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# META procedure

Combines estimates from individual trials (R.W. Payne & S. Senn).

### Options

`PRINT` = string tokens Controls output (`estimates`, `overalltest`, `heterogeneity`, `confidenceplot`, `radialplot`, `monitoring`); default `esti`, `over`, `hete`, `conf` Which combined estimates to include in the output (`fixed`, `random`); default `fixe`, `rand` How to form the random estimate (`maxlikelihood`, `maxremllikelihood`, `moments`, `reml`); default `reml` Label for the x-axis of the confidence plot; default `'treatment effect'` How to set the sizes of symbols on the confidence plot (`equal`, `inversese`); default `inve` Probability level to use for the confidence intervals; default 0.95 Method to use for calculating the confidence interval for random estimates formed by maximum likelihood or REML (`approximate`, `profile`); default `prof` Type of test to use for the overall probability values (`greaterthan`, `lessthan`, `twosided`); default `grea` Maximum number of iterations to use with `RMETHOD` settings `maxlikelihood` and `maxremllikelihood`; default 100 Convergence criterion to use with `RMETHOD` settings `maxlikelihood` and `maxremllikelihood`; default 10-6

### Parameters

`ESTIMATES` = variates Supplies the estimates to combine Specifies the standard errors of the estimates Labels to use for each variate of `ESTIMATES` in the output Saves the combined estimate for each variate of `ESTIMATES`, treating them as fixed effects Saves the standard error of the combined estimate for each variate of `ESTIMATES`, treating them as fixed effects Saves the probability of the combined estimate for each variate of `ESTIMATES`, treating them as fixed effects Saves the combined estimate for each variate of `ESTIMATES`, treating them as random effects Saves the standard error of the combined estimate for each variate of `ESTIMATES`, treating them as random effects Saves the probability of the combined estimate for each variate of `ESTIMATES`, treating them as random effects Saves the statistic Q for the test of heterogeneity across trials Saves the degrees of freedom of the statistic Q Saves the random effect variance Saves lower values of the confidence interval Saves upper values of the confidence interval

### Description

`META` produces a combined estimate of a parameter that has been estimated in several separate trials, using the methods described in Chapter 4 of Whitehead (2002).

The estimates to be combined in the meta analyses must be supplied, in a variate, using the `ESTIMATES` parameter. Their standard errors must be supplied similarly, using the `SEESTIMATES` parameter. The `LABELS` parameter can supply a text with a label to be used for each estimate in the output; if this is not supplied, the default is to use the integers 1, 2 and so on.

Printed output is controlled by the `PRINT` option, with settings:

    `estimates` table with the estimates from the individual trials and the combined estimates, with standard errors and confidence intervals; overall tests using the combined estimates; test for heterogeneity of the estimates across trials (Whitehead 2002, Section 4.2.3); plot of the individual and combined estimates, and their confidence intervals; plot of the standardized estimates against their precision i.e. the reciprocal of the standard error (also known as a Galbraith plot; see Whitehead 2002 Section 7.3.2); monitoring information from the estimation with `RMETHOD` settings `maxlikelihood`, `maxremllikelihood` and `reml` (see below).

By default `PRINT=esti,over,hete,conf`.

The `SELECTION` option controls which combined estimates are presented in the output:

    `fixed` presents combined estimates formed assuming that the `ESTIMATES` are fixed (see Whitehead 2002, Section 4.2); presents combined estimates formed assuming that the `ESTIMATES` are random.

By default `SELECTION=fixe,rand`.

The method to use to form the combined estimates formed assuming that the `ESTIMATES` are random, is specified by the `RMETHOD` option:

    `maxlikelihood` estimates the variance component of the random effects using maximum likelihood (Hardy & Thompson 1996, also see Whitehead 2002, Section 4.3.8); estimates the variance component of the random effects by maximizing the REML likelihood (Whitehead 2002, Section 4.3.8); estimates the variance component of the random effects using the method of moments (DerSimonian & Laird 1986, also see Whitehead 2002, Section 4.3.3); estimates the variance component of the random effects using the `REML` directive (Whitehead 2002, Section 4.3.8).

The `maxremllikelihood` setting is based on the same criterion as the `reml` setting, but it programs the maximization explicitly, in a FOR loop. It thus provides an alternative to use if the `REML` directive experiences convergence problems. By default `RMETHOD=reml`.

The `CIMETHOD` option specifies how to calculate the confidence interval for a random estimate formed by maximum likelihood or REML. The default is to use profile likelihood (c.f. Hardy & Thompson 1996), but you can set `CIMETHOD=approximate` to use a Normal approximation instead.

The `XLABEL` option can supply a label for the x-axis of the confidence plots; the default is `'treatment effect'`. By default the sizes of the symbols used to plot the estimates on the confidence plots are inversely proportional to their standard errors, but you can set option `SMETHOD=equal` to use equal sizes. The `CIPROBABILITY` option specifies the probability level to use for the confidence intervals; (default 0.95 i.e. 95%). The `PRMETHOD` option specifies the type of test to use for the overall probability values: `greaterthan`, `lessthan` or `twosided`; the default is `greaterthan`.

The `MAXCYCLE` option specifies the maximum number of iterations to use with `RMETHOD` settings `maxlikelihood` and `maxremllikelihood` (default 100). The `TOLERANCE` option specifies the convergence criterion (default 10-6).

The combined estimate formed assuming that the `ESTIMATES` are fixed can be saved, in a scalar, using the `FIXEDESTIMATE` parameter. Its standard error and probability can be saved, each in a scalar, using `SEFIXEDESTIMATE` and `PRFIXEDESTIMATE` parameters. Similarly, the combined estimate formed assuming that the `ESTIMATES` are random can be saved using the `RANDOMESTIMATE` parameter, and the `SERANDOMESTIMATE` and `PRRANDOMESTIMATE` parameters can save its standard error and probability. The `QSTATISTIC` and `QDF` parameters can save the statistic Q for the test of heterogeneity across trials and its number of degrees of freedom, again in scalars. The `RVARIANCE` parameter can save a scalar containing the random effect variance. Finally, the `LOWER` and `UPPER` parameters can save variates containing the lower and upper values of the confidence interval.

Options: `PRINT`, `SELECTION`, `RMETHOD`, `XLABEL`, `CIPROBABILITY`, `CIMETHOD`, `PRMETHOD`, `MAXCYCLE`, `TOLERANCE`.

Parameters: `ESTIMATES`, `SEESTIMATES`, `LABELS`, `FIXEDESTIMATE`, `SEFIXEDESTIMATE`, `PRFIXEDESTIMATE`, `RANDOMESTIMATE`, `SERANDOMESTIMATE`, `PRRANDOMESTIMATE`, `QSTATISTIC`, `QDF`, `RVARIANCE`, `LOWER`, `UPPER`.

### Method

`META` uses the algorithms described in Chapter 4 of Whitehead (2002).

### Action with `RESTRICT`

`ESTIMATES`, `SEESTIMATES` or `LABELS` can be restricted to form combined estimates using only a subset of those in `ESTIMATES`.

### References

DerSimonian, R. & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177-188.

Hardy, R.J. & Thompson, S.G. (1996). A likelihood approach to meta-analysis with random effects. Statistics in Medicine, 15, 619-629.

Whitehead, A. (2002). Meta-Analysis of Controlled Clinical Trials. Wiley, Chichester.

Directives: `REML`, `VRESIDUAL`.
Procedures: `VRMETAMODEL`, `VMETA`.

Commands for: REML analysis of linear mixed models.

### Example

```CAPTION  'META example',!t('Recovery time after anaethesia',\
'(Whitehead 2002, Tables 4.24, 4.30 & 4.33)'); STYLE=meta,plain
TEXT     study
'Centre 1'  0.864  0.477
'Centre 2'  0.646  0.318
'Centre 3'  0.272  0.244
'Centre 4'  0.916  0.345
'Centre 5'  0.867  0.350
'Centre 6'  0.819  0.311
'Centre 7'  0.809  0.304
'Centre 8'  1.212  0.477
'Centre 9' -0.273  0.250 :
META     [PRINT=estimates,overalltest,heterogeneity,confidenceplot,\