Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences

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As we have seen, we compute V.R. in situations of this type by placing the among

groups mean square in the numerator and the within groups mean square in the denom-

inator, so that the numerator degrees of freedom is equal to the number of

groups minus 1, and the denominator degrees of freedom value is equal to

The ANOVA Table

The calculations that we perform may be summarized and

displayed in a table such as Table 8.2.2, which is called the ANOVA table.

8. Statistical decision. To reach a decision we must compare our computed V.R. with

the critical value of F, which we obtain by entering Appendix Table G with

numerator degrees of freedom and denominator degrees of freedom.

If the computed V.R. is equal to or greater than the critical value of F, we reject the null

hypothesis. If the computed value of V.R. is smaller than the critical value of F, we do

not reject the null hypothesis.

Explaining a Rejected Null Hypothesis There are two possible expla-

nations for a rejected null hypothesis. If the null hypothesis is true, that is, if the two

sample variances are estimates of a common variance, we know that the probability of

getting a value of V.R. as large as or larger than the critical F is equal to our chosen

level of signiﬁcance. When we reject we may, if we wish, conclude that the null

hypothesis is true and assume that because of chance we got a set of data that gave rise

to a rare event. On the other hand, we may prefer to take the position that our large com-

puted V.R. value does not represent a rare event brought about by chance but, instead,

reﬂects the fact that something other than chance is operative. This other something we

conclude to be a false null hypothesis.

It is this latter explanation that we usually give for computed values of V.R. that

exceed the critical value of F. In other words, if the computed value of V.R. is greater

than the critical value of F, we reject the null hypothesis.

N - k

k - 1

j =1

- 12= a

j =1

b- k = N - k

1k - 12,

8.2 THE COMPLETELY RANDOMIZED DESIGN 317

TABLE 8.2.2 Analysis of Variance Table for the Completely Randomized Design

Source of Degrees Variance

Variation Sum of Squares of Freedom Mean Square Ratio

Among samples

Within samples

Total N - 1SST =

j =1

i=1

- x

MSW = SSW>1N - k2N - kSSW =

j =1

i =1

- x

V.R. =

MSA

MSW

MSA = SSA>1k - 12k - 1SSA =

j = 1

- x

It will be recalled that the original hypothesis we set out to test was

Does rejection of the hypothesis about variances imply a rejection of the hypothesis of

equal population means? The answer is yes. A large value of V.R. resulted from the fact

that the among groups mean square was considerably larger than the within groups mean

square. Since the among groups mean square is based on the dispersion of the sample

means about their mean (called the grand mean), this quantity will be large when there

is a large discrepancy among the sizes of the sample means. Because of this, then, a sig-

niﬁcant value of V.R. tells us to reject the null hypothesis that all population means are

equal.

9. Conclusion. When we reject we conclude that not all population means are

equal. When we fail to reject we conclude that the population means may all

be equal.

10. Determination of p value.

EXAMPLE 8.2.1

Game meats, including those from white-tailed deer and eastern gray squirrels, are

used as food by families, hunters, and other individuals for health, cultural, or per-

sonal reasons. A study by David Holben (A-1) assessed the selenium content of meat

from free-roaming white-tailed deer (venison) and gray squirrel (squirrel) obtained

from a low selenium region of the United States. These selenium content values were

also compared to those of beef produced within and outside the same region. We want

to know if the selenium levels are different in the four meat groups.

Solution:

1. Description of data. Selenium content of raw venison (VEN), squirrel

meat (SQU), region-raised beef (RRB), and nonregion-raised beef (NRB),

in of dry weight, are shown in Table 8.2.3. A graph of the data

in the form of a dotplot is shown in Figure 8.2.4. Such a graph highlights

mg>100 g

: m

= m

318 CHAPTER 8 ANALYSIS OF VARIANCE

TABLE 8.2.3 Selenium Content, in of Four Different Meat Types

Meat Type

VEN SQU RRB NRB

26.72 14.86 37.42 37.57 11.23 15.82 44.33

28.58 16.47 56.46 25.71 29.63 27.74 76.86

29.71 25.19 51.91 23.97 20.42 22.35 4.45

26.95 37.45 62.73 13.82 10.12 34.78 55.01

10.97 45.08 4.55 42.21 39.91 35.09 58.21

21.97 25.22 39.17 35.88 32.66 32.60 74.72

14.35 22.11 38.44 10.54 38.38 37.03 11.84

Mg /100 g,

(

Continued

)

8.2 THE COMPLETELY RANDOMIZED DESIGN 319

Meat Type

VEN SQU RRB NRB

32.21 33.01 40.92 27.97 36.21 27.00 139.09

19.19 31.20 58.93 41.89 16.39 44.20 69.01

30.92 26.50 61.88 23.94 27.44 13.09 94.61

10.42 32.77 49.54 49.81 17.29 33.03 48.35

35.49 8.70 64.35 30.71 56.20 9.69 37.65

36.84 25.90 82.49 50.00 28.94 32.45 66.36

25.03 29.80 38.54 87.50 20.11 37.38 72.48

33.59 37.63 39.53 68.99 25.35 34.91 87.09

33.74 21.69 21.77 27.99 26.34

18.02 21.49 31.62 22.36 71.24

22.27 18.11 32.63 22.68 90.38

26.10 31.50 30.31 26.52 50.86

20.89 27.36 46.16 46.01

29.44 21.33 56.61 38.04

24.47 30.88

29.39 30.04

40.71 25.91

18.52 18.54

27.80 25.51

19.49

Source: David H. Holben, Ph.D. Used with permission.

the main features of the data and brings into clear focus differences in sele-

nium levels among the different meats.

2. Assumptions. We assume that the four sets of data constitute independ-

ent simple random samples from the four indicated populations. We

assume that the four populations of measurements are normally distrib-

uted with equal variances.

FIGURE 8.2.4 Selenium content of four meat types. VEN  venison,

and NRB = nonregion-raised beef.RRB = region-raised beef,

SQU = squirrel,

Meat type

VEN

SQU

RRB

Selenium content (mg/100 g of dry weight)

NRB

0 20406080100120140

3. Hypotheses.

(On average the four meats have the same

selenium content.)

Not all are equal (At least one meat yields an average selenium

content different from the average selenium content of at least one other

meat.)

4. Test statistic. The test statistic is

5. Distribution of test statistic. If is true and the assumptions are met,

the V.R. follows the F distribution with numerator degrees of

freedom and denominator degrees of freedom.

6. Decision rule. Suppose we let The critical value of F from

Appendix Table G is The decision rule, then, is reject if the

computed V.R. statistic is equal to or greater than 3.95.

7. Calculation of test statistic. By Equation 8.2.2 we compute

By Equation 8.2.4 we compute

The results of our calculations are displayed in Table 8.2.4.

8. Statistical decision. Since our computed F of 27.00 is greater than 3.95

we reject

9. Conclusion. Since we reject we conclude that the alternative

hypothesis is true. That is, we conclude that the four meat types do not

all have the same average selenium content.

10. p value. Since for this test. ■

A Word of Caution The completely randomized design is simple and, therefore,

widely used. It should be used, however, only when the units receiving the treatments are

homogeneous. If the experimental units are not homogeneous, the researcher should con-

sider an alternative design such as one of those to be discussed later in this chapter.

In our illustrative example the treatments are treatments in the usual sense of the

word. This is not always the case, however, as the term “treatment” as used in experi-

mental design is quite general. We might, for example, wish to study the response to the

27.00 7 3.95, p 6 .01

SSW = 58009.05560 - 21261.82886 = 36747.22674

SSA = 21261.82886

SST = 58009.05560

6 3.95.

a = .01.

144 - 4 = 140

4 - 1 = 3

V.R. = MSA>MSW.

m’sH

: m

= m

320 CHAPTER 8 ANALYSIS OF VARIANCE

TABLE 8.2.4 ANOVA Table for Example 8.2.1

Source SS df MS F

Among samples 21261.82886 3 7087.27629 27.00

Within samples 36747.22674 140 262.48019

Total 58009.05560 143

same treatment (in the usual sense of the word) of several breeds of animals. We would,

however, refer to the breed of animal as the “treatment.”

We must also point out that, although the techniques of analysis of variance are

more often applied to data resulting from controlled experiments, the techniques also

may be used to analyze data collected by a survey, provided that the underlying assump-

tions are reasonably well met.

Computer Analysis Figure 8.2.5 shows the computer procedure and output for

Example 8.2.1 provided by a one-way analysis of variance program found in the

MINITAB package. The data were entered into Columns 1 through 4. When you com-

pare the ANOVA table on this printout with the one given in Table 8.2.4, you see that the

printout uses the label “factor” instead of “among samples.” The different treatments are

referred to on the printout as levels. Thus level

and so on. The printout gives the four sample means and standard deviations as well as

level 2 = treatment 2,1 = treatment 1,

8.2 THE COMPLETELY RANDOMIZED DESIGN 321

FIGURE 8.2.5 MINITAB procedure and output for Example 8.2.1.

Dialog box: Session command:

Stat ➤ ANOVA ➤ Oneway (Unstacked) MTB>AOVONEWAY C1-C4

Type C1-C4 in responses (in separate columns)

Click OK.

Output:

One-way ANOVA: NRB, RRB, SQU, VEN

Analysis of Variance for Selenium

Source DF SS MS F P

Meat Typ 3 21262 7087 27.00 0.000

Error 140 36747 262

Total 143 58009

Individual 95% CIs For Mean

Based on Pooled StDev

Level N Mean StDev -------+---------+--------+----------

NRB 19 62.05 31.15 (----*----)

RRB 53 29.08 10.38 (--*--)

SQU 30 43.25 19.51 (---*---)

VEN 42 25.88 8.03 (--*---)

-------+---------+--------+----------

Pooled StDeV = 16.20 30 45 60

the pooled standard deviation. This last quantity is equal to the square root of the error

mean square shown in the ANOVA table. Finally, the computer output gives graphic rep-

resentations of the 95 percent conﬁdence intervals for the mean of each of the four pop-

ulations represented by the sample data.

Figure 8.2.6 contains a partial SAS

printout resulting from analysis of the data

of Example 8.2.1 through use of the SAS

statement PROC ANOVA. SAS

computes

some additional quantities as shown in the output. This quan-

tity tells us what proportion of the total variability present in the observations is

accounted for by differences in response to the treatments. (root MSE/selen

mean). Root MSE is the square root of MSW, and selen mean is the mean of the 18

observations.

Note that the test statistic V.R. is labeled differently by different statistical soft-

ware programs. MINITAB, for example, uses F rather than V.R. SAS

uses the label

F Value.

A useful device for displaying important characteristics of a set of data analyzed

by one-way analysis of variance is a graph consisting of side-by-side boxplots. For each

sample a boxplot is constructed using the method described in Chapter 2. Figure 8.2.7

shows the side-by-side boxplots for Example 8.2.1. Note that in Figure 8.2.7 the vari-

able of interest is represented by the vertical axis rather than the horizontal axis.

Alternatives

If the data available for analysis do not meet the assumptions for one-

way analysis of variance as discussed here, one may wish to consider the use of the

Kruskal-Wallis procedure, a nonparametric technique discussed in Chapter 13.

Testing for Signiﬁcant Differences Between Individual Pairs

of Means

When the analysis of variance leads to a rejection of the null hypothe-

sis of no difference among population means, the question naturally arises regarding just

which pairs of means are different. In fact, the desire, more often than not, is to carry

C.V. = 100

R-Square = SSA>SST.

322 CHAPTER 8 ANALYSIS OF VARIANCE

FIGURE 8.2.6 Partial SAS

printout for Example 8.2.1.

The SAS System

Analysis of Variance Procedure

Dependent Variable: selen

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 3 21261.82886 7087.27629 27.00 <.0001

Error 140 36747.22674 262.48019

Corrected Total 143 58009.05560

R-Square Coeff Var Root MSE selen Mean

0.366526 45.70507 16.20124 35.44736

out a signiﬁcance test on each and every pair of treatment means. For instance, in Exam-

ple 8.2.1, where there are four treatments, we may wish to know, after rejecting

which of the six possible individual hypotheses should be

rejected. The experimenter, however, must exercise caution in testing for signiﬁcant dif-

ferences between individual means and must always make certain that the procedure is

valid. The critical issue in the procedure is the level of signiﬁcance. Although the prob-

ability, of rejecting a true null hypothesis for the test as a whole is made small, the

probability of rejecting at least one true hypothesis when several pairs of means are tested

is, as we have seen, greater than There are several multiple comparison procedures

commonly used in practice. Below we illustrate two popular procedures, namely Tukey’s

HSD test and Bonferroni’s method. The interested student is referred to the books by

Hsu (7) and Westfall et al. (8) for additional techniques.

Tukey’s HSD Test Over the years several procedures for making multiple com-

parisons have been suggested. A multiple comparison procedure developed by Tukey (9)

is frequently used for testing the null hypothesis that all possible pairs of treatment means

are equal when the samples are all of the same size. When this test is employed we select

an overall signiﬁcance level of The probability is then, that one or more of the null

hypotheses is false.

Tukey’s test, which is usually referred to as the HSD (honestly signiﬁcant differ-

ence) test, makes use of a single value against which all differences are compared. This

value, called the HSD, is given by

(8.2.9)

where is the chosen level of signiﬁcance, k is the number of means in the experiment,

N is the total number of observations in the experiment, n is the number of observations

in a treatment, MSE is the error or within mean square from the ANOVA table, and q is

obtained by entering Appendix Table H with and N - k.a, k,

HSD = q

a,k,N -k

MSE

a,a.

: m

= m

8.2 THE COMPLETELY RANDOMIZED DESIGN 323

FIGURE 8.2.7 Side-by-side boxplots for Example 8.2.1.

NRB

Meat type

Selenium (mg/100 g)

RRB SQU VEN

150

100

The statistic q, tabulated in Appendix Table H, is known as the studentized range

statistic. It is deﬁned as the difference between the largest and smallest treatment means

from an ANOVA (that is, it is the range of the treatment means) divided by the error

mean square over n, the number of observations in a treatment. The studentized range

is discussed in detail by Winer (10).

All possible differences between pairs of means are computed, and any differ-

ence that yields an absolute value that exceeds HSD is declared significant.

Tukey’s Test for Unequal Sample Sizes When the samples are not all

the same size, as is the case in Example 8.2.1, Tukey’s HSD test given by Equation

8.2.9 is not applicable. Tukey himself (9) and Kramer (11), however, have extended the

Tukey procedure to the case where the sample sizes are different. Their procedure,

which is sometimes called the Tukey-Kramer method, consists of replacing MSE/n in

Equation 8.2.9 with where and are the sample sizes of

the two groups to be compared. If we designate the new quantity by HSD*, we have

as the new test criterion

(8.2.10)

Any absolute value of the difference between two sample means that exceeds

HSD* is declared signiﬁcant.

Bonferroni’s Method Another very commonly used multiple comparison

test is based on a method developed by C. E. Bonferroni. As with Tukey’s method,

we desire to maintain an overall significance level of for the total of all pair-wise

tests. In the Bonferroni method, we simply divide the desired significance level by

the number of individual pairs that we are testing. That is, instead of testing at a sig-

nificance level of , we test at a significance level of where k is the number of

paired comparisons. The sum of all terms cannot, then, possibly exceed our stated

level of . For example, if one has three samples, A, B, and C, then there are

pair-wise comparisons. These are and If we choose a

significance level of then we would proceed with the comparisons and use

a Bonferroni-corrected significance level of Therefore, our p value must

be no greater then .017 in order to reject the null hypothesis and conclude that two

means differ.

Most computer packages compute values using the Bonferroni method and pro-

duce an output similar to the Tukey’s HSD or other multiple comparison procedures. In

general, these outputs report the actual corrected p value using the Bonferroni method.

Given the basic relationship that then algebraically we can multiply both sides

of the equation by k to obtain In other words, the total is simply the sum of

all of the pk values, and the actual corrected p value is simply the calculated p value

multiplied by the number of tests that were performed.

aa = pk.

p = a>k,

a>3 = .017

a = .05,

= m

, m

= m

k = 3a

a>k

a>k,a

HSD* = q

a,k,N -k

MSE

1MSE >2211>n

+ 1>n

324

CHAPTER 8 ANALYSIS OF VARIANCE

EXAMPLE 8.2.2

Let us illustrate the use of the HSD test with the data from Example 8.2.1.

Solution: The first step is to prepare a table of all possible (ordered) differences

between means. The results of this step for the present example are dis-

played in Table 8.2.5.

Suppose we let Entering Table H with and we

ﬁnd that The actual value is which can be obtained from SAS

In Table 8.2.4 we have

The hypotheses that can be tested, the value of HSD*, and the statistical decision

for each test are shown in Table 8.2.6.

SAS

uses Tukey’s procedure to test the hypothesis of no difference between

population means for all possible pairs of sample means. The output also contains

MSE = 262.4802.

q = 3.667,q 6 3.68.

N - k = 140,a = .05, k = 4,a = .05.

8.2 THE COMPLETELY RANDOMIZED DESIGN 325

TABLE 8.2.5 Differences Between Sample

Means (Absolute Value) for Example 8.2.2

VEN RRB SQR NRB

VEN — 3.208 17.37 36.171

RRB — 14.163 32.963

SQU — 18.801

NRB —

Table 8.2.6 Multiple Comparison Tests Using Data of Example 8.2.1 and HSD*

Hypotheses HSD* Statistical Decision

HSD* = 3.677

262.4802

b= 12.32H

SQU

= m

NRB

HSD* = 3.677

262.4802

= 11.23H

RRB

= m

NRB

HSD* = 3.677

262.4802

= 9.60H

RRB

= m

SQU

HSD* = 3.677

262.4802

b= 11.61H

VEN

= m

NRB

HSD* = 3.677

262.4802

b= 10.04H

VEN

= m

SQU

HSD* = 3.677

262.4802

= 8.68H

VEN

= m

RRB

Do not reject

since 3.208 6 8.68

Reject since

17.37 7 10.04

Reject since

36.171 7 11.61

Reject since

14.163 7 9.60

Reject since

32.963 7 11.23

Reject since

18.801 7 12.32

confidence intervals for the difference between all possible pairs of population means.

This SAS output for Example 8.2.1 is displayed in Figure 8.2.8.

One may also use SPSS to perform multiple comparisons by a variety of meth-

ods, including Tukey’s. The SPSS outputs for Tukey’s HSD and Bonferroni’s method

for the data for Example 8.2.1 are shown in Figures 8.2.9 and 8.2.10. The outputs con-

tain an exhaustive comparison of sample means, along with the associated standard

errors, p values, and 95% conﬁdence intervals. ■

326 CHAPTER 8 ANALYSIS OF VARIANCE

FIGURE 8.2.8 SAS

multiple comparisons for Example 8.2.1.

The SAS System

Analysis of Variance Procedure

Tukey’s Studentized Range (HSD) Test for selen

NOTE: This test controls the Type I experimentwise error rate.

Alpha 0.05

Error Degrees of Freedom 140

Error Mean Square 262.4802

Critical Value of Studentized Range 3.67719

Comparisons signiﬁcant at the 0.05 level are indicated by ***.

Difference

type Between Simultaneous 95%

Comparison Means Conﬁdence Limits

NRB - SQU 18.801 6.449 31.152 ***

NRB - RRB 32.963 21.699 44.228 ***

NRB - VEN 36.171 24.524 47.818 ***

SQU - NRB -18.801 -31.152 -6.449 ***

SQU - RRB 14.163 4.538 23.787 ***

SQU - VEN 17.370 7.300 27.440 ***

RRB - NRB -32.963 -44.228 -21.699 ***

RRB - SQU -14.163 -23.787 -4.538 ***

RRB - VEN 3.208 -5.495 11.910

VEN - NRB -36.171 -47.818 -24.524 ***

VEN - SQU -17.370 -27.440 -7.300 ***

VEN - RRB -3.208 -11.910 5.495