How to impute study-specific standard deviations in meta-analyses of skewed continuous endpoints?

Table 2 Example of SAS code to simulate a meta-analysis on 15 datasets with 15 records generated from a Gamma distribution (alpha = 2 and beta = 5 vs alpha = 2 and beta = 7 for the treatment and control groups, respectively)

* q is the assigned library;

**************************************************;

* SIMULATIONS;

**************************************************;

%let s = gamma;

%let ndset = 15;

* Simulation of n = 15 dataset using the Gamma distributions;

%macro simul;

%do q = 1 %to &ndset;

%let seed = %sysevalf(1234567 + &q);

%let num_i = %sysevalf(&ndset);

%let v = %sysevalf(0 + &q);

data s&q;

k = &q;

%do i = 1%to &num_i;

var1 = 5*rangam(&seed,2);

var2 = 7*rangam(&seed,2);

output;

%end;

run;

%end;

* Dataset combining;

data simul_&s;

set

%do w = 1%to &ndset;

s&w

%end;

;

run;

%mend;

%simul;

* Descriptive statistics for each dataset;

ods trace on;

ods output summary = summary_&s;

proc means data = simul_&s mean std median q1 q3;

class k;

var var1 var2;

run;

ods trace off;

data summary_&s;

set summary_&s;

l1 = (var1_Median-var1_Q1)/0.6745;

l2 = (var2_Median-var2_Q1)/0.6745;

u1 = (var1_Q3-var1_Median)/0.6745;

u2 = (var2_Q3-var2_Median)/0.6745;

if l1 > u1 then MeSD_v1_cons=l1; else MeSD_v1_cons=u1;

if l2 > u2 then MeSD_v2_cons=l2; else MeSD_v2_cons=u2;

if l1 > u1 then MeSD_v1_prec=u1; else MeSD_v1_prec=l1;

if l2 > u2 then MeSD_v2_prec=u2; else MeSD_v2_prec=l2;

MeSD_v1_mean=(var1_Q3-var1_Q1)/1.349;

MeSD_v2_mean=(var2_Q3-var2_Q1)/1.349;

* Median difference;

MeD = var1_Median-var2_Median;

*1 conservative estimate of standard deviation;

a1sd = ((MeSD_v1_cons)**2)/NObs;

b1sd = ((MeSD_v2_cons)**2)/NObs;

MeSD_cons=sqrt(a1sd + b1sd);

*2 less conservative estimate of standard deviation;

a2sd = ((MeSD_v1_prec)**2)/NObs;

b2sd = ((MeSD_v2_prec)**2)/NObs;

MeSD_prec = sqrt(a2sd + b2sd);

*3 mean estimate of standard deviation;

a3sd = ((MeSD_v1_mean)**2)/NObs;

b3sd = ((MeSD_v2_mean)**2)/NObs;

MeSD_mean = sqrt(a3sd + b3sd);

*4 Interquartile range;

a4sd = ((var1_Q3-var1_Q1)**2)/NObs;

b4sd = ((var2_Q3-var2_Q1)**2)/NObs;

MeSD_iqr = sqrt(a4sd + b4sd);

* Mean difference and pooled standard deviation;

MD = var1_Mean-var2_Mean;

asd = ((var1_StdDev)**2)/NObs;

bsd = ((var2_StdDev)**2)/NObs;

SD = sqrt(asd + bsd);

drop l1 l2 u1 u2 asd bsd a1sd b1sd a2sd b2sd a3sd b3sd a4sd b4sd;

run;

*************************;

* Meta-analyses;

data sum_&s;

set summary_&s;

keep k NObs MeD MeSD_cons MeSD_prec MeSD_mean MeSD_iqr MD SD qq;

run;

*1 Median and conservative estimate of standard deviation;

data meta_&s.1;

set sum_&s;

model = "Conservative SD";

MDz = MeD;

SDz = MeSD_cons;

w = 1/(SDz**2);

MDw = MDz*w;

keep model k NObs MDz SDz w MDw;

run;

*2 Median and less conservative estimate of standard deviation;

data meta_&s.2;

set sum_&s;

model = "Less Conservative SD";

MDz = MeD;

SDz = MeSD_prec;

w = 1/(SDz**2);

MDw = MDz*w;

keep model k NObs MDz SDz w MDw;

run;

*3 Median and mean estimate of standard deviation;

data meta_&s.3;

set sum_&s;

model = "Mean SD";

MDz = MeD;

SDz = MeSD_mean;

w = 1/(SDz**2);

MDw=MDz*w;

keep model k NObs MDz SDz w MDw;

run;

*4 Median and interquartile range;

data meta_&s.4;

set sum_&s;

model = "IQR";

MDz = MeD;

SDz = MeSD_iqr;

w = 1/(SDz**2);

MDw = MDz*w;

keep model k NObs MDz SDz w MDw;

run;

*Mean and standard deviation (reference);

data meta_&s.5;

set sum_&s;

model = "Reference";

MDz = MD;

SDz = SD;

w = 1/(SDz**2);

MDw = MDz*w;

keep model k NObs MDz SDz w MDw;

run;

proc format;

value model

1 = "conservative SD"

2 = "Less Conservative SD "

3 = "Mean SD "

4 = "IQR"

5 = "Reference"

;

run;

*** Fixed effect model meta-analysis - Inverse of Variance method;

%macro meta_iv;

%do i = 1%to 5;

ods output Summary = somme&i;

proc means data = meta_&s&i sum;

var MDw w;

run;

data somme&i;

set somme&i;

model = &i;

format model model.;

theta = MDw_Sum/w_Sum;

se_theta = 1/(sqrt(w_sum));

lower = theta - (se_theta*1.96);

upper = theta + (se_theta*1.96);

mtheta = sqrt(theta**2);

CV = se_theta/mtheta;

keep model theta se_theta lower upper cv;

run;

%end;

data aaMeta_&s;

set

%do w = 1% to 5;

somme&w

%end;

;

run;

title "distr = &s - k = &ndset";

proc print; run;

%mend;

%meta_iv;