1. Statistics Chi-Square

2. 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) ?
? ?

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

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

5. 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

6. 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

7. 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

8. 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

9. … 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

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

11. 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

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

13. 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

14. 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

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

16. (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

17. (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

18. (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

19. (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

20. 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

21. Tests for Count Data
Open the spreadsheet

22. In the row called “P Chi-Sq” Move your curser to the cell next to it
Tests for Count Data
PROJECT QUESTION
In the row called “P Chi-Sq” Move your curser to the cell next to it

23. Go to: “Formulas” “More Functions” “Statistical” “CHISQ.TEST”
Tests for Count Data
PROJECT QUESTION
Go to: “Formulas” “More Functions” “Statistical” “CHISQ.TEST”

24. Excel calls observed data “actual” (nobody else does…)
Tests for Count Data
PROJECT QUESTION
Excel calls observed data “actual” (nobody else does…)

25. Don’t include the total column
Tests for Count Data
PROJECT QUESTION
Don’t include the total column

26. There’s your probability value! Do you reject H0?
Tests for Count Data
PROJECT QUESTION
There’s your probability value! Do you reject H0?

27. Do you reject H0? Nope, the values are not different enough
Tests for Count Data
PROJECT QUESTION
Do you reject H0? Nope, the values are not different enough

28. Do you reject H0? We “fail to reject H0”
Tests for Count Data
PROJECT QUESTION
Do you reject H0? We “fail to reject H0”

29. Tests for Count Data
PROJECT QUESTION
What could we do?

30. What could we do? Increase n
Tests for Count Data
PROJECT QUESTION
What could we do? Increase n

31. 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

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

33. 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!

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

35. Tests for Count Data
How different do the data have to be to be “statistically significant” (allow you to reject the hypothesized data) Add 1 to the “Brothers” and subtract 1 from the “Sisters”

36. Tests for Count Data
PROJECT QUESTION
9.6% - not yet! Try again!

37. Tests for Count Data
PROJECT QUESTION
5.2% - so close!!!! Try again!

38. Tests for Count Data
PROJECT QUESTION
2.7% - finally !!!!

39. Tests for Count Data
The data have to be: Before they are significantly different!
Sibling Study
Brothers
Sisters
Total
Observed # 34 18 52
Expected # 26

40. Tests for Count Data
What could you do to make it more likely that you would find a significant difference?

41. Tests for Count Data
What could you do to make it more likely that you would find a significant difference? Use a bigger sample size “n”

42. Questions?

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

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

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

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

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

48. 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?  

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

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

51. 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

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

53. 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

54. Died or suffered a heart attack Did not die or suffer a heart attack
Chi-Square
PROJECT QUESTION
Here are Dr. Topol’s results:
Died or suffered a heart attack
Did not die or suffer a heart attack
Directional Atherectomy 44
468
Balloon Angioplasty 23
477

55. 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

56. Died or suffered a heart attack Did not die or suffer a heart attack
Chi-Square
PROJECT QUESTION
How would you calculate the expected values???
Died or suffered a heart attack
Did not die or suffer a heart attack
Directional Atherectomy 44
468
Balloon Angioplasty 23
477

57. First calculate row and column totals and the grand total:
Tests for Count Data
PROJECT QUESTION
First calculate row and column totals and the grand total:

58. First calculate row and column totals and the grand total:
Tests for Count Data
PROJECT QUESTION
First calculate row and column totals and the grand total:

59. Calculate expected values using: Exp = (RowTot)(ColTot)/GrandTot
Tests for Count Data
PROJECT QUESTION
Calculate expected values using: Exp = (RowTot)(ColTot)/GrandTot

60. Exp(DirAth/Died) = (RowTot)(ColTot)/GrandTot = (512)(67)/1012
Tests for Count Data
PROJECT QUESTION
Exp(DirAth/Died) = (RowTot)(ColTot)/GrandTot = (512)(67)/1012

61. Fill in the other cells:
Tests for Count Data
PROJECT QUESTION
Fill in the other cells:

62. The expected values are:
Tests for Count Data
PROJECT QUESTION
The expected values are:

63. Now we need a Chi- Square:
Tests for Count Data
PROJECT QUESTION
Now we need a Chi- Square:

64. Now we need a Chi-Square:
Tests for Count Data
PROJECT QUESTION
Now we need a Chi-Square:

65. Now we need a Chi-Square:
Tests for Count Data
PROJECT QUESTION
Now we need a Chi-Square:

66. Can Dr. Topol reject the null hypothesis?
Tests for Count Data
PROJECT QUESTION
Can Dr. Topol reject the null hypothesis?

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

68. Questions?

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

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

71. Tests for Count Data
Or a 3D column chart

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

73. How Excel Should Be (but isn’t…)
Antarctic Cyclones Enter the observed data in blue cells in the table:
40-49S
50-59S
60-79S
Fall
370
526
980
Winter
452
624
1200
Spring
273
513
995
Summer
422
1059
1751

74. Tests for Count Data
PROJECT QUESTION
Are there significant differences in cyclone count for different latitude and season categories?

75. Tests for Count Data
PROJECT QUESTION

76. Questions?