Meta-analysis and effect size

Chong-ho Yu, Ph.Ds.

Power analysis and meta-analysis

The previous lesson mentioned that power is a function of effect size, alpha level and sample size. What is effect size? How is it related to power analysis? How can we determine effect size? Answers to these questions are provided below. It is important to point out that although power analysis requires the effect size yielded from meta-analysis, meta-analysis does not rely on power analysis. This could be a standalone method on its own right.

"Meta" is Greek prefix meaning "after" or "beyond." Meta-analysis (MA) is a secondary analysis after other researchers had done their own analyses and the meta-analyzer can go beyond what had been accomplished in the past. Simply put, MA is analysis of analyses. Even if you might never done MA before, you have already done quasi-meta-analysis, namely, literature review, which is typically presented in the following way:

  • According to So and So (2009), Program X can effectively reduce the risk of colon cancer.
  • Smith and Smith (2010) found that patients were unresponsive to Treatment X. It is a waste of your money.
  • Prior research found that the therapeutic effect of Program X is minimal (Doe and Doe, 2014).
What can be done with the preceding heterogeneous or even contradictory findings? The answer is: One must go beyond literature review by looking into MA and effect size.

What is effect size?

Before discussing the effect size, I would like to introduce a broader concept: comparison in terms of a standard. Many statistical formulas seem to be difficult to follow. Indeed, many of them are nothing more than a standardized comparison. Take comparing wealth as a metaphor. How could we compare the net assets of American IBM corporation and Japanese Sony corporation? The simplest way is to compare them in US dollars, the standard currency for international trade. By the same token, a t-test is a mean comparison in terms of the standard deviation. Many statistics follow this thread of logic.

Effect size can be conceptualized as a standardized difference. In the simplest form, effect size, which is denoted by the symbol "d", is the mean difference between groups in standard score form i.e. the ratio of the difference between the means to the standard deviation. This concept is derived from a school of methodology named Meta-analysis, which was developed by Glass (1976).

P value: Not zero!

Conventionally researchers draw a conclusion based on the p value alone. If the p value is less than .05 or .01, the effect, difference, or relationship is believed to be significant. The p value, by definition, is the probability of correcting rejecting the null hypothesis (no effect, no difference, or no relationship) assuming that the null is true. However, is it sufficient? Consider the following scenarios:

  • When the US President asks a four-star general to estimate the potential casualty of a possible war, he replies, "Not zero. Some soldiers will die."
  • Your boss wants you to give a raise and asks you how much you expect, you answer, "Not zero. I deserve more compensation."
  • After you have gone through a rigorous fitness program, your wife asks you how much weight you lost. You answer, "Not zero. I lost a few pounds."
Do you think the above are satisfactory answers? Of course, not. However, this type of answer is exactly what we can get from conventional hypothesis testing. When the null hypothesis is rejected, at most the conclusion is: "The null hypothesis is false! Not null! The effect, difference, or correlation is not zero." The p value tells you how unlikely the statistics can be observed in the long run by chance alone; it says nothing about the degree of treatment effectiveness, the magnitude of the association, or the distance of the performance gap. The following figure shows that the difference, indicated by red arrows, can be anything.

When the null hypothesis is reject: Not zero

Researchers should be concerned with not only whether a null hypothesis is false or not, but also how false it is. In other words, if the difference is not zero, how large the difference one should expect? By specifying an effect size, which is the minimum difference that is worth research attention, researcher could design a study with optimal power rather than wasting resources on trivial effects. The larger the effect size (the difference between the null and alternative means) is, the greater the power of a test is.

effect size

Ideally, power analysis employs the population effect size. However, in practice the effect size must be estimated from sample data.

How can we determine effect size?

Gene Glass's approach

There are several ways to calculate effect size. The three most popular approaches are Gene Glass's approach, Hunter-schmidt's approach, and Cohen's d. The basic formula of Glass's approach is:

Mean of control group - Mean of treatment group

Standard deviation of the control group

The control group's standard deviation is used because it is not affected by the treatment (Glass, McGraw, & Smith, 1981).

Hunter-schmidt's approach

Hunter and Schmidt (1990) suggested using a pooled within-group standard deviation because it has less sampling error than the control group standard deviation under the condition of equal sample size. In addition, Hunter and Schmidt corrected the effect size for measurement error by dividing the effect size by the square root of the reliability coefficient of the dependent variable:

Effect size
Measurement error correction =

Square root of r

Concepts of measurement error and reliability coefficient will be discussed in the section "Measurement."

Cohen's d

When there are two independent groups (e.g. control and treatment), Cohen's d can be obtained by the following formula:

Treatment group mean - Control group mean

SQRT(Treatment group variance + Control group variance)/2)

When a study reports a Chi-square test result with one degree of freedom (n=2), the following formula can be employed to approximate Cohen's d:

abs(d) = 2*SQRT(Chi-square/N - Chi-square)

where N is the total sample size

When a study reports a hit rate (percentage of success after taking the treatment or no treatment), the following formula can be used:

d = arscine(p1) + arscine(p2)

where p1 and p2 are the hit rates of the two groups (e.g. control and treatment) (Poston & Hanson, 2010)

Conventional values

The conventional values of effect size (Cohen, 1962) are:

Smalld = .20
Mediumd = .40
Larged = .60

Other researchers may have different values for small, medium, and large effect size. The magnitude of effect size depends on the subject matter. For example, in medical research d = .05 may consider a large effect size i.e. if the drug can save even five more lives, further research should be considered.

It is important to point out that Cohen defined .40 as the medium effect size because it was close to the average observed effect size based on his literature review using Journal of Abnormal and Social Psychology during the 1960s. The so-called small, medium, and large effect sizes are specific to a particular domain (abnormal and social psychology) and thus they should not be treated as the universal guideline (Aguinis, & Harden, 2009). Because different subject matters might have different effect sizes, Welkowitz, Ewen, Cohen (1982) explicitly stated that one should not use conventional values if one can specify the effect size that is appropriate to the specific problem. Moreover, Wilkinson and Task Force (1999) gave the following advice, "Because power computations are most meaningful when done before data are collected and examined, it is important to show how effect-size estimates have been derived from previous research and theory in order to dispel suspicions that they might have been taken from data used in the study or, even worse, constructed to justify a particular sample size."

It is a common practice for researchers to collect articles in their fields and catalog them in EndNote for future citation. It may be more beneficial to use this collection to calculate and constantly update the effect size of the subject matter to be studied.

In practice, it may be difficult to find past research studies related to your topic, especially when the topic is fairly new. To rectify this situation, Glass, McGraw, and Smith (1981) suggested to look at studies in similar domains. For example, if you are not able to locate enough research papers on Web-based instruction, you can use studies on hypertext and multimedia. Before the introduction of World Wide Web, hypertext and multimedia have been widely employed in computer-based instruction programmed in HyperCard, Authorware, and Director. Concepts related to Web-based instruction such as collaboration in chat sessions and mailing lists can be found in research on collaboration in other instructional settings.

It is noteworthy that not all research studies can be included in your collection for meta-analysis. Only well-designed studies which conform to the standards established by Campbell and Stanley (1963) and Cook and Campbell (1979) should be considered. Criteria of well-designed studies will be discussed in the section "Design of experiment"

Applications of meta-analysis

As discussed in the section concerning power analysis, computing effect size is essential to sample size determination. Nevertheless, meta analysis can not only be used for synthesizing results of past research, but also for new research studies. For example, Baker and Dwyer (2000) conducted eight studies regarding visualization as an instructional variable (n=2000). If all subjects are used for one analysis, the study will be over-powered. Instead, the effect size is computed in each study individually. The findings of eight studies are pooled to draw inferences as to the meaning of a collective body of research.

Besides the risk of overpowering, using all data in one test may lead to the Simpson's paradox. Simpson's Paradox is a phenomenon that the conclusion drawn from the aggregate data is opposite to the conclusion drawn from the contingency table based upon the same data.

The following example is given by Schwarz (1998). A university conducted a study to examine whether there is a sex bias in admission. The admission data of the MBA program and the law school were analyzed. The first table shows the MBA data:


MBA Program






480 (80%)

120 (20%)

600 (100%)


180 (90%)

20 (10%)

200 (100%)

By looking at the MBA data only, it seems that females are admitted at a slightly higher rate than males in the MBA program. The same pattern can be found in the law school data.


Law School






10 (10%)

90 (90%)

100 (100%)


100 (33%)

200 (66%)

300 (100%)

Interestingly enough, when the two data sets are pooled, females seem to be admitted at a lower rate than males!


MBA and Law School








700 (100%)




500 (100%)

To avoid the Simpson Paradox, Olkin (2000) recommends researchers to employ meta-analysis rather than pooling. In pooling, data sets are first combined and then the groups are compared. In meta-analysis, groups in different data sets are compared first and then the comparisons are combined.

This “divide and conquer” strategy is also known as Split/Analyze/Meta-Analyze (SAM) approach. In the previous example the data set is partitioned by logical categories (e.g. school or college). When there is no obvious way to split the data, Cheung and Jak (2016) recommended partitioning the data randomly if the data set is very big. Next, these data are analyzed as separate studies and at the end all these results are combined by meta-analysis.

Merits of meta-analysis 

  • A single study might lack statistical power due to a small sample size. Nevertheless, when many prior stuides are combined together, statistical power increases.
  • An individual study might over-estimate or under-estimate the efect size. Again, when many studies are pooled together, the precision of the estimation can be substantially improved.
  • A single study might have a very narrow focus. Meta-analysis can answer questions not posed by those scattering studies.
  • When diverse or even conflicting results are found in previous studies, a meta-analyst is able to resolve the dispute by looking at the forest instead of the trees. However, the meta-analyzer must take variation across studies into account (Higgins & Green, 2008).

Limitations of meta-analysis

Ioannides (2016) criticized hat meta-analysis implemented in biomedical science is often "redundant, misleading, and conflicted" (p.485). Like every methodology, meta-analysis also has certain limitations and weaknesses:

Assumption of standardized effects

It is important to point out that in some branches of meta-analysis computation of effect size is based upon a pooled variance or an adjusted variance. In response to this practice, Berk and Freedman (2003) are skeptical to the merit of meta-analysis. In their view, the claimed merit of meta-analysis is illusory. First, many meta-analyses use studies from both randomized experiments and observational studies. In the former, it is usually the case that subjects are not drawn at random from populations with a common variance. In observational studies there is no randomization at all. Thus, it is gratuitous to assume that standardized effects are constant across studies.

While this criticism is valid to some degree, the shortcoming can be easily fixed by setting a higher bar in the inclusion and exclusion criteria. For example, in the meta-analysis regarding the effect of intercessory prayer on the effectiveness of social workers, Hodges (2007) included only randomized controlled trials. Studies that used less rigorous designs, such as single-case studies and non-randomized studies, were excluded. In a similar study, Thompson (2007) started with 150 potential candidates but at the end only 23 studies that employed true experiments were retained in his meta-analysis. Retrospective observational studies that were deigned as quasi-experiments were removed from his meta-analysis.

Social dependence

Further, Berk and Freedman (2003) questioned the assumed independence of studies for meta-analysis. Researchers are trained in similar ways, read the same papers, talk to each other, write proposals for the same funding agencies, and publish the findings to the same pool of peer-review journals. Earlier studies lead to later studies in the sense that each generation of doctoral students trains the next. They questioned whether this social dependence compromises statistical independence.

It is true that in some cultures the mentees tend to follow the exact footstep of the mentor. In this case, close relationships among researchers might be a threat against the validity of meta-analysis. Nonetheless, today it is very common for researchers to think independently and to challenge each other. As a matter of fact, divergence, instead of conformity, is the norm of the academic community.

Publication bias

Another common problem of meta-analysis is publication bias, also know as the file-drawer effect (today we should call it "hard-drive effect" because we store our manuscripts digitally): Publication bias leads to the censoring of studies with non-significant results. As a remedy, Keng and Beretvas (2005) developed methodology to quantify the effect that publication bias can have on correlation estimation. The most common methods to check publication bias are funnel plots and Egger's test (Steme & Egger, 2001).

In response to this challenge, Schmidt and Hunter (2015) counter-argued that publication bias is only one of many sources of research biases. Other sources include researcher's personal bias, sampling error, confirmation bias, and reviewer bias.

To minimize possible bias, one can purposefully look for unpublished research studies, such as dissertations, theses, and conference presentations. One can go even further by soliciting unpublished manuscripts from researchers in the field (e.g. Stajkovic & Lee, 2001). 

Logic of courtroom

Root (2003) challenged the merits of meta-analysis at the philosophical level. According to Root, standard hypothesis testing is based upon the logic of physical sciences, in which the researcher must gamble with the unknown future, in the sense that the prediction derived from the hypothesis may not be in alignment to the proposed theory. However, meta-analysis is implicitly tied to the logic of courtroom, in which collected evidence is used to explain past events. In a retrospective methodology such as meta-analysis, the synthesizer has the luxury of choosing what past studies to be included. Using gambling as an analogy, Root pointed out that computing probabilities based on known facts is like betting money in a game after the result is known.

Subjective selection

The result of meta-analysis is tied to the selection criteria set by the researcher. For example, while Smith, Glass and Miller's (1980) meta-analysis showed a positive effect of psychotherapy, Prioleau, Murdoch and Brody (1983) produced a different conclusion. In an attempt to resolve the debate concerning whether mammography can reduce the mortality rate of breast cancer, a research team utilizing meta-analysis found that there was no reliable evidence to support the claim that mass screening for breast cancer had a positive effect for any women. On the contrary, the US Preventive Service Task Force that employed meta-analysis, too, concluded that use of mammogram significantly enhanced the survival rate of women from 40-74 years of age. Aschengeau and Seage III (2007) asserted that the preceding contradiction is a result of different criteria for selecting the literature.

Nonetheless, discrepancy occurs in many studies utilizing other methodologies. If one rejects meta-analysis on this ground, then one should also be opposed to almost all methodologies.

Superrealization bias

"Superrealization bias," the term coined by Cronbach et al. (1980) is germane to effect size and meta-analysis. Superrealization bias refers to the phenomenon that in a small-scale study, experimenters are able to monitor the quality of implementation or create unrealistic conditions, but these ideal conditions could never be realized on a large scale study. Slavin (2008), and Slavin and Smith (2008) asserted that small studies are not inherently biased, but a collection of small studies tend to be biased. Thus, Slavin warned against reporting average effect sizes using a cluster of low n studies.

Varying conditions across studies

It is important to point out that quite a few controversial conclusions in medical research arise from meta-analyses. For example, based on meta-analyses, the medical research community asserted that antidepressants are not more effective than placebos. But critics charged that not all the studies included in the meta-analyzes used the same protocols, definitions, types of patients and doses. The alleged safety of Avandia is another example. A meta-analysis from the combined trials showed that only 55 people in 10,000 had heart attacks when using Avandia whereas 59 people per 10,000 had heart attacks in comparison groups. However, after a series of statistical manipulations, this conclusion was reversed. It was argued that a meta-analysis synthesizing many small-scale studies is not a good substitute for a single trial with a large sample size (Siegfried, 2010).

No scientific breakthrough was made through meta-analysis

Sohn (1995, 1996, 1997) questioned whether meta-analysis, as a form of literature review, can be a good tool for scientific discovery. In a similra vein, Skeptics and New Atheism authors tried to discredit meta-analysis because this method was used for studying supernatural and paranormal phenomena. For example, Stenger (2007) wrote:

This procedure (meta-analysis) is highly questionable. I am unaware of any extraordinary discovery in all of science that was made using meta-analysis. If several, independent experiments do not find significant evidence for a phenomenon, we surely cannot expect a purely mathematical manipulation of the combined data to suddenly produce a major discovery. No doubt parapsychologists and their supporters will dispute my conclusions. But they cannot deny the fact that after one hundred and fifty years of attempting to verify a phenomenon, they have failed to provide any evidence that the phenomenon exists that has caught the attention of the bulk of the scientific community. We safely conclude that, after all this effort, the phenomenon very likely does not exist (Kindle Locations 824-830).

In a similar vein, the Skeptic's Dictionary (2012) website defines meta-analysis as the following:

A meta-analysis is a type of data analysis in which the results of several studies, none of which need find anything of statistical significance, are lumped together and analyzed as if they were the results of one large study.

Is the above the correct definition of meta-analysis? Is it a common practice that a meta-analyst puts together the results from studies that show no significant effects and then "mathematically manipulate" the data to prove a point? Did the illustration of the Simpson's Paradox clearly indicate that it is possible to yield opposite conclusions when one analysis combines all data and the other one partitions the data set? In response to Stenger's attack, Bartholomew (2011) replied that it is not sure what the words “mathematical manipulation” and “suddenly” mean. The so-called manipulation in meta-analysis is no more mathematical than other statistical procedures, such as hypothesis testing. When the criticism, no matter how sophisticated it sounds, is misguided by the wrong definition and poor statistical knowledge, it is nothng more than attacking a straw man (Yu, 2012)

Software for meta-analysis

You can use either all-purpose stat programs or specialized programs to conduct meta-analysis. SAS is an example of all-purpose stat programs that can perform meta-analysis (Wang & Bushman, 1999). For specialized programs, one can use BioStat (2017) or Devilly (2005). On one hand StatDirect (2014) is considered an all-purpose stat application because it can perform meta-analysis as well as other statistical procedures, but on the other hand it can also be viewed as a specialized package because its features are made for biomedical, public health, and epidemiological research. The image below is a screenshot of StatDirect.


The following is a screenshot of Effect size Generator written by Grant Devilly:

Further reading

To get a quick overview of effect size, I recommend reading a book chapter on effect size written by Tatsuoka (1993) in A Handbook for data analysis in the behavioral sciences (pp. 461-479), edited by Gideon Keren, Charles Lewis and published by Hillsdale, N.J. : L. Erlbaum Associates.

For learning the procedure of conducting meta-analysis, please look at Liao (1998) as an example.


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Last update: 2017

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