table: two-way frequency
Serum Cholesterol
Systolic Blood pressure
<127
167+
Total
< 200
117
121 47 22
307 85 98 43 20
246
115
209 68
439
260+ 67 99 46 33
245
388
527
204
118
1237

30. Example This comes from the drug use data. The two variables are:
Age (C) and
Antidepressant Use (R)
measured for a sample of n = 33,957 subjects.

31. Two-way Frequency Table
Percentage antidepressant use vs Age

33. The c2 statistic for measuring dependence amongst two categorical variables
Define
= Expected frequency in the (i,j) th cell in the case of independence.

34. Columns 1 2 3 4 5
Total
x11
x12
x13
x14
x15 R1
x21
x22
x23
x24
x25 R2
x31
x32
x33
x34
x35 R3
x41
x42
x43
x44
x45 R4 C1 C2 C3 C4 C5 N

35. Columns 1 2 3 4 5
Total
E11
E12
E13
E14
E15 R1
E21
E22
E23
E24
E25 R2
E31
E32
E33
E34
E35 R3
E41
E42
E43
E44
E45 R4 C1 C2 C3 C4 C5 n

36. Justification Proportion in column j for row i
overall proportion in column j 1 2 3 4 5
Total
E11
E12
E13
E14
E15 R1
E21
E22
E23
E24
E25 R2
E31
E32
E33
E34
E35 R3
E41
E42
E43
E44
E45 R4 C1 C2 C3 C4 C5 n

37. and Proportion in row i for column j overall proportion in row i 1 2 34 5
Total
E11
E12
E13
E14
E15 R1
E21
E22
E23
E24
E25 R2
E31
E32
E33
E34
E35 R3
E41
E42
E43
E44
E45 R4 C1 C2 C3 C4 C5 n

38. The c2 statistic
Eij= Expected frequency in the (i,j) th cell in the case of independence.
xij= observed frequency in the (i,j) th cell

39. Example: studying the relationship between Systolic Blood pressure and Serum Cholesterol
In this example we are interested in whether Systolic Blood pressure and Serum Cholesterol are related or whether they are independent.
Both were measured for a sample of n = 1237 cases

40. Systolic Blood pressure
Observed frequencies
Serum Cholesterol
Systolic Blood pressure
<127
167+
Total
< 200
117
121 47 22
307 85 98 43 20
246
115
209 68
439
260+ 67 99 46 33
245
388
527
204
118
1237

41. Systolic Blood pressure
Expected frequencies
Serum Cholesterol
Systolic Blood pressure
<127
167+
Total
< 200
96.29
130.79
50.63
29.29
307
77.16
104.8
40.47
23.47
246
137.70
187.03
72.40
41.88
439
260+
76.85
104.38
40.04
23.37
245
388
527
204
118
1237
In the case of independence the distribution across a row is the same for each row The distribution down a column is the same for each column

43. Standardized residuals
The c2 statistic

44. Example This comes from the drug use data. The two variables are:
Role (C) and
Antidepressant Use (R)
measured for a sample of n = 33,957 subjects.

45. Two-way Frequency Table
Percentage antidepressant use vs Role

47. Calculation of c2
The Raw data
Expected frequencies

48. The Residuals
The calculation of c2

49. Example
In this example n = individuals who had been victimized twice by crimes
Rows = crime of first vicitmization
Cols = crimes of second victimization

52. Brief introduction to Statistical Packages
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Brief introduction to Statistical Packages