The maximum likelihood estimates to model parameters are obtained by maximizing the likelihood function (

3). We propose a method to use the existing computing packages to maximize the likelihood function and obtain the MLEs of the model parameters. The core step here is to treat the reported intercepts and slopes as observations for outcome variable

*Y* for these studies from which only these summary statistics are available. This approach is justified based on the fact that the estimated intercepts and slopes are unbiased estimates of the relevant study-specific intercepts and slopes and asymptotically normally distributed. More specifically, for study

*i*(1≤

*i* ≤

*s*), if the only published summary statistic is the estimated slope

_{1}_{i} with the estimated variance

, then we treat the estimated slope as an observed outcome from an individual in study

*i* and let

*y*_{1}_{i} =

_{1}_{i}. Given

*β*_{1}_{i},

_{1}_{i} is unbiased estimate of

*β*_{1}_{i} and asymptotically normal with estimated variance

. Across the studies,

*β*_{1}_{i} is assumed normal with mean

*β*_{1} and variance

. It follows that

where

*u*_{i}_{1} = 0,

*x*_{i}_{1} = 1,

*e*_{1}_{i} is distributed as

,

*ε*_{1}_{i} is distributed as

, and

*e*_{1}_{i} and

*ε*_{i}_{1} are independent. Model (

4) is exactly the same as model (

2) when

*u*_{ij} = 0 and

*x*_{ij}= 1 for study

*i*(1≤

*i* ≤

*s*) as if there was only one observation from one individual for the outcome of the study.

If published summary statistics contain both estimates to the intercept and slope from study

*i*(1≤

*i* ≤

*s*), then both the slope estimate

_{1}_{i} and a linear combination of

can be treated as two independent observations for the outcome variable from two different individuals in the study. To see this, we let

This linear combination of

in

*y*_{i}_{2} is based on the well-known technique of Gram-Schmidt orthogonalization

^{24} so that, conditional on

,

*y*_{i}_{1} from (

4) and

*y*_{i}_{2} are not only uncorrelated but also with the same variance

. All these make

*y*_{i}_{1} and

*y*_{i}_{2} satisfy model (

1) for study

*i*. Across the studies,

follows a bivariate normal distribution with mean vector

and covariance matrix

*A*. It follows that

and

follows a bivariate normal distribution with mean vector 0 and covariance matrix

*A*, and is independent of

*ε*_{i}_{2}. Therefore, model (

5) and (

4) together are exactly the same as model (

2) for study

*i*(1≤

*i* ≤

*s*) as if there were only two independent observations from two individuals for the outcome of the study. Mathematically, the information from the reported intercept and slope

is exactly the same as that from

because of the one-to-one linear transformation. Statistically, these two vectors are equivalent too because they provide exactly the same contribution to the likelihood function for the entire meta-analysis. This same reason also warrants the invariance of the resulting statistical inferences when different linear one-to-one transformations are used to transform data. The reason that

is used for study

*i* is the fact that it conforms to model (

2) with the rest of studies and therefore facilitates the use of standard computing packages to implement the meta-analysis.

When

,

*i* = 1,2,…,

*s*, are assumed known, the maximization to the likelihood function (

3) can be easily implemented in SAS (R) MIXED procedure

^{25} through appropriate manipulation of the input SAS data set. In order for SAS (R) MIXED procedure

^{25} to implement model (

2) simultaneously for the studies with individual patient data and for the other studies that have only the summary statistics available, we need to first prepare an augmented SAS data set which contains all available data from both studies with individual patient data and those with only summary statistics. More specifically, for studies that individual patient data are available, this augmented data set (called AUGDATA) contains the study identification, the subject identification within each study, the response variable, and two covariates

*u*_{ij} (i.e., = 1) and

*x*_{ij}. For studies that only summary statistics are available, this augmented data set contains a single observation of these variables from a single individual with

*u* = 0 and

*x* = 1 as given in (

4) for a study that reported only the treatment effect (i.e., the slope) and two observations from two individuals as given by (

4) and (

5) for a study that reported both the intercept and slope estimates. Similar data codes have also been used by Houwelingen et al.

^{26} when SAS (R) MIXED procedure

^{25} was used to perform the traditional meta-analyses when only summary statistics were available from all trials. Another SAS data set (called COVAR) is also needed for SAS (R) MIXED procedure

^{25} to implement model (

2) simultaneously for the studies with individual patient data and for the others that have only the summary statistics available. This data set contains either the initial estimates or the permanent estimates (i.e., for these studies with only summary statistics available) for the variance and covariance parameters involved in the random components of the model. More specifically, the first three observations of COVAR should be the initial estimates to the three parameters in the covariance matrix

of the random vector

. Next observations of COVAR should be all within-study variances

’s from the error term

*ε*_{ij}’

*s* in model (

2). Notice that for these studies (i.e., the first

*s* studies) with only summary statistics available,

is permanently estimated by (or treated as known as)

in this analysis, whereas for these studies with individual patient data, initial estimates for these variance parameters can be specified for the maximization program of SAS (R) MIXED procedure

^{25} to proceed. With these data sets AUGDATA and COVAR, the following SAS codes can be used to implement the proposed meta-analysis (in italics):

Proc mixed data=AUGDATA method=; class study id;

Model y=u x/noint s cl ddfm=;

Random u x/sub=study type=un;

Repeated study/sub=id group=study;

Parms/parmswdata=COVAR eqcon=4 to 3+s;

Run;

In the above SAS code, the MODEL statement fits model (

2) when both reported summary statistics and individual patient data are used in the analysis. Option METHOD= specifies either the MLE or the restricted maximum likelihood estimation (REML) in the maximization of function (

3). Option NOINT makes sure that model (

2) is fitted with covariates

*u* and

*x* and without an additional intercept. Option CL gives the 95% confidence interval for the expected slope and intercept across the studies. Option DDFM specifies the method for computing the denominator degrees of freedom for the test of fixed effects (e.g., SATTERTH, KR). The RANDOM statement specifies the fact that the study-specific vector of intercept and slope

follows a bivariate normal distribution across studies. The REPEATED statement specifies the subject-specific error term in model (

2) which allows different variances across different studies. The PARMS statement specifies the initial values for the covariance and variance parameters in model (

2) while treating the variances of the summary statistics for the studies with only summary statistics available as permanently given by their reported estimates. Alternatively, instead of providing initial estimates to variance/covariance estimates in COVAR, one can also estimate the fixed effects and the variance/covariance parameters with repeated calls to SAS (R) MIXED procedure until they converge. This iterative approach has also recently been used in meta-analyses combining entire survival curves over multiple studies

^{27}^{,}^{28}.