1. Welcome to .
Week 13 Tues .
MAT135 Statistics

2. In-Class Project
Cephalic Index Use the giant calipers to measure the width of your head from side to side – be sure to use the inches measurement inside the calipers Then measure the depth of your skull from front to back

3. In-Class Project
To find your cephalic index, divide the width by the depth and multiply by 100

4. In-Class Project
If your cephalic index is greater than 83, it is called “brachycephalic” If your cephalic index is less than 75, it is called “dolichocephalic” Between these two numbers, you are “mesocephalic”

5. In-Class Project
These measurements are used to investigate genealogy – please record your ancestry on the sheet, too Be sure to keep a copy of your results for later!

6. Hypothesis Tests
So far we’ve tested hypotheses about means: μ = value use a z-test μ1= μ2 use a t-test μ1= μ2 = μ3 use an F-test (ANOVA) ?
? ?

7. WHAT ABOUT OTHER TYPES OF DATA AND OTHER HYPOTHESES ???
Hypothesis Tests
WHAT ABOUT OTHER TYPES OF DATA AND OTHER HYPOTHESES ???

8. Tests for Count Data
Another type of data would be counts (frequencies) in categories:
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52

9. Tests for Count Data
The hypothesis you would be testing would not be about mean values…
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52

10. Tests for Count Data
An alternate hypothesis might be: Are brothers and sisters equally likely? Ha: #bro ≠ #sis
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52

11. Tests for Count Data
The hypothesis can be based on science or history or some other “educated guess”
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52

12. Tests for Count Data
A null hypothesis you might have about this data would be: H0: #bro = #sis
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52

13. … We’ll use a test called “Chi-Square” “X 2”
Tests for Count Data
… We’ll use a test called “Chi-Square” “X 2”
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52

14. Tests for Count Data
A Chi-Square is shaped like an F distribution (both are squared)

15. Tests for Count Data
A Chi-Square needs the original data and some “hypothesized” data
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Hypothesized # 26

16. Tests for Count Data
The “hypothesized” data are called “expected” values
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26

17. Tests for Count Data
The hypothesized values must add up to the original total count
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26

18. Tests for Count Data
They will come from the null (want-to-disprove) hypothesis H0: #bro = #sis
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26

19. Tests for Count Data
To calculate the ChiSq, we use the formula: (𝑶−𝑬) 𝟐 𝑬

20. (𝑶−𝑬) 𝟐 𝑬 Calculate O-E Sibling Study Brothers Sisters Total
Chi-Square
PROJECT QUESTION
(𝑶−𝑬) 𝟐 𝑬 Calculate O-E
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
O-E 5 -5

21. (𝑶−𝑬) 𝟐 𝑬 Square the O-E values
Chi-Square
PROJECT QUESTION
(𝑶−𝑬) 𝟐 𝑬 Square the O-E values
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2 25

22. (𝑶−𝑬) 𝟐 𝑬 Divide them by E Sibling Study Brothers Sisters Total
Chi-Square
PROJECT QUESTION
(𝑶−𝑬) 𝟐 𝑬 Divide them by E
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

23. (𝑶−𝑬) 𝟐 𝑬 Add them up: 25/26 + 25/26 = 1.923076923
Chi-Square
PROJECT QUESTION
(𝑶−𝑬) 𝟐 𝑬 Add them up: 25/ /26 =
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

24. Ok, so our CHiSq statistic is 1.923076923 We need a probability!
Chi-Square
PROJECT QUESTION
Ok, so our CHiSq statistic is We need a probability!
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

25. Using “2nd” “0” X2cdp(1.923076923,9999,1) The p value is: .1655178562
Chi-Square
PROJECT QUESTION
Using “2nd” “0” X2cdp( ,9999,1) The p value is:
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

26. The p value is: .1655178562 Do we reject H0?
Chi-Square
PROJECT QUESTION
The p value is: Do we reject H0?
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

27. Chi-Square
PROJECT QUESTION
The p value is: Do we reject H0? Nope, the values are not different enough
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

28. Chi-Square
PROJECT QUESTION
The p value is: Do we reject H0? Nope, the values are not different enough What could we do?
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

29. Chi-Square
PROJECT QUESTION
The p value is: Do we reject H0? Nope, the values are not different enough What could we do? Increase n
Sibling Study
Brothers
Sisters
Total
Observed # 31 21 52
Expected # 26
(O-E)2/E
25/26

30. Tests for Count Data
The t-test and ANOVA F-test were designed to be powerful (reject H0 a lot) even with small sample sizes

31. Tests for Count Data
A Chi-Square test is not very powerful It only rejects the hypothesis when the data are very VERY different

32. Tests for Count Data
This means it is a very conservative test – nobody is going to think you cheated if you use a Chi-square!

33. Tests for Count Data
It also means we don’t have to set a level of practical significance…

34. Questions?

35. Tests for Count Data
Most Chi-Squared tests don’t have specific hypothesized values

36. Tests for Count Data
Most Chi-Squared tests don’t have specific hypothesized values The expected values come from the table of observed data

37. Tests for Count Data
Ha: p1 ≠ p2 H0: p1 = p2

38. Tests for Count Data
To compare two ways to removing plaque clogging arteries, Dr. Eric J. Topol and colleagues conducted a study

39. Tests for Count Data
They randomly assigned 1,012 heart patients to have either directional coronary atherectomy or balloon angioplasty

40. Tests for Count Data
Is there evidence of a significant difference in the two approaches in the proportion of deaths or heart attacks within 6 months of treatment?  

41. What would be Dr. Topol’s alpha-level?
Chi-Square
PROJECT QUESTION
What would be Dr. Topol’s alpha-level?

42. What would be his alternate hypothesis?
Chi-Square
PROJECT QUESTION
What would be his alternate hypothesis?

43. Chi-Square
PROJECT QUESTION
What would be his alternate hypothesis? Ha: p death or heart attack for directional atherectomy ≠ p death or heart attack for balloon angioplasty

44. What would be his null hypothesis?
Chi-Square
PROJECT QUESTION
What would be his null hypothesis?

45. Chi-Square
PROJECT QUESTION
What would be his null hypothesis? H0: p death or heart attack for directional atherectomy = p death or heart attack for balloon angioplasty

46. Died or suffered a heart attack Did not die or suffer a heart attack
Chi-Square
PROJECT QUESTION
Do you think there is a practically significant difference?
Died or suffered a heart attack
Did not die or suffer a heart attack
Directional Atherectomy 44
468
Balloon Angioplasty 23
477

47. Click “MATRX” Go to “EDIT” Make sure it’s on “[A]” Hit “ENTER”
Chi-Square
PROJECT QUESTION
Click “MATRX” Go to “EDIT” Make sure it’s on “[A]” Hit “ENTER”

48. It should look like: MATRIX[A] 1 ×1 [ 0 ]
Chi-Square
PROJECT QUESTION
It should look like: MATRIX[A] 1 ×1 [ 0 ]

49. Chi-Square
PROJECT QUESTION
It wants to know how big our observed data “matrix” is MATRIX[A] 1 ×1 [ 0 ]

50. The size of a matrix is: #rows × #columns MATRIX[A] 1 ×1 [ 0 ]
Chi-Square
PROJECT QUESTION
The size of a matrix is: #rows × #columns MATRIX[A] 1 ×1 [ 0 ]

51. Died or suffered a heart attack Did not die or suffer a heart attack
Chi-Square
PROJECT QUESTION
How big is the matrix? (ignore labels and totals) MATRIX[A] 1 ×1 [ 0 ]
Died or suffered a heart attack
Did not die or suffer a heart attack
Directional Atherectomy 44
468
Balloon Angioplasty 23
477

52. Died or suffered a heart attack Did not die or suffer a heart attack
Chi-Square
PROJECT QUESTION
MATRIX[A] 2 ×2 [ 0 0 ]
Died or suffered a heart attack
Did not die or suffer a heart attack
Directional Atherectomy 44
468
Balloon Angioplasty 23
477

53. Died or suffered a heart attack Did not die or suffer a heart attack
Chi-Square
PROJECT QUESTION
Go to the data row and enter the data values: MATRIX[A] 2 ×2 [ ] [ ]
Died or suffered a heart attack
Did not die or suffer a heart attack
Directional Atherectomy 44
468
Balloon Angioplasty 23
477

54. The TI will calculate the expected values for you!!
Chi-Square
PROJECT QUESTION
The TI will calculate the expected values for you!!

55. Click “STAT” “TESTS” “X2-Test” “ENTER”
Chi-Square
PROJECT QUESTION
Click “STAT” “TESTS” “X2-Test” “ENTER”

56. Chi-Square
PROJECT QUESTION
X2-Test Observed: [A] Expected: [B] Calculate Draw Highlight “Calculate” “ENTER”

57. Chi-Square
PROJECT QUESTION
Can we reject H0?

58. Can we reject H0? At 5% yes, at 1% no…
Chi-Square
PROJECT QUESTION
Can we reject H0? At 5% yes, at 1% no…

59. Chi-Square
PROJECT QUESTION
Can we reject H0? At 5% yes, at 1% no… Remember, the Chi-Square is a VERY conservative test

60. Chi-Square
PROJECT QUESTION
What could Dr. Topol do to make it more likely that he would find a significant difference?

61. Sometimes you want to know what the expected values were for your data
Chi-Square
PROJECT QUESTION
Sometimes you want to know what the expected values were for your data

62. Chi-Square
PROJECT QUESTION
Sometimes you want to know what the expected values were for your data Click “MATRX” “EDIT” Go to [B] “ENTER”

63. Chi-Square
PROJECT QUESTION

64. Questions?

65. Tests for Count Data Suppose we had a larger matrix: Senate Voting Dem
Rep
Indep
Favor . 25 4 2
Oppose . 20 42 1

66. Tests for Count Data It works the same way! Senate Voting Dem Rep
Indep
Favor . 25 4 2
Oppose . 20 42 1

67. Tests for Count Data p = 5.58756E-6 = 0.0000058756 Senate Voting Dem
Rep
Indep
Favor . 25 4 2
Oppose . 20 42 1

68. Tests for Count Data
so you reject H0 and conclude…

69. Tests for Count Data
…there is a statistically significant difference between the way the parties vote on this issue

70. Tests for Count Data
How would we tell which are different?

71. Tests for Count Data
The hi-lo-close graph!

72. Tests for Count Data
Or the 3d column chart

73. Tests for Count Data
Note: Excel won’t handle an expected value of “0” – you must leave these out of your analysis

74. Questions?

75. In-Class Project
Which cephalic are you? European people are believed to be more dolichocephalic African and Middle-Eastern people are more often mesocephalic East Asian, Hispanic and native American people are more often brachycephalic

76. In-Class Project
Does your head reflect your genealogy?

77. In-Class Project It’s not just for people!
Dolichocephalic Mesocephalic Brachycephalic

78. You survived!
Turn in your homework! Don’t forget your homework due next class! See you Thursday!