1 Fixed-Effects and Random-Effects in Meta-Analysis

1.1 Two Datasets from Clinical Studies

We will use to data

  • Data for Cochrane Collaboration Logo: Binary Data
  • Clinical Studies on Amlodipine: Continuous Data

These data are shown below

Binary Data:

##           name ev.trt n.trt ev.ctrl n.ctrl
## 1     Auckland     36   532      60    538
## 2        Block      1    69       5     61
## 3        Doran      4    81      11     63
## 4        Gamsu     14   131      20    137
## 5     Morrison      3    67       7     59
## 6 Papageorgiou      1    71       7     75
## 7      Tauesch      8    56      10     71

Continuous Data:

##           study n.amlo mean.amlo var.amlo n.plac mean.plac var.plac
## 1  Protocol 154     46    0.2316   0.2254     48   -0.0027   0.0007
## 2  Protocol 156     30    0.2811   0.1441     26    0.0270   0.1139
## 3  Protocol 157     75    0.1894   0.1981     72    0.0443   0.4972
## 4 Protocol 162A     12    0.0930   0.1389     12    0.2277   0.0488
## 5  Protocol 163     32    0.1622   0.0961     34    0.0056   0.0955
## 6  Protocol 166     31    0.1837   0.1246     31    0.0943   0.1734
## 7 Protocol 303A     27    0.6612   0.7060     27   -0.0057   0.9891
## 8  Protocol 306     46    0.1366   0.1211     47   -0.0057   0.1291

1.2 Review of Fixed- and Random-Effects Models in M-a

1.2.1 Hypotheses and Effect Size

We commonly degfine hypothesis as follows \[ \begin{aligned} &H_0: \text{No difference, or efficacy of a new drug is not different} \\ &H_a: \text{There emerge difference, or efficacy of a new drug is different} \end{aligned} \]

Also, let \(\delta\) is a comparative function of the efficacy response measure in each treatment group. There are more than one option, such as drug-control or drug/control. Thus, We can state hypothesis by such a function \[ \begin{aligned} &H_0: \delta = 0 \\ &H_a: \delta > 0 \end{aligned} \]

Each study has its own treatment effect \(\widehat{\delta}_i\) and the within variance \(\widehat{\sigma}^2_i\), where \(i = 1,2,\dots,K\).

There are two meta-analysis approaches, namely fixed- and random-effects. While the assumption that the underlying \(\delta\) across all studies are the same is required, it is relaxed in random-effect modelling. This is, \(\{\widehat{\delta}_i|i=1,2,\dots,K\}\) is the estimate of \(\{\delta_i|i=1,2,\dots,K\}\). So, Random-effects meta-analysis model can incorporate both within-study variability and between-study variability.

1.2.2 Fixed-effects Model

Again, the underlying assumption of fixed-effects model is that all studies share a common overal effect size \(\delta\). Thus \[ \widehat{\delta}_i= \delta+\epsilon_i \tag{1.1} \] where \(\epsilon_i \sim N(0,\widehat{\delta}^2_i)\), so that \(\widehat{\delta}_i \sim N(\delta,\widehat{\sigma}^2_i)\).

The main idea is that the global \(\delta\) is then estimated by combining the individual estimates by some wrighting scheme in order to obtain the most precise estimate of the global effect. Let \(w_i\) be the weight of \(\widehat{\delta}_i\), we obtain \[ \widehat{\delta} = \sum_{i=1}^Kw_i\widehat{\delta}_i \tag{1.2} \] \[ \widehat{\sigma}^2 = Var(\widehat{\delta}) = \sum_{i=1}^Kw^2_i\widehat{\delta}^2_i \tag{1.3} \]

Then, it is easily to obtain confidence interval \[ \widehat{\delta} \pm1.96\sqrt{\widehat{\sigma}^2} \tag{1.4} \]

1.2.2.1 The Weighting Schemes

The important assumption is \(\sum_{i=1}^Kw_i =1\), and typical choices of \(w_i\) are

  1. Weighting by the number of studies as \[ w_i = \frac{1}{K} \tag{1.5} \]
  2. Weighting by the number of patients in each study as: \[ w_i = \frac{N_i}{N} \tag{1.6} \] where \(N_i\) is the number of patients in study \(i\), and \(N = \sum_{i=1}^KN_i\)
  3. Weighting by the number of patients from each study and each treatment as: \[ w_i = \frac{N_{iD}N_{iP}}{N_{iD}+N_{iP}}\frac{1}{w} \tag{1.7} \] where \(N_{iD}\) and \(N_{iP}\) are the number of patients in the new drug \((D)\) and placebo \((P)\) groups respectively at study \(i\). Also \(w = \sum w_i\)
  4. Weighting by the inverse variance \[ w_i=\frac{1}{\widehat{\sigma}^2_i}\frac{1}{w} \tag{1.8} \]

1.2.3 Random-Effects Meta-Analysis Model: DerSimonian Laird

1.2.3.1 Random-Effects Model

As mentioned above, the underlying true treatment effect \(\{\delta_{iR}|i=1,2,\dots,K\}\) are not the same. \[ \begin{aligned} &\widehat{\delta}_{iR} \sim N(\delta_{iR},\sigma^2_i) \\ &\delta_{iR} \sim N(\delta, \tau^2) \end{aligned} \tag{1.9} \]

So, the random-effects model can be described as follows \[ \widehat{\delta}_{iR} = \delta + \nu_i+\epsilon_i \tag{1.10} \] where \(\nu_i \sim N(0,\tau^2)\). Under assumption \(\nu_i \perp\!\!\!\!\perp \epsilon_i\), We can write \[ \widehat{\delta}_{iR} \sim N(\delta, \sigma^2_i+\tau^2). \tag{1.11} \]

Similar to fixed-effects model, we can obtain \[ \widehat{\delta}_R = \frac{\sum_{i=1}^Kw_{iR}\widehat{\delta}_{iR}}{\sum_{i=1}^Kw_{iR}} \tag{1.12} \] with S.E estimated as \[ SE(\widehat{\delta}_R) = \bigg[\frac{1}{\sum w_{iR}}\bigg]^{1/2} \tag{1.13} \] where \[ \widehat{w}_{iR} = \frac{1}{\widehat{\sigma}^2_i+\widehat{\tau}^2} \tag{1.14} \] \[ SE(\widehat{\delta}_R) = \sqrt{\frac{1}{\sum w_{iR}}} \tag{1.15} \]

The question that arise so far is how to estimate of \(\tau\). There are several methods to obtain \(\widehat{\tau}^2\). The most commonly used estimate is from DerSimonian and Laird, used the method of moments. \[ \widehat{\tau}^2 = \frac{Q-(K-1)}{U} \tag{1.16} \] if \(Q>K-1\), otherwise, \(\widehat{\tau}^2=0\) where \[ Q = \sum^Kw_i(\widehat{\delta}_i - \widehat{\delta})^2 \tag{1.17} \] \[ U = \sum^Kw_i-\frac{\sum^Kw_i^2}{\sum^Kw_i} \tag{1.18} \]

Note that statistic \(Q\) is used for testing the statistical significance of heterogeneity across studies.

1.2.3.2 Derivation of DerSimonian-Laird Estimator of \(\boldsymbol{\tau^2}\)

\[ \begin{aligned} Q &= \sum^Kw_i(\widehat{\delta}_i-\widehat{\delta})^2 = \sum^Kw_i\Big[(\widehat{\delta}_i-\delta)(\widehat{\delta}-\delta)\Big] \\ &= \sum^Kw_i(\widehat{\delta}_i-\delta)^2 - 2\sum^Kw_i(\widehat{\delta}_i-\delta)(\widehat{\delta}-\delta)+\sum^Kw_i(\widehat{\delta}-\delta)^2 \end{aligned} \]

Thus,

\[ \begin{aligned} E(Q) &= \sum^Kw_iE(\widehat{\delta}_i-\delta) - \Big(\sum^Kw_i\Big)E(\widehat{\delta}-\delta)^2 \end{aligned} \]

Expectation of middle term is zero since \[ \begin{aligned} E\Big[(\widehat{\delta}_i-\delta)(\widehat{\delta}-\delta)\Big] &= E\Big\{ E\Big[(\widehat{\delta}_i-\delta)(\widehat{\delta}-\delta)\big|\widehat{\delta}_i=d_i\Big] \Big\}\\ &=E\Big[(d_i-\delta)E(\widehat{\delta}-\delta)\Big] \\ &= 0 \end{aligned} \]

Now, \[ \begin{aligned} E(Q) &= \sum^Kw_iVar(\widehat{\delta}_i) - \Big(\sum^Kw_i\Big)E(\widehat{\delta}-\delta)^2\\ &= \sum^Kw_i(w_i^{-1}+\tau^2) - \Big(\sum^Kw_i\Big)Var\bigg(\frac{\sum^Kw_i\widehat{\delta}_i}{\sum^Kw_i}\bigg)\\ &= \sum^Kw_i(w_i^{-1}+\tau^2) - \Big(\sum^Kw_i\Big)\frac{\sum^Kw_i^2Var(\widehat{\delta}_i)}{\Big(\sum^Kw_i\Big)^2} \\ &= \sum^Kw_i(w_i^{-1}+\tau^2) - \Big(\sum^Kw_i\Big)\frac{\sum^Kw_i^2(w_i^{-1}+\tau^2) }{\Big(\sum^Kw_i\Big)^2}\\ &= \sum^Kw_i(w_i^{-1}+\tau^2) - \Big(\sum^Kw_i\Big)\Bigg[\frac{1}{\sum^Kw_i}+\frac{\tau^2\sum^Kw_i^2}{\Big(\sum^Kw_i\Big)^2}\Bigg]\\ &= (K-1) + \tau^2\bigg[\sum^Kw_i - \frac{\sum^Kw_i^2}{\sum^Kw_i} \bigg]. \end{aligned} \]

So, \[ \widehat{\tau}^2 = \frac{Q- (K-1)}{\sum^Kw_i-\frac{\sum^Kw_i^2}{\sum^Kw_i}} \]

1.3 Data Analysis in R