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On the Practice of Dichotomization of Quantitative Variables

Robert C. MacCallum, Shaobo Zhang, Kristopher J. Preacher, and Derek D. Rucker

Ohio State University

The authors examine the practice of dichotomization of quantitative measures,

wherein relationships among variables are examined after 1 or more variables have

been converted to dichotomous variables by splitting the sample at some point on

the scale(s) of measurement. A common form of dichotomization is the median

split, where the independent variable is split at the median to form high and low

groups, which are then compared with respect to their means on the dependent

variable. The consequences of dichotomization for measurement and statistical

analyses are illustrated and discussed. The use of dichotomization in practice is

described, and justifications that are offered for such usage are examined. The

authors present the case that dichotomization is rarely defensible and often will

yield misleading results.

We consider here some simple statistical proce-

dures for studying relationships of one or more inde-

pendent variables to one dependent variable, where all

variables are quantitative in nature and are measured

on meaningful numerical scales. Such measures are

often referred to as individual-differences measures,

meaning that observed values of such measures are

interpretable as reflecting individual differences on

the attribute of interest. It is of course straightforward

to analyze such data using correlational methods. In

the case of a single independent variable, one can use

simple linear regression and/or obtain a simple corre-

lation coefficient. In the case of multiple independent

variables, one can use multiple regression, possibly

including interaction terms. Such methods are rou-

tinely used in practice.

However, another approach to analysis of such data

is also rather widely used. Considering the case of one

independent variable, many investigators begin by

converting that variable into a dichotomous variable

by splitting the scale at some point and designating

individuals above and below that point as defining

two separate groups. One common approach is to split

the scale at the sample median, thereby defining high

and low groups on the variable in question; this ap-

proach is referred to as a median split. Alternatively,

the scale may be split at some other point based on the

data (e.g., 1 standard deviation above the mean) or at

a fixed point on the scale designated a priori. Re-

searchers may dichotomize independent variables for

many reasons—for example, because they believe

there exist distinct groups of individuals or because

they believe analyses or presentation of results will be

simplified. After such dichotomization, the indepen-

dent variable is treated as a categorical variable and

statistical tests then are carried out to determine

whether there is a significant difference in the mean of

the dependent variable for the two groups represented

by the dichotomized independent variable. When

there are two independent variables, researchers often

dichotomize both and then analyze effects on the de-

pendent variable using analysis of variance

(ANOVA).

There is a considerable methodological literature

examining and demonstrating negative consequences

of dichotomization and firmly favoring the use of re-

gression methods on undichotomized variables. Nev-

ertheless, substantive researchers often dichotomize

independent variables prior to conducting analyses. In

this article we provide a thorough examination of the

practice of dichotomization. We begin with numerical

examples that illustrate some of the consequences of

dichotomization. These include loss of information

about individual differences as well as havoc with

Robert C. MacCallum, Shaobo Zhang, Kristopher J.

Preacher, and Derek D. Rucker, Department of Psychology,

Ohio State University.

Shaobo Zhang is now at Fleet Credit Card Services, Hor-

sham, Pennsylvania.

Correspondence concerning this article should be ad-

dressed to Robert C. MacCallum, Department of Psychol-

ogy, Ohio State University, 1885 Neil Avenue, Columbus,

Ohio 43210-1222. E-mail: maccallum.1@osu.edu

Psychological Methods

2002, Vol. 7, No. 1, 19–40

Copyright 2002 by the American Psychological Association, Inc.

1082-989X/02/$5.00DOI: 10.1037//1082-989X.7.1.19

19

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regard to estimation and interpretation of relationships

among variables. We then examine the dichotomiza-

tion approach in terms of issues of measurement of

individual differences and statistical analysis. We

next review current practice, providing evidence of

common usage of dichotomization in applied research

in fields such as social, developmental, and clinical

psychology, and we examine and evaluate justifica-

tions offered by users and defenders of this procedure.

Overall, we present the case that dichotomization of

individual-differences measures is probably rarely

justified from either a conceptual or statistical per-

spective; that its use in practice undoubtedly has se-

rious negative consequences; and that regression and

correlation methods, without dichotomization of vari-

ables, are generally more appropriate.

Numerical Examples

Example Using One Independent Variable

We begin with a series of numerical examples us-

ing simulated data to illustrate and distinguish be-

tween the regression approach and the dichotomiza-

tion approach. Raw data for these numerical examples

can be obtained from Robert C. MacCallum’s Web

site (http://quantrm2.psy.ohio-state.edu/maccallum/).

Let us first consider the case of one independent vari-

able, X, and one dependent variable, Y. We defined a

simulated population in which the two variables fol-

lowed a bivariate normal distribution with a correla-

tion of ?XY? .40. Using a procedure described by

Kaiser and Dickman (1962), we then drew a random

sample (N ? 50 observations) from this population.

Sample data were scaled such that MX? 10, SDX?

2, MY? 20, and SDY? 4. A scatter plot of the

sample data is displayed in Figure 1. We assessed the

linear relationship between X and Y by obtaining the

sample correlation coefficient, which was found to be

rXY? .30, with a 95% confidence interval of (.02,

.53). The squared correlation (r2

that about 9% of the variance in Y was accounted for

by its linear relationship with X. A test of the null

hypothesis that ?XY? 0 yielded t(48) ? 2.19, p ?

.03, leading to rejection of the null hypothesis and the

conclusion that there is evidence in the sample of a

nonzero linear relationship in the population. Of

course, the outcome of this test was foretold by the

confidence interval, because the confidence interval

did not contain the value of zero.

We then conducted a second analysis of the rela-

tionship between X and Y, beginning by converting X

into a dichotomous variable. We split the sample at

the median of X, yielding high and low groups on X,

each containing 25 observations. The resulting di-

chotomized variable is designated XD. A scatter plot

of the data following dichotomization of X is shown in

Figure 2. The relationship between X and Y was then

evaluated by testing the difference between the Y

means for the high and low groups on X. Those means

were found to be 21.1 and 19.4, respectively, and a

XY? .09) indicated

Figure 1. Scatter plot of raw data for example of bivariate relationship.

MACCALLUM, ZHANG, PREACHER, AND RUCKER

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test of the difference between them yielded t(48) ?

1.47, p ? .15, resulting in failure to reject the null

hypothesis of equal population means. From a corre-

lational perspective, the correlation after dichotomi-

zation was rXDY? .21 (r2

confidence interval of (−.07, .46). Again, the confi-

dence interval implies the result of the significance

test, this time indicating a nonsignificant relationship

because the interval did include the value of zero. A

comparison of the results of analysis of the associa-

tion between X and Y before and after dichotomization

of X shows a distinct loss of effect size and loss of

statistical significance; these issues are addressed in

detail later in this article.

The dichotomization procedure can be taken one

step further by splitting both X and Y, thereby yielding

a 2 × 2 frequency table showing the association be-

tween XDand YD. In the present example, this ap-

proach yielded frequencies of 13 in the low–low and

high–high cells and frequencies of 12 in the low–high

and high–low cells. The corresponding test of asso-

ciation ?2

1(1, N ? 50) ? 0.08, p ? .78, showed a

nonsignificant relationship between the dichotomized

variables, which was also indicated by the small cor-

relation between them, rXDYD? .06 with a 95% con-

fidence interval of (−.22, .33). Note that the dichoto-

mization of both X and Y has further eroded the

strength of association between them.

XDY? .04), with a 95%

Example Using Two Independent Variables

We next consider an example where there are two

independent variables, X1and X2. In practice such

data would appropriately be treated by multiple re-

gression analysis. However, a common approach in-

stead is to convert both X1and X2into dichotomous

variables and then to conduct ANOVA using a 2 × 2

factorial design. We provide an illustration of both

methods. Following procedures described by Kaiser

and Dickman (1962), we constructed a sample (N ?

100 observations) from a multivariate normal popu-

lation such that the sample correlations among the

three variables would have the following pattern: rX1Y

? .70, rX2Y? .35, and rX1X2? .50. To examine the

relationship of Y to X1and X2, we first conducted

multiple regression analyses. Without loss of gener-

ality, regression analyses were conducted on stan-

dardized variables, ZY, Z1, and Z2. Using a linear re-

gression model with no interaction term, we obtained

a squared multiple correlation of .49 with a 95% con-

fidence interval of (.33, .61).1This squared

1Confidence intervals for squared multiple correlations

are useful but are not commonly provided by commercial

software for regression analysis. A free program offered by

James Steiger, available from his Web site (http://

Figure 2. Scatter plot for example of bivariate relationship after dichotomization of X. (Note

that there is some overlapping of points in this figure.)

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

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multiple correlation was statistically significant, F(2,

97) ? 46.25, p < .01. The corresponding standardized

regression equation was

ZˆY? .70(Z1) + .00(Z2).

The coefficient for Z1was statistically significant, ?1

? .70, t(1) ? 8.32, p < .01, whereas the coefficient

for Z2obviously was not significant, ?2? .00, t(1) ?

0.00, p ? 1.00, indicating a significant linear effect of

Z1on ZY, but no significant effect of Z2. Inclusion of

an interaction term Z3? Z1Z2in the regression model

increased the squared multiple correlation slightly to

.50. The corresponding regression equation was

ZˆY? .72(Z1) − .01(Z2) − .07(Z3).

The standardized regression coefficient for Z1was

statistically significant, ?1? .72, t(1) ? 8.38, p <

.01, whereas the other two coefficients were not, ?2

? −.01, t(1) ? −0.12, p ? .96, and ?3? −.07, t(1)

? −1.11, p ? .27, indicating no significant linear

effect of Z2and no significant interaction.

We then conducted a second analysis on the same

data, beginning by dichotomizing X1and X2by split-

ting both at the median to create X1Dand X2D. Results

of a two-way ANOVA yielded a significant main ef-

fect for X1D, F(1, 96) ? 42.50, p < .01; a significant

main effect for X2D, F(1, 96) ? 5.26, p ? .02; and a

nonsignificant interaction, F(1, 96) ? 0.19, p ? .67.

Total variance accounted for by these effects was .40.

Of special note in these results is the presence of a

significant main effect of X2Dthat was not present in

the regression analyses but arose only after both in-

dependent variables were dichotomized. This phe-

nomenon is discussed further later in this article. It is

also noteworthy that total variance accounted for was

reduced from .50 prior to dichotomization to .40 after

dichotomization.

Summary of Examples

From these few examples it should be clear that

there exist potential problems when quantitative inde-

pendent variables are dichotomized prior to analysis

of their relationship to dependent variables. The ex-

ample with one independent variable showed a loss of

effect size and of statistical significance following

dichotomization of X. The example with two indepen-

dent variables showed a significant main effect fol-

lowing dichotomization that did not exist prior to di-

chotomization. Although many other cases and

examples could be examined, these simple illustra-

tions reveal potentially serious problems associated

with dichotomization. If phenomena such as those just

illustrated would be common in practice, then di-

chotomization of variables probably should not be

done unless rigorously justified. In the following sec-

tion we examine these phenomena closely, focusing

on the impact of dichotomization on measurement and

representation of individual differences as well as on

results of statistical analyses.

Measurement and Statistical Issues Associated

With Dichotomization

Representing Individual Differences

We first consider the impact of dichotomization on

the measurement and representation of individual dif-

ferences associated with a variable of interest. Sup-

pose we have a single independent variable, X, mea-

sured on a quantitative scale, and we observe a sample

of individuals who vary along that scale, and suppose

the resulting distribution is roughly normal. A distri-

bution of this type is illustrated in Figure 3a, with four

specific individuals (A, B, C, D) indicated along the

x-axis. Assuming approximately interval properties

for the scale of X, these individuals can be compared

with each other with respect to their standing on X.

For instance, Individuals B and C are more similar to

each other than are Individuals A and D. Individuals

A and B are different from each other, and that dif-

ference is greater than the difference between B and

C. When observed in empirical research, such differ-

ences and comparisons among individuals would

seem to be relevant to the understanding of such a

variable and its relationship to other variables. Re-

searchers in psychology typically invest great effort in

the development of measures of individual differences

on characteristics and behaviors of interest. Instru-

ments are developed so as to have adequate levels of

reliability and validity, implying acceptable levels of

precision in measuring individuals, and implying in

turn that individual differences such as those just de-

scribed are meaningful.

Now suppose X is dichotomized by a median split

as illustrated in Figure 3b. Such dichotomization al-

ters the nature of individual differences. After di-

chotomization, Individuals A and B are defined as

equal as are Individuals C and D. Individuals B and C

www.interchg.ubc.ca/steiger/homepage.htm) computes

such confidence intervals as well as other useful statistical

information associated with correlation coefficients.

MACCALLUM, ZHANG, PREACHER, AND RUCKER

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are different, even though their difference prior to

dichotomization was smaller than that between A and

B, who are now considered equal. Following dichoto-

mization, the difference between A and D is consid-

ered to be the same as that between B and C. The

median split alters the distribution of X so that it has

the form shown in Figure 3c. Clearly, most of the

information about individual differences in the origi-

nal distribution has been discarded, and the remaining

information is quite different from the original. Such

an altering of the observed data must raise questions.

What was the purpose of measuring individual differ-

ences on X only to discard much of that information

by dichotomization? What are the consequences for

the psychometric properties of the measure of X? And

what is the impact on results of subsequent analyses

of the relationship of X to other variables?

It seems that to justify such discarding of informa-

tion, one would need to make one of two arguments.

First, one might argue that the discarded information

is essentially error and that it is beneficial to eliminate

such error by dichotomization. The implication of

such an argument would be that the true variable of

interest is dichotomous and that dichotomization of X

produces a more precise measure of that true di-

chotomy. An alternative justification might involve

recognition that the discarded information is not error

but that there is some benefit to discarding it that

compensates for the loss of information. Both of these

perspectives are discussed further later in this article.

Impact on Results of Statistical Analyses

Review of types of correlation coefficients and their

relationships.Dichotomization of quantitative vari-

ables affects results of statistical analyses involving

those variables. To examine these effects, it is neces-

sary to understand the meaning of and relationships

among several different types of correlation coeffi-

cients. A correlation between two quantitatively mea-

sured variables is conventionally computed as the

common Pearson product–moment (PPM) correla-

tion. A correlation between one quantitative and one

dichotomous variable is a point-biserial correlation,

and a correlation between two dichotomous variables

is a phi coefficient. The point-biserial and phi coeffi-

cients are special cases of the PPM correlation. That

is, if we apply the PPM formula to data involving one

quantitative and one dichotomous variable, the result

will be identical to that obtained using a formula for

a point-biserial correlation. Similarly, if we apply the

PPM formula to data involving two dichotomous vari-

ables, the result will be identical to that obtained using

a formula for a phi coefficient. The point-biserial and

phi coefficients are typically used in practice for

analyses of relationships involving variables that are

true dichotomies. For example, one could use a point-

biserial correlation to assess the relationship between

gender and extraversion, and one could use a phi co-

efficient to measure the relationship between gender

and smoking status (smoker vs. nonsmoker).

Some variables that are measured as dichotomous

variables are not true dichotomies. Consider, for ex-

ample, performance on a single item on a multiple

choice test of mathematical skills. The measured vari-

able is dichotomous (right vs. wrong), but the under-

lying variable is continuous (level of mathematical

knowledge or ability). Special types of correlations,

specifically biserial and tetrachoric correlations, are

used to measure relationships involving such artificial

dichotomies. Use of these correlations is based on the

assumption that underlying a dichotomous measure is

a normally distributed continuous variable. For the

case of one quantitative and one dichotomous vari-

Figure 3. Measurement of individual differences before

and after dichotomization of a continuous variable.

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

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able, a biserial correlation provides an estimate of the

relationship between the quantitative variable and the

continuous variable underlying the dichotomy. For

the case of two dichotomous variables, the tetrachoric

correlation estimates the relationship between the two

continuous variables underlying the measured di-

chotomies. Biserial correlations could be used to es-

timate the relationship between a quantitative mea-

sure, such as a measure of neuroticism as a

personality attribute, and the continuous variable that

underlies a dichotomous test item, such as an item on

a mathematical skills test. Tetrachoric correlations are

commonly used to estimate relationships between

continuous variables that underlie observed dichoto-

mous variables, such as two test items.

Note that for the case of one quantitative and one

dichotomous variable, one could calculate a point-

biserial correlation, to measure the observed relation-

ship, or a biserial correlation, to estimate the relation-

ship involving the continuous variable underlying the

dichotomous measure. The biserial correlation will be

larger than the corresponding point-biserial correla-

tion, because of the assumed gain in measurement

precision inherent in the former. In the population,

the relationship between these two correlations is

given by

?pb= ?b?

h

?pq?,

(1)

where p and q are the proportions of the population

above and below the point of dichotomization, and h

is the ordinate of the normal curve at that same point

(Magnusson, 1966). Values of h for any point of di-

chotomization can be found in standard tables of nor-

mal curve areas and ordinates (e.g., Cohen & Cohen,

1983, p. 521).

For the case of two dichotomous variables, one

could compute either a phi coefficient to measure the

observed relationship or the tetrachoric correlation to

estimate the relationship between the underlying di-

chotomies. Again because of the assumed gain in

measurement precision, the tetrachoric correlation is

higher than the corresponding phi coefficient. Al-

though the general relationship between a phi coeffi-

cient and a tetrachoric correlation is quite complex, it

can be defined for dichotomization at the mean as

follows:

?phi= 2?arcsin(?tetrachoric)???

(2)

(Lord & Novick, 1968, p. 346). If the assumptions

inherent in the biserial and tetrachoric correlations are

valid, then the corresponding point-biserial correla-

tion and phi coefficient can be seen to underestimate

the relationships of interest because of their failure to

account for the artificial nature of the dichotomous

measures. Given this background on correlation coef-

ficients, we now turn to an examination of how di-

chotomization of a quantitative variable impacts mea-

sures of association between variables.

Analyses of effects of one independent variable.

Various aspects of the impact of dichotomization on

results of statistical analysis have been examined and

discussed in the methodological literature for many

years. Here we review and examine in detail the most

important issues in this area, beginning with the sim-

plest case of dichotomization of a single independent

variable. Basic issues associated with this case were

discussed by Cohen (1983). Suppose that X and Y

follow a bivariate normal distribution in the popula-

tion with a correlation of ?XY; variance in Y accounted

for by its linear relationship with X is then ?2

dichotomized at the mean to produce XD, then the

resulting population correlation between XDand Y can

be designated ?XDY. (Note that for a normally distrib-

uted variable, dichotomization at the mean and the

median are equivalent in the population.) To under-

stand the impact of dichotomization on the relation-

ship between the two variables, we must examine the

relationship between ?XYand ?XDY. This relationship

can be seen to correspond to the theoretical relation-

ship between a biserial and point-biserial correlation.

That is, ?XDYcorresponds to a point-biserial correla-

tion, representing the association between a dichoto-

mous variable and a quantitative variable, and ?XYis

equivalent to the corresponding biserial correlation,

where X is the continuous, normally distributed vari-

able that underlies XD. The relationship between a

point-biserial and biserial correlation was given in

Equation 1. Given this relationship, the effect of di-

chotomization on ?XYcan then be represented as

?XDY= ?XY?

XY. If X is

h

?pq?.

(3)

The value of h/√pq can be viewed as a constant, to be

designated d, representing the effect of dichotomiza-

tion under normality. For example, if X is dichoto-

mized at the mean to produce XD, then p ? .50, q ?

.50, h ? .399, yielding d ? .798. The effect of di-

chotomization on the correlation is then given by ?XDY

MACCALLUM, ZHANG, PREACHER, AND RUCKER

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? (.798)?XY, with shared variance being reduced by

?2

XY. To represent the effect of dichoto-

mization at points other than the mean, Figure 4

shows the value of d ? h/√pq as a function of p, the

proportion of the population above the point of di-

chotomization. It can be seen that as the point of

dichotomization moves further from the mean, the

impact on the correlation coefficient increases.

Clearly there is substantial loss of effect size in the

population due to dichotomization at any point. This

quantification of the loss is consistent with the sub-

jective impression conveyed by Figures 1 and 2,

where the linear relationship between X and Y appears

to be weakened by dichotomization of X.

The results in our numerical example reported ear-

lier are consistent with these theoretical results. Our

sample was drawn from a population where ?XY? .40

and ?2

XY? .16. Following dichotomization, these

population values would become ?XDY? (.798)(.40)

? .32 and ?2

XDY? (.637)(.16) ? .10. In our sample

we found that dichotomization reduced rXY? .30 to

rXDY? .21, and the corresponding squared correlation

from .09 to .04. Thus, the proportional reduction of

effect size in our sample was slightly larger than

would have occurred in the population.

Of course, there would be sampling variability in

the degree of change from rXYto rXDY. If we were to

generate a new sample (N ? 50) for our illustration,

XDY? (.637)?2

we would obtain different values of rXYand rXDY. The

impact of dichotomization would vary from sample to

sample. An interesting question is whether dichoto-

mization could cause rXYto increase, even under nor-

mality, simply because of sampling error. This point

is relevant because researchers sometimes justify di-

chotomization because of a finding that it yielded a

higher correlation. To examine this issue, we con-

ducted a small-scale simulation study. We defined

five levels of population correlation, ?XY? .10, .30,

.50, .70, .90. We then generated repeated random

samples from bivariate normal populations at each

level of ?XY, using six different levels of sample size,

N ? 50, 100, 150, 200, 250, 300. We generated

10,000 such samples for each combination of levels of

sample size and ?XY. In each sample we computed rXY,

then dichotomized X at the median and computed

rXDY. For each combination of ?XYand sample size, we

then simply counted the number of times, out of

10,000, that rXDY> rXY, that is, the number of times

where dichotomization resulted in an increase in the

correlation. Results are shown in Table 1. Of interest

is the fact that when sample size and ?XYwere rela-

tively small, it was not unusual to find that dichoto-

mization resulted in an increase in the correlation be-

tween the variables, simply due to sampling error.

That is, even though, under bivariate normality, di-

chotomization must cause the population correlation

Figure 4. Proportional effect of dichotomization of X on correlation between X and Y as a

function of point of dichotomization; p and q are the proportions of the population above and

below the point of dichotomization, and h is the ordinate of the normal curve at the same point.

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

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to decrease, application of this approach in small

samples or when ?XYis relatively small can easily

result in an increase in the sample correlation. Un-

doubtedly in some cases such increases would cause a

correlation that was not statistically significant prior

to dichotomization to become so after dichotomiza-

tion. These results must raise a caution about potential

justification of dichotomization in practice. A finding

that rXDY> rXYmust not be taken as evidence that

dichotomization was appropriate or beneficial. In fact,

under conditions typically found in psychological re-

search, dichotomization will cause the population cor-

relation to decrease; an observation of an increase in

the sample correlation in practice is very possibly

attributable to sampling error. Failure to understand

this phenomenon could easily cause inappropriate

substantive interpretations.

The loss of effect size in the population following

dichotomization, and corresponding expected loss in

the sample, can affect the outcome of tests of statis-

tical significance. In our earlier example, the t statistic

for testing the significance of r dropped from 2.19

prior to dichotomization to 1.47 after, and statistical

significance was lost. This loss of statistical signifi-

cance can be attributed directly to loss of statistical

power. Considering the power of the test of the null

hypothesis of zero correlation, we note that in our

example, prior to dichotomization power was .84,

based on ?XY? .40, N ? 50, ? ? .05, two-tailed test.

After dichotomization of X, power was reduced to .63,

based on ?XDY? (.798)(.40) ? .32. Such a loss of

power would become more severe as the point of

dichotomization moves away from the mean, because

the loss of effect size would be greater (see Figure 4).

As noted by Cohen (1983), the loss of power caused

by dichotomization can be viewed alternatively as an

effective loss of sample size. For instance, in our ex-

ample, prior to dichotomization, power of .63 could

have been achieved with a sample size of only 32.

Thus, the reduction in power from .84 to .63 due to

dichotomization was equivalent to reducing sample

size from 50 to 32, or discarding 36% of our sample.

For a two-tailed test of the null hypothesis of zero

correlation, using ? ? .05, this effective loss of

sample size resulting from a median split will be con-

sistently close to 36%. It will deviate from this level

when any of these aspects is altered, in particular

becoming greater when the point of dichotomization

deviates from the mean.

Let us next consider the case where both the inde-

pendent variable X and the dependent variable Y are

dichotomized, thereby converting a correlation ques-

tion into analysis of a 2 × 2 table of frequencies. We

can examine the impact of double dichotomization by

focusing on the relationship between ?XYand ?XDYD,

the correlations before and after double dichotomiza-

tion. The relationship between these values corre-

sponds to the relationship between a phi coefficient

and the corresponding tetrachoric correlation, assum-

ing bivariate normality. The value ?XDYDis a phi co-

efficient, a correlation between two dichotomous vari-

ables, and the value ?XYis the corresponding

tetrachoric correlation, the correlation between nor-

mally distributed variables, X and Y, that underlie the

two dichotomies, XDand YD. For dichotomization at

the mean, the relationship between the phi coefficient

and the tetrachoric correlation was given in Equation

2. In the present context, this relationship becomes

?XDYD? 2[arcsin(?XY)]/?

(4)

and thus represents the impact of double dichotomi-

zation on the correlation of interest.2This relationship

is shown in Figure 5, indicating the association be-

tween population correlations obtained before di-

chotomization (analogous to tetrachoric correlation,

on horizontal axis) and after dichotomization (analo-

gous to phi coefficient, on vertical axis). For instance,

the value of ?XY? .40 in our example would be

reduced to ?XDYD? .26. Our sample result showed an

even larger reduction, from rXY? .30 to rXDYD? .06.

2For the case of dichotomization of both variables, Co-

hen (1983) incorrectly assumed that the effect on ?XYwould

be the square of the effect of single dichotomization; for

example, ?XDYD? (.798)2?XYfor dichotomization at the

mean. This same error occurs in Peters and Van Voorhis

(1940) and was recognized by Vargha et al. (1996).

Table 1

Frequency of Increase in Correlation Coefficient After

Dichotomization of Independent Variable for Various

Levels of Population Correlation (?XY) and Sample Size

?XY

N

50100 150200250300

.10

.30

.50

.70

.90

4,126

2,430

1,015

173

3,760

1,620

371

3,435

1,109

127

3,277

776

3,015

612

2,882

434

56

1

0

25

0

0

2

0

0

123

000

Note.

rXDY> rXY.

Entries indicate the number of trials out of 10,000 in which

MACCALLUM, ZHANG, PREACHER, AND RUCKER

26

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In general, loss of effect size is greater when both

variables are dichotomized than when only one is di-

chotomized. As the point of dichotomization moves

away from the mean, the relationship between the phi

coefficient and tetrachoric correlation becomes much

more complex (Kendall & Stuart, 1961; Pearson,

1900), and the difference between the two coefficients

becomes greater.

As in the case of dichotomization of only X, the

loss of effect size under double dichotomization can

be represented as a loss of statistical power. In our

example, power of the test of zero correlation would

be reduced from .84 prior to dichotomization to only

.45 after dichotomization of both X and Y. And as

before, such loss of power could be represented as an

effective discarding of a portion of the original

sample. Loss of power, or effective loss of sample

size, becomes more severe if either or both of the

variables are dichotomized at points away from the

mean.

It is important to keep in mind that the effects of

dichotomization of X, or of both X and Y, just de-

scribed are based on the assumption of bivariate nor-

mality of X and Y. It may be tempting to conclude that

in empirical populations these effects would not hold.

However, Cohen (1983) emphasized that these phe-

nomena would be altered only marginally by nonnor-

mality. It would require extreme skewness, hetero-

scedasticity, or nonlinearity to substantially alter these

consequences of dichotomization, conditions that

probably are rather uncommon in social science data.

When such conditions are present, that situation sim-

ply means that the formulas provided above for the

influence of dichotomization on correlations might

not hold closely. Such a complication does not pro-

vide justification for dichotomization of variables.

Rather, it would still be advisable not to dichotomize,

but instead to retain information about individual dif-

ferences and to consider resolving the extreme skew-

ness, heteroscedasticity, or nonlinearity by use of

transformations of variables or nonlinear regression.

The issue of nonlinearity merits special attention.

Suppose that the original relationship between X and

Y were nonlinear, such that a scatter plot such as that

in Figure 1 revealed a clear nonlinear association.

Such a relationship could be represented easily using

nonlinear regression, and the investigator could obtain

and present a clear picture of the association between

the variables. However, if X, or both X and Y, were

dichotomized, that nonlinear relationship would be

completely obscured. Presentation of results based on

analyses conducted after such dichotomization would

be misleading and invalid. Such errors can easily oc-

cur accidentally if researchers dichotomize variables

without examining the nature of the association be-

tween the original variables.

Analyses of effects of two independent variables.

We next examine statistical issues for the case of two

independent variables, X1and X2, and one dependent

variable, Y. The linear relationship of X1and X2to Y

can be studied easily using regression methods. Stan-

dard linear regression can be extended to investigate

interactive effects of the independent variables by in-

troducing product terms (e.g., X3? X1X2) into the

regression model (Aiken & West, 1991). However, in

practice it is not uncommon for investigators to di-

chotomize both X1and X2prior to analysis and to use

ANOVA rather than regression. Over a period of

more than 30 years, a number of methodological pa-

pers have examined the impact of such an approach

on statistical results and conclusions. Humphreys and

colleagues investigated several issues in this context

in a series of articles. Humphreys and Dachler (1969a,

1969b) discussed an approach that they called a

pseudo-orthogonal design in which individuals are

selected in high and low groups on both X1and X2so

as to produce a 2 × 2 design with equal sample sizes

in each cell. This approach does not involve dichoto-

mization of X1and X2after data have been collected

but does involve treating individual-differences mea-

sures as if they were categorical with only two levels.

Such a design had been used by Jensen (1968) in a

study of the relationship of intelligence (X1) and so-

Figure 5. Relationship between phi and tetrachoric corre-

lation for dichotomization of X and Y at their means.

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

27

Page 10

cioeconomic status (X2) to a measure of rote learning

(Y). Humphreys and Dachler (1969a, 1969b) pointed

out that this approach forces the independent variables

into an orthogonal design when in fact the original X1

and X2may well be correlated. They showed that an

ensuing ANOVA would yield biased estimates of dif-

ferences among means as a result of ignoring the cor-

relation between X1and X2. Humphreys and Fleish-

man (1974) provided further discussion of potentially

misleading results from a pseudo-orthogonal design

and also examined the approach wherein measures of

X1and X2are dichotomized after data are gathered.

Humphreys and Fleishman focused on the fact that

this approach will generally yield unequal sample

sizes in the resulting 2 × 2 design, and they reviewed

various ways to analyze such data using ANOVA.

They showed how ANOVA results would be related

to regression results and described the expected loss

of effect size attributable to dichotomization, as well

as the potential occurrence of spurious interactions. In

yet another article on this matter, Humphreys (1978a),

commenting on an applied article by Kirby and Das

(1977), again cautioned against dichotomization of

independent variables to construct a 2 × 2 ANOVA

design and reiterated the impact in terms of loss of

effect size and power as well as distortion of effects.

Throughout this series of articles Humphreys and

colleagues repeated the general theme of negative

consequences associated with dichotomization of con-

tinuous independent variables, either by selection of

high and low groups or by dichotomization of col-

lected data. They emphasized the loss of information

about individual differences and the bias in estimates

of effects. They argued that ANOVA methods are

inappropriate (“unnecessary, crude, and misleading”;

Humphreys, 1978a, p. 874) when independent vari-

ables are individual-differences measures and that it is

preferable to use regression and correlation methods

in such situations so as to retain information about

individual differences and avoid negative conse-

quences incurred by dichotomization (Humphreys,

1978b).

The case of dichotomization of two independent

variables was examined further by Maxwell and

Delaney (1993). After citing numerous empirical

studies that followed such a procedure, Maxwell and

Delaney showed that the impact of dichotomization

on main effects and interactions depends on the pat-

tern of correlations among independent and dependent

variables. Although under many conditions dichoto-

mization of two independent variables will result in

loss of effect size for main effects and interaction, it

was shown that under some conditions dichotomiza-

tion can yield a spurious main effect. The reader will

recall that our numerical example presented earlier

exhibited such a phenomenon. Our regression analy-

ses showed a near zero effect for one of the indepen-

dent variables, but ANOVA using dichotomized in-

dependent variables yielded a significant main effect

for that same variable. Maxwell and Delaney showed

that when the partial correlation of one independent

variable with the dependent variable is near zero and

the independent variables are correlated with each

other, a spurious significant main effect is likely to

occur after dichotomization of both predictors. Our

earlier numerical example had this property: The two

predictors were substantially correlated (.50), and the

partial correlation of X2with Y was zero. The regres-

sion analysis properly revealed no effect of X2on Y,

whereas ANOVA after dichotomization of X1and X2

yielded a spurious main effect of X2. Maxwell and

Delaney demonstrated that there would be highly in-

flated Type I error rates for tests of main effects in

such situations and that these spurious effects were a

result of bias in estimating population effects and

were not attributable to sampling error. Finally, Max-

well and Delaney also showed that spurious signifi-

cant interactions can occur when two independent

variables are dichotomized. Such an event can occur

when there are direct nonlinear effects of one or both

of X1and X2on Y but no interaction in the regression

model. After dichotomization of X1and X2a subse-

quent ANOVA will often yield a significant interac-

tion simply as a misrepresentation of the nonlinearity

in the effect of X1and/or X2.

Vargha, Rudas, Delaney, and Maxwell (1996) ex-

tended the work of Maxwell and Delaney (1993) by

further examining the case of two independent vari-

ables and one dependent variable. They carefully de-

lineated the loss of effect size or the likely occurrence

of spurious significant effects under various combi-

nations of dichotomized and nondichotomized vari-

ables, showing that the impact of dichotomization de-

pends on the pattern of correlations among the three

variables.

In some instances where effects of two quantitative

independent variables are to be investigated, data are

analyzed by dichotomizing only one of the two vari-

ables, leaving the other intact. Such an approach has

been used to study moderator effects. For instance, if

it is hypothesized that the influence of X1on Y de-

pends on the level of X2, the researcher might dichoto-

MACCALLUM, ZHANG, PREACHER, AND RUCKER

28

Page 11

mize X2and then conduct separate regression analyses

of Y on X1for each level of X2. Moderation is then

assessed by testing the difference between the two

resulting values of rX1Y, with a significant difference

supposedly indicating a moderator effect. It is impor-

tant to note that most methodologists would advise

against such an approach for investigating moderator

effects and would recommend instead the use of stan-

dard regression methods that incorporate interactions

of quantitative variables (Aiken & West, 1991).

Bissonnette, Ickes, Bernstein, and Knowles (1990)

conducted a simulation study to compare the dichoto-

mization approach to the regression approach for ex-

amining moderator effects. When no moderator effect

was present in the population, they found high Type I

error rates under the dichotomization approach, indi-

cating common occurrence of spurious interactions.

Under the regression approach, Type I error rates

were nominal. When moderator effects were present

in the population, they found a higher rate of correct

detection of such effects using the regression ap-

proach than the dichotomization approach. Given the

distortion of information incurred by dichotomization

along with the inflated Type I error rates or loss of

power, as well as the straightforward capacity of stan-

dard regression to test for moderator effects, the use

of the dichotomization method in this context seems

unwarranted and risky.

Finally, it should be noted that although our pre-

sentation here has been limited to the cases of one or

two independent variables, the issues and phenomena

we have examined are not limited to those cases. The

same issues apply for designs with three or more in-

dependent variables, although further complexities

arise depending on the pattern of correlations among

the variables and how many variables are dichoto-

mized.

Aggregation and comparison of results across stud-

ies. Regardless of the number of independent vari-

ables, dichotomization raises issues regarding com-

parison and aggregation of results across studies.

Allison, Gorman, and Primavera (1993) cautioned

that dichotomization may introduce a lack of compa-

rability of measures and results across studies. For

example, groups defined as high or low after dichoto-

mization may not be comparable between studies, es-

pecially if the point of dichotomization is data depen-

dent (e.g., the median). That is, the point of

dichotomization may vary considerably between stud-

ies, thus making groups not comparable. Even when

dichotomization is conducted using a predefined scale

point, resulting groups may differ considerably de-

pending on the nature of the population from which

the sample was drawn.

Hunter and Schmidt (1990) discussed problems

caused by dichotomization when results from differ-

ent studies are to be aggregated using meta-analysis.

They showed that dichotomization of one or more

independent variables will tend to cause downward

distortion of aggregated measures of effect sizes as

well as upward distortion of measures of variation of

effect sizes across studies. They suggested methods

for correcting these distortions in meta-analytic stud-

ies. However, those methods do not resolve the prob-

lem because they rely on their own assumptions and

estimation methods. Such corrections are necessary

only because of the persistent use of dichotomization

in applied research.

Summary of Impact of Dichotomization on

Measurement and Statistical Analyses

In this section we have reviewed literature on the

variety of negative consequences associated with di-

chotomization. These include loss of information

about individual differences, loss of effect size and

power, the occurrence of spurious significant main

effects or interactions, risks of overlooking nonlinear

effects, and problems in comparing and aggregating

findings across studies. To our knowledge, there have

been no findings of positive consequences of dichoto-

mization. Given this state of affairs, it would seem

that use of dichotomization in applied research would

be rather rare, but that is not at all the case. We now

examine such usage in selected areas of applied re-

search in psychology.

The Use of Dichotomization in Practice

Method

We selected six journals publishing research ar-

ticles in clinical, social, personality, applied, and de-

velopmental psychology. The specific journals se-

lected were Journal of Personality and Social

Psychology, Journal of Consulting and Clinical Psy-

chology, Journal of Counseling Psychology, Develop-

mental Psychology, Psychological Assessment, and

Journal of Applied Psychology. These journals are

known for publishing research articles of high quality.

The rationale behind selecting leading journals is

simple. Leading journals are known for their high

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

29

Page 12

standards, especially when statistical and method-

ological considerations are present. If we were to find

common use of dichotomization in high-quality jour-

nals, this would suggest not only that uses of dichoto-

mization must appear in other journals as well but also

that leading researchers, as well as editors and review-

ers, may be relatively unaware of the consequences

associated with the use of dichotomization.

For each journal, we set out to examine all articles

published from January 1998 through December

2000. This time interval was selected so as to reflect

current practice. We limited our review to articles

containing empirical studies, and we examined each

such article to determine if any measured variables

had been dichotomized prior to statistical analyses.

An examination of articles published in 1998 in these

six journals showed relatively frequent use of dichoto-

mization in three of the journals—Journal of Person-

ality and Social Psychology, Journal of Consulting

and Clinical Psychology, and Journal of Counseling

Psychology—but relatively rare usage in the other

three—Developmental Psychology, Psychological As-

sessment, and Journal of Applied Psychology. (Al-

though Developmental Psychology contained rela-

tively few uses of dichotomization per se, it did

contain an abundance of examples wherein subjects

were divided into several groups based on chronolog-

ical age.) Therefore, the full 3-year literature review

was conducted only on the former set of three jour-

nals. In our review, we tabulated information about

various forms of dichotomization, including median

splits, mean splits, and other splits at selected scale

points. We also noted other types of splits, such as

tertiary splits or selection of extreme groups, although

such splits are not examined directly in this article.

For each instance of dichotomization we noted the

variable that was split. In addition, we noted whether

or not a justification for the split was given and what

that justification was.

Results

In the three journals examined, there were a total of

958 articles. Of the 958 articles we found that a total

of 110 articles, or 11.5%, contained at least one in-

stance of dichotomization of a quantitatively mea-

sured variable. For those 110 articles a total of 159

instances of dichotomization were identified. For

practical reasons, we did not count multiple median

splits performed on the same variable within a given

article. Summary information about our literature sur-

vey is presented in Table 2.

As stated earlier, we also examined justifications

offered for use of dichotomization. Of the 110 cases

in which dichotomization was conducted, only 22 of

those cases (20%) were accompanied by any justifi-

cation. Thus, dichotomization seems to be most often

used without any explicit justification. Some of the

justifications offered for dichotomization included (a)

following practices used in previous research, (b) sim-

plification of analyses or presentation of results, (c)

gaining a capability for examining moderator effects,

(d) categorizing because of skewed data, (e) using

clinically significant cutpoints, and (f) improving sta-

tistical power. In most cases, however, as noted

above, no justification at all was offered, and it is not

clear that this matter was even considered. Variables

that were split were most often self-rated psychologi-

cal scales. Scores on instruments assessing depres-

sion, anxiety, marital satisfaction, self-monitoring, at-

titudes, self-esteem, need for cognition, narcissism,

and so forth were frequently dichotomized, although

these examples represent only a small subset of the

dichotomized variables encountered. Chronological

age was the subject of frequent dichotomization, al-

though it was more often segmented into several “age

groups” based on arbitrary cutpoints.

Potential Justifications for Dichotomization

We now consider the question of why there seems

to be persistent use of dichotomization in applied re-

search in psychology in spite of the methodological

case against it. We examine here a variety of possible

reasons or justifications for such usage and offer our

own assessment of each. Some of these justifications

are extracted from explicit statements in published

studies, whereas others are drawn from numerous dis-

Table 2

Summary of Literature Search

Journal

No. of

articles

No. of

articles

with splitsa

Percentage

of articles

with splits

No. of

articles with

double splits

JPSP

JCCP

JCP

Total

518

312

128

958

82 (123)

20 (27)

8 (9)

110 (159)

15.8

6.4

6.3

11.5

7

1

1

9

Note.

lished in JPSP, JCCP, and JCP from January 1998 through Decem-

ber 2000.

JPSP ? Journal of Personality and Social Psychology; JCCP ?

Journal of Consulting and Clinical Psychology; JCP ? Journal of

Consulting Psychology.

aNumbers in parentheses refer to total number of variables split.

The literature search was conducted on all articles pub-

MACCALLUM, ZHANG, PREACHER, AND RUCKER

30

Page 13

cussions with colleagues and applied researchers over

a period of many years. Many researchers who use

dichotomization are eager to defend the practice in

such discussions, and in the following sections we try

to provide a fair representation of such defenses.

Lack of Awareness of Costs

Some researchers who use dichotomization may

simply be unaware of its likely costs and negative

consequences as delineated earlier in this article.

When made aware of such costs, some of these indi-

viduals may be eager to use more appropriate regres-

sion methods and thereby avoid the costs and enhance

the measurement and statistical aspects of their stud-

ies. We hope that the present article will have such an

impact on researchers.

Perceiving Costs as Benefits

Some investigators acknowledge the costs but ar-

gue that those very costs provide a justification for

dichotomization. This argument is that because di-

chotomization typically results in loss of measure-

ment information as well as effect size and power, it

must yield a more conservative test of the relationship

between the variables of interest. Therefore, a finding

of a statistically significant relationship following di-

chotomization is more impressive than the same find-

ing without dichotomization; the relationship must be

especially strong to still be found even when effect

size and power have been reduced. Essentially, this

argument reduces to the position that a more conser-

vative statistical test is a benefit if it yields a signifi-

cant result.

Let us consider this defense carefully. The argu-

ment focuses on results of a test of statistical signifi-

cance and also rests on the premise that dichotomiza-

tion will make such tests and corresponding measures

of effect size more conservative. The focus on statis-

tical significance is unfortunate, and the premise is

false. First, regarding a focus on statistical signifi-

cance, the American Psychological Association

(APA) Task Force on Statistical Inference (Wilkinson

and the Task Force on Statistical Inference, 1999)

urged researchers to pay much less attention to ac-

cept–reject decisions and to focus more on measures

of effect size, stating that “reporting and interpreting

effect sizes . . . is essential to good research” (p. 599).

In addition, the Publication Manual of the American

Psychological Association (5th ed.; APA, 2001) em-

phasized that significance levels do not reflect the

magnitude of an effect or the strength of a relationship

and stated the following with respect to reporting re-

sults: “For the reader to fully understand the impor-

tance of . . . [a researcher’s] findings, it is almost al-

ways necessary to include some index of effect size or

strength of relationship in . . . [the article’s] Results

section” (p. 25). With regard to the present issue, this

perspective implies that it is misguided to focus

merely on whether or not a statistical test yields a

significant result after dichotomization. Rather, it is

important to consider the size of the effect of interest.

Our review of methodological issues presented earlier

emphasized that dichotomization can play havoc with

measures of effect size, generally reducing their mag-

nitude in bivariate relationships and potentially reduc-

ing or increasing them in analyses involving multiple

independent variables. Second, the belief that statis-

tical tests will always be more conservative after di-

chotomization is mistaken. Although this will tend to

be true in the case of dichotomization of a single

independent variable, it is by no means always true.

Our earlier results (see Table 1) showed that dichoto-

mization may cause an increase in effect size simply

due to sampling error, thus producing a less conser-

vative test. In addition, statistical tests may not be

more conservative when two or more independent

variables are dichotomized. In that case, as illustrated

earlier, spurious significant main effects or interac-

tions may arise depending on the pattern of intercor-

relations between the variables. Thus, it is not at all

the case that dichotomization will always make a sta-

tistical test or a measure of effect size more conser-

vative.

Even if dichotomization did routinely yield more

conservative statistical tests and measures of effect

size, such a defense of its usage is highly suspect.

Researchers typically design and conduct studies so as

to enhance power and measures of effect size, as re-

flected in efforts to use reliable measures, obtain large

samples, and use moderate alpha levels in hypothesis

testing. If in fact more conservative tests were more

desirable and defensible, then perhaps researchers

should use less reliable measures, smaller samples,

and small levels of alpha. One might then argue that

if significant effects were still found, those effects

must be quite strong and robust. Of course, such an

approach to research would be counterproductive as

are most uses of dichotomization. Our general view is

that this defense of dichotomization is not supportable

on statistical grounds and is inconsistent with basic

principles of research design and data analysis.

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

31

Page 14

Lack of Awareness of Proper Methods

of Analysis

Over a period of many years, in discussions about

dichotomization, we have encountered numerous re-

searchers who simply have been unaware that re-

gression/correlation methods are generally more ap-

propriate for the analysis of relationships among

individual-differences measures. This is especially

true of individuals whose primary training and expe-

rience in the use of statistical methods is limited to or

heavily emphasizes ANOVA. Of course, ANOVA is

an important tool for data analysis in psychological

research. However, difficulties and serious conse-

quences arise when a problem that is best handled by

regression/correlation methods is transformed into an

ANOVA problem by dichotomization of independent

variables. We are convinced that many researchers are

not sufficiently familiar with or aware of regression/

correlation methods to use them in practice and there-

fore often force data analysis problems into an

ANOVA framework. This seems especially true in

situations where there are multiple independent vari-

ables and the investigator is interested in interactions.

We have repeatedly encountered the argument that

dichotomization is useful so as to allow for testing of

interactions, under the (mistaken) belief that interac-

tions cannot be investigated using regression meth-

ods. In fact, it is straightforward to incorporate and

test interactions in regression models (Aiken & West,

1991), and such an approach would avoid problems

described earlier in this article regarding biased mea-

sures of effect size and spurious significant effects.

Furthermore, regression models can easily incorpo-

rate higher way interactions as well as interactions of

a form other than linear × linear—for example, a lin-

ear × quadratic interaction. Thus, we encourage re-

searchers who use dichotomization out of lack of fa-

miliarity with regression methods to invest the modest

amount of time and effort necessary to be able to

conduct straightforward regression and correlational

analyses when independent variables are individual-

differences measures.

A related issue, or underlying cause of this lack of

awareness of appropriate statistical methods, involves

training in statistical methodology in graduate pro-

grams in psychology. Results of surveys of such train-

ing (Aiken, West, Millsap, & Taylor, 2000; Aiken,

West, Sechrest, & Reno, 1990) indicate that education

in statistical methods is somewhat limited for many

graduate students in psychology and often focuses

heavily on ANOVA methods. Although training in

regression methods is certainly available in many pro-

grams, it receives less emphasis and is less often part

of a standard program. Clearly, enhanced training in

basic methods of regression analysis could increase

awareness and usage of appropriate methods for

analysis of measures of individual differences and

help to avoid some of the problems resulting from

overreliance on ANOVA, such as the common usage

of dichotomization.

Dichotomization Resulting in

Higher Correlation

It is not rare for an investigator to dichotomize an

independent variable, X, and to find that the correla-

tion between X and the dependent variable, Y, is

higher after dichotomization than before dichotomi-

zation; that is, rXDY> rXY. Under such circumstances it

may be tempting for the investigator to believe that

such a finding justifies the use of dichotomization. It

does not. In our earlier discussion of the impact of

dichotomization on results of statistical analyses, we

reviewed how population correlations would gener-

ally be reduced. We also showed by means of a simu-

lation study that this effect will not always hold in

samples (see the results in Table 1). That is, simply

because of sampling error, sample correlations may

increase following dichotomization, especially when

sample size is small or the sample correlation is small.

Such an occurrence in practice may well be a chance

event and does not provide a sound justification for

dichotomization.

Dichotomization as Simplification

A common defense of dichotomization involves, in

one form or another, an argument that analyses and

results are simplified. In terms of analyses, this argu-

ment rests on the premise that an analysis of group

differences using ANOVA is somehow simpler than

an analysis of individual differences using regression/

correlation methods. We suggest that proponents of

this position may simply be more familiar and com-

fortable with ANOVA methods and that the analyses

and results themselves are not simpler. Both ap-

proaches can be applied in routine fashion, and results

can be presented in standard ways using statistics or

graphics. For instance, from our example of a bivari-

ate relationship presented early in this article, corre-

lational results showed rXY? .30, r2

2.19, p ? .03. Figure 1 shows a scatter plot of the raw

data, which conveys an impression of strength of as-

sociation. Results after dichotomization would be pre-

XY? .09, t(48) ?

MACCALLUM, ZHANG, PREACHER, AND RUCKER

32

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sented in terms of a difference between group means.

Means of Y for the high and low groups on X were

21.1 and 19.4, respectively. The test of the difference

between these means yielded t(48) ? 1.47, p ? .15.

Both sets of results are straightforward to present, and

we see no real gain in simplicity by using ANOVA

instead of regression.

For designs with two independent variables, the

same principle holds. ANOVA results are typically

presented in terms of tables or graphs of cell means.

Regression results would be presented in terms of

regression coefficients and associated information.

An extremely useful mechanism for presenting inter-

actions in a regression analysis is to display a plot of

several regression lines. For example, a finding of a

significant interactive effect of X1and X2on Y could

be portrayed in terms of three regression lines. The

first represents the regression of Y on X1for an indi-

vidual low on X2(e.g., 1 standard deviation below the

mean), the second represents the regression of Y on X1

for an individual at the mean of X2, and the third

represents the regression of Y on X1for an individual

high on X2(e.g., 1 standard deviation above the

mean). All three lines can be displayed in a single

plot, which can provide a simple and clear basis for

understanding the nature of the interaction. This ap-

proach to presenting interactions is described and il-

lustrated in detail by Aiken and West (1991). Again,

we see no loss in simplicity of analysis or results for

regression versus ANOVA.

We do recognize that for some individuals it may

be conceptually simpler to view results in terms of

group differences rather than individual differences.

However, this conceptual simplification, to whatever

extent it is real, is achieved only at a high cost—loss

of information about individual differences, havoc

with effect sizes and statistical significance, and so

forth. Furthermore, the acceptance of such a simpli-

fied perspective will be misleading if the “groups”

yielded by dichotomization are not real but are simply

an artifact of arbitrarily splitting a sample and dis-

carding information about individual differences. We

consider the notion of groups shortly.

In considering the argument that dichotomization

provides simplification, Allison et al. (1993) coun-

tered with the view that in fact dichotomization intro-

duces complexity. They emphasized complications

arising when two or more independent variables are

dichotomized. Such a procedure typically creates a

nonorthogonal ANOVA design, with its associated

problems of analysis and interpretation, and also plays

havoc with effect sizes and interpretations as de-

scribed earlier. In fact, regression analyses are much

simpler and allow the researcher to avoid these prob-

lems and the associated chance of being misled by

results.

One additional perspective regarding the defense of

simplification should be considered. Some research-

ers may dichotomize variables for the sake of the

audience, believing that the audience will be more

receptive to and will more easily understand analyses

and results conveyed in terms of group differences

and ANOVA. Although this perspective may be par-

tially valid at times, we urge researchers and authors

to avoid this trap. Clearly such a concession has nega-

tive consequences that far outweigh any perceived

gains, very possibly including the drawing of incor-

rect conclusions about relationships among variables.

No real interests are served if researchers use methods

known to be inappropriate and problematic in the be-

lief that the target audience will better understand

analyses and results, especially when the results may

be misleading and proper methods are simple and

straightforward. All parties would be better served by

obtaining and implementing the basic knowledge nec-

essary for selection and use of appropriate statistical

methods as well as for reading and understanding re-

sults of corresponding analyses.

Representing Underlying Categories

of Individuals

Perhaps the most common defense of dichotomiza-

tion is that there actually exist distinct groups of in-

dividuals on the variable in question, that a dichoto-

mized measure more appropriately represents those

groups, and that analyses should be conducted in

terms of group differences rather than individual dif-

ferences. Such an argument is potentially controver-

sial both statistically and conceptually and must be

examined closely.

There seem to be two distinct perspectives regard-

ing the construction or existence of distinct groups of

individuals on psychological attributes. These views

have been discussed in the personality research litera-

ture. Block and Ozer (1982) discussed the type-as-

label versus type-as-distinctive form perspectives.

Drawing a similar distinction, Gangestad and Snyder

(1985) described phenetic versus genetic bases for

classes and also expressed the view that some psy-

chological variables are dimensional, wherein indi-

vidual differences are a matter of degree, whereas

others are discrete, wherein individual differences

DICHOTOMIZATION OF QUANTITATIVE VARIABLES

33