A
meta-analysis that simultaneously examines multiple predictors of the relationship
between two variables is called meta-regression. It is roughly analogous to linear
regression. The IVs are moderator
variables. That is, they are the
variables you believe may predict the size of the effect. The DV is the effects sizes that you are
meta-analyzing.
The
general approach is to
1)
obtain the studies (not discussed here)
2)
calculate the effect sizes and variances by hand (not discussed
here),
3)
enter the data into SAS,
4)
conduct the analyses without predictor variables, to estimate
the overall effect,
5)
examine whether there is significant heterogeneity of the
effect (not discussed here), and
6)
examine moderators.
The
DV is the effect size that characterizes the relationship of X and Y. Values of the DV are usually z (the Fisher transform of r), the log-odds (the natural log of the
odds ratio) or d. The variance of each effect size is
calculated outside of SAS in the usual way.
To
enter the data into SAS, follow the format below.
DATA test1;
INPUT trial ln_or
est year year_m cohort
design country mos;
CARDS;
1 0.348254219 0.121887759 2003 3 1
1 2 2
3 0.055663868 0.007594335 1999 -1 0
1 1 1
4 -0.968310623
0.000455847 2003 3
0 1 3 1
8
-0.199289901 0.001559349 2002
2 0
1 2 1
9
-0.768254653 0.078953618 2003
3 0
1 2 1
32 0.284104251 0.199718382 1998 -2 0
1 2 2
33 0.071692745 0.020341215 2001 1 0
1 2 1
39 0.435497395 0.067556427 1991 -9 0
1 1 2
40
-0.947515921 0.019046944 1996
-4 0 1
3 1
42 0.445625551 0.040364599 2000 0 0
0 1 2
46 0.249910566 0.001183328 2003 3 0
1 1 1
; run;
trial is just an arbitrary name for the study. BUT, the
list of studies must be sorted in ascending numerical order, otherwise SAS
messes up the stats. This is an undocumented bug.
ln_or is the name for the
effect size (but you can choose any name). In this case, it is the log
odds.
est is the variance for
the effect size. Note that this is, in essence, the inverse of the sample
size. This variable must be called est.
The
syntax below will yield an estimate of the pooled effect size, log odds in this
example. This is a fixed effects model.
PROC MIXED method = ml data=test1;
class trial;
model ln_or = /s cl;
repeated / group = trial;
parms/ parmsdata=test1 eqcons
= 1 to 11;
run;
s requests an estimate of the pooled effect size and cl asks for confidence intervals.
The
parms command tells SAS where to get the variances;
it looks for the variable named est.
The
output:
Effect
Estimate Error
DF t Value Pr >
|t| Alpha
Lower Upper
Intercept
-0.5197 0.01585
10 -32.78
<.0001 0.05
-0.5550 -0.4843
The
intercept is the estimate of the pooled effect size. Thus, the pooled log
odds is -.52 and is significantly different from zero. (Recall that the natural log of OR = 1 is
zero.)
If
you want to conduct a meta-regression, then start adding predictors to the
model. To examine moderators, one must have heterogeneity of variance, a
topic not addressed here but easily understood from relevant texts on
meta-analysis.
Note
that the predictors will start changing the intercept and you have to think
carefully about whether the intercept will still be interpretable. For
example, using continuous covariate such as year of study will then
estimate the intercept holding year constant at zero (a useless
estimate). So mean centering some variables or other transformations may
make sense.
Another
example is using a categorical covariate. The intercept will now be
estimated for the highest level of the categorical covariate. Next, SAS
will estimate the relative effect size for the other levels of the IV, how many
times larger or smaller the other effect sizes are than the referent.
PROC MIXED method = ml data=test1;
class trial
country;
model ln_or = country mos /s cl;
repeated / group = trial;
parms/ parmsdata=test1 eqcons
= 1 to 11;
run;
Type 3 Tests of Fixed Effects
Num Den
Effect
DF DF F Value
Pr > F
country
2 7
532.87 <.0001
mos
1 7
4.39 0.0743
Effect country Estimate
Error DF t Value Pr > |t|
Alpha Lower Upper
Intercept
-1.2613 0.1416 7
-8.91 <.0001 0.05
-1.5961 -0.9265
country
1 1.1886
0.03819 7 31.12
<.0001 0.05 1.0983
1.2789
country
2 0.7813
0.04299 7 18.18
<.0001 0.05 0.6797
0.8830
country
3
0 .
. .
.
.
. .
mos
0.2935 0.1400
7 2.10 0.0743
0.05 -0.03762 0.6246
In
this output, country moderates the effect size but mos does not. The
estimate of the log odds is -1.2613 for country 3 (
For
the log-odds ratio, multiply the log (estimate) for the intercept by the log
(estimate) for any level of a categorical covariate to find the estimate of the
OR for that level.
For
z, add the estimate for the intercept
to the estimate) for any level of a categorical covariate to find the estimate
of the z for that level.
--------------------------------------
Most
statisticians who write on this topic argue strongly for moving beyond fixed
effects models, as shown above, to random effects models.
--------------------------------------
Easy
to read intro material:
Wolf, F.M. (1986). Meta-Analysis: Quantitative Methods for
Research Synthesis.
A
bit harder but definitive:
Cooper, H., & Hedges L. V.. (1994). The handbook of research synthesis.
Over
the top unless you have a serious stats background, but it gives relatively clear
SAS code:
Normand, S. L. (1999). Meta-analysis: formulating, evaluating,
combining, and reporting. Statistics in Medicine, 18(3), 321-59.
van Houwelingen, H. C., Arends, L. R.,
& Stijnen, T. (2002). Advanced methods in meta-analysis: multivariate
approach and meta-regression. Statistics
in Medicine, 21(4), 589-624.
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