While you wait: Make sure to have the data used with example 12-1 in your calculators. (ie...
-
Upload
leticia-fugler -
Category
Documents
-
view
214 -
download
1
Transcript of While you wait: Make sure to have the data used with example 12-1 in your calculators. (ie...
While you wait:
Make sure to have the data used with example 12-1 in your calculators. (ie medication vs exercise vs diet)
Make sure to have your assignment for section 12-1 available also.
Bluman, Chapter 12 1
KATU Coverage of King School
Bluman, Chapter 12 2
12-2 The Scheffé Test and the Tukey Test
When the null hypothesis is rejected using the F test, the researcher may want to know where the difference among the means is.
The Scheffé test and the Tukey test are procedures to determine where the significant differences in the means lie after the ANOVA procedure has been performed.
Bluman, Chapter 12 3
The Scheffé Test In order to conduct the Scheffé test,
one must compare the means two at a time, using all possible combinations of means.
For example, if there are three means, the following comparisons must be done:
Bluman, Chapter 12 4
1 2 1 3 2 3 versus versus versus X X X X X X
Formula for the Scheffé Test
where and are the means of the samples being compared, and are the respective sample sizes, and the within-group variance is .
Bluman, Chapter 12 5
2
2 1 1
i j
S
W i j
X XF
s n n
iX jX
in jn
2Ws
F Value for the Scheffé Test
To find the critical value F for the Scheffé test, multiply the critical value for the F test by k 1:
There is a significant difference between the two means being compared when Fs is greater than F.
Bluman, Chapter 12 6
1 C.V. F k
The value of can be found on:1) ANOVA Summary Table2) Calculator
Compare the calculator results with the values on the chart.
7
Source Sum of Squares
d.f. MeanSquares
F
Between
Within (error)
160.13
104.80
2
12
80.07
8.73
9.17
Total 264.93 14
Chapter 12Analysis of Variance
Section 12-2Example 12-3
Page #641
Bluman, Chapter 12 8
Example 12-3: Lowering Blood PressureUsing the Scheffé test, test each pair of means in Example 12–1 to see whether a specific difference exists, at α = 0.05.
Bluman, Chapter 12 9
1 2
2
1 2
21 2
a. For versus ,
1 1
S
W
X X
X XF
s n n
211.8 3.8
18.338.73 1 5 1 5
2 3
2
2 3
22 3
b. For versus ,
1 1
S
W
X X
X XF
s n n
23.8 7.6
4.148.73 1 5 1 5
Example 12-3: Lowering Blood PressureUsing the Scheffé test, test each pair of means in Example 12–1 to see whether a specific difference exists, at α = 0.05.
Bluman, Chapter 12 10
1 3
2
1 3
21 3
c. For versus ,
1 1
S
W
X X
X XF
s n n
211.8 7.6
5.058.73 1 5 1 5
Example 12-3: Lowering Blood Pressure
Bluman, Chapter 12 11
The critical value for the ANOVA for Example 12–1 was F = 3.89, found by using Table H with α = 0.05, d.f.N. = 2, and d.f.D. = 12.
In this case, it is multiplied by k – 1 as shown.
Since only the F test value for part a ( versus ) is greater than the critical value, 7.78, the only significant difference is between and , that is, between medication and exercise.
3.89F
1X 2X
1X 2X
An Additional Note
Bluman, Chapter 12 12
On occasion, when the F test value is greater than the critical value, the Scheffé test may not show any significant differences in the pairs of means. This result occurs because the difference may actually lie in the average of two or more means when compared with the other mean. The Scheffé test can be used to make these types of comparisons, but the technique is beyond the scope of this book.
The Tukey Test The Tukey test can also be used after
the analysis of variance has been completed to make pairwise comparisons between means when the groups have the same sample size.
The symbol for the test value in the Tukey test is q.
Bluman, Chapter 12 13
Formula for the Tukey Test
where and are the means of the samples being compared, is the size of the sample, and the within-group variance is .
Bluman, Chapter 12 14
2
i j
W
X Xq
s n
iX jXn
2Ws
Chapter 12Analysis of Variance
Section 12-2Example 12-4
Page #642
Bluman, Chapter 12 15
Example 12-4: Lowering Blood PressureUsing the Tukey test, test each pair of means in Example 12–1 to see whether a specific difference exists, at α = 0.05.
Bluman, Chapter 12 16
1 2
1 2
2
a. For versus ,
W
X X
X Xq
s n
11.8 3.86.06
8.73 5
1 3
1 3
2
b. For versus ,
W
X X
X Xq
s n
11.8 7.63.18
8.73 5
Example 12-3: Lowering Blood PressureUsing the Tukey test, test each pair of means in Example 12–1 to see whether a specific difference exists, at α = 0.05.
Bluman, Chapter 12 17
3.8 7.62.88
8.73 5
2 3
2 3
2
c. For versus ,
W
X X
X Xq
s n
Example 12-3: Lowering Blood Pressure
Bluman, Chapter 12 18
To find the critical value for the Tukey test, use Table N.The number of means k is found in the row at the top, and the degrees of freedom for are found in the left column (denoted by v). Since k = 3, d.f. = 12, and α = 0.05, the critical value is 3.77.
Example 12-3: Lowering Blood Pressure
Bluman, Chapter 12 19
Hence, the only q value that is greater in absolute value than the critical value is the one for the difference between and . The conclusion, then, is that there is a significant difference in means for medication and exercise.
These results agree with the Scheffé analysis.
1X 2X
Homework
Read section 12.2 Sec 12.2 page 646 #1,2,3,5, 11
Bluman, Chapter 12 20