/ )

T.chuicaI Paper 292 

/ AD

AN EMPIRICAL INVESTIGATION OF THE EFFECT OF

HETEROSCEDASTICITY AND HETEROGENEITY OF

VARIANCE ON THE ANALYS IS OF COVARIANCE

AND THE JO HNSON-NEYMAN TECHNIQUE

Joyce Lee Shields

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INDIVIDUAL TRAINING & SKILL EVALUATION TECHNICAL AREA

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U. S. Army

Research Institute for the Behavioral and Social Sciences

Jul y 1978

Approved for public rele ase , d islru butlon unlimited.

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U. S. ARMY RESEARCH INSTITUTE

FOR THE BEHAVIORAL AND SOCIAL SCIENC ES

A Field Operating Agency under the Jurisdiction of the

Deputy Chief of Staff for Personnel

WILLIAM L. HAUSER

JOSEPH ZEIDNER Colonel , U S Army

Technical Director (Designate) Commander

NOTICES

DISTRIBUTION: Primary distribution of this report has been made by ARI. Please address correspondence concerning distribution of reports to: U. 5. Army Research Institute for the Behavioral and Social Sciencn, ATTN: PERI-P , 5001 Eisenhov~

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the U. S. Army Research Institut, for the Behavioral and Social Sciences.

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Q The findings in this report are not to be construed as en official Department of the Army position , unless o d•signited by other authorized documents.

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IN P

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PIRICAL

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4NVESTIGATION OF THE EFFECT O~.-’ L. Final,

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chnical

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IETEROSCEDAS CITY AND ~ETEROGENEITY OF ~ ARIANC ‘t)N THE ~~~ALYSIS OF COVAIÜANCE AND THE)OHNSON- S_ PERFORMING ORG. REP

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and Social Sciences , 5001 Eisenhower Avenue, 2Q762722A777

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II. CONTROLLING OFFICE NAME AND ADDRESS U.- REP,.,..T 9J ’L I&_

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Ia. SUPPLEMENTARY NOTES

This publication is based upon a dissertation submitted by Joyce L. Shield

in partial fulfillment of the requirement for the Ph.D.

19. KEY WORDS (Continue an rever e. dde if n.ce.ear y end identity by block number)

Analysis of Covariance (ANCOVA) Psychological Statistics

Johnson—Neyman Technique Statistical Models

Monte Carlo Program Regression Analysis

25. ABST RACT’ ~~~~~~~~ en ,evr.e e .

~* if nen.ewy wd Identity by block nienb.ev)

‘4The robustness of the Johnson—Neyman technique and analysis of covar iance

(ANCOVA) to violations of assumptions of homoscedasticity and homogeneity of

variance was tested through the use of Monte Carlo computer procedures. The

study simulated a one—way, fixed—effects analysis with two treatment groups , one criterion , and one covariate. Five fixed values of the covariate were

selected with zero mean and unit variance, while the values of Y were varied

randomly with a constant regression coefficient of .75. Four combinations —‘

—-- >r ’ r

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~~~~ ~~ U73 EDtTION OF 5 NOV 65 IS OBSO&ETE

UNCLA..S~~I1flE D CLASS,FICATIO,, OF THIS PAGE (ISien Del. Entered)

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UNCLASSIFIED

SICUNITY CLASSIFICATION OF THIS PAGC(Phen Date Entered)

20. ABSTRACT (Continued)

of group sizes (10 ,10;10 ,20;20, 1O;20 ,20) , five combinations of group var iances

(1 ,l; 1 ,2; 2,1;1 ,5;5 ,1), and five forms of heteroscedasticity (combined in

18 different pairs), were studied . These conditions were combined to produce

186 different simulated experimental conditions. For each simulated condition

3000 pseudo—random samples were generated and sampling distributions relevant

to the Johnson—Neyman technique and ANCOVA were coInPiled.r

~~~~~,,,~~,~~~,, Results indicated that ANCOVA is robust to violations of assumptiori~~~f

homoscedasticity and homogeneity of variance , bo th singly and in combina tion ,

when group sizes were equal. For cases of different group sizes and heterogeneous variances a predictable bias was observed. When the larger variance

was combined with the smaller group size the bias was conservative . When the

pairings were reversed the bias was nonconservative . The Johnson—Neyman

technique was sensitive to violation of the assumption of homoscedasticity

for both equal and unequal group sizes. The effect of heteroscedasticity

was to ord er the probability that any f ixed value of X would be included in

a reg ion of significance in a sequence parallel to the form of heteroscedas—

ticity. That is, in general , as the variance for a fixed value of X

increased , the probability of including that value of the covariate in a

region of significance increased.

I sus SlIlls tectIw

Self lectIsi /3

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Techulcal Pap.r 292

AN EMPIRICAL INVESTIGATIO N OF. THE EFFECT OF

HETEROSCEDASTIC ITY AND HETEROGENEITY

OF VARIA NCE ON THE ANALYSIS OF COYARIANCE

AND THE JO HNSON-NEYMAN TECHNI QUE

Joyce Lee Shields

Submitted by:

Milton S. Katz, Chief

INDIVIDUAL TRAI NI NG & SKILL EV ALU ATION TECHNI CAL AREA

Technica l accuracy and Approved by:

completeness certified by:

E. Ralph Dusek, Director

______________________ IN DIVIDUAL TR AINING & PERFORM AN CE

Milton S. Katz, RESEARCH LABORATORY

Technica l Area Chief

Joseph Zeidner

TECHNICAL DIRECTOR (DESIGNATE)

U.S. ARMY RESEARCH INSTITUTE FOR THE BEHAVIORAL AND SOCIAL SCIENCES

5001 Eisenhower Avenue. Alexandria, Virginia 22333

Office , Deputy Chief of Staff for Personnel

Department of the Army

July 1978

Army Project Number Individual Training Technology

20762722A777

Approwed for public release; distribution unlimited. 
~ 

~~

--~~~~~~~~~~ --—~~--—--~~~~~~~~~ ~ ---

~~~~ -~~~~~ -- ~~~~~~~~ --~~~~~~~~ ~~~~~~~~ . -

ARI Research Reports and Technical Papers are intended for sponsors of

R&D tasks and other research and military agencies. Any findings ready for

implementation at thc time of publication are presented in the latter part of

the Brief. Upon completion of a major phase of the task, formal recommen- dations for official action normally are conveyed to appropriate military

agencies by briefing or Disposition Form.

- -.- --- ~~ - —~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ,-- - .

~~~~~~ -- - - -, —r n. -~~~~~~~~ 
r

~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

FOREWORD

In the course of conducting research , the U.S. Army Research

Institute for the Behavioral and Social Sciences (ARI) makes use

of many statistical models and techniques suited to a wide

variety of information gathering and hypothesis testing . The

nature of conditions in the field , where the best Army da ta can

be collected , often results in violations of assumptions assumed

to be critical for the validity of specific statistical

operations or models. The information provided in this Techn ical Paper is useful, not only in the Individual Training and

Skill Evaluation Technical Area but to experimenters and

analysts in other areas of behavior science research who need

to determine the appropriateness of using Analysis of Covariance

(ANCOVA) or the Johnson—Neyman technique.

The entire research is responsive to requirements of RDT&E

Project PE62722A777, Individual Training Technology, FT 1978

Work Program , and to special requirements of the Deputy Chief of

Staff for Personnel.

eph Leç~~ler

Te hnical Director (Designa te) 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
AN E!.

~ IRICPJ~ INVESTIGATION OF THE EFFECT OF HETEROSCEDASTI CITY AND HETEROGENEITY OF VARIAN CE ON THE ANALYSIS OF COVARIANCE AND THE JOHNSON-NEYMAN

TECHNIQUE.

BRIEF

Requirement :

To determine the effects of violating the assumptions of homoscedas—

ticity and homogeneity of variance on significance tests associated with

ANCOVA and the Johnson-Neyman technique .

Procedure:

The robustness of the Johnson—Neyman technique and analysis of c~vari—

ance (ANCOVA ) to violations of assumptions of homoscedasticity and homogeneity

of variance was tested through use of Monte Carlo computer procedures . The

study simulated a one—way , fixed—effects analysis with two treatment groups ,

one criterion , Y , and one covariate , X. Five fixed values of the covariate

were selected with zero mean and unit variance , while the values of Y were

va~ied randomly with a constant regression coefficient of .75. Four combinations of group sizes (lO ,1O;lO ,20;20,lO;20 ,20), five combinations of

group variances (l,l;l ,2;2 ,l;l ,5;5 ,l), and five forms of heteroscedasticity

(combined in 18 different pairs), were studied. These conditions were

combined to produce 186 different simulated experimental conditions. For

each simulated condition , 3000 pseudo-random samples were generated and

sampling distributions relevant to the Johnson-Neyman technique and

ANCOVA were compiled.

Findings :

Results indicated that ANCOVA is robust to violations of assumptions

of homoscedasticity and homogeneity of variance , both singly and in combination , wt~

en group sizes were equal. For cases of different group sizes and

heterogeneous variances a predi ctable bias was observed. When the larger

variance was combined with the larger group size the bias was conservative .

When the pairings were reversed the bias was non—conservative . The

Johnson—Neyman technique was sensitive to violation of the assumption

homoscedasticity for both equal and unequal group sizes. The effect of

heteroscedasticity was to order the probability that any fixed value of

X would be included in a region of significance in a sequence parallel

to the form of heteroscedasticity. That is , in general , as the variance

for a fixed value of X increased , the probability of including that

value of the covariate in a region of si gnificance increased. As observed

with ANCOVA , the Johnson—Neyman techni que was robust to heterogeneity of

variance when group sizes were equal. However , when group sizes were

~

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,

not equal the empirical probabilities were biased in a non—conservative

direction when the larger variance was combined with the smaller group

size , and in a conservative direction when the larger variance and

larger group size we re combined.

Utilization of Findings :

In many empirical situations such as practical field experiments

conducted within the Army it is not possible to meet all the assumptions

of statistical models. If one or more of the assumptions is violated ,

the user has the choice of abandoning the model or proceeding with the

analysis at some risk. The results of this study may be used by the

investigator to estimate the degree of bias in the test of significance

associated with either ANCOVA or the Johnson—Neyman technique .

1
F: - - -

AN E!

~

TIRICAL INVESTIGATION OF THE EFFECT OF HETEROSCEDASTICITY AND

~ETERO.GENEITY OF VARIANCE ON THE ANALYSIS OF COVARIANCE AND THE

JOHNSON-NEYMAN TECHNI QUE

CONTENTS

Page

REVIEW OF RELATED LITERATURE 3

PROCEDURE

Heterogeneity of Variance 7

Heteroscedasticity 8

The Simulation Procedure 11

Goodness-of— fit Procedure 11

RESULTS 12

The Analysis of Covariance 12

The Johnson-Neyman Technique 12

DIS CUSSION 26

SUMMARY 28

REFERENCES 30

TABLES

Table 1. Comparison of Actual and Nominal Levels of

Significance in Simulations of “True”

Experiments Obtained by Peckham (1968),

Hamilton (1972), and McClaren (1972) 6

2. Values of a and p where a =a (1— ) and

a~~1, = .75 

y x y

3. Values for a where a =1,2, or 5 , for the

yx . 4 y x

Five Forms of He~eroscedasticity for the Five

Fixed Values of the Concomitant Variable , X 9

~ Results of the JColrnogorov—Smi rnov Goodness—of—fit

Tests of the Empirical to the Theoretical F

Distribution for Each of the Simulated Experimental

Conditions 13
- - ;—. —----—---.. —.-.

—

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—

~

--—-——.“.,—.-

~

. 

— -.—- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —. .-.

AN E!

~~IRICAL INVESTIGATION OF THE EFFECT OF HETERO SCEDASTICITY AND

HETEROGENEITY OF VARIAN CE ON THE ANALYSIS OF COVAR IANCE AND THE

JO}INSON-NEYMAN TECHNIQUE

CONTENTS

Page

TAB LES ( CONTINUED)

Table 5. Empirical Alpha Levels Corresponding to Nominal

Alpha Levels of .10, .05 , .02, and .01 for Each

Simulated Experimental Condition i~

4

6. Empirical Probability of Inclusion of Each

Fixed X in ‘i Region of Significance at Nominal

Alpha Levels of .10 , .05 , .025, and .01 for Each

Experimental Condition 17

7. Empirical Probability of Obtaining a Johnson—

Neyman Region of Significance at Nominal Alpha

Equal to .10, .05, .025, and .01 for Each Simulated

Experimental ConditIon 23

8. Empirical Results Obtained Using Potthoff’s

Simultaneous Con fidence Bounds When All Assumptions

Are Met and Group Size Equals 10 28

___ 

-__
_______

AN E~~ IRICAL INVESTIGATION OF THE EFFECT OF HETEROSCEDASTICITY AND

HETEROGENEITY OF VARIANCE ON THE ANALYSIS OF COVARIANCE LND THE JOHNSON—

NEYMAN TEC}TN IQUE

The effect on Type I error rates and power of vi olations of

assumptions of mathematical models underlying statistical analyses has

been studied for some time. Although many of the theoretical consequences

have been derived , it was only with the advent of high speed computers

that the empirical consequences could be determined with facility . As

poirted. out in a recent review by Glass et al. (1972), “the assumptions

of most mathematical models are always false to a greater or lesser

degree. ” The purpose of this study was to compare the empirical results

of certain reasonable violations of two of the assumptions underlying

the mathematical models associated with the analysis of covariance

(Fisher, 1932) and the Johnson—Neyman technique (Johnson and Neyman .

1936).

Although both the analysis of covariance (ANCOVA ) and the

Johnson—Neyman technique incorporate regression methods in order to

increase the precision of an experimental design , ANCOVA assumes

homogeneity of regression whereas the Johnson—Neyman technique is not

based on this assumption. Furthermore , there are two current areas of

great interest in educational psycho1o~ r research where heterogeneity

of regression might be expected (e.g. , Aptitude Treatment Interaction

(Bracht , 1970) and Moderator Variables (Bartlett et al., 1969; Ohiselli ,

1963)). Therefore, the Johnson—Neyman technique becomes an important

alternative to ANCOVA .

The Johnson—Neyman technique defines a region along the covariate

where significant treatment differences exist between two groups.

Unlike ANCOVA , the Johnson—Neyman technique does not test a hypothesis ,

but yields a confidence interval. The region not included in the

confidence interval , or the region of significance , may be a single

continuous region or two distinct regions where one group is

significantly better on the criterion in one region , and signifi cantly

inferior in the other region .

The mathematical model for both ANCOVA and the Johnson—Neyman

technique is:

~kj = + 

~j

X

kj + e

ki

where

= Y intercept of the Y—on—X regression line for group j.

= regression slope of the Y—on—X regression line for group j.

e

kJ = error of estimate for score k in group j.

~

- - - -- - ----- --- - - .

~ 

.
The e

ki are assumed to be independently and normally distributed with

O mean and homogeneous variance , a

2 = a

2 . That is , it is assumed

yJx ij

that errors of estimate are homogeneous for each fixed value of the

concomitant variable , X4, both within a treatment group (homoscedasticity),

and between treatment g~oups (homogeneity of variance).

In addition to the assumption of constant error variance , both

models assume : a linear relationship between the criterion and concomitant

variable and that values of the concomitant variable are fixed and

measured without error. ANCOVA makes the additional assumption of — homogeneity of regression , i.e., ~j~~w for all groups .

In many empirical situations it is not possible to meet all the

assumptions of statistical models. If one or more of the assumptions is

violated , the user has the choice of abandoning the modU or proceeding

with the analysis at some risk. Theoretical discussi~~ s of a technique

often enable the investigator to determine whether his test is biased

in a conservative or liberal direction , while empirical investigations

of violations often enable the investigator to estimate the degree of

bias.

The purpose of this study was to compare , by Monte Carlo methods ,

the effects of violating assumptions of homoscedasticity and homogeneity

of variance on significance tests associated with ANCOVA and the

Johnson-Neyman technique . The research questions asked are :

1. Are the Johnson.-Neyman technique and ANCOVA robust to the

violation of the assumption of homoscedasticity ?

2. Are the Johnson-Neyman technique and ANCOVA robust to the

violation of the assumption of homogeneity of variance?

3. Are the Johnson—Neyman technique and ANCOVA robust to the

simultaneous violation of the assumptions of homoscedasticity and

homogeneity of variance?

—2— 
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REVIEW OF RELATED LITERATURE

ANCOVA is an extension of analysis of variance (ANOVA ) and

regression analysis. As such , assumptions underlying ANCOVA include

all those associated with ANOVA and regression analysis as well as the

assumption of homogeneity of regression . Therefore, results from studies

of the robustness of ANOVA Co violations of some of its assumptions

might be expected to extend to ANCOVA . Particularly pertinent to the

current study is research on violation of the assumption of homogeneity

of variance. The effect of heterogeneous variances on Type I error

rates has been investigated theoretically (Box , ]‘)514; Scheff (, 1959) and

empirically (Norton , as reported in Lindquist , 1953).

Scheff~ (1959) reports the work of Hsu who calculated the exact

probability of Type I error rates for a two—tailed t test at the .05

level for three di fferent pairs of2sa~ple sizes (15,5;5, 3 and 7,7) and

9 ratios of population variances a1/a2 (O ,.l,.2 ,.5,l,2 ,5,lO ,~ ).

Scheff ~ (1959) extended Hsu ’s data to large samples in the two—group

case and studied 14 rat i

~ s ~f sample si z es (l ,2,5,oo) and 7 ratios of

population variances , a1/a2 (O,.2,.5 ,l,2,5, oc

~

).

Scheff~ concluded that inequality of variances has little effect

on Type I error rate when sample sizes are equal , but has serious effects

when sample sizes are not equal. In general , when the larger variance is

associated with the smaller sample size the true probability of Type I

error was found to exceed the nominal value , and when the larger

variance and sample size were paired the Type I error rate was found to

be less than the nominal value.

Box (19514) calculated exact probabilities of Type I errors for

fixed—effect ANOVA F tests at a nominal 5 percent level. The results

were obtained for 3, 5, and 7 groups , variance ratios of 1:2:3; 1:1:3;

1:1:1:1:3; and 1:1:1:1:1:1:7, and for 11 combinations of equal and

unequal group sizes. In general, it was observed that the F test is

robust to the violation of homogeneity of variance for equal sample

sizes , but not for unequal sample sizes. When the largest variance

was associated with the smallest sample size , the actual Type I error

rate was greater than the nominal value ; when the largest variance was

associated with the largest sample size , the value of ~ was smaller than

nominal.

The robustness of the F test to heterogeneous variances in balanced

designs reported by Scheff~ (1959) and Box (19514) was consistent with

an empirical investi gation by Norton (as reported in Lindquist , 1953).

Using a Monte Carlo simulation technique , he constructed (by hand)

populations of 10,000 cases each and sampled from these populations

in order to create empirical sampling distributions . In the phase

of his study in which he investigated heterogeneity of variance ,

populations were normal with equal means , but different variances

—3—

____ __________ -- -

~~~~ - - -
. V . - —.— ~~~~~~~~~~~~~~~~~~~~~~~~~~~

(a~ = 25, 100 , and 225). Marked heterogeneity of variance (1:14:9)

resulted in a small but predictable bias in the Type I errors and

empirical values were generally greater than nominal values .

Little research on the effect of heterogeneity of variance on

robustness of ANCOVA has been conducted. A theoretical paper by

Potthoff (1965) showed that the sensitivity of ANCOVA to heterogeneous

variances depended on the ratio n1

a /n2

a where n is sample size -

x

l x

2

and a is the standard error of the covariate. Three empirical studies

have ~een conducted concerning effects of violating the assumption of

homogeneity of variance while simultaneously violating the assumption

of homogeneity of regression (Peckham , 1968; Hamilton , 1972; and

Mc~1aren , 1972). In all studies , var~ance of the criterion measure

(a ) and variance of the covariate (a ) were held constant while

va~ying population regr~ssion s1ope~ ~~), pere~y2producing a

concomitant change in a (where a = a — a ~ ). In the model for

ANCOVA the assumptio~ o~ ~omogeneit~ ~f va~ianc~ applies to the variance

error of estimate (a ) .

y x

Pe~

ckharn (1968) varied regression slopes (and a

2

1 ) , number of

groups and sample size , although sample size was eq~A~ between groups

for all of his comparisons. Values of the covariate were fixed and

were chosen to conform as closely as possible to a normal distribution .

Peckham found that there were very small discrepancies in the actual

and empirical significance levels. In general , he observed that as

the degree of heterogeneity of the regression slopes (and heterogeneity

of variance) increased , the empirical rate of Type I errors was less

than the nominal value.

In McClaren ’s (1972) study , the number of treatment groups were

2, 3, and 5, and the sample sizes were 20, 30, 140 , 100, or 200, and

regression slopes were .l,.2,.3,.14,.5,.6,.7,.8, and .9. The average

slope across treatments was held constant at .5 for 180 out of the

183 simulated conditions . The values of the concomitant variable

were fixed , with zero mean and unit variance . For equal group sizes ,

he found that as degree of heterogeneity of regression increased (and

heterogeneity of variance increased), the empirical level of significance

became more conservative. For unequal sample sizes , the results parallel

the effect of violating the assumption of homogeneity of variance

reported by Box (19514) and Scheff~ (1959). That is , when the smallest

regression coefficient and the largest variance were combined with the

smallest sample size , the empirical significance levels were biased in

a non—conservative direction and when the pairings were reversed the

test was conservative.

Hamilton (1972 ) restricted the number of groups to two and varied

sizes , with distributions of the criterion and the covariate being

bivariate normal. Hamilton’s results , in general , parallel the results

of the effect of violating the assumption of homogeneity of variance
reported by Box (19514) and Scheff~ (1959). Hamilton found ANCOVA

robust to the violation of homogeneity of regression (and variance)

for equal sample sizes , but observed large discrepancies in empirical

and nominal alpha levels for unequal ~ampl~ sizes. When the larger

sample si~e occurred with the larger regression slope , and therefore

V smaller a , the empirical alpha levels were greater than corresponding

nominal a~~~a levels. When the smalle~ group size was paired with the

larger regression slope (and smaller a he observed that empirical

alpha levels were less than correspond~~g nominal levels. Hamilton ’s

study appears to present evidence for the generalizability of results

from the study of the effect of violating the assumptions associated

with ANOVA , as suggest ed by Cochran (1957) and Winer (1971).

In one condition , Hamilton studied the same combination of equal

sample sizes , number of groups , and regression coefficients as Peckham

and as McClaren , but failed to replicate their results; a comparison

is presented in Table 1. Whereas Hamilton ’s values were close to nominal

alpha, Peckham and McClaren observed a conservative bias in empirical

alpha levels where group sizes were equal and regression slopes

heterogeneous. It is difficult to resolve the discrepancies in the

results of these studies. Although it is impossible to determine simple

effects of violating the assumption of homogeneity of regression or

variance from the results of Hamilton (1972), McClaren (1972), and Peckham

(1968), an analytical study by Atiquallah (19614) suggests that the F

test of ANCOVA is robust to the violation of the assumption of

homogeneity of regression when sample size is large and the means of the

concomitant variable are equal ; otherwise , the test is biased in a

conservative direction .

There is no study examining the unique effect of heteroscedast.i citv

~n the robustness of ANCOVA. For an overview of ANCOVA and the effects

of other violations on ANOVA and A.NCOVA, conmrehensive reviews by

Glass et al. (1971) and Elashoff (1969) are available.

The Johnson—Neyman technique has not received as much attention

as ANCOVA and little is known concerning the effects of violating

assumptions underlying this statistical method. As originally presented , the Johnson—Neyman technique was designed for the case of two predictor

variables , two treatment groups, and one criterion (Johnson and Neyman , 1936), and papers on the technique have concerned extension to cases of “n” predictor variables and “k” groups (Abelson , 1953; Potthoff , 19614).

—5 - .

~~~~~~~~~~~~~~~~~~~~
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~-

~~~L L ~~~~~~ ±~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

— ----_ ..

--

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.- - -—---,- ———----—_ .—_--

Table 1

Comparison of Actual and Nominal Levels of Significance in Simulations

of “True” Experiments Obtained by Peckham (1968), Hamilton (1972), and McClaren (1972 )

Group Sizes

n 1=lO ,n 2

=lO n1

=20 ,n2~2O 

—

Regression

Coefficients Nominal Alpha Nominal Alpha

.10 .05 .01 .10 .05 .01

Peckham

.5,.5 .0914 .052 .013 .102 .0149 .013

.14 ,.6 .0 96 .050 .010 .089 .0 145 .009

.3,.7 .09 1 .o14~ .0 11 .097 .05 1 .009

.2 ,.8 .076 .0 39 .006 .076 .0 38 .008

.1,.9 .055 .0 29 .0014 .055 .027 .005

Hamilton

.5,.5 .098 .0149 .013 .115 .057 .015

.099 .053 .011 .103 .052 .011

.1014 .051 .010 .100 .0514 .012

.2,.8 .105 .058 .013 .103 .051 .009

.1,.9 .109 .057 .015 .103 .056 .015

McClaren

.105 .O1~7 .012

.14 ,.6 .099 .0149 .009

.090 .049 .011

.2,.8 .0714 .037 .010

.060 .022 .0014

— 6 —

I

_____________ - - -V
- V- — - — -

~~~~~ -~~~ V

V -- - V. V — V . - ~~~~~ _ - - -. - - - - - -- ——

PROCEDURE

The effect of violating the following assumptions on F tests

ass ociated with ANCOVA , and regions of significance associated with

the Johnson—Neyman technique were investigated:

1. heteroscedasticity

2. heterogeneity of variance

3. heterogeneity of variance and heteroscedasticity

The study simulated a simple one—way , fixed—effects analysis with

two treatment groups , one criterion , and one covariate . Five fixed

values of X, the covariate , assumed to be measured without error , with

zero mean and unit variance , were selected. Four combinations of group

sizes were used: 1O ,1O;20 ,2O;1O ,20; and 20,10; with an equal number

of cases at each fixed value of X. All assumptions underlying the two

methods were met except those unde r study . The value of the regression

coefficient was held constant at B .75, and the expected values of

~ (

~ = BX + c

~

) for both groups were —0.81485, — 0.142143, 0.0000% +0.142143.

+0.81485. Nominal significance levels of .01 to .99, increasing in

steps of .01, were used, for comparison with empirical c~ levels.

Heterogeneity of Variance

Three values of heterogeneity of variance were studied; a

~ I~ was

set at 3 values: 1, 2, and 5. As a

2

1 changes, w~ile B and a~ are held

constant , there is a concomitant cha~é~ in ~ and. a

y 

; resulting values

are shown in Table 2. = 1 was paired with a

2

~ = 1 (homogeneity of

variance ), a

2

1 = 2 , and ~

2

l 5. These pairs combined wi~h each of the

pairs ~f gro~~ Xsizes produ~~~ 11 experimental conditions (a ‘ x = 1 paired

with a

y~~ = 1 and group size combinations 10,20 and 20,10 a~~ equivalent).

Table 2

Values of a

2 and p where

and a

~~1,8=.75

2 2

a a P

yx y

1 i.~ 6 0.60

2 2.56 0.147

5 5.56 0.32

—7—

___ -- - V~~~~~~~~~~~~~ . - -
Heteros cedasti city

Five forms of heteroscedasticity were studied. The average

value of a was held constant at2l , 2 , or 5, while values of the

error vari~i~~e for each fixed X (a ) were distributed over the

y x.j

5 fixed values of X so that the following forms were approximated:

a. Equal a2 

~ for all X~~ , homoscedasticit y (form A) .

ii

b. Greatest a

2 

~ 

in the center with gradually decreasing

~ ij

a to either end point (form B). Y x ~j

c. Greatest a

2 

~ 

at the largest value of X and gradually

~ ij

decreasing to the smallest value of X ( form C).

d. Greatest a

2 

at the smallest value of X and gradually y x ij

increasing to the largest value of X (form D).

e. Smallest a

2 

x in the center gradually increasing to either

y ij

end point (form E).

Table 3 gives a

2 

at each fixed point of X for the 5 forms where

2 2

a

2 = 1, a = 2, and a = 5, and a graphic representation of the

yx yx y x

forms of heteroscedasticity is presented in Figure 1. When combined

in pairs , the 5 forms of heteroscedasticity produce 25 combinations

(including the null case). The pairings of form D with forms A , B , C ,

and E are the reverse repetitions of pairs CA , CB , CD , and CE and ,

hence , these were not investi gated. Thus , there were 18 combinations

of heteroscedasticity which were combined with 11 combinations of

group. sizes and variance ratios . Where group sizes and variances were

equal, mirror images of combinations were not investigated. Therefore,

the total number of simulated experimental conditions was reduced to

186.

— 8—
___________ V

Table 3

Values for a

2 where a

2 

x~~ ’

2’ or 5, for the Five Forms

y x .j Y

of Heteroscedasticity for the Five Fixed Values of

the Concomitant Variable , X

X1,~ Fixed Value Form of Heteroscedasticity

of X A B C D E

2

a = 1

- 

y x

-.1.141142 1.00 0.145 0.35 1.75 1.50

—0.7071 1.00 0.925 0.70 1.20 0.85

0.0000 1.00 2.25 1.00 1.00 0.30

+0.7071 1.00 0.925 1.20 0.70 0.85

+1.141142 1.00 0.145 1.75 0.35 1.50

2

a = 2

y Ix

~l.141142 2.00 0.90 0.70 3.50 3.00

—0.7071 2.00 1.85 1.140 2.140 1.70

0.0000 2.00 14.50 2.00 2.00 0.60

+0.7071 2.00 1.85 2.140 1.140 1.70

+1.141142 2.00 0.90 3.50 0.70 3.00

2

a = 5

y Ix

—1.141142 5.00 2.25 1.75 8.75 7.50

—0.7071 5.00 14.625 3.50 6.00 14.25

0.0000 5.00 11.25 5.00 5.00 1.50

+0.7071 5.00 14.625 6.00 3.50 14.25

+1.141142 5.00 2.25 8.75 1.75 7.50

—9—

_____ ~~~Vt

~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - 
B

Ftt

Fig. 1. Forms of heteros cedasticity studied (form A is

homoscedastic). Each vertical line is in standard deviation units,

and the avei age unit is 1.

— 10 —

—- -. —_-V,._------ --- — — ~~~~~~~~~~ __, . ‘ -- ,-.-.
~ - -.-- ----- -- -—----—- - —--.- — 

The Simulation Procedure

For each of the 186 simulated conditions , the following

parameters were set :

1. The fixed set of values for X, the concomitant variable .

2. The number of observations per fixed X for each group .

3. The expectation of the dependent variable Y. for each fixed

x . . 

1

ii

4. The standard deviation , a , of each fi xed X . fo r each

yx. ij

group . ij

5. The number of samples , which was set equal to 3000.

UNIVAC 1108 Math—Pak ( 1970) subroutines RANDN and RANDU were

used to generate random numbers. RANDU computes uniformly distributed

pseudo—random real numbers between 0 and 1, whe reas , RANDN p roduces

sets of pseudo—random numbers which are normally distributed with a

specified mean and standard deviation. Studies (UNIVAC 1108 Math—Pak

Programmer ’s Reference Manual, UP—7542, section l1

~.3) have shown that

the initialization number for the random number generator is critical

for ensuring properties of randomness. Therefore , RANDU was used to

supply a new starting number to RANDN for each of the 3000 samples in

each run . A check on randomness of the generated sequences was made

for each of the 186 initialization numbers. Using each of the

initialization numbers an empirical F distribution was created for

the case where no assumptions were violated and group sizes were each

equal to 10. The Kolmogorov—Smirnov test was used to assess the

goodness—of— fit of these empirical F distributions to theoretical F

distributions ; at a = .05, the test failed to reject the hyopthesis

of no difference between the nominal and empirical distributions for

all initialization numbers used. In addition , checks of the randomness

and normality of the samples generated by RANDN have been run and

satisfactory results have been reported by Hamilton (1972).

Goodness—of—fi t Procedure

A total of 186 different goodness—of—fit testing situations

were simulated in this study . Both ANCOVA and the Johnson—Neyman

technique were carried out for each experimental condition . Ninety—

nine F values were computed for each simulation at significance levels

ranging from .01 to .99 in steps of .01. The goodness—of—fit of the

empirical F distributions for ANCOVA to the theoretical F distributions

was tested using the Kolmogorov—Smirnov one—sample goodness—of—fit

test at- ct = .05 (Siegel, 1956). In addition , the number of Type I

errors was computed at each of the significance levels.

— 11 — 
~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For the Johnson—Neyman technique , the probability that each fixed

X would be in cluded in a reg ion of si gn ifican ce , as well as the total

probability of obtaining any region of sign ificance , and probabilities

of obtaining central versus tail regions at nominal significance levels

of .01, .025, .05, and .10 were computed.

RESULT S

The Analysis of Covariance

The results of the goodness—of—fit tests in each of the 186 simulated

conditions are presented in Table 4. The symbol NS means that the

Kolmogorov—Smirnov test failed to reject the null hypothesis that there was

no difference between the empirical and nominal F distributions ; the symbol

S means that the Kolmogorov—Smirnov test rejected the null hypothesis.

The forms of heteroscedasticity are represented by letters A ,B,C ,D, and

E ( see Figure 1, or Table 3). The empirical significance levels corre— V

sporiding to the nominal levels of .lO ,.05,.02 , and .01 for all experimental

combinations are shown in Table 5.

The Johnson—Neyman Technique

The empirical probability of the inclusion of each fixed value of

X in a region of significance at nominal alpha levels of .lO ,.05 ,.O25, and

.01 under each of the 186 simulated experimental conditions is reported in

Table 6. Table 7 shows empirical probabilities of obtaining any region of

significance , when the nominal alpha level was set at .l0 ,.O5,.O25, and

.01.

— 12 —

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DISCUSSION

ANCO VA appears to be robust to the violation of the assumptions

of homoscedasticity and homogeneity of variance both singly and in

combination , when group sizes are equal . In eve ry condition where

group sizes and variances were heterogeneous the goodness—of—fit

hypothesis was rejected. When the larger variance was combined with

the larger group size ) the empirical significance levels of AN COVA were

conservative , and when the larger variance was combined with the smaller

group size , the empirical alpha levels were nonconservative. The effect

of,.the violation of the assumption of homogeneity of variance on the

empiri cal F distribution of ANCOVA pa rallels the results obtained by

Norton (as reported in Lindquist , 1953) and Box (19514) when investigating

the effect of heterogeneity of variance on the empirical F distribution

of M1OVA. In addition , the results obtained by Hamilton (1972),

McClaren (1972), and Peckham ( 1968), where they investigated the

simultaneous violation of the assumptions of homogeneity of variance and

regre ssion , are replicated.

The failure to find any condition where heteroscedasticity alone

was responsible for the rejection of the goodness—of—fit hypothesis

leads one to the conclusion that the assumption of homoscedasticity is

not import ant to ANCOVA . Thus, if variances are homogeneous , the transformation of heteroscedastic data before using ANCOVA , as was suggest ed

by Elashoff (1969), may be unnecessary .

The Johnson—Neyman technique is robust to the vi olation of the

assumption of homogeneity of variance when group sizes are equal . However ,

when group sizes are unequal heterogeneity of variance biases the

probabilities of obtaining regions of significance in the same direction

as found with ANCOVA. That is , the larger variance combined with the

smaller group size produces a non—conservative bias and the larger variance

combined with the larger group size produces a conservative bias .

The probability that a fixed X would ‘be included in a region of

significance was consistently determined by the form of heteroscedasticity .

If the shapes of the probability distributions shown in Table 6 are

compare d to the shapes of heteroscedasticity in Figure 1, the following

conclusions emerge :

1. When a

2 is constant across X. , the probability that X . y x .j ij

is included in a region of significance is constant and equal to the nominal

alpha level.

2. When is greatest for the central value of X. and smallest

y l x ij lj

for the tails , the probability that X is included in a region of

significance is greatest for the centrtl X. and smallest for the tails ,

and in general , the probabilities are conse~’vative .

— 26 —
--—- ~ -- -- ~ 

3. When the a

2 is greatest at either end value and yIx ij -

progressively decreases to the opposi te end value , the probability that

~~~ will be in cluded in a region of significance follows the size of

The average significance level is close to the nominal level of

significance.

14. When a

2 is smallest for the center X . and largest at

y x 1,4

either end , the proba~ility that ~~ will be included in a region of

significance is greatest at the end values of X and smallest for the

central X1~~. In general , the empirical probabilities are nonconservative .

The effect of the form of heteroscedasticity when èombined with

a different form of heteroscedasticity is partially determined by the

average variance of the group . The form of heteroscedasticity combined

with the larger variance has a greater influence on the form of the

probability distribution for fixed values of X.

Although the magnitude of the empirical values of the probabilities

are further influenced by the condition of unequal group sizes and

heterogeneity of varian ce , the effects of the form of heteroscedasticity

outlined above hold constant. That is , the relative differences in the

probabilities associated with each X1 ,~ is fixed by the form of

heteroscedasticity, but the size of the probabilities are biased by the

combination of unequal group sizes and heterogeneity of variance.

The probability of finding any region of significance greatly

exceeded the nominal significance level for all forms of heteroscedasticity.

As explicated by Potthoff’ (19614) the Johnson—Neyman technique may be used

to specify the region of significance , but does not set simultaneous

confidence bounds. For example , one can say with 95 percent confidence

that for any specifi c point P within the region of significance that

there is a true di fference between the two groups when nominal alpha

equals .05. One cannot say with 95 percent confidence that for all

points within the region of significance there is a true difference in

performance for the two groups.

Although not directly related to the research questions of the

dissertation , it is of int erest to inquire how to set a simultaneous

confidence bound equal to the nominal alpha level. Potthoff suggests a

method for setting simultaneous confidence bounds which is basically

identical with the defining inequality for the Johnson—Neyman region of

significance with the exception that F1 + 1 4 • is replaced by (r+l)F ,n1 n2— ,a

where r is the number of concomitant variables. When this r+l,n1

+n2— ;

~

procedure was tested for the simu lated exp~ rixnent~ l condition of equal

group sizes (10,10 and equal variances (

~~ x l )0y~x l the probability

of finding a region of significance matched the nominal a level as shown

in Table 8.

— 27 —
_________ - 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Table 8

Empirical Results Obtained Using Potthoff’s Simultaneous.

Confidence Bounds When All Assumptions Are Met

and Group Si z e Equals 10

Nominal Alpha

.10 .05 .025 .01

Simultaneous Confidence

Coefficient .099 .0147 .025 .011

Therefore , it is recommended that Potthoff’s procedure for obtaining

simultaneous confidence bounds for the Johnson—Neyman technique be utilized

when one wishes to obtain a region where one can state with 1—ce confidence

that there is a treatment effect simultaneously for all points within

the region .

A second approach to maintaining the experiment—wise ~rpe I error

rate at the nominal significance level is to follow the procedure

suggested by Johnson and Jackson (1959) and Abelson (1953). First , test

for homogeneity of regression ; if this hypothesis is rejected proceed with

the Johnson—Neyman technique , otherwise , use ANCOVA.

The marked effect of heteroscedasticity on the Johnson—Neyman

technique suggests that when heteroscedasticity is observed in the data

that the use of the Johnson—Neyman technique is not appropriate if the

heteroscedasticity cannot be eliminated or minimized. If forms C and D

are present in the data it may be possible to apply a va riance—

stabilizing transformation (Dayton , 1970).

2 The problem of estimating

and testing regression coefficients when a is a function of X. yx.~ ij

has been discussed by Rutemiller and Bowers (1968) and Levenbach (1973).

~ovever, the proper way of’ dealing with heteroscedasticity was a problem

presented in the original J(~

hnson and Neyman article (1936) and it has

not been solved to date.

SUM?vIARY

The robustness of the Johnson—Neyman technique and analysis of

covariance (ANCOVA ) to violations of assumptions of homoscedasticity -J

and homogeneity of variance was tested through use of Monte Carlo

computer procedures. The study simulated a one—way , fixed—effects analysis

with two treatment groups , one criteri on , Y , and one covariate , X . Five

fixed values of the covariate were selected with zero mean and unit

variance , while the values of Y were varied randomly with a constant

regression coefficient of .75. Four combinations of group sizes (10,10;

lO ,20;2O ,lO;20,20), five combinations of group variances (1,l;l ,2;2,1;l ,

5;5 ,l), and five forms of heteroscedasticity (combined in 18 different

- 28 - 

~~- -~~~~~~~~~-~~ -~~~- - ~~~~~~ - - ‘ ~~~~ --‘-~~~~~~~~~~~~~ -- ----- -
pairs), were studied. These conditions were combined to produce i86

different simulated experimental conditions. For each simulated

condition , 3000 pseudo—random samples were generated and sampling

distributions relevant to the Johnson—Neyman techni que and ANCOVA

were compiled.

Results indicated that ANCOVA is robust to violations of

assumptions of homoscedasticity and homogeneity of variance , both

singly and in combination , when group sizes were equal . For cases

of different group sizes and heterogeneous variances a predictable

bias was observed. When the larger variance was combined with the

larger group size the bias was conservative . When the pairings were

reversed the bias was non—conservative . The Johnson—Neyman technique

was sensitive to violation of the assumption of homoscedasticity for

both equal and unequal group sizes. The effect of heteroscedasticity

was to order the probability that any fixed value of X would be included

in a region of significance in a sequence parallel to the form of

heteroscedasticity. That is , in general , as the variance for a fixed

value of X increased , the probability of including that value of .the

covariate in a region of significance increased. As observed with

ANCOVA , the Johnson—Neyman technique was robust to heterogeneity of

variance when group sizes were equal. However, when group sizes were

not equal the empirical probabilities were biased in a non—conservative

direction when the larger variance was combined with the smaller group

size , and in a conservative direction when the larger variance and

larger group size were combined.

— 29 —

__________ ___ 
- - 

____________ -

~~~~~ - -~~ -~~- -- ~~-~~--~~~~-- ~~~~ 

~~~

-

~~

---

~~

--

~~~~~~

-

REFE HEN CES

Abelson , H. P. A note on the Neyman—Johnson technique .

Psychometrika, 1953, 18, 213—218.

Atiqullah , M. The robustness of the covariance analysis of a one-way

classification . Biometrika, 19614, 51, 365—372.

Bartlett , C. J., and O’Leary , B. A differential prediction model to

moderate the effects of heterogeneous gr’-ups in personnel selection

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