In this section, I de ne the type of study-level data required for meta-analysis. Ideally, this can be extracted directly from published papers, but this isn't always possible. Methods for meta-analysis using these data are given from section 1.5 onwards.
Required study-level data usually consist of an estimate of some parameter and its variance, denoted by ˆθi and ˆσi2 respectively for a given study i. This parameter is
commonly referred to as an eect size. In a health research setting, this parameter often represents a measure of the dierence between two groups, such as an active treatment groups and a control/placebo group. For example, a study may meas- ure the risk of myocardial infarction in a group of patients receiving intravenous magnesium and in a control group [114]. In this case, ˆθi represents an estimated
dierence in the risk between these groups [46]. To increase generality, I refer to them as groups one and two in this thesis. A number of measures can be used to calculate ˆθi depending on the type of study outcome, such as an odds ratio for a
binary outcome, or a standardised mean dierence for a continuous outcome. I show how these measures, among others, are calculated in the following two sections.
1.3.1 Summarising continuous outcome data
In studies with a continuous outcome, data from each group can be summarised by its mean, standard deviation and sample size. ˆµ1iand ˆµ2idenote the observed means,
ˆ
sd1i and ˆsd2i denote the observed standard deviations and n1i and n2i denote the
sample sizes in groups 1 and 2 respectively. The mean dierence and the standardised mean dierence are two commons ways to measure the dierence in ˆµ1iand ˆµ2i, which
1.3.1.1 Mean dierence (MD)
The mean dierence (MD) can be calculated by [8]:
ˆ
θi = ˆµ1i− ˆµ2i
If we assume the equal variances of ˆµ1i and ˆµ2i, the variance of ˆθi is:
ˆ σi2 = n1i+ n2i n1in2i · S2 i where Si2 = (n1i− 1) ˆsd 2 1i+ (n2i− 1) ˆsd22i n1i+ n2i− 2 (1.1) Alternatively, without making the equal-variances assumption:
ˆ σi2 = ˆ sd21i n1i + ˆ sd22i n2i
1.3.1.2 Standardised mean dierence (SMD)
If MDs are comparable but on dierent scales, it is not advisable to combine them in a meta-analysis in their current form. For example, the continuous outcome of physical functioning could be measured in rehabilitation studies using measures based on dierent questionnaires with dierent scoring methods [130]. To address this problem and allow studies to be pooled more meaningfully, we calculate standardised mean dierences (SMD) [8]: ˆ θi = ˆ µ1i− ˆµ2i Si
where Siis the estimated standard deviation of the mean dierence (see formula 1.1)
an assumes equal variances between groups.
When ˆθi is on the SMD scale, a good approximation of its variance is:
ˆ σi2 = n1i+ n2i n1in2i + ˆ θ2 i n1i+ n2i
Hedges [43] proved that the SMD measure has positive bias in studies with small sample sizes and therefore suggested applying a correction factor; the bias corrected ˆ
θi becomes Ji· ˆθi and ˆσ2i becomes Ji2· ˆσi2where Ji = 1−3/ (4 (n1i+ n2i− 2) − 1). The
correction factor has since become widely used and suggested in many meta-analysis texts [8, 19, 42, 128].
Continuous study outcomes are usually on the same scale, so the unstandardised mean dierence (MD) is more commonly used in practice [19, 88].
1.3.2 Summarising binary outcome data
In studies with a binary outcome, data can be presented in the form of a contingency table (e.g. table 1.1). From this data, we can derive measures that compare the event probability between groups such as the relative risk or odds ratio.
Event No event Total
Group 1 ai bi n1i = ai + bi
Group 2 ci di n2i = ci+ di
Total ai+ ci bi+ di Ni = ai+ bi+ ci + di
Table 1.1: Standard contingency table notations for a study i with a binary outcome
1.3.2.1 Relative risk (RR)
The risk of an event in groups one and two are ai/n1i and ci/n2i. The relative
[8]. This measure is transformed onto the log scale for meta-analysis to make the eect estimate within a given study conform approximately to a normal sampling distribution. Normality of study estimates is one of the assumptions in the standard meta-analysis models introduced in section 1.5.1. The log RR (ˆθi) and its variance
(ˆσ2
i) in each study are therefore:
ˆ θi = log ai/n1i ci/n2i ˆ σi2 = 1 ai − 1 n1i + 1 ci − 1 n2i ˆ
θi and ˆσi2 cannot be calculated in the above formulae when zero events are observed
in one or both groups. Throughout this thesis, a continuity correction is applied when required by adding 0.5 to ai, bi, ci and di [10]. Other methods are available for
dealing with zero events [113].
1.3.2.2 Odds ratio (OR)
The odds of an event in study groups one and two are ai/bi and ci/di. From this,
the odds ratio (OR) is (ai/bi) / (ci/di). As with RRs, ORs are transformed onto the
log scale: ˆ θi = log ai/bi ci/di ˆ σi2 = 1 ai + 1 bi + 1 ci + 1 di
The OR measure has superior statistical properties over RR [5, 19]. First, estimated ORs don't change when the cause and eect is reversed (i.e. if the study groups become the event of interest and vice versa). Second, logistic regression methodology of binary data has developed using the odds ratio measure, so this gives added convenience when conducting a logistic regression analysis [5]. Finally, ORs follow the normal distribution more closely than RRs when transformed to the log scale. These are perhaps some of the reasons why it is the most commonly used measure in systematic reviews of health research (see chapter 4 for a summary of outcome measures used in meta-analyses from Cochrane reviews).