Tests for the Equality of Several Variances

13.4 Tests for the Equality of Several Variances

Although the f-ratio obtained from the analysis-of-variance procedure is insensitive to departures from the assumption of equal variances for the k normal populations when the samples are of equal size, we may still prefer to exercise caution and run a preliminary test for homogeneity of variances. Such a test would certainly

be advisable in the case of unequal sample sizes if there was a reasonable doubt concerning the homogeneity of the population variances. Suppose, therefore, that we wish to test the null hypothesis

H 0 :σ 2 1 2 =σ 2 2 =···=σ k

against the alternative

H 1 : The variances are not all equal.

The test that we shall use, called Bartlett’s test, is based on a statistic whose sampling distribution provides exact critical values when the sample sizes are equal. These critical values for equal sample sizes can also be used to yield highly accurate approximations to the critical values for unequal sample sizes.

First, we compute the k sample variances s 2 ,s 1 2 2 ,...,s 2 k from samples of size

n 1 ,n 2 ,...,n k , with

n i = N . Second, we combine the sample variances to give

i=1

the pooled estimate

k ) n k −1 ] 1/(N −k)

b=

is a value of a random variable B having the Bartlett distribution. For the special case where n 1 =n 2 =···=n k = n, we reject H 0 at the α-level of significance if

b<b k (α; n),

13.4 Tests for the Equality of Several Variances 517 where b k (α; n) is the critical value leaving an area of size α in the left tail of the

Bartlett distribution. Table A.10 gives the critical values, b k (α; n), for α = 0.01 and 0.05; k = 2, 3, . . . , 10; and selected values of n from 3 to 100.

When the sample sizes are unequal, the null hypothesis is rejected at the α-level of significance if

b<b k (α; n 1 ,n 2 ,...,n k ),

where n 1 b k (α; n 1 )+n 2 b k (α; n 2 )+···+n k b k (α; n k )

b k (α; n 1 ,n 2 ,...,n k )≈

As before, all the b k (α; n i ) for sample sizes n 1 ,n 2 ,...,n k are obtained from Table

A.10. Example 13.3: Use Bartlett’s test to test the hypothesis at the 0.01 level of significance that the

population variances of the four drug groups of Example 13.2 are equal. Solution : We have the hypotheses

H 0 :σ 2 1 =σ 2 2 2 2 =σ 3 =σ 4 ,

H 1 : The variances are not equal, with α = 0.01.

Critical region: Referring to Example 13.2, we have n 1 = 20, n 2 = 9, n 3 = 9, n 4 = 7, N = 45, and k = 4. Therefore, we reject when

b<b 4 (0.01; 20, 9, 9, 7) (20)(0.8586) + (9)(0.6892) + (9)(0.6892) + (7)(0.6045)

Computations: First compute s 2 = 662.862, s 1 2 = 2219.781, s 2 2 2 3 = 2168.434, s 4 = 946.032, and then

Decision: Do not reject the hypothesis, and conclude that the population variances of the four drug groups are not significantly different.

Although Bartlett’s test is most often used for testing of homogeneity of vari- ances, other methods are available. A method due to Cochran provides a compu- tationally simple procedure, but it is restricted to situations in which the sample

518 Chapter 13 One-Factor Experiments: General sizes are equal. Cochran’s test is particularly useful for detecting if one variance

is much larger than the others. The statistic that is used is

largest S 2

G=

i=1

and the hypothesis of equality of variances is rejected if g > g α , where the value of

g α is obtained from Table A.11. To illustrate Cochran’s test, let us refer again to the data of Table 13.1 on moisture absorption in concrete aggregates. Were we justified in assuming equal variances when we performed the analysis of variance in Example 13.1? We find that

s 2 = 12,134, s 1 2 2 = 2303, s 2 3 = 3594, s 2 4 = 3319, s 2 5 = 3455. Therefore,

which does not exceed the table value g 0.05 = 0.5065. Hence, we conclude that the assumption of equal variances is reasonable.

Exercises

13.1 Six different machines are being considered for Tablet use in manufacturing rubber seals. The machines are

A B C D E being compared with respect to tensile strength of the

5.2 9.1 3.2 2.4 7.1 product. A random sample of four seals from each ma-

4.7 7.1 5.8 3.4 6.6 chine is used to determine whether the mean tensile

8.1 8.2 2.2 4.1 9.3 strength varies from machine to machine. The follow-

6.2 6.0 3.1 1.0 4.2 ing are the tensile-strength measurements in kilograms

3.0 9.1 7.2 4.0 7.6 per square centimeter × 10 −1 :

Machine 13.3 In an article “Shelf-Space Strategy in Retailing,” 1 2 3 4 5 6 published in Proceedings: Southern Marketing Associa- 17.5 16.4 20.3 14.6 17.5 18.3 tion, the effect of shelf height on the supermarket sales 16.9 19.2 15.7 16.7 19.2 16.2 of canned dog food is investigated. An experiment was 15.8 17.7 17.8 20.8 16.5 17.5 conducted at a small supermarket for a period of 8 days 18.6 15.4 18.9 18.9 20.5 20.1 on the sales of a single brand of dog food, referred to

Perform the analysis of variance at the 0.05 level of sig- as Arf dog food, involving three levels of shelf height: nificance and indicate whether or not the mean tensile knee level, waist level, and eye level. During each day, strengths differ significantly for the six machines.

the shelf height of the canned dog food was randomly changed on three different occasions. The remaining

13.2 The data in the following table represent the sections of the gondola that housed the given brand number of hours of relief provided by five different were filled with a mixture of dog food brands that were brands of headache tablets administered to 25 subjects both familiar and unfamiliar to customers in this par- experiencing fevers of 38 ◦

C or more. Perform the anal- ticular geographic area. Sales, in hundreds of dollars, ysis of variance and test the hypothesis at the 0.05 level of Arf dog food per day for the three shelf heights are of significance that the mean number of hours of relief given. Based on the data, is there a significant differ- provided by the tablets is the same for all five brands. ence in the average daily sales of this dog food based Discuss the results.

on shelf height? Use a 0.01 level of significance.

Exercises 519 Shelf Height

State University, was designed to assess the ability of Knee Level

Waist Level

Eye Level

this enzyme to undergo conformation or shape changes. 77 88 85 Changes in the specific activity of the enzyme caused 82 94 85 by variations in the concentration of NADP could be

86 93 87 interpreted as supporting the theory of conformational 78 90 81 change. The enzyme in question is located in the in- 81 91 80 ner membrane of the tapeworm’s mitochondria. Tape- 86 94 79 worms were homogenized, and through a series of cen- 77 90 87 trifugations, the enzyme was isolated. Various con- 81 87 93 centrations of NADP were then added to the isolated

enzyme solution, and the mixture was then incubated 13.4 Immobilization of free-ranging white-tailed deer in a water bath at 56 ◦

C for 3 minutes. The enzyme by drugs allows researchers the opportunity to closely was then analyzed on a dual-beam spectrophotometer, examine the deer and gather valuable physiological in- and the results shown were calculated, with the specific formation. In the study Influence of Physical Restraint activity of the enzyme given in nanomoles per minute and Restraint Facilitating Drugs on Blood Measure- per milligram of protein. Test the hypothesis at the ments of White-Tailed Deer and Other Selected Mam-

0.01 level that the average specific activity is the same mals, conducted at Virginia Tech, wildlife biologists for the four concentrations. tested the “knockdown” time (time from injection to

NADP Concentration (nm) immobilization) of three different immobilizing drugs.

360 Immobilization, in this case, is defined as the point

where the animal no longer has enough muscle control 11.01 11.38 11.02 6.04 10.31 to remain standing. Thirty male white-tailed deer were

12.09 10.67 10.67 8.65 8.30 randomly assigned to each of three treatments. Group

10.55 12.33 11.50 7.76 9.48 A received 5 milligrams of liquid succinylcholine chlo-

11.26 10.08 10.31 10.13 8.89 ride (SCC); group B received 8 milligrams of powdered

9.36 SCC; and group C received 200 milligrams of phency-

clidine hydrochloride. Knockdown times, in minutes, 13.6 A study measured the sorption (either absorp- were recorded. Perform an analysis of variance at the tion or adsorption) rates of three different types of or-

0.01 level of significance and determine whether or not ganic chemical solvents. These solvents are used to the average knockdown time for the three drugs is the clean industrial fabricated-metal parts and are poten- same.

tial hazardous waste. Independent samples from each type of solvent were tested, and their sorption rates

Group were recorded as a mole percentage. (See McClave, A B C Dietrich, and Sincich, 1997.)

5 7 4 Aromatics

Chloroalkanes Esters 14 16 6 1.06 0.95 1.58 1.12 0.29 0.43 0.06 7 7 3 0.79 0.65 1.45 0.91 0.06 0.51 0.09 10 7 5 0.82 1.15 0.57 0.83 0.44 0.10 0.17 7 5 6 0.89 1.12 1.16 0.43 0.55 0.53 0.17 23 10 8 1.05 0.61 0.34 0.60

11 6 7 Is there a significant difference in the mean sorption rates for the three solvents? Use a P-value for your 11 12 3 conclusions. Which solvent would you use?

13.5 The mitochondrial enzyme NADPH:NAD 13.7 It has been shown that the fertilizer magnesium transhydrogenase of the common rat tapeworm (Hy- ammonium phosphate, MgNH 4 PO 4 , is an effective sup-

menolepiasis diminuta) catalyzes hydrogen in the plier of the nutrients necessary for plant growth. The transfer from NADPH to NAD, producing NADH. compounds supplied by this fertilizer are highly solu- This enzyme is known to serve a vital role in the ble in water, allowing the fertilizer to be applied di- tapeworm’s anaerobic metabolism, and it has recently rectly on the soil surface or mixed with the growth been hypothesized that it may serve as a proton ex- substrate during the potting process. A study on the change pump, transferring protons across the mito- Effect of Magnesium Ammonium Phosphate on Height chondrial membrane. A study on Effect of Various of Chrysanthemums was conducted at George Mason Substrate Concentrations on the Conformational Vari- University to determine a possible optimum level of ation of the NADPH:NAD Transhydrogenase of Hy- fertilization, based on the enhanced vertical growth re- menolepiasis diminuta, conducted at Bowling Green sponse of the chrysanthemums. Forty chrysanthemum

520 Chapter 13 One-Factor Experiments: General seedlings were divided into four groups, each containing erage attained height of chrysanthemums? How much

10 plants. Each was planted in a similar pot containing MgNH 4 PO 4 appears to be best? a uniform growth medium. To each group of plants an

13.8 For the data set in Exercise 13.7, use Bartlett’s grams per bushel, was added. The four groups of plants test to check whether the variances are equal. Use were grown under uniform conditions in a greenhouse α = 0.05. for a period of four weeks. The treatments and the re- spective changes in heights, measured in centimeters,

increasing concentration of MgNH 4 PO 4 , measured in

13.9 Use Bartlett’s test at the 0.01 level of signifi- are shown next.

cance to test for homogeneity of variances in Exercise Treatment

13.5 on page 519.

50 g/bu 100 g/bu 200 g/bu 400 g/bu 13.2 12.4 16.0 12.6 7.8 14.4 21.0 14.8 13.10 Use Cochran’s test at the 0.01 level of signifi- 12.8 17.2 14.8 13.0 20.0 15.8 19.1 15.8 cance to test for homogeneity of variances in Exercise

13.0 14.0 14.0 23.6 17.0 27.0 18.0 26.0 13.4 on page 519.

15.0 20.0 22.2 24.4 20.2 23.2 25.0 18.2 cance to test for homogeneity of variances in Exercise 13.11 Use Bartlett’s test at the 0.05 level of signifi- Can we conclude at the 0.05 level of significance that

13.6 on page 519.

different concentrations of MgNH 4 PO 4 affect the av-