1 The Common Metrics: Correlation, Standardized Mean Difference, and Odds Ratio
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CODING INDIVIDUAL STUDIES
Imagine that a researcher, Dr. A, wishes to investigate whether two
groups (male versus female, two treatment groups, etc.) differ on a particular
variable X. So she collects data from five individuals in each group (N = 10).
She finds that Group 1 members have scores of 4, 4, 3, 2, and 2, for a mean of
3.0 and (population estimated) standard deviation of 1.0, whereas Group 2
members have scores of 6, 6, 5, 4, and 4, for a mean of 5.0 and standard deviation of 1.0. Dr. A performs a t-test and finds that t(8) = 3.16, p = .013. Finding
that Group 2 was significantly higher than Group 1 (according to traditional
criteria of a = .05), she publishes the results.
Further imagine that Dr. B reads this report and is skeptical of the results.
He decides to replicate this study, but collects data from only three individuals in each group (N = 6). He finds that individuals in Group 1 had scores of
4, 3, and 2, for a mean of 3.0 and standard deviation of 1.0, whereas Group 2
members had scores of 6, 5, and 4, for a mean of 5.0 and standard deviation
of 1.0. His comparison of these groups results in t(4) = 2.45, p = .071. Dr. B
concludes that the two groups do not differ significantly and therefore that
the findings of Dr. A have failed to replicate.
Now we have a controversy on our hands. Graduate student C decides
that she will resolve this controversy through a definitive study involving 10
individuals in each group (N = 20). She finds that individuals in Group 1 had
scores of 4, 4, 4, 4, 3, 3, 2, 2, 2, and 2, for a mean of 3.0 and standard deviation of 1.0, whereas individuals in Group 2 had scores of 6, 6, 6, 6, 5, 5, 4, 4,
4, and 4, for a mean of 5.0 and a standard deviation of 1.0. Her inferential test
is highly significant, t(18) = 4.74, p = .00016. She concludes that not only do
the groups differ, but also the difference is more pronounced than previously
thought!
This example illustrates the limits of relying on the Null Hypothesis
Significance Testing Framework in comparing results across studies. In each
of the three hypothetical studies, individuals in Group 1 had a mean score
of 3.0 and a standard deviation of 1.0, whereas individuals in Group 2 had a
mean score of 5.0 and a standard deviation of 1.0. The hypothetical researchers’ focus on significance tests led them to inappropriate conclusions: Dr. B’s
conclusion of a failure to replicate is inaccurate (because it does not consider
the inadequacy of statistical power in the study), as is Student C’s conclusion
of a more pronounced difference (which mistakenly interprets a low p value
as informing the magnitude of an effect). A focus on effect sizes would have
alleviated the confusion that arose from a reliance only on statistical significance and, in fact, would have shown that these three studies provided perfectly replicating results. Moreover, if the researchers had considered effect
sizes, they could have moved beyond the question of whether the two groups
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differ to consider also the question of how much the two groups differ. These
limitations of relying exclusively on significance tests have been the subject
of much discussion (see, e.g., Cohen, 1994; Fan, 2001; Frick, 1996; Harlow,
Mulaik, & Steiger, 1997; Meehl, 1978; Wilkinson & the Task Force on Statistical Significance, 1999), yet this practice unfortunately persists.
For the purposes of most meta-analyses, I find it useful to define an effect
size as an index of the direction and magnitude of association between two variables.1 As I describe in more detail later in this chapter, this definition includes
traditional measures of correlation between two variables, differences between
two groups, and contingencies between two dichotomies. When conducting
a meta-analysis, it is critical that effect sizes be comparable across studies. In
other words, a useful effect size for meta-analysis is one to which results from
various studies can be transformed and therefore combined and compared. In
this chapter I describe ways that you can compute the correlation (r), standardized mean difference (g), or odds ratio (o) from a variety of information commonly reported in primary studies; this is another reason that these indexes
are useful in summarizing or comparing findings across studies.
A second criterion for an effect size index to be useful in meta-analysis
is that it must be possible to compute or approximate its standard error.
Although I describe this importance more fully in Chapter 8, I should say a
few words about it here. Standard errors describe the imprecision of a sample-based estimate of a population effect size; formally, the standard error
represents the typical magnitude of differences of sample effect sizes around
a population effect size if you were to draw multiple samples (of a certain size
N) from the population. It is important to be able to compute standard errors
of effect sizes because you generally want to give more weight to studies that
provide precise estimates of effect sizes (i.e., have small standard errors) than
to studies that provide less precise estimates (i.e., have large standard errors).
Chapter 8 of this book provides further description of this idea.
Having made clear the difference between statistical significance and
effect size, I next describe three indices of effect size that are commonly used
in meta-analyses.
5.1.2 Pearson Correlation
The Pearson correlation, commonly represented as r, represents the association between two continuous variables (with variants existing for other
forms, such as r pb when one variable is dichotomous and the other is continuous, φ when both are dichotomous). The formula for computing r (the sample
estimate of the population correlation, ρ) within a primary data set is:
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Equation 5.1: Computing r
r
£ x
i
− x yi − y
N −1sx sy
£Z
X
ZY
N
• xi and yi are scores of individual i on the two variables.
• x and y are the sample means of the two variables.
• N is the sample size.
• sx and sy are the population estimated standard deviations of the
two variables.
• ZX and ZY are standardized scores, computed as ZX = (xi – x)/sx.
The conceptual meaning of positive correlations is that individuals who
score high on X (relative to the sample mean on X) also tend to score high on
Y (relative to the sample mean on Y), whereas individuals who score low on X
also tend to score low on Y. The conceptual meaning of negative correlations
is that individuals who score high on one variable tend to score low on the
other variable. The rightmost portion of Equation 5.1 provides an alternative
representation that illustrates this conceptual meaning. Here, Z scores (standardized scores) represent high values as positive and low values as negative,
so a preponderance of high scores with high scores (product of two positive)
and low scores with low scores (product of two negative) yields a positive
average cross product, whereas high scores with low scores (product of positive and negative) yield a negative average cross product.
You are probably already familiar with the correlation coefficient, but perhaps are not aware that it is an index of effect size. One interpretation of the correlation coefficient is in describing the proportion of variance shared between
two variables with r2. For instance, a correlation between two variables of r =
.50 implies that 25% (i.e., .502) of the variance in these two variables overlaps.
It can also be kept in mind that the correlation is standardized, such that correlations can range from 0 to ± 1. Given the widespread use of the correlation
coefficient, many researchers have an intuitive interpretation of the magnitude
of correlations that constitute small or large effect sizes. To aid this interpretation, you can consider Cohen’s (1969) suggestions of r = ± .10 representing small
effect sizes, r = ± .30 representing medium effect sizes, and r = ± .50 representing
large effect sizes. However, you should bear in mind that the typical magnitudes
of correlations found likely differ across areas of study, and one should not be
dogmatic in applying these guidelines to all research domains.
In conclusion, Pearson’s r represents a useful, readily interpretable index
of effect size for associations between two continuous variables. In many
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89
meta-analyses, however, r is transformed before effect sizes are combined or
compared across studies (for contrasting views see Hall & Brannick, 2002;
Hunter & Schmidt, 2004; Schmidt, Hunter, & Raju, 1988). Fisher’s transformation of r, denoted as Zr, is commonly used and shown in Equation 5.2.
Equation 5.2: Fisher’s transformation of r
¤ 1 r ³
Z r ½ln ¥
´
¦ 1
r µ
• Zr is Fisher’s transformation of r.
• r is the correlation coefficient.
The reason that r is often transformed to Zr in meta-analyses is because
the distribution of sample r’s around a given population ρ is skewed (except in
sample sizes larger than those commonly seen in the social sciences), whereas
a sample of Zr s around a population Zr is symmetric (for further details see
Hedges & Olkin, 1985, pp. 226–228; Schulze, 2004, pp. 21–28).2 This symmetry is desirable when combining and comparing effect sizes across studies.
However, Zr is less readily interpretable than r both because it is not bounded
(i.e., it can have values greater than ±1.0) and simply because it is unfamiliar to many readers. Typical practice is to compute r for each study, convert
these to Zr for meta-analytic combination and comparison, and then convert
results of the meta-analysis (e.g., mean effect size) back to r for reporting.
Equation 5.3 converts Zr back to r.
Equation 5.3: Converting Zr back to r
r
e2Zr – 1
e2Zr 1
• r is the correlation coefficient.
• Zr is Fisher’s transformation of r.
Although I defer further description until Chapter 8, I provide the equation for the standard error here, as you should enter these into your metaanalytic database during the coding process. The standard error of Zr is
shown in Equation 5.4.
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Equation 5.4: Standard error of Zr
SEZr
1
N
3
• N is the sample size of the study.
This equation reveals an obvious relation inherent to all standard errors:
As sample size (N) increases, the denominator of Equation 5.4 increases and
so the standard error decreases. A desirable feature of Zr is that its standard
error depends only on sample size (as I describe later, standard errors of some
effects also depend on the effect sizes themselves).
5.1.3 Standardized Mean Difference
The family of indices of standardized mean difference represents the magnitude of difference between the means of two groups as a function of the
groups’ standard deviations. Therefore, you can consider these effect sizes to
index the association between a dichotomous group variable and a continuous variable.
There are three commonly used indices of standardized mean difference
(Grissom & Kim, 2005; Rosenthal, 1994).3 These are Hedges’s g, Cohen’s d,
and Glass’s index (which I denote as gGlass),4 defined by Equations 5.5, 5.6,
and 5.7, respectively:
Equations 5.5–5.7: Computing standardized mean differences
5.5: Hedges’s g:
g=
5.6: Cohen’s d:
d=
5.7: Glass’s index: gGlass =
M1 – M2
spooled
M1 – M2
sdpooled
M1 – M2
s1
• M1 and M2 are the means of Groups 1 and 2.
• spooled is the pooled estimate of the population standard deviation.
• sdpooled is the pooled sample standard deviation.
• s1 is the estimate of the population standard deviation from Group
1 (control group).
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91
These three equations are identical in consisting of a raw (unstandardized) difference of means as their numerators. The difference among them is
in the standard deviations comprising the denominators (i.e., the standardization of the mean difference). The equation (5.5) for Hedges’s g uses the pooled
estimates5 of the population standard deviations of each group, which is the
familiar s xi
x 2 n
1. The equation (5.6) for Cohen’s d is similar but
instead uses the pooled sample standard deviations, sd xi
x 2 n . This
sample standard deviation is a biased estimation of the population standard
deviation, with the underestimation greater in smaller than larger samples.
However, with even modestly large sample sizes, g and d will be virtually
identical, so the two indices are not always distinguished.6 Often, people
describe both indices as Cohen’s d, although it is preferable to use precise
terminology in your own writing.7
The third index of standardized mean difference is gGlass (sometimes
denoted with ∆ or g′), shown in Equation 5.7. Here the denominator consists
of the (population estimate of the) standard deviation from one group. This
index is often described in the context of therapy trials, in which the control
group is said to provide a better index of standard deviation than the therapy
group (for which the variability may have also changed in addition to the
mean). One drawback to using gGlass in meta-analysis is that it is necessary
for the primary studies to report these standard deviations for each group;
you can use results of significance tests (e.g., t-test values) to compute g or d,
but not gGlass. Reliance on only one group to estimate the standard deviation
is also less precise if the standard deviations of the two populations are equal
(i.e., homoscedastic; see Hedges & Olkin, 1985). For these reasons, metaanalysts less commonly rely on gGlass relative to g or d. On the other hand, if
the population standard deviations of the groups being compared differ (i.e.,
heteroscedasticity), then g or d may not be meaningful indexes of effect size,
and computing these indexes from inferential statistics reported (e.g., t-tests,
F-ratios) can be inappropriate. In these cases, reliance on gGlass is likely more
appropriate if adequate data are reported in most studies (i.e., means and
standard deviations from both groups).8 If it is plausible that heteroscedasticity might exist, you may wish to compare the standard deviations (see Shaffer, 1992) of two groups among the studies that report these data and then
base the decision to use gGlass versus g or d depending on whether or not
(respectively) the groups have different variances.
Examining Equations 5.5–5.7 leads to two observations regarding the
standardized mean difference as an effect size. First, these can be either positive or negative depending on whether the mean of Group 1 or 2 is higher. This
is a desirable quality when your meta-analysis includes studies with potentially
conflicting directions of effects. You need to be consistent in considering a par-
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ticular group (e.g., treatment vs. control, males vs. females) as Group 1 versus
2 across studies. Second, these standardized mean differences can take on any
values. In other words, they are not bounded by ± 1.00 like the correlation
coefficient r. Like r, a value of 0 implies no effect, but standardized mean differences can also have values greater than 1. For example, in the hypothetical
example of three researchers given earlier in this chapter, all three researchers
would have found g = (3 – 5)/1 = –2.00 if they considered effect sizes.
Although not as commonly used in primary research as r, these standardized mean differences are intuitively interpretable effect sizes. Knowing
that the two groups differ by one-tenth, one-half, one, or two standard deviations (i.e., g or d = 0.10, 0.50, 1.00, or 2.00) provides readily interpretable
information about the magnitude of this group difference.9 As with r, Cohen
(1969) provided suggestions for interpreting d (which can also be used with
g or gGlass), with d = 0.20 considered a small effect, d = 0.50 considered a
medium effect, and d = 0.80 considered a large effect. Again, it is important
to avoid becoming fixated on these guidelines, as they do not apply to all
research situations or domains. It is also interesting to note that transformations between r and d (described in Section 5.5) reveal that the guidelines for
interpreting each do not directly correspond.
As I did for the standard error of Zr, I defer further discussion of weighting until Chapter 8. However, the formulas for the standard errors of the
commonly used standardized mean difference, g, should be presented here:
Equation 5.8: Standard error of g
SE g
n1 n2
4 g2
g2
y
2n1 n2 Ntotal equal sample sizes
n1n2
• n1 and n2 are the sample sizes of Groups 1 and 2.
• Ntotal is the total sample size of the study (assuming equal sample
size per group).
I draw your attention to two aspects of this equation. First, you should use
the first equation when sample sizes of the two groups are known; the second
part of the equation is a simplified version that can be used if group sizes are
unknown, but it is reasonable to assume approximately equal group sizes. Second, you will notice that the standard errors of estimates of the standardized
mean differences are dependent on the effect size estimates themselves (i.e., the
effect size appears in the numerators of these equations). In other words, there
is greater expected sampling error when the magnitudes (positive or negative)
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93
of standardized mean differences are large than when they are small. As discussed later (Chapter 8), this means that primary studies finding larger effect
sizes will be weighted relatively less (given the same sample size) than primary
studies with smaller effect sizes when results are meta-analytically combined.
Before ending this discussion of standardized mean difference effect
sizes, it is worth considering a correction that you should use when primary
study sample sizes are small (e.g., less than 20). Hedges and Olkin (1985)
have shown that g is a biased estimate of the population standardized mean
differences, with the magnitude of overestimation becoming nontrivial with
small sample sizes. Therefore, if your meta-analysis includes studies with
small samples, you should apply the following correction of g for small sample size (Hedges & Olkin, 1985, p. 79; Lipsey & Wilson, 2001, p. 49):
Equation 5.9: Small sample adjustment of g
gadjusted = g –
3g
4(n1 + n2) – 9
• n1 and n2 are the sample sizes of Groups 1 and 2.
5.1.4Odds Ratio
The odds ratio, which I denote as o (OR is also commonly used), is a useful
index of effect size of the contingency (i.e., association) between two dichotomous variables. Because many readers are likely less familiar with odds ratios
than with correlations or standardized mean differences, I first describe why
the odds ratio is advantageous as an index of association between two dichotomous variables.10
To understand the odds ratio, you must first consider the definition of
odds. The odds of an event is defined as the probability of the event (e.g.,
of scoring affirmative on a dichotomous measure) divided by the probability of the alternative (e.g., of scoring negative on the measure), which can
be expressed as odds = p / (1 – p), where p equals the proportion in the
sample (which is an unbiased estimate of population proportion, Π) having
the characteristic or experiencing the event. For example, if you conceptualized children’s aggression as a dichotomous variable of occurring versus not
occurring, you could find the proportion of children who are aggressive (p)
and estimate the odds of being aggressive by dividing by the proportion of
children who are not aggressive (1 – p). Note that you can also compute odds
for nominal dichotomies; for example, you could consider biological sex in
terms of the odds of being male versus female, or vice versa.
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The next challenge is to consider how you can compare probabilities or
odds across two groups. This comparison actually indexes an association
between two dichotomous variables. For instance, you may wish to know
whether boys or girls are more likely to be aggressive, and our answer would
indicate whether, and how strongly, sex and aggression are associated. Several ways of indexing this association have been proposed (see Fleiss, 1994).
The simplest way would be to compute the difference between probabilities
in two groups, p1 – p2 (where p1 and p2 represent proportions in each group;
in this example, these values would be the proportions of boys and girls who
are aggressive). An alternative might be to compute the rate ratio (sometimes
called risk ratio), which is equal to the proportion experiencing the event
(or otherwise having the characteristics) in one group divided by the proportion experiencing it in the other, RR = p1 / p2. A problem with both of
these indices, however, is that they are highly dependent on the rate of the
phenomenon in the study (for details, see Fleiss, 1994). Therefore, studies in
which different base rates of the phenomenon are found (e.g., one study finds
a high prevalence of children are aggressive, whereas a second finds that very
few are aggressive) will yield vastly different differences and risk ratios, even
given the same underlying association between the variables. For this reason,
these indices are not desirable effect sizes for meta-analysis.
The phi (f) coefficient is another index of association between two
dichotomous variables. It is estimated via the following formula (where f is
the estimated population association):
Equation 5.10: Computing f
ˆ n00n11
n01n10
F
n0u n1u nu 0nu1
• n00 is the number of participants who scored negative on dichotomous variables X and Y.
• n01 is the number of participants who scored negative on X and
positive on Y.
• n10 is the number of participants who scored positive on X and
negative on Y.
• n11 is the number of participants who scored positive on X and Y.
• n0• and n1• are the total number of participants who scored negative and positive (respectively) on X (i.e., the marginal sums).
• n•0 and n•1 are the total number of participants who scored negative and positive on Y.
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95
Despite the lack of similarity in appearance between this equation and
Equation 5.1, f is identical to computing r between the two dichotomous
variables, X and Y. In fact, if you are meta-analyzing a set of studies involving
associations between two continuous variables in which a small number of
studies artificially dichotomize the variables, it is appropriate to compute f
and interpret this as a correlation (you might also consider correcting for the
attenuation of correlation due to artificial dichotomization; see Chapter 6).
However, f (like the difference in proportions and rate ratio) also suffers
from the limitation that it is influenced by the rates of the variables of interest (i.e., the marginal frequencies). Thus studies with different proportions
of the dichotomies can yield different effect sizes given the same underlying
association (Fleiss, 1994; Haddock, Rindskopf, & Shadish, 1998). To avoid
this problem, the odds ratio (o) is preferred when one is interested in associations between two dichotomous variables. The odds ratio in the population
is often represented as omega (ω) and is estimated from sample data using
the following formula:
Equation 5.11: Computing odds ratio, o
o=
n00n11
n01n10
• n00 is the number of participants who scored negative on X and
Y.
• n01 is the number of participants who scored negative on X and
positive on Y.
• n10 is the number of participants who scored positive on X and
negative on Y.
• n11 is the number of participants who scored positive on X and Y.
Although this equation is not intuitive, it helps to consider that it represents the ratio of the odds of Y being positive when X is positive [(n11/n1•) /
(n10/n1•)] divided by the odds of Y being positive when X is negative [(n01/
n0•) / (n00/n0•)], algebraically rearranged.
The odds ratio is 1.0 when there is no association between the two
dichotomous variables, ranges from 0 to 1 when the association is negative,
and ranges from 1 to positive infinity when the association is positive. Given
these ranges, the distribution of sample estimates around a population odds
ratio is necessarily skewed. Therefore, it is common to use the natural log
transformation of the odds ratio when performing meta-analytic combina-
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tion or comparison, expressed as ln(o). The standard error of this logged
odds ratio is easily computed (whereas computing the standard error of an
untransformed odds ratio is more complex; see Fleiss, 1994), using the following equation:
Equation 5.12: Standard error of ln(o)
SE ln o
1
1
1
1
n00 n01 n10 n11
• n00 is the number of participants who scored negative on X and
Y.
• n01 is the number of participants who scored negative on X and
positive on Y.
• n10 is the number of participants who scored positive on X and
negative on Y.
• n11 is the number of participants who scored positive on X and Y.
5.2Computing r from Commonly Reported Results
You can compute r from a wide range of results reported in primary studies.
In this section, I describe how you can compute this effect size when primary
studies report correlations, inferential statistics (i.e., t-tests or F-ratios from
group comparisons, c2 from analyses of contingency tables), descriptive data,
and probability levels of inferential tests. I then describe some more recent
approaches to computing r from omnibus test results (e.g., ANOVAs with
more than two groups). Table 5.1 summarizes the equations that I describe
for computing r, as well as those for computing standardized mean differences (e.g., g) and odds ratios (o).
5.2.1From Reported Correlations
In the ideal case, primary reports would always report the correlations
between variables of interest. This reporting certainly makes our task much
easier and reduces the chances of inaccuracies due to computational errors
or rounding imprecision. When studies report correlation coefficients, these
are often in a tabular form.