1. Comparing k Populations
Means – One way Analysis of Variance (ANOVA)

2. The F test

3. The F test – for comparing k means
Situation
We have k normal populations
Let mi and s denote the mean and standard deviation of population i.
i = 1, 2, 3, … k.
Note: we assume that the standard deviation for each population is the same.
s1 = s2 = … = sk = s

4. We want to test
against

5. To test
against
use the test statistic

6. the statistic
is called the Between Sum of Squares and is denoted by SSBetween
It measures the variability between samples
k – 1 is known as the Between degrees of freedom and
is called the Between Mean Square and is denoted by MSBetween

7. the statistic
is called the Error Sum of Squares and is denoted by SSError
is known as the Error degrees of freedom and
is called the Error Mean Square and is denoted by MSError

8. then

9. The Computing formula for F:
Compute 1) 2) 3) 4) 5)

10. Then 1) 2) 3)

11. The critical region for the F test
We reject if
Fa is the critical point under the F distribution with n1 = k - 1degrees of freedom in the numerator and n2 = N – k degrees of freedom in the denominator

12. Example
In the following example we are comparing weight gains resulting from the following six diets
Diet 1 - High Protein , Beef
Diet 2 - High Protein , Cereal
Diet 3 - High Protein , Pork
Diet 4 - Low protein , Beef
Diet 5 - Low protein , Cereal
Diet 6 - Low protein , Pork

14. Hence

15. Thus
Thus since F > we reject H0

16. A convenient method for displaying the calculations for the F-test
The ANOVA Table
A convenient method for displaying the calculations for the F-test

17. Anova Table Mean Square F-ratio Between k - 1 SSBetween MSBetween
Source
d.f.
Sum of Squares
Mean Square
F-ratio
Between
k - 1
SSBetween
MSBetween
MSB /MSE
Within
N - k
SSError
MSError
Total
N - 1
SSTotal

18. The Diet Example Mean Square F-ratio Between 5 4612.933 922.587 4.3
Source
d.f.
Sum of Squares
Mean Square
F-ratio
Between 5
4.3
Within 54
(p = )
Total 59

19. Using SPSS
Note: The use of another statistical package such as Minitab is similar to using SPSS

20. Assume the data is contained in an Excel file

21. Each variable is in a column
Weight gain (wtgn)
diet
Source of protein (Source)
Level of Protein (Level)

22. After starting the SSPS program the following dialogue box appears:

23. If you select Opening an existing file and press OK the following dialogue box appears

24. The following dialogue box appears:

25. If the variable names are in the file ask it to read the names
If the variable names are in the file ask it to read the names. If you do not specify the Range the program will identify the Range:
Once you “click OK”, two windows will appear

26. One that will contain the output:

27. The other containing the data:

28. To perform ANOVA select Analyze->General Linear Model-> Univariate

29. The following dialog box appears

30. Select the dependent variable and the fixed factors
Press OK to perform the Analysis

31. The Output

32. Comments
The F-test H0: m1 = m2 = m3 = … = mk against HA: at least one pair of means are different
If H0 is accepted we know that all means are equal (not significantly different)
If H0 is rejected we conclude that at least one pair of means is significantly different.
The F – test gives no information to which pairs of means are different.
One now can use two sample t tests to determine which pairs means are significantly different

33. Fishers LSD (least significant difference) procedure:
Test H0: m1 = m2 = m3 = … = mk against HA: at least one pair of means are different, using the ANOVA F-test
If H0 is accepted we know that all means are equal (not significantly different). Then stop in this case
If H0 is rejected we conclude that at least one pair of means is significantly different, then follow this by
using two sample t tests to determine which pairs means are significantly different

34. Hypothesis testing and Estimation
Linear Regression
Hypothesis testing and Estimation

35. Assume that we have collected data on two variables X and Y. Let
(x1, y1) (x2, y2) (x3, y3) … (xn, yn)
denote the pairs of measurements on the on two variables X and Y for n cases in a sample (or population)

36. The Statistical Model

37. Each yi is assumed to be randomly generated from a normal distribution with
mean mi = a + bxi and
standard deviation s.
(a, b and s are unknown) yi
a + bxi s xi
Y = a + bX
slope = b a

38. The Data The Linear Regression Model
The data falls roughly about a straight line.
Y = a + bX
unseen

39. Fitting the best straight line to “linear” data
The Least Squares Line
Fitting the best straight line
to “linear” data

40. Let
Y = a + b X
denote an arbitrary equation of a straight line.
a and b are known values.
This equation can be used to predict for each value of X, the value of Y.
For example, if X = xi (as for the ith case) then the predicted value of Y is:

41. The residual
can be computed for each case in the sample,
The residual sum of squares (RSS) is
a measure of the “goodness of fit of the line
Y = a + bX to the data

42. The optimal choice of a and b will result in the residual sum of squares
attaining a minimum.
If this is the case than the line:
Y = a + bX
is called the Least Squares Line

43. The equation for the least squares line
Let

44. Computing Formulae:

45. Then the slope of the least squares line can be shown to be:

46. and the intercept of the least squares line can be shown to be:

47. The residual sum of Squares
Computing formula

48. Estimating s, the standard deviation in the regression model :
Computing formula
This estimate of s is said to be based on n – 2 degrees of freedom

49. Sampling distributions of the estimators

50. The sampling distribution slope of the least squares line :
It can be shown that b has a normal distribution with mean and standard deviation

51. Thus
has a standard normal distribution, and
has a t distribution with df = n - 2

52. (1 – a)100% Confidence Limits for slope b :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom

53. Testing the slope
The test statistic is:
- has a t distribution with df = n – 2 if H0 is true.

54. The Critical Region
Reject
df = n – 2
This is a two tailed tests. One tailed tests are also possible

55. The sampling distribution intercept of the least squares line :
It can be shown that a has a normal distribution with mean and standard deviation

56. Thus
has a standard normal distribution and
has a t distribution with df = n - 2

57. (1 – a)100% Confidence Limits for intercept a :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom

58. Testing the intercept
The test statistic is:
- has a t distribution with df = n – 2 if H0 is true.

59. The Critical Region
Reject
df = n – 2

60. Example

61. The following data showed the per capita consumption of cigarettes per month (X) in various countries in 1930, and the death rates from lung cancer for men in   TABLE : Per capita consumption of cigarettes per month (Xi) in n = 11 countries in 1930, and the death rates, Yi (per 100,000), from lung cancer for men in   Country (i) Xi Yi Australia Canada Denmark Finland Great Britain Holland Iceland Norway Sweden Switzerland USA  

63. Fitting the Least Squares Line

64. Fitting the Least Squares Line
First compute the following three quantities:

65. Computing Estimate of Slope (b), Intercept (a) and standard deviation (s),

66. 95% Confidence Limits for slope b :to
t.025 = critical value for the t-distribution with 9 degrees of freedom

67. 95% Confidence Limits for intercept a :
-4.34 to 17.85
t.025 = critical value for the t-distribution with 9 degrees of freedom

68. Y = (0.228)X
95% confidence Limits for slope to
95% confidence Limits for intercept to 17.85

69. Testing the positive slope
The test statistic is:

70. The Critical Region
Reject
df = 11 – 2 = 9
A one tailed test

71. we reject
and conclude

72. Confidence Limits for Points on the Regression Line
The intercept a is a specific point on the regression line.
It is the y – coordinate of the point on the regression line when x = 0.
It is the predicted value of y when x = 0.
We may also be interested in other points on the regression line. e.g. when x = x0
In this case the y – coordinate of the point on the regression line when x = x0 is a + b x0

73. y = a + b x
a + b x0 x0

74. (1- a)100% Confidence Limits for a + b x0 :
ta/2 is the a/2 critical value for the t-distribution with n - 2 degrees of freedom

75. Prediction Limits for new values of the Dependent variable y
An important application of the regression line is prediction.
Knowing the value of x (x0) what is the value of y?
The predicted value of y when x = x0 is:
This in turn can be estimated by:.

76. The predictor
Gives only a single value for y.
A more appropriate piece of information would be a range of values.
A range of values that has a fixed probability of capturing the value for y.
A (1- a)100% prediction interval for y.

77. (1- a)100% Prediction Limits for y when x = x0:
ta/2 is the a/2 critical value for the t-distribution with n - 2 degrees of freedom

78. Example
In this example we are studying building fires in a city and interested in the relationship between:
X = the distance of the closest fire hall and the building that puts out the alarm
and
Y = cost of the damage (1000$)
The data was collected on n = 15 fires.

79. The Data

80. Scatter Plot

81. Computations

82. Computations Continued

83. Computations Continued

84. Computations Continued

85. 95% Confidence Limits for slope b :
4.07 to 5.77
t.025 = critical value for the t-distribution with 13 degrees of freedom

86. 95% Confidence Limits for intercept a :
7.21 to 13.35
t.025 = critical value for the t-distribution with 13 degrees of freedom

87. Least Squares Line
y=4.92x+10.28

88. (1- a)100% Confidence Limits for a + b x0 :
ta/2 is the a/2 critical value for the t-distribution with n - 2 degrees of freedom

89. 95% Confidence Limits for a + b x0 :

90. 95% Confidence Limits for a + b x0

91. (1- a)100% Prediction Limits for y when x = x0:
ta/2 is the a/2 critical value for the t-distribution with n - 2 degrees of freedom

92. 95% Prediction Limits for y when x = x0

93. 95% Prediction Limits for y when x = x0

94. Linear Regression Summary
Hypothesis testing and Estimation

95. (1 – a)100% Confidence Limits for slope b :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom

96. Testing the slope
The test statistic is:
- has a t distribution with df = n – 2 if H0 is true.

97. (1 – a)100% Confidence Limits for intercept a :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom

98. Testing the intercept
The test statistic is:
- has a t distribution with df = n – 2 if H0 is true.

99. (1- a)100% Confidence Limits for a + b x0 :
ta/2 is the a/2 critical value for the t-distribution with n - 2 degrees of freedom

100. (1- a)100% Prediction Limits for y when x = x0:
ta/2 is the a/2 critical value for the t-distribution with n - 2 degrees of freedom

101. Comparing k Populations
Proportions
The c2 test for independence

102. The c2 test for independence

103. 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

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

105. Table: two-way frequency

106. The c2 test for independence
Define
= Expected frequency in the (i,j) th cell in the case of independence.

107. H0: R and C are independent
Then to test
H0: R and C are independent
against
HA: R and C are not independent
Use test statistic
Eij= Expected frequency in the (i,j) th cell in the case of independence.
xij= observed frequency in the (i,j) th cell

108. Sampling distribution of test statistic when H0 is true
- c2 distribution with degrees of freedom n = (r - 1)(c - 1)
Critical and Acceptance Region
Reject H0 if :
Accept H0 if :

110. Standardized residuals
Test statistic
degrees of freedom n = (r - 1)(c - 1) = 9
Reject H0 using a = 0.05

111. Another Example
This data comes from a Globe and Mail study examining the attitudes of the baby boomers.
Data was collected on various age groups

112. One question with responses
Are there differences in weekly consumption of alcohol related to age?

113. Table: Expected frequencies

114. Table: Residuals
Conclusion: There is a significant relationship between age group and weekly alcohol use

115. Examining the Residuals allows one to identify the cells that indicate a departure from independence
Large positive residuals indicate cells where the observed frequencies were larger than expected if independent Large negative residuals indicate cells where the observed frequencies were smaller than expected if independent

116. Another question with responses
In an average week, how many times would you surf the internet?
Are there differences in weekly internet use related to age?

117. Table: Expected frequencies

118. Table: Residuals
Conclusion: There is a significant relationship between age group and weekly internet use

119. Echo (Age 20 – 29)

120. Gen X (Age 30 – 39)

121. Younger Boomers (Age 40 – 49)

122. Older Boomers (Age 50 – 59)

123. Pre Boomers (Age 60+)