1. Meta-Analysis
Glass (Education Researcher, 5:3-8; 1976) suggested that there
were three types of analyses:
Primary analysis tests the hypothesis for which the data were collected.
Secondary analysis re-analyzes data already collected by someone else to answer
questions raised in primary analysis or to address new questions.
Meta-analysis combines evidence across studies.

2. Most often meta-analysis is based on summary measures derived from
the literature :
difference in risk
difference in means
odds ratios …
Meta-analyses appearing in the medical literature have most often dealt with summarizing treatment effects.

3. Some Formalization We have summary measures from k studies
and variances of these summary measures
We assume the are known constants

4. Fixed Effects Model

5. A test statistic
If the fixed effects model is true, then:
and

6. libname meta "&path\metaiml";
/*the pre-eclampsia data
Pre-eclampsia is high bp and urine protein
after 20 weeks of pregnancy */
proc print data=meta.preeclampsia;
run;

7. %let data=meta.preeclampsia;
%let effect=lnor;
%let var=varlnor;
%let outdata=tmp;
proc sql noprint;
select SUM(&effect*(1/&var))/SUM(1/&var) , 1/SUM(1/&var)
into :fixedmn,:fixedvar
from &data;
select sum( 1/&var*(&effect-&fixedmn)**2),
count(*)-1 into :qstat, :qstat_df
quit;
data &outdata;
fixedeffectsmean=&fixedmn;
fixedeffectsvar=&fixedvar;
qstat=&qstat;
q_df=&qstat_df;
q_pval=1-cdf("chisq",qstat,q_df);
lower95_ci=fixedeffectsmean-1.96*sqrt(fixedeffectsvar);
upper95_ci=fixedeffectsmean+1.96*sqrt(fixedeffectsvar);
run;
proc print data=tmp;run;

8. The fixed effects macro
%macro fixedeff(effect=,var=,data=&syslast,outdata=fixedresults);
/*======
Parameters:
effect=variable with estimated effect size
var=variable with variance of estimated effect size
data=data set containing the effect and its variance
outdata=name of dataset to contain results */
proc sql;
select SUM(&effect*(1/&var))/SUM(1/&var) , 1/SUM(1/&var)
into :fixedmn,:fixedvar
from &data;
select sum( 1/&var*(&effect-&fixedmn)**2),
count(*)-1 into :qstat, :qstat_df
quit;
data &outdata;
fixedeffectsmean=&fixedmn;
fixedeffectsvar=&fixedvar;
qstat=&qstat;
q_df=&qstat_df;
q_pval=1-cdf("chisq",qstat,q_df);
lower95_ci=fixedeffectsmean-1.96*sqrt(fixedeffectsvar);
upper95_ci=fixedeffectsmean+1.96*sqrt(fixedeffectsvar);
run;
%mend;

9. %fixedeff(effect=lnor,var=varlnor,data=meta.preeclampsia)
proc print data=fixedresults;
run;