In the one-way, or completely randomized, ANOVA of the previous section, treat- ments are randomly assigned to all of the persons or other test units in the ex- periment. As a result, the composition of the treatment groups may be such that certain kinds of people or test units are overrepresented in some treatment groups and underrepresented in others, simply by chance. If the characteristics of the participants or test units have a strong influence on the measurements we obtain, we may be largely measuring the differing group compositions rather than the effects of the treatments.
For example, let’s assume we have randomly selected 12 citizens from a small community and these persons are to participate in an experiment intended to compare the night-vision effectiveness of four different headlamp designs. If we have treatment groups of equal size and randomly assign the treatments, as shown in part A of Table 12.5, it’s likely that the representation of older drivers would not be exactly the same in all four groups. This would reduce our ability to compare the headlamp designs, since night vision tends to decrease with age.
In this situation, the value of the variable that we really want to measure (i.e., the distance at which a headlamp enables a suburban traffic sign to be read) is being strongly influenced by another variable (age category) that has not been considered in the experiment.
In the randomized block design presented in this section, persons or test units are first arranged into similar groups, or blocks, before the treatments are as- signed. This allows us to reduce the amount of error variation. For example, in
( 12.4 )
firm. Interpret the results summarized in the following Minitab output.
Analysis of Variance Source DF SS MS F p Factor 2 69.9 35.0 2.45 0.092 Error 92 1311.2 14.3 Total 94 1381.1 Individual 95% CIs for Mean Based on Pooled StDev
Level N Mean StDev ---+---+---+--- DEPT1 30 17.640 3.458 (---*---) DEPT2 25 15.400 3.841 (---*---)
DEPT3 40 16.875 3.956 (---*---) ---+---+---+--- Pooled StDev = 3.775 15.0 16.5 18.0
12.34 A large investment firm claims that no discrepancy exists between the average incomes of its male and female investment counselors. Random samples consisting of 15 male counselors and 17 female counselors have been selected, and the results examined through the use of ANOVA. In terms of this situation, interpret the components of the following Minitab output.
Analysis of Variance
Source DF SS MS F p Factor 1 142.8 142.8 2.70 0.111 Error 30 1586.0 52.9 Total 31 1728.8 Individual 95% CIs for Mean Based on Pooled StDev
Level N Mean StDev ---+---+---+--- MALES 15 47.654 6.963 (---*---) FEMALES 17 43.420 7.530 (---*---)
---+---+---+--- Pooled StDev = 7.271 42.0 45.5 49.0
12.35 Given the summary information in Exercise 12.33, verify the calculation of the 95% confidence interval for each of the treatment means.
12.36 Given the summary information in Exercise 12.34, verify the calculation of the 95% confidence interval for each of the treatment means.
12.37 In general, how do the assumptions, procedures, and results of one-way ANOVA compare with those for the pooled-variances t-test of Chapter 11?
12.38 Use the pooled-variances t-test of Chapter 11 in comparing the sample means for the data of
Exercise 12.28. Is the conclusion consistent with the one reached in that exercise? Explain.
12.39 Use the pooled-variances t-test of Chapter 11 in comparing the sample means for the data of Exercise 12.30. Is the conclusion consistent with the one reached in that exercise? Explain.
TABLE 12.5 In the one-way, or completely
randomized, design of part A, our ability to compare the night-vision effectiveness of headlamp treatments may be reduced as the result of chance differences in the age- category compositions of the groups. The randomized block design in part B ensures that groups will be comparable in terms of the age categories of their members.
Measured: The distance (yards) at which a traffic sign can be read at night.
Treatments: Four headlamp designs, 1, 2, 3, and 4.
Age categories: Y, driver is under 30 years; M, 30–60 years; O, .60 years.
A. Completely Randomized Approach
1. Twelve drivers are randomly selected from the community (e.g., 5 Y, 3 M, and 4 O).
2. One of the four treatments is randomly assigned to each person. With random assignment, the treatment groups could end up like this:
B. Randomized Block Approach
1. Four members from each age group are randomly selected from the community (4 Y, 4 M, and 4 O).
2. For each age category, treatments are randomly assigned to the members. Each treatment group will include one person from each age category:
Y M Y O Y O M Y O O Y M
For night-vision measurement, treatment 2 would be at a distinct disadvantage and treatment 3 would have a distinct advantage.
Members of Treatment Group 1 2 3 4
Y Y Y Y M M M M O O O O Members of Treatment Group 1 2 3 4 4 Y, treatments randomly assigned →
4 M, treatments randomly assigned → 4 O, treatments randomly assigned →
the night-vision experiment just described, use of the randomized block design would ensure that the treatment groups are comparable in terms of the age cat- egories of their members. Exerting this control over the age-category variable (now referred to as a blocking variable) allows us to better compare the effec- tiveness of the headlamp designs, or treatments. The resulting experiment would have a format like that shown in part B of Table 12.5.
Although we are controlling, or blocking, one variable, our primary concern lies in testing whether the population means could be the same for all of the treat- ment groups. Accordingly, the null and alternative hypotheses are
H0: 1525 . . . 5t for treatments 1 through t H1: The population means are not equal.
Model and Assumptions
As with the one-way ANOVA of the previous section, the null and alternative hypotheses can also be expressed in terms of an equation in which each indi- vidual observation is considered to be the sum of several components. For the randomized block design, these components include both treatment and block effects:
In this model, a measurement (xij) associated with block i and treatment j is the sum of the actual population mean for all of the treatments (), the effect of treatment j (j), the effect of block i (i), and a sampling error (ij). We are only controlling for the effect of the blocking variable, not attempting to examine its influence. Thus, the equivalent null and alternative hypotheses are expressed only in terms of the treatments:
H0: j 5 0 for treatments j 5 1 through t (Each treatment has no effect.) H1: jị 0 for at least one of the j 5 1 through t treatments.
(One or more treatments has an effect.) In the randomized block design, there is just one observation or measure- ment for each block-treatment combination. For example, as part B of Table 12.5 shows, headlamp treatment 1 would be administered to one person in the under- 30 category, one person age 30–60, and one person over 60. Thus, each combina- tion of block level and treatment represents a sample of one person or test unit.
The randomized block design assumes (1) for each block-treatment combi- nation (i.e., each combination of the values of i and j), the sample of size 1 has been randomly selected from a population in which the xij values are normally distributed; (2) the variances are equal for the xij values in these populations; and (3) there is no interaction between the blocks and the treatments. In the random- ized block design, interaction is present when the effect of a treatment depends on the block to which it has been administered. For example, in the headlamp experiment in part B of Table 12.5, the performance of a given headlamp design relative to the others should be approximately the same, regardless of which age group is considered. The presence or absence of such interactions will be the subject of a later discussion. As with one-way ANOVA, an alternative method is available if one or more of these assumptions cannot be made. In the case of the randomized block design, this is the Friedman test of Chapter 14.
Procedure
The procedure for carrying out the randomized block design, which is summarized in Table 12.6, is generally similar to the one-way ANOVA described in Table 12.2. The following descriptions correspond to parts A through D of Table 12.6.
Part A: The Null and Alternative Hypotheses
The null and alternative hypotheses are expressed in terms of the equality of the population means for all of the treatment groups.
xij 5 1 j 1 i 1 ij
The effect of treatment j The effect of block i
The overall population mean for all of the treatments
Random error associated with the sampling process An individual observation
or measurement, this is the observation in the ith block for treatment j.
Part B: The Format of the Data to Be Analyzed
The data can be listed in tabular form, as shown, with a separate column for each of the t treatments and a separate row for each of the n blocks. Each cell (com- bination of a block i and a treatment j) contains just one observation. As with one-way, or completely randomized, ANOVA, the grand mean ( 5x ) is the mean of all of the observations that have been recorded.
TABLE 12.6 Summary of the randomized
block design.
A.
B.
H0: 1525 ... 5t
H1: The population means are not all the same.
C.
D.
Source of Degrees of
Variation Sum of Squares Freedom Mean Square F-Ratio Treatments,
TR SSTR 5n o
j 5 1t ( x }.j2 }}x )2 t 2 1 MSTR 5 SSTR
wwwt 2 1 F 5 MSTR wwwMSE
Blocks, B SSB 5 t o
i 5 1 n
( x }i.2 }}x )2 n 2 1 MSB 5 SSB
wwwn 2 1 F 5 MSB wwwMSE
Sampling SSE 5 SST 2 SSTR (t 2 1)(n 2 1) MSE 5 SSE wwwwww(t 2 1)(n 2 1) Error, E 2 SSB
Total, T SST 5 o
j 5 1t o
i5 1n (x ij2}}x )2 tn 2 1
If F 5 MSTR
wwwMSE is . F [, (t 2 1), (t 2 1)(n 2 1)], reject H0 at the level.
Blocks, i 5 1 to n
Treatments, j 5 1 to t
1 2 3 t
x11 x12 x13 . . . x1t x }1.
x21 x22 x23 . . . x2t x }2.
x31 x32 x33 . . . x3t x }3.
. . . . . . . . . . . . . . . . . . . . . xn1 xn2 xn3 . . . xnt x }n.
}x .1 }x .2 }x .3 . . . x }.t
Block means
Treatment means
xij5 the observation for the ith block and the jth treatment }}x 5 grand mean,
mean of all the observations
}}x 5 j 5o 1
t
o
i 5 1 n
xij wwwN
In addition to listing a format for the xij values, part B of Table 12.6 shows the block means and the treatment means. For example, x }1. is the mean for block 1, with the dot (.) in the subscript indicating the mean is across all of the treat- ments. Likewise, x }.3 is the mean for treatment 3, with the dot portion of the subscript indicating the mean is across all of the blocks.
Part C: The Calculations for the Randomized Block Design
Part C of Table 12.6 describes the specific computations for the randomized block design; each quantity is associated with a specific source of variation within the sample data. They correspond to the xij 5 1 j 1 i 1 ij model discussed previously, and their calculation and interpretation are similar to their counter- parts in the one-way, or completely randomized, design.
The Sum of Squares Terms: Quantifying the Sources of Variation
• Treatments, TR SSTR is the sum of squares that reflects the amount of variation between the treatment means and the grand mean, 5x . SSB is the sum of squares that reflects the amount of variation between the block means and the grand mean, 5x .
• Sampling error, E SSE is the sum of squares that reflects the total amount of variation that is due to sampling error. It is most easily calculated by first determining SST then subtracting SSTR and SSB.
• Total variation, T SST is the sum of squares that reflects the total amount of variation in the data, with each data value being compared to the grand mean, then the differences are squared and summed.
Making the Amounts from the Sources of Variation Comparable. MSTR, MSB, and MSE are the mean squares for treatments, blocks, and error, respectively.
As part C of Table 12.6 shows, each is obtained by dividing the corresponding sum of squares by an appropriate value for df.
Part D: The Test Statistic, the Critical Value, and the Decision Rule
The Test Statistic. The test statistic is MSTRyMSE. As with the one-way or completely randomized design, MSTR has estimated the common variance (2) based on variation between the treatment means, while MSE has estimated the common variance based on variance within the treatment groups. The test statis- tic is the ratio of these separate estimates of the common variance.
The Critical Value and the Decision Rule. The test is right-tail and, for a given level of significance (), we will reject H0: 1 5 2 5 . . . 5 t if the calculated value of F is greater than F[, (t 2 1), (t 2 1)(n 2 1)]. In referring to the F dis- tribution table, we use the v1 5 (t 2 1) column and the v2 5 (t 2 1)(n 2 1) row in identifying the critical value.
EXAMPLE
Randomized Block ANOVA Procedure
To illustrate the application of the randomized block design, we will use an ex- ample that corresponds to our introductory discussion for this procedure. Com- puter outputs and their interpretation will then be presented.
EXAMPLE EXAMP L
The new-product development team for an automotive headlamp firm has four different headlamp designs under consideration. A test is set up to compare their effectiveness in night-driving conditions, and the measurement of interest is the distance at which a suburban traffic sign can be read by the driver. Recogniz- ing that younger drivers tend to have better night vision than older drivers, the team has planned the experiment so that age group will be a blocking variable.
Four persons from each age group (or block) are selected, then the treatments are randomly assigned to members from each block. When each person is subjected to one of the headlamp designs, the distance at which the traffic sign can be read is measured, with the results in Table 12.7. At the 0.05 level, could the headlamp designs be equally effective? The data are also in file CX12LITE.
SOLUTION
For this test, the null hypothesis is H0: 1 5 2 5 3 5 4 for the four headlamp designs, or treatments. According to the null hypothesis, the treatment population means are the same—i.e., the four headlamp designs are equally effective. The alter- native hypothesis holds that the population means are not equal and that the head- lamp designs are not equally effective. Preliminary calculations, such as the block means, treatment means, and the grand mean are shown in Table 12.7. We will now proceed with the remaining calculations, described in part C of Table 12.6.
Treatment Sum of Squares, SSTR
The treatment sum of squares is calculated in a manner analogous to that of the one-way ANOVA method. In the randomized block design, the treatment groups are of equal size, and in this experiment each group has n 5 3 persons.
EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE E
TABLE 12.7 The data and the results of
preliminary calculations for the randomized block design to determine whether four headlamp designs could be equally effective. The blocks are according to age group, with i 5 1 for the four under-30 drivers, i 5 2 for the four 30–60 drivers, and i 5 3 for the four over-60 drivers.
Data are the distance (in yards) at which a suburban traffic sign can be read.
84.00 80.00 85.67 80.33 90 87 93 85 86 79 87 83 76 74 77 73 Treatments, headlamps j 5 1 to 4
1 2 3 4
Blocks, age groups i 5 1 to 3
88.75 83.75 75.00
The grand mean is the mean of all the data values, or }}
x 5 90 1 86 1 76 1 . . . 1 83 1 73
wwwwwwwwwwwwww12 5 82.50 yards
Block means
Treatment means The mean for block i 5 3 is x }3.5 76 1 74 1 77 1 73
wwwwwwwww4 5 75.00 yards The mean for treatment group j 5 2 is
}x .2 5 87 1 79 1 74
wwwwwww3 5 80.00 yards
Chapter 12: Analysis of Variance Tests 435
SSTR 5 n o
j 5 1t ( }x .j 2 5x )2
5 3 (84.00 2 82.50)2 1 (80.00 2 82.50)2
1(85.67 2 82.50)2 1 (80.33 2 82.50)2 and SSTR 5 69.77 Block Sum of Squares, SSB
The sum of squares comparing block means with the overall mean is calculated as SSB 5 t o
i 5 1 n
( x }i.2 5x )2
5 4[(88.75 2 82.50)2 1 (83.75 2 82.50)2 1 (75.00 2 82.50)2] and
SSB 5 387.50 Total Sum of Squares, SST
SST considers the individual data values. In this sum of squares, each xij value is compared to the grand mean, 5x :
SST 5 o
j 5 1t o
i 5 1
n
(xij 2 5x )2
5 (90 2 82.50)2 1(86 2 82.50)2 1(76 2 82.50)2
1 (87 2 82.50)2 1(79 2 82.50)2 1(74 2 82.50)2 1 (93 2 82.50)2
1(87 2 82.50)2 1(77 2 82.50)2
1 (85 2 82.50)2 1(83 2 82.50)2 1(73 2 82.50)2
and SST 5 473.00
Error Sum of Squares, SSE
This quantity is most easily computed by first calculating the preceding quanti- ties, then using the relationship SST 5 SSTR 1SSB 1SSE, or
SSE 5 SST 2 SSTR 2SSB 5 473.00 2 69.77 2 387.50 and SSE 5 15.73
Treatment Mean Square (MSTR) and Error Mean Square (MSE )
SSTR and SSE are now divided by their respective degrees of freedom so that (1) they will be comparable and (2) each will be a separate estimate of the com- mon variance that the treatment group populations are assumed to share. Recall that we have t 5 4 treatments and n 5 3 blocks.
The estimate of 2 that is based on the between-treatment variation is MSTR 5 SSTR
wwwt 2 1 5 69.77
www4 2 1 and MSTR 5 23.26
The estimate of 2 that is based on the within-treatment, or sampling-error variation is
MSE 5 SSE
wwwwwww(t 2 1)(n 2 1) 5 15.73
wwwwwww(4 2 1)(3 2 1) and MSE 5 2.62
PLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMP
The Test Statistic, F
The test statistic is the ratio of the two estimates for 2, or F 5 MSTR
wwwMSE 5 23.26
www2.62 and F 5 8.88
We have now generated the information described in part C of Table 12.6, and the results are summarized in the randomized block ANOVA table shown in Table 12.8. Note that Table 12.8 includes the listing of the mean square [MSB 5 SSBy(n 2 1) 5 193.75] and the F-ratio (F 5 MSByMSE 5 73.95) for the block- ing variable, neither of which was calculated previously. These were omitted from our previous calculations because our intent in the randomized block design is only to control for the effect of the blocking variable, not to investigate its effect.
In Section 12.5, we will examine the simultaneous effects of two independent variables in a method called two-way analysis of variance.
The Critical Value of F and the Decision
The test is being carried out at the 5 0.05 level, and the critical value of F can be found from Appendix A as
Critical value of F 5 F(, v1, v2) The df associated with the numerator of F is
v1 5 (t 2 1) or (4 2 1) 5 3 The df associated with the denominator of F is
v2 5 (t 2 1)(n 2 1) 5 (4 2 1)(3 2 1) 5 6
For 5 0.05, v1 5 3 and v2 5 6, the critical F is F(0.05, 3, 6) 5 4.76. The calculated value (F 5 8.88) exceeds the critical value, and, at the 0.05 level, we are able to reject the null hypothesis that the population means are equal. At this level of significance, our conclusion is that the headlamp treatments are not equally effective.
Using the F distribution tables in Appendix A, we can narrow down the p-value for the test. For the 0.05, 0.025, and 0.01 levels, the respective critical values are 4.76, 6.60, and 9.78, and the calculated F (F 5 8.88) falls between the critical values for the 0.025 and 0.01 levels. Based on our statistical tables, the most accurate statement we can make about the p-value for the test is that it is somewhere between 0.025 and 0.01.
EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE EXAMPLE
TABLE 12.8 From the data in Table 12.7 and the computations in the text, we can construct this standard table describing the results of the randomized block ANOVA test. The format is similar to that shown in part C of Table 12.6. Because our calculations were rounded to two decimal places, the results shown here will differ slightly from those in the corresponding computer printouts.
Variation Degrees of
Source Sum of Squares Freedom Mean Square F
Treatments 69.77 3 23.26 8.88
Blocks 387.50 2 193.75 73.95
Error 15.73 6 2.62
Total 473.00 11
Figure 12.2 is a multiple plot of the measurements according to the treat- ments and blocks. Each line connects the distances measured for persons in one of the age-group blocks. Regardless of the headlamp design, the distance at which the traffic sign could be read was longest for the youngest person exposed to that treatment, and shortest for the oldest person exposed to the treatment. This further demonstrates the appropriateness of using the randomized block design with age group as a blocking variable.
Testing the Effectiveness of the Blocking Variable
As mentioned earlier, our primary intent is to control for the effect of the block- ing variable, not to measure its effect. If we do wish to examine the effectiveness of the blocking variable, we can use this procedure:
FIGURE 12.2 This multiple plot visually summarizes the data in Table 12.7. The display helps show why blocking according to age group would be appropriate for this type of study.
1
1
1
1 2 2
2 2
3
3
1 70 75 80 85 90
Number of yards at which sign could be read
95
2
Headlamp treatment
3 4
3
3 Lines for blocks 1, 2, and 3
1 = Driver under 30 2 = 30 to 60 years 3 = Driver over 60
Hypotheses for testing the effectiveness of the blocking variable:
H0: The levels of the blocking variable are equal in their effect.
H1: At least one level has an effect different from the others.
Calculated value of F 5 MSBMSE, computed as shown in Table 12.6 Critical value of F 5 F (, v1, v2)
where v15 (n 2 1), df associated with the numerator, and v25 (t 2 1)(n 2 1), df associated with the denominator and n 5 number of blocks, t 5 number of treatments Decision rule: If calculated F F (, v1, v2), reject H0 at the level.