Short abstract

Need for consistency

Assessment of the consistency of effects across studies is an essential part of meta-analysis. Unless we know how consistent the results of studies are, we cannot determine the generalisability of the findings of the meta-analysis. Indeed, several hierarchical systems for grading evidence state that the results of studies must be consistent or homogeneous to obtain the highest grading.^2-4

Tests for heterogeneity are commonly used to decide on methods for combining studies and for concluding consistency or inconsistency of findings.^5,6 But what does the test achieve in practice, and how should the resulting P values be interpreted?

Testing for heterogeneity

A test for heterogeneity examines the null hypothesis that all studies are evaluating the same effect. The usual test statistic (Cochran's Q) is computed by summing the squared deviations of each study's estimate from the overall meta-analytic estimate, weighting each study's contribution in the same manner as in the meta-analysis.⁷ P values are obtained by comparing the statistic with a χ² distribution with k-1 degrees of freedom (where k is the number of studies).

The test is known to be poor at detecting true heterogeneity among studies as significant. Meta-analyses often include small numbers of studies,^6,8 and the power of the test in such circumstances is low.^9,10 For example, consider the meta-analysis of randomised controlled trials of amantadine for preventing influenza (fig 1).¹¹ The treatment effects in the eight trials seem inconsistent: the reduction in odds vary from 16% to 93%, with some of the confidence intervals not overlapping. But the test of heterogeneity yields a P value of 0.09, conventionally interpreted as being non-significant. Because the test is poor at detecting true heterogeneity, a non-significant result cannot be taken as evidence of homogeneity. Using a cut-off of 10% for significance¹² ameliorates this problem but increases the risk of drawing a false positive conclusion (type I error).¹⁰

Open in a separate window

Fig 1

Eight trials of amantadine for prevention of influenza.¹¹ Outcome is cases of influenza. Summary odds ratios calculated with random effects method

Conversely, the test arguably has excessive power when there are many studies, especially when those studies are large. One of the largest meta-analyses in the Cochrane Database of Systematic Reviews is of clinical trials of tricyclic antidepressants and selective serotonin reuptake inhibitors for treatment of depression.¹³ Over 15 000 participants from 135 trials are included in the assessment of comparative drop-out rates, and the test for heterogeneity is significant (P = 0.005). However, this P value does not reasonably describe the extent of heterogeneity in the results of the trials. As we show later, a little inconsistency exists among these trials but it does not affect the conclusion of the review (that serotonin reuptake inhibitors have lower discontinuation rates than tricyclic antidepressants).

Since systematic reviews bring together studies that are diverse both clinically and methodologically, heterogeneity in their results is to be expected.⁶ For example, heterogeneity is likely to arise through diversity in doses, lengths of follow up, study quality, and inclusion criteria for participants. So there seems little point in simply testing for heterogeneity when what matters is the extent to which it affects the conclusions of the meta-analysis.

Quantifying heterogeneity: a better approach

We developed an alternative approach that quantifies the effect of heterogeneity, providing a measure of the degree of inconsistency in the studies' results.¹⁴ The quantity, which we call I², describes the percentage of total variation across studies that is due to heterogeneity rather than chance. I² can be readily calculated from basic results obtained from a typical meta-analysis as I² = 100%×(Q - df)/Q, where Q is Cochran's heterogeneity statistic and df the degrees of freedom. Negative values of I² are put equal to zero so that I² lies between 0% and 100%. A value of 0% indicates no observed heterogeneity, and larger values show increasing heterogeneity.

Examples of values of I²

The principal advantage of I² is that it can be calculated and compared across meta-analyses of different sizes, of different types of study, and using different types of outcome data. Table 1 gives I² values for six published meta-analyses along with 95% uncertainty intervals. The upper limits of these intervals show that conclusions of homogeneity in meta-analyses of small numbers of studies are often unjustified.^11,13,15-19

The tamoxifen and streptokinase meta-analyses, in which all the included studies found similar effects,^16,17 have I² values of 3% and 19% respectively. These indicate little variability between studies that cannot be explained by chance. For the review comparing drop-outs on selective serotonin reuptake inhibitors with tricyclic antidepressants, I² is 26%, indicating that although the heterogeneity is highly significant, it is a small effect.

The reviews of trials of magnesium after myocardial infarction (I² = 63%) and case-control studies investigating the effects of electromagnetic radiation on leukaemia (69%) both included studies with diverse results. The high I² values show that most of the variability across studies is due to heterogeneity rather than chance. Although no significant heterogeneity was detected in the review of amantadine,¹¹ the inconsistency was moderately large (I² = 44%).

Figure 2 shows the observed values of I² from 509 meta-analyses in the Cochrane Database of Systematic Reviews. Almost half of these meta-analyses (250) had no inconsistency (I² = 0%). Among meta-analyses with some heterogeneity, the distribution of I² is roughly flat.

Open in a separate window

Fig 2

Distribution of observed values of I² based on odds ratios from 509 meta-analyses of dichotomous outcomes in the Cochrane Database of Systematic Reviews. Data are from the first subgroup (if any) in the first meta-analysis (if any) in each review, if it involved a dichotomous outcome and at least two trials with events. Meta-analyses conducted with metan in STATA¹⁵

Further applications of I²

I² can also be helpful in investigating the causes and type of heterogeneity, as in the three examples below.

Methodological subgroups

Figure 3 shows the six case-control studies of magnetic fields and leukaemia broken down into two subgroups based on assessment of their quality.¹⁹ If heterogeneity is identified in a meta-analysis a common option is to subgroup the studies. Because of loss of power, non-significant heterogeneity within a subgroup may be due not to homogeneity but to the smaller number of studies. Here, the P values for the heterogeneity test are higher for the two subgroups (P = 0.3 and P = 0.009) than for the complete data (P = 0.007), which suggests greater consistency within the subgroups. However, the values of I² show that the three low quality studies are more inconsistent (I² = 79%) than all six (I² = 69%) (table 2). Substantially less inconsistency exists among the high quality studies (I² = 15%), although uncertainty intervals for all of the I² values are wide.

Open in a separate window

Fig 3

Meta-analyses of six case-control studies relating residential exposure to electromagnetic fields to childhood leukaemia.¹⁹ Summary odds ratio calculated by random effects method

Table 2

More advanced applications of l² for assessing heterogeneity in three published meta-analyses. Meta-analyses were conducted with either meta or metan in STATA¹⁵

				Heterogeneity test
Topic	Outcome/analysis	Effect measure	No of studies	Q	df	P	I2 (95% uncertainty intervals)*
Magnetic fields and leukaemia19	All studies	Odds ratio	6	15.9	5	0.007	69 (26 to 87)
	High quality	Odds ratio	3	2.4	2	0.31	17 (0 to 91)
	Low quality	Odds ratio	3	9.4	2	0.009	79 (32 to 94)
Human albumin for critically ill20	Death	Risk ratio	24†	15.3	23	0.88	0 (0 to 17)
	Death	Risk difference	30	36.7	29	0.15	21 (0 to 50)
Tamoxifen to prevent recurrence of breast cancer17	All studies	Peto odds ratio	55	108.2	54	0.00002	50 (32 to 63)
	Total within groups‡	Peto odds ratio	—	59.9	52	0.21	13 (0 to 39)
	Between groups‡	Peto odds ratio	3 groups	48.3	2	<0.00001	96 (91 to 98)

Open in a separate window

df=degrees of freedom,

^*Values of l² are percentages. 95% uncertainty intervals are calculated as proposed by Higgins and Thompson.¹⁴

^†Studies with no events in either treatment group do not contribute to this analysis.

^‡Subgroup defined by duration of tamoxifen treatment.

How much is too much heterogeneity?

A naive categorisation of values for I² would not be appropriate for all circumstances, although we would tentatively assign adjectives of low, moderate, and high to I² values of 25%, 50%, and 75%. Figure 2 shows that about a quarter of meta-analyses have I² values over 50%. Quantification of heterogeneity is only one component of a wider investigation of variability across studies, the most important being diversity in clinical and methodological aspects. Meta-analysts must also consider the clinical implications of the observed degree of inconsistency across studies. For example, interpretation of a given degree of heterogeneity across several studies will differ according to whether the estimates show the same direction of effect.

An alternative quantification of heterogeneity in a meta-analysis is the among-study variance (often called τ²), calculated as part of a random effects meta-analysis. This is more useful for comparisons of heterogeneity among subgroups, but values depend on the treatment effect scale. We believe, I² offers advantages over existing approaches to the assessment of heterogeneity (box). Focusing on the effect of heterogeneity also avoids the temptation to perform so called two stage analyses, in which the meta-analysis strategy (fixed or random effects method) is determined by the result of a statistical test. Such strategies have been found to be problematic.^23,24 We therefore believe that I² is preferable to the test of heterogeneity when assessing inconsistency across studies.

Notes

We thank Keith O'Rourke and Ian White for useful comments.

Contributors: The authors all work as statisticians and have extensive experience in methodological, empirical and applied research in meta-analysis. JH, JD, and DA are coconvenors of the Cochrane Statistical Methods Group. The views expressed in the paper are those of the authors. All authors contributed to the development of the methods described. JH and ST worked more closely on the development of I². JH is guarantor.

Funding: This work was funded in part by MRC Project Grant G9815466.

Competing interests: None declared.

References