**Introduction**

**Systematic Reviews**

**Effect sizes**

_{A}– M

_{B})/S],

_{A}is the mean reduction in blood pressure on treatment, A, and M

_{B}is the mean reduction in blood pressure on treatment, B and S is the pooled standard deviation from the both samples. The pooled standard deviation is defined as

_{A}-1)(s

_{A})

^{2}+ (n

_{B}-1)(s

_{B})

^{2}]/ (n

_{A}+ n

_{B}-2)}

^{1/2},

_{A }and s

_{B }are the standard deviations (2) from samples A and B, respectively, and n

_{A}and n

_{B}are the sample sizes for A and B, respectively. The value c(m) is a correction factor (3) for the bias in estimating the true ES due to small sample sizes. It is defined as

_{A}+ n

_{B}-2. One can see that as the sample size increases c(m) converges to the value 1. The variance of the study ES is sometimes estimated as (3)

_{A}+ n

_{B})/ n

_{A}n

_{B}] + (ES)

^{2}/(2T-3.94),

_{A}+ n

_{B}. The approximate constant 3.94 is a correction factor (3) in this type of calculation.

_{1}, ES

_{2},……..,ES

_{k}resulting in a mean effect size, ES

_{mean},

_{mean}= ∑

_{i=1,k}w

_{i}ES

_{i}/∑

_{i=1,k}w

_{i }for i=1,…,k.

_{i}for i = 1,…,k. These are weights. In a meta-analysis, not all studies should be weighted equally. For example, a study containing 500 subjects should be weighted more than a study containing 80 subjects due to the added information. Weights can be determined in other ways such as the amount of follow up in longitudinal studies. A study in which the endpoint is time dependent such as survival and where subjects are followed for 10 years should de weighted more than a study in which subjects are only followed for 5 years. The actual calculations for determining weights can be quite complicated. One common technique is to allow the weight to be the inverse variance of the effect size (3) which we defined above i.e,

_{i}= 1/variance(ES

_{i)}, i=1,…,k.

_{i}is the total number in study i, i = 1,…..,k, then N = N

_{1}+ N

_{2}+….+N

_{k.}We then define w

_{i}= N

_{i}/N for i = 1,…,k, as the weight of the ith study. One of the assumptions often made in a meta-analysis is that the between studies variation do not differ. Thus the weights defined above do not account for the between studies variation (4). This is called a fixed effects meta-analysis. To allow for the randomness between the studies or a random effects model we let

_{i}= 1/ [variance(ES

_{i }+ τ

^{2})], i=1,…,k,

_{mean}. The null hypothesis is that there is no difference between A and B in the combined analysis or H

_{0:}d = 0 versus the alternative hypothesis, H

_{A:}d ¹ 0. A 95% confidence interval (5) on the population mean is:

_{mean}-1.96τ, ES

_{mean}+1.96τ].

_{mean}= 0.086, t =0.109 for the fixed effects model and t= 0.243 for the random effects model. The fixed effects 95% confidence interval (CI) = (-0.127-0.299) and the random effects 95% CI = (-0.387-0.563). In both cases the interval covers 0 and there is no benefit to A from the 9 studies. One can also see that the random effects CI is wider than the fixed effects CI as it takes into account the added variation between the studies which we attribute to heterogeneity or some inherent differences among the studies. Clearly this is so as about half of the studies showed some superiority due to treatment A and the others showed an advantage due to treatment B.

*Figure 1. Forest plot of ES with their 95% confidence intervals.***Discussion of heterogeneity**

_{0}: Q=0 versus the alternative hypothesis, H

_{A}: Q¹ 0 implying there is significant heterogeneity. The critical value of the 9 studies for a chi square distribution on 9-1=8 df at an alpha level of 0.05 is 16.919. The calculated value of Q from our data is 38.897. Clearly the calculated Q is much greater than the critical value – over twice as much. Therefore we reject the null hypothesis and determine there is significant heterogeneity among our studies. This does not change our conclusion that there is no significant effect due to A. However, it does alert us that we should try to determine the source of the heterogeneity. Some possible sources of heterogeneity may be the clinical differences among the studies such as patient selection, baseline disease severity, management of inter current outcomes (toxicity), patient characteristics or others. One may also investigate the methodological differences between the studies such as mechanism of randomization, extent of withdrawals and lost to follow up or heterogeneity still can be due to chance alone and the source not easily detected. Quantifying heterogeneity can be one of the most troublesome aspects of meta-analysis. It is important because it can affect the decision about the statistical model to be selected (fixed or random effects). If significant heterogeneity is found then potential moderator variables can be found to explain this variability. It may require concentrating the meta–analysis on a subset of studies that are homogeneous within themselves. Let’s pursue this idea of heterogeneity one step further. Let’s examine the I

^{2 }index (6) which quantifies the extent of heterogeneity from a collection of effect sizes.

^{2 }index can easily be interpreted as the percentage of heterogeneity in the system or basically the amount of the total variation accounted for by the between studies variance, τ

^{2}. In our example I

^{2 }= 79.433% or 79.4% of the variance or heterogeneity is due to τ

^{2}or the between studies variance.

**Publication bias**

*a*+

*b x*precision, where a is the intercept and b is the slope). Since precision depends largely on sample size, small trials will be close to zero on the horizontal axis and vice versa for larger trials. The purpose is to not reject the null hypothesis that the line goes through the origin, i.e. intercept = 0. For our nine studies the funnel plot (not shown here) is symmetrical but rather flat as there are too few studies to result in a good plot. However, the symmetry is preserved as shown by the regression results. The test of the null hypothesis that the intercept equal to zero is not significant at the 0.05 alpha level indicates that the line goes through the origin. The actual p-value is 0.7747 indicating that there is strong evidence in the data to not support rejection of the null hypothesis of zero intercept. Thus there is no publication bias. Figure 2 is a funnel plot of 14 studies comparing two anti hypertensive therapies. The horizontal axis is the ES of the standard difference in means that we described above and the vertical axis is the standard error. One can see the symmetry. The studies with larger sample size or most precision are the dots or points located at the top of the graph and the studies with decreasing sample size or less precision (larger standard error) scatter towards the bottom of the plot. The diamond at the bottom of the plot is the means diamond for the overall meta-analysis comparing the two treatments and one sees it is close to zero. The statistical test for the intercept = 0 or the line goes through the origin is P = 0.730. The line is seen as the middle vertical line on the graph. The two angled lines give an outline of the pattern of points and one can see the funnel shape of the plot. Detailed discussions of funnel plots and test of symmetry can be found throughout the literature (8,9) pertaining to meta-analyses.

*Figure 2. Funnel Plot of 14 studies with the effect size (Std diff in means) on the horizontal and the standard error on the vertical.*