/*========================================================= File name: 5_k_independent_parameters.txt Written by: Karl L. Wuensch, WuenschK@ECU.edu Date: 28-August-2012. =========================================================== To test the null hypothesis that k independent parameters all have the same value, one can use the chi-square test of heterogenity that is commonly used in meta-analysis. See Fleiss's article on meta-analysis for details. ---------------------------------------------------------- Fleiss, JL. The statistical basis of meta-analysis. Statistical Methods in Medical Research, 1993, 2: 121-145. ---------------------------------------------------------- For this method, one needs k independent effect size estimates (called Y) along with their variances (i.e., the square of their standard errors). Each one of those estimates appears on a separate row in the data file. The INPUT data are: Group - consecutive integers starting with 1 Y - the measure of effect size seY - the SE of Y n - the sample size Note - a string variable with a note. In some cases, Y and se.Y may have to be computed from other variables -- e.g., when testing the heterogeneity of correlation coefficients, we compute Y = the Fisher r-to-z transformed value of Pearson r. Users can replace the data lines (between "CARDS" and "%MACRO ComputeQ") with their own data. *Create macro for computing Q. */ %MACRO ComputeQ; proc means noprint; var Group w wy; output out=sums sum=x1 sumW sumWY max=S x2 x3; data sums2; set sums; Do Group=1 to S; sw=sumw; swy=sumWY; output; end; keep group S sw swy; data Qgroup; merge xyzzy sums2; by Group; Ybar=swy/sw; Qgroup=w*(Y-Ybar)**2; proc means noprint; var Group Qgroup; output out=Q sum=nix1 Q max=S nix2; data p; set Q; df=S-1; p=1-probchi(Q,df); proc print; var Q df p; ID; %MEND ComputeQ; * Test the heterogeneity of the INTERCEPTS in Table 2; data xyzzy; input Group Y seY n Note \$35.; varY=seY*seY; w=1/varY; wy=w*y; CARDS; 1 142.011 10.664 22 Intercept, Burbank 2 148.053 11.142 47 Intercept, Lancaster 3 144.038 18.250 17 Intercept, Long Beach 4 130.445 10.228 56 Intercept, Glendora ; %ComputeQ title 'Comparing Intercepts, k Independent Groups'; run; * Now test the heterogeneity of the SLOPES in Table 2; data xyzzy; input Group Y seY n Note \$35.; varY=seY*seY; w=1/varY; wy=w*y; *GROUPS MUST BE CONSECUTIVE INTEGERS STARTING WITH 1; CARDS; 1 4.179 1.105 22 Slope, Burbank 2 3.709 1.177 47 Slope, Lancaster 3 3.749 1.866 17 Slope, Long Beach 4 5.689 1.044 56 Slope, Glendora ; %ComputeQ title 'Comparing Slopes, k Independent Groups'; run; /**** Test equivalence of correlations *** In the case where one has k independent Pearson correlations, Y = r-prime, the Fisher r-to-z transformed correlation. The variance of Y = 1/(n-3). We illustrate using correlations between FHEIGHT and FWEIGHT in 4 different areas -- see Table 1 in the article. */ data xyzzy; input Group r n; Y = 0.5*log(abs((1+r)/(1-r))); varY=1/(n-3); w=1/varY; wy=w*y; *GROUPS MUST BE CONSECUTIVE INTEGERS STARTING WITH 1; CARDS; 1 .628 24 2 .418 49 3 .438 19 4 .589 58 ; %ComputeQ title 'Comparing Correlation Coefficients, k Independent Groups'; run; /* Below, we run this code on the 2 independent correlations * between FWT and MHT in Lancaster and in Glendora. * The Q-value we obtain below should be equal to the square of * the z-value (2.63183**2 = 6.9265) obtained earlier and the one-tailed p from Q * should be identical to the two-tailed p (.00849) from z. */ data xyzzy; input Group r n; Y = 0.5*log(abs((1+r)/(1-r))); varY=1/(n-3); w=1/varY; wy=w*y; CARDS; 1 -.181 49 2 .330 58 ; %ComputeQ title 'Comparing Correlation Coefficients, k=2 Independent Groups'; run;