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.

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KATU Coverage of King School

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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.

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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 .

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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.

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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

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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.

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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.

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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

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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

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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.

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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 .

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2

i j

W

X Xq

s n

iX jXn

2Ws

Chapter 12Analysis of Variance

Section 12-2Example 12-4

Page #642

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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.

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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.

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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

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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

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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

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