JDMS 24:155–162 May/June 2008 155

Measures of
Association
How to Choose? Researchers in sonography, as well as other areas,
often wish to measure the strength of relationship
HARRY KHAMIS, PhD or association between two variables. For exam-
ple, one may wish to determine if, on the average,
total cholesterol level increases as age increases
for adult American men. However, there are a
very large number of measures or coefficients
(i.e., a number that indicates the strength of the
relationship between two variables) from which
to choose. It is not infrequent to find a researcher
selecting an incorrect coefficient to measure a
given association, thereby possibly rendering a
false or misleading conclusion. The choice of the
proper measure of association is based on, among
other things, the characteristics of each of the two
variables involved. This article enumerates every
case that can be encountered by the researcher
and provides an appropriate measure of associa-
tion that can be used.
Key words: coefficient, relationship, ordinal,
nominal, continuous

Very often, as a part of the general analyses of a

set of data, a researcher wishes to determine the
strength of the relationship between two variables
using a single coefficient or measure of association—
namely, a number (often between –1 and +1 or
between 0 and 1) that is used as a measure of how
strongly the two variables are related. In such
instances, it is important that the appropriate meas-
ure is used to assess the strength of the relation-
ship. If an inappropriate measure is used, then the
resulting value is meaningless, and misleading results
From Wright State University, Dayton, Ohio.
Correspondence: Harry Khamis, PhD, Wright State University, may be concluded. For example, the commonly
130 Mathematical and Microbiological Sciences Bldg, 3640 Col. used Pearson correlation coefficient is not neces-
Glenn Hwy, Dayton, OH 45435. E-mail: harry.khamis@wright.edu. sarily the correct measure of association in every
DOI: 10.1177/8756479308317006 instance. To decide on the appropriate measure of
156 JOURNAL OF DIAGNOSTIC MEDICAL SONOGRAPHY May/June 2008 VOL. 24, NO. 3

association, one must identify the level of meas- only a few possible values. Examples of discrete-
urement (defined and discussed below) of each valued variables are gender (with values male,
variable being studied. Based on this information, female), severity of disease (with values mild,
an appropriate measure of association can be iden- moderate, severe), type of operation (with values
tified as outlined below. Indeed, for any given sit- standard, modified, laparoscopic), New York bor-
uation, there may be several different measures of ough (with values Manhattan, Queens, etc.).
association that are valid. Rather than providing an Because of the way these variables are defined, it
exhaustive list of all possible such measures, this is not possible to observe a value between two
article provides recommendations of just one or given values. For example, each patient is either
two measures for each practical situation encoun- male or female with no other possible designation.
tered by the researcher. Ample references are In the case of discrete variables, there are two
provided, directing the researcher to sources of subcategories of measurement scale, ordinal and
further reading. Several real-data examples are nominal. An ordinal variable is a discrete variable
provided, illustrating the measures discussed. having an order associated with its levels. For
In the next section, levels of measurement are example, severity of disease is an ordinal variable
defined and discussed. In subsequent sections, the because the “moderate” level represents a some-
process of selecting an appropriate measure of what more severe disease state than the “mild”
association for any given situation is outlined, and level, and the “severe” level corresponds to a more
a summary and conclusion are provided. severe condition than the “moderate” level. If the
levels of a discrete variable do not have any order
Levels of Measurement associated with them, then the variable is called a
discrete nominal variable. Type of operation is a
The level of measurement (or measurement nominal variable because the three different types
scale) of a variable is its designation as continuous of operation do not have any order associated with
or discrete. them. You might argue that indeed there is an order
A continuous-valued variable has values that, at associated with these three levels—for instance,
least theoretically, come from a continuum of the one type of operation may be more expensive than
real number line. For such variables, there are, the- another type of operation. However, this is a dif-
oretically at least, no gaps in the possible values of ferent variable from the one defined earlier. If the
the variable. Examples of continuous-valued vari- variable of interest is cost of operation, with levels
ables are gestational age, blood pressure, body inexpensive, moderate, and expensive, then indeed
mass index, left ventricular ejection fraction (cal- this would be an ordinal variable. However, type
culated as the percentage of blood expelled in a of operation is a nominal variable.
cycle), size of rotator cuff calcification (in cm),
and coracoacromial distance (in cm). Consider Measures of Association—How to Choose
gestational age: it is possible that there is a fetus
with a gestational age of 60.32 days, even though Suppose you wish to study the relationship
the recorded value is 60 days. The reason that the between two variables by using a single measure
more precise measurement, 60.32 days, is not or coefficient. There are many considerations that
recorded is due perhaps to the lack of accuracy of go into selecting an optimum measure of associa-
the measuring instrument. tion; these are briefly discussed in the Summary
A discrete-valued variable has values that are section. For simplification, we base our selection
discrete or “separated.” For such variables, the of an appropriate measure exclusively on what
values do not come from a continuum of the real type of variable we are measuring (i.e., its level of
number line; rather, there are gaps between the measurement)—namely, whether it is (1) continu-
values of the variable. Usually, such variables have ous, (2) discrete ordinal, or (3) discrete nominal.
 MEASURES OF ASSOCIATION / Khamis 157

On the basis of this information, we choose an (i.e., knowledge of the value of one variable in no way
appropriate measure of association with which to improves the ability to predict the value of the other
analyze the relationship between the two vari- variable), (2) the two variables are highly variable, or
ables. From a practical point of view, the six pos- (3) the two variables have a nonlinear relationship.
sible combinations of variables encountered by There is no universal rule for interpreting a given
researchers are as follows: value of r, but informal guidelines have been given. For
most studies involving medical, biomedical, biological,
1. Continuous-continuous health care, sociological, educational, and psychologi-
2. Continuous-ordinal cal data, the following guidelines are appropriate:4
3. Continuous-nominal
4. Ordinal-ordinal r Interpretation of Linear Relationship
5. Ordinal-nominal 0.8 Strong positive
6. Nominal-nominal 0.5 Moderate positive
0.2 Weak positive
For each of these combinations of variables, one or 0.0 No relationship
more measures of association that accurately assess –0.2 Weak negative
–0.5 Moderate negative
the strength of the relationship between the two vari- –0.8 Strong negative
ables are discussed below. The following is not an
exhaustive list of all possible measures of association
but rather the most commonly used and practically The above guidelines are generally in agreement
useful measures. Most of these measures are available with Cohen’s5 recommended guidelines:
on most statistical software packages. The mathemat-
ical formulas and more extensive details for these |r| < 0.3 → Weak relationship
measures can be found in many statistics texts; three 0.3 ≤ |r| ≤ 0.5 → Moderate relationship
|r| > 0.5 → Strong relationship
basic references for such measures are Goodman and
Kruskal,1 Liebetrau,2 and Khamis.3
Suppose one or both of the continuous variables
1. CONTINUOUS-CONTINUOUS
of interest have extreme values (sometimes called
outliers). Annual income may be such a variable. For
Consider two continuous variables. In most example, although the vast majority of workers
instances, the Pearson correlation coefficient (also within a given organization have annual incomes
called the Pearson product-moment correlation coef- relatively close to the average annual income, there
ficient), r, is appropriate for measuring the strength of are a few individuals with positions at the highest
the linear relationship between them. The value of r levels of the administrative hierarchy (CEO, presi-
lies between –1 and +1. Values close to –1 indicate a dent, vice presidents, etc.) whose incomes are
strong negative linear relationship (as the value of extremely high. In this case, a more appropriate
one variable increases, the value of the other variable measure of a linear relationship is the Spearman
decreases; e.g., consider age and physical stamina for rank correlation coefficient. This is especially true if
adult Americans—as people get older, they have less a test of the statistical significance of the strength of
physical stamina in general). Values close to +1 indi- the relationship is desired. Its values also range from
cate a strong positive linear relationship (as the value –1 to +1, and it is interpreted in the same way as r.
of one variable increases, so does the value of the For further reading on these coefficients, refer
other variable; e.g., consider years of experience and to almost any standard statistics text—for exam-
annual income among professional sonographers). ple, Sokal and Rohlf6 or Zar.7
Values of r close to zero indicate no linear relation-
ship between the two variables. A value of r close to Example 1. In a study to determine whether left
zero can occur (1) if the two variables are independent atrial size, pressure, and ejection fraction are
158 JOURNAL OF DIAGNOSTIC MEDICAL SONOGRAPHY May/June 2008 VOL. 24, NO. 3

useful in diagnosing patients with left ventricle Center obtained data on 52 subjects. One of the
diastolic dysfunction through noninvasive means,8 questions posed in the survey was as follows:
the Pearson correlation coefficient between (1) the “Have you had a lot of energy in the last four
left atrial pressure evaluated through pulmonary weeks?” The possible responses were 1 = none of
wedge pressure and (2) the E/A wave velocity the time, 2 = a little of the time, 3 = some of the
ratio is r = 0.77. This can be characterized as a time, 4 = a good bit of the time, 5 = most of the time,
“strong” positive linear relationship between the and 6 = all of the time. This researcher wished to
two variables. assess the strength of relationship between this
ordinal variable and the age of the subject. For
2. CONTINUOUS-ORDINAL these data, Kendall’s τb is 0.08. There appears to
be no strong relationship between these two vari-
If one variable is continuous and the other is
ables. The Spearman rank correlation coefficient
ordinal, then an appropriate measure of associa-
for these data, with the coding given above, is
tion is Kendall’s coefficient of rank correlation
0.10, very close to the value of Kendall’s τb, ren-
tau-sub-b, τb. If the two variables are denoted by X
dering the same conclusion. This illustrates the
(continuous) and Y (ordinal), then consider the
closeness of Kendall’s τb with the Spearman rank
levels of Y to be numerically coded according to
correlation coefficient mentioned above.
the order of the levels (e.g., assign 1, 2, 3, . . . to
the levels). Then Kendall’s τb uses the numerical
values of X and the coded numerical values of Y to 3. CONTINUOUS-NOMINAL
render a number (coefficient) between –1 and +1
that measures the strength of relationship between If one variable is continuous and the other is
X and Y. For further reading, see Liebetrau.2 nominal with just two categories, then use the
If the ordinal variable, Y, has a large number of point-biserial correlation coefficient. This coef-
levels (say, five or six or more), then one may use the ficient ranges between –1 and +1. Values close
Spearman rank correlation coefficient to measure the to ±1 indicate a strong positive/negative rela-
strength of association between X and Y. In doing so, tionship, and values close to zero indicate no
one must be careful in numerically coding the levels relationship between the two variables. If the
of Y in a practically meaningful way, keeping in mind nominal variable has more than two levels, then
that a metric is being imposed by the coding scheme. one can calculate the point-biserial correlation
See Chatfield9(p45) and Luce and Narens10 for further between the continuous variable and all possible
discussion. A typical example of this treatment is if Y pairs of levels of the nominal variable; this
represents degree of agreement, with the following k (k−1)
would result in such coefficients,
levels: 1 = very strongly agree, 2 = strongly agree, 2
3 = agree, 4 = neutral, 5 = disagree, 6 = strongly dis- where k represents the number of levels of the nom-
agree, and 7 = very strongly disagree. Another exam- inal variable. For further reading, see Tate.11,12
ple involves income ranges where each level is coded The calculation of the point-biserial correlation
with the midpoint between the lowest and highest coefficient is accomplished by coding the two lev-
income of the range (e.g., a range of $50,000–$75,000 els of the binary variable “0” and “1” and obtain-
would be coded 62,500). ing the Pearson correlation coefficient between the
The performance of the Spearman rank correla- continuous variable and this coded binary variable.
tion coefficient is comparable to that of Kendall’s
τb, with the former being somewhat better for large Example 3. The following data show the absolute
sample sizes (see Zar7(p392)). value of the difference in labor time from six hours
(median labor time) for expectant mothers and
Example 2. A medical researcher who was a client whether they receive analgesia (Y, N); see Kotz
at the Wright State University Statistical Consulting and Johnson.13(p279)
 MEASURES OF ASSOCIATION / Khamis 159

Y 14.8 12.4 10.1 7.1 6.1 4.6 3.2 3.0 2.4 2.3 2.1 can be used to measure association. The same dis-
0.8 0.1 cussion as for the continuous-nominal case (see
N 13.8 5.8 4.3 3.5 3.3 2.8 2.8 2.5 1.7 1.7 1.5 above) applies here. See Cureton.15
1.3 1.3 1.2 1.2 1.1 0.7 0.6 0.5 0.2 0.2
Example 5. In the study discussed in example
The point-biserial correlation coefficient for 3, the rank-biserial correlation coefficient is 0.43.
these data is 0.36. There is a “moderately” strong This value is somewhat larger than the point-
association between extreme labor time (either biserial correlation coefficient of 0.36. Because
very short or very long) and the use of analgesia in there may be one or more outliers in the data set
expectant mothers. Specifically, receiving analge- (e.g., 13.8 in the “N” group appears to be an out-
sia is associated with more extreme labor time. lier), the rank-biserial correlation coefficient is
more appropriate. The general conclusion is the
4. ORDINAL-ORDINAL same as that of example 3: there is a moderately
strong association between extreme labor time
If both variables are ordinal, then an appropriate and use of analgesia.
measure of association is Kendall’s τb. If both ordinal
variables have a large number of levels, then an appro-
priate numerical coding scheme can be used and the 6. NOMINAL-NOMINAL
Spearman rank correlation coefficient calculated. See
the discussion of the continuous-ordinal case above. Consider two discrete nominal variables whose
association is of interest. Suppose that both vari-
Example 4. The following data come from a study ables have just two levels; the resulting data dis-
of economic voting behavior by Kuklinski and West:14 play is in the form of a two-by-two contingency
table. One common way of measuring association
Expected Financial in such a table is to use the phi coefficient, ϕ.
Well-Being Values of ϕ lie between 0 and 1. Values of ϕ close
Better Same Worse to 0 indicate very little association, and values
close to 1 indicate nearly perfect predictability.
Present Financial Well-Being Fleiss16 provides, as a rule of thumb, that any value
Better 70 85 15
Same 10 134 41
of ϕ less than 0.30 or 0.35 may be taken to indi-
Worse 27 60 100 cate no more than trivial association. See Fleiss16
for further discussion.
For two nominal variables in which at least one
The value of Kendall’s τb for these data is 0.39.
of the variables has more than two levels, a useful
This represents a “moderately” strong association
measure of association is Goodman and Kruskal’s
between these two variables: better present financial
lambda, λ. The value λ is the relative decrease in
well-being tends to correspond to better expected
the probability of error in guessing the level of one
financial well-being.
of the variables as between the level of the other
It is interesting to note that, if the levels of each
variable known and unknown. The value λ lies
variable are coded 1, 2, and 3 for worse, same, and
between 0 and 1; values close to 1 correspond to a
better, respectively, then the Spearman rank correla-
strong association. For further details, see Goodman
tion coefficient is 0.42. This value is very close to
and Kruskal.1
the value for τb, once again illustrating the closeness
of these two measures.
Example 6. The following contingency table
5. ORDINAL-NOMINAL
represents a cross-classification of hair color and
eye color in males (see Kendall17(p300)). Is there
If one variable is nominal and the other is ordi- an association between hair and eye color in
nal, then the rank-biserial correlation coefficient males?
160 JOURNAL OF DIAGNOSTIC MEDICAL SONOGRAPHY May/June 2008 VOL. 24, NO. 3

Hair Color
Example 7. For the data in example 2, where the
sample size is n = 52, the SE of the estimated
H1 H2 H3 H4 Kendall’s τb is 0.0991. Then, with 95% confidence,
Eye color the interval 0.08 ± 0.1982 or [–0.12, 0.28] contains
E1 1768 807 189 47 the population coefficient value. Because this inter-
E2 946 1387 746 53 val contains zero, we cannot be highly confident
E3 115 438 288 16 that this coefficient is not zero. In fact, the data do
not adequately support the claim that the coefficient
The value of λ for these data is 0.21. The reduc- differs from zero. Consequently, we cannot con-
tion in the probability of error in predicting the level clude that age is related to “recent energy level” for
of one factor is 0.21 by knowing the level of the the population from which these 52 individuals
other factor, compared with not knowing the level were selected.
of the other factor. That is, you can eliminate about
20% of your errors in predicting the level of one of Example 8. For the data of example 4, the SE of
the factors if you know the level of the other factor. Kendall’s τb coefficient is 0.04. The 95% confi-
dence interval is approximately 0.39 ± 0.08 or
[0.31, 0.47]. There is strong statistical evidence
Reliability of the Estimated Coefficient
that this coefficient is not zero because zero is not
Because a given coefficient of correlation or contained in the interval. Note that the SE is much
association is calculated from a sample, it is meas- smaller here than for the data in example 7; that is
ured with a certain margin of error. Generally, the because the sample size is much larger: n = 542.
smaller the sample size, the larger the margin of
error. Your statistical software may provide a stan- Summary
dard error (SE) along with the estimate of the coef-
ficient. Very often, the margin of error can be In this article, appropriate measures of associa-
calculated as approximately twice the standard tion have been provided based on the types (or lev-
error. Then, it can be concluded with approxi- els of measurement) of the variables involved. For
mately 95% confidence that the interval, any given situation, there may be a wide variety of
measures of association to choose from. The rec-
estimated coefficient ± 2*SE, ommendations given in this article can be summa-
rized as follows:
contains the “true” or “population” coefficient value.
See Goodman and Kruskal1 for further details. Variable X
By the true or population value, we mean the
following: consider a population of subjects that is Variable Y Nominal Ordinal Continuous
under study, such as all 50- to 60-year-old Nominal ϕ or λ Rank biserial Point biserial
Caucasian Americans. In practice, one obtains a Ordinal Rank biserial τb or Spearman τb or Spearman
random sample from this population (e.g., ran- Continuous Point biserial τb or Spearman Pearson or
domly select 100 subjects from the population), Spearman
obtains values of two variables of interest (e.g., ϕ = phi coefficient, λ = Goodman and Kruskal’s lambda,
age and cholesterol level), and then calculates the τb = Kendall’s τb.
coefficient of interest (e.g., the Pearson correlation
coefficient). This value is the estimated coefficient. For any pair of variables, say X and Y, whose
If the coefficient value had been calculated from association is to be studied, identify the type of
the entire population instead of only 100 randomly variable for each (namely, nominal, ordinal, or
selected subjects, then the true or population value continuous) and choose the measure of association
of the coefficient would have been obtained. by referring to the table above. For example, if one
 MEASURES OF ASSOCIATION / Khamis 161

variable is discrete ordinal and the other is contin- are several different procedures for handling ties,
uous (it does not matter which variable is called X so it is possible for a given coefficient to give rise
and which is called Y), then an appropriate measure to slightly different values for the same data set.
for assessing the strength of association is Kendall’s Most statistical software packages will have a
τb or the Spearman correlation coefficient—notice default procedure for handling ties.
the intersection of the row corresponding to “ordi-
nal” and the column corresponding to “continuous”
Conclusion
or, equivalently, the row corresponding to “continu-
ous” and the column corresponding to “ordinal.” It is important that the measure or coefficient
For more details about the selection of the measure used to assess the strength of association between
of association for this case, see the discussion two variables is appropriate for the data involved. A
under “continuous-ordinal” above. coefficient that is designed to measure the strength
Several cautionary notes need to be made as of association between two continuous variables,
follows. such as the Pearson correlation coefficient, should
not be used to assess the strength of relationship
• Establishment of a strong relationship between between two ordinal variables or between an ordi-
two variables does not necessarily imply a nal variable and a nominal variable, for example.
cause-effect relationship. Although a strong This article provides a simple way of choosing an
association is necessary for establishment of a appropriate measure of association for a given pair
cause-effect relationship, it is not sufficient. of variables by classifying the variables according
• Establishment of a strong relationship between to their levels of measurement. By using these rec-
two variables does not necessarily imply agree- ommendations, it is hoped that research in sonogra-
ment between the two variables, nor does it nec- phy, as well as in other areas of medical research,
essarily imply high reliability. The notions of will lead to more accurate and more reliable results.
agreement and reliability are quite different from
association. For further reading about agreement
between two variables, see Bland and Altman18 References
and Cohen19; for reliability, see Fleiss.20 1. Goodman LA, Kruskal WH: Measures of Association for
• When assessing the strength of association Cross Classifications. New York, Springer-Verlag, 1979.
between two variables, it is important to adjust 2. Liebetrau AM: Measures of Association. Beverly Hills,
for the effects of important confounders. This CA, Sage, 1983.
can be done with the use of partial correlation 3. Khamis HJ: Measures of association, in Armitage P,
Colton T (eds): Encyclopedia of Biostatistics. 2nd ed.
coefficients. See Fisher and van Belle21 for further
New York, John Wiley, 2004.
discussion. 4. Newton RR, Rudestam KE: Your Statistical Consultant.
• If one of the two variables under study is antecedent Thousand Oaks, CA, Sage, 1999.
to the other, or there is an a priori cause-effect rela- 5. Cohen J: Statistical Power Analysis for the Behavioral
tionship, or there is a dependent-independent vari- Sciences. 2nd ed. Hillsdale, NJ, Lawrence Erlbaum, 1988.
able structure, then the variables are handled 6. Sokal RR, Rohlf FJ: Biometry. New York, W. H.
Freeman, 1995.
asymmetrically. In this case, prediction is often the 7. Zar JH: Biostatistical Analysis. Upper Saddle River, NJ,
research goal, and special measures of association Prentice Hall, 1996.
are recommended. See Goodman and Kruskal1 for 8. Valentine AL, Pope J, Read T: Evaluation of left atrial
further details. size and pressure as it relates to left ventricle diastolic
• When nominal or ordinal variables are involved, measurements in patients with left ventricle hypertrophy.
J Diagn Med Sonography 2003;19:73–79.
then there may be several ties in the data. That is,
9. Chatfield C: Problem Solving: A Statistician’s Guide.
several subjects may have the same X and Y val- 2nd ed. New York, Chapman & Hall, 1995.
ues. For instance, in example 6, many subjects 10. Luce RD, Narens L: Measurement scales on the contin-
(1768) had hair color H1 and eye color E1. There uum. Science 1987;236:1527–1532.
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