preserve.

set printback=on.

****************************************************************************

*** Meta-Analysis: Fixed and Random Effects Models

*** Valentim R. Alferes (University of Coimbra, Portugal)

*** valferes@fpce.uc.pt

**

** This syntax does a meta-analysis on a set of studies comparing two

** independent means. It produces results for both fixed and random effects

** models, using Cohen's d statistic, with or without Hedges' correction.

**

** The user has TEN MODES FOR ENTERING SUMMARY DATA (see PART 1):

**

** Mode 1 - Study No., N1, M1, SD1, N2, M2 SD2.

** Mode 2 - Study No., N1, M1, N2, M2 SD_POOL.

** Mode 3 - Study No., Direction of Effect, Difference, N1, SD1, N2, SD2.

** Mode 4 - Study No., Direction of Effect, Difference, N1, N2, SD_POOL.

** Mode 5 - Study No., DF, M1, SD1, M2 SD2.

** Mode 6 - Study No., DF, M1, M2, SD_POOL.

** Mode 7 - Study No., Direction of Effect, DF, Difference, SD1, SD2.

** Mode 8 - Study No., Direction of Effect, DF, Difference, SD_POOL.

** Mode 9 - Study No., Direction of Effect, N1, N2, T_OBS.

** Mode 10 - Study No., Direction of Effect, DF, T_OBS.

**

** There are no limits for the number of studies to be analyzed and the user

** can input data simultaneously in the ten modes or enter all the studies

** only in one mode. In the modes not used, the lines of data have to be

** cleared, but not the corresponding command lines.

**

** If the input are means, the program assumes that Group 1 is the

** experimental or focus group and Group 2 is the control or comparison

** group.

**

** If the input are differences between group means or observed Ts, they

** are registered in absolute values (DIF=|M1-M2| or T_OBS=|Tobs|) and the

** user specifies the direction of effect in a different variable (DIRECT):

** +1 (if the effect is in the expected direction: Group 1 mean greater

** than Group 2 mean) and -1 (if the effect is reversed: Group 1 mean lesser

** than Group 2 mean).

**

** When the input are degrees of freedom, the syntax asssumes equal Ns if df

** are even, and N2=N1-1 if they are odd.

**

** When the data are selected from two contrasting ANOVA treatments, the

** user can input them in modes 2 or 4 and let the pooled standard deviation

** (SD_POOL) equals the squared root of the original ANOVA MS Error.

**

** By default the measure of effect size is Hedges' correction to Cohen's d.

** If you want to use d statistic without correction, you can change the

** default in the corresponding command line.

**

** The OUTPUT is organized in nine tables:

**

** Table 1 - User's data

**

** Table 2 - Program imputations

**

** Table 3 - Individual T Tests and observed power

** - N1, N2, degrees of freedom (DF), difference between group means (DIF),

** observed T (T_OBS), two-tailed probability (P_TWO), and one-tailed

** probability (P_ONE);

** - Alfa (ALFA), Harmonic N (N_HARM), noncentrality parameter (NCP), and

** observed power (OPOWER).

** [for algorithm, see Borenstein et al., 2001]

**

** Table 4 - Measures of Effect Size and Nonoverlap

** Measures of effect size:

** - Cohen's d (D);

** [Cohen, 1988, p. 20]

** - Hedges' correction (D_H);

** [D_H = d, in Hedges & Olkin, 1985; D_H = d*, in Hunter & Schmidt,

** 1990; see Cortina & Nouri, 2000, p. 9];

** - r point biserial (R);

** - Squared r point biserial (R2);

** - Binomial Effect Size Display (BESD_LO and BESD_UP).

** [see formulas in Rosenthal et al. 2000, pp. 8-19]

**

** Measures of nonoverlap:

** - U1 (percent of nonoverlap between the two distributions);

** - U2 (the highest percent in Group 1 that exceeds the same lowest

** percent in Group 2);

** - U3 (percentile standing = percentile of the Group 2 distribution

** corresponding to the 50th percentile of Group 1 distribution);

** [see formulas in Cohen, 1988, pp. 21-23]

**

** Table 5 - Non weighted effect size - Descriptive statistics

** - Number of studies (NSTUDIES), Cohen's d (D), and Hedges' correction

** (D_H) (minimun, maximun, mean, sem, and sd).

**

** Table 6 - Fixed effects model

** - Weighted average effect size (EF_SIZE), VARIANCE, and standard error

** (SE);

** - z Test (z), two-tailed probability (P_TWO), and one-tailed probability

** (P_ONE);

** - Confidence level (CL), and lower (CI_LOWER) and upper (CI_UPPER)

** interval confidence limits.

** [see formulas in Shadish & Haddock, 1994, pp. 265-268]

**

** Table 7 - Chi-square Test for homogeneity of effect size:

** - Q statistic, degrees of freedom (K), and two-tailed probability

** (P_CHISQ)

** [see formula in Shadish & Haddock, 1994, p. 266]

**

** Table 8 - Random Variance Component

** - V0 [see formula in Lipsey & Wilson, 2001, p. 134].

**

** Table 9 - Random effects model

** - Weighted average effect size (EF_SIZE), VARIANCE, and standard error

** (SE);

** - z Test (z), two-tailed probability (P_TWO), and one-tailed probability

** (P_ONE);

** - Confidence level (CL), and lower (CI_LOWER) and upper (CI_UPPER)

** interval confidence limits.

** [see formulas and procedures in Lipsey & Wilson, 2001, pp. 134-135]

**

** For calculating observed power of individual studies, the syntax assumes

** alfa = 0.05. For calculating the confidence interval of weighted effect

** sizes, the syntax assumes confidence level = 95%. If you want, you can

** modify these values in the corresponding lines (see PART 2).

**

** After running the syntax, the user can have access to Tables 2, 3 and 4

** in SPSS active file, so that he may handle the data for other meta-

** analytic procedures based on different effect size measures or exact

** probabilities (see other syntaxes in this site).

**

** In the example, we have 20 studies and we have used the ten input data

** modes.

set printback=off.

Table 1 - User's data
	Study	Direct	N1	N2	DF	M1	M2	DIF	SD1	SD2	SD_POOL	T_OBS
1	1	1	17	16	.	7.46	6.23	.	1.98	2.45	.	.
2	2	1	15	15	.	5.34	4.47	.	2.14	2.51	.	.
3	3	-1	14	16	.	7.32	8.23	.	.	.	2.67	.
4	4	1	23	27	.	6.20	4.47	.	.	.	2.21	.
5	5	1	10	11	.	.	.	1.04	3.04	2.98	.	.
6	6	-1	12	12	.	.	.	2.25	2.63	2.21	.	.
7	7	-1	34	33	.	.	.	1.32	.	.	2.44	.
8	8	1	20	20	.	.	.	1.25	.	.	3.09	.
9	9	1	.	.	34	7.46	6.33	.	1.69	2.98	.	.
10	10	-1	.	.	33	5.34	5.46	.	2.94	2.31	.	.
11	11	1	.	.	27	7.76	5.29	.	.	.	2.77	.
12	12	1	.	.	28	6.30	4.21	.	.	.	2.41	.
13	13	1	.	.	40	.	.	3.07	1.77	2.87	.	.
14	14	-1	.	.	37	.	.	2.11	2.62	2.21	.	.
15	15	-1	.	.	23	.	.	2.22	.	.	1.88	.
16	16	1	.	.	34	.	.	3.17	.	.	1.94	.
17	17	1	20	20	.	.	.	.	.	.	.	4.74
18	18	-1	14	15	.	.	.	.	.	.	.	3.17
19	19	1	.	.	54	.	.	.	.	.	.	5.46
20	20	-1	.	.	49	.	.	.	.	.	.	2.27

Table 2 - Program imputations
	Study	Direct	N1	N2	DF	M1	M2	DIF	SD1	SD2	SD_POOL	T_OBS
1	1	1	17	16	31	7.46	6.23	1.23	1.98	2.45	2.22	1.59
2	2	1	15	15	28	5.34	4.47	.87	2.14	2.51	2.33	1.02
3	3	-1	14	16	28	7.32	8.23	-.91	.	.	2.67	-.93
4	4	1	23	27	48	6.20	4.47	1.73	.	.	2.21	2.76
5	5	1	10	11	19	.	.	1.04	3.04	2.98	3.01	.79
6	6	-1	12	12	22	.	.	-2.25	2.63	2.21	2.43	-2.27
7	7	-1	34	33	65	.	.	-1.32	.	.	2.44	-2.21
8	8	1	20	20	38	.	.	1.25	.	.	3.09	1.28
9	9	1	18	18	34	7.46	6.33	1.13	1.69	2.98	2.42	1.40
10	10	-1	18	17	33	5.34	5.46	-.12	2.94	2.31	2.65	-.13
11	11	1	15	14	27	7.76	5.29	2.47	.	.	2.77	2.40
12	12	1	15	15	28	6.30	4.21	2.09	.	.	2.41	2.37
13	13	1	21	21	40	.	.	3.07	1.77	2.87	2.38	4.17
14	14	-1	20	19	37	.	.	-2.11	2.62	2.21	2.43	-2.71
15	15	-1	13	12	23	.	.	-2.22	.	.	1.88	-2.95
16	16	1	18	18	34	.	.	3.17	.	.	1.94	4.90
17	17	1	20	20	38	.	.	.	.	.	.	4.74
18	18	-1	14	15	27	.	.	.	.	.	.	-3.17
19	19	1	28	28	54	.	.	.	.	.	.	5.46
20	20	-1	26	25	49	.	.	.	.	.	.	-2.27

TABLE 3 - Individual T Tests and observed power
	Study	Direct	DIF	DF	T_OBS	P_TWO	P_ONE	ALFA	N_HARM	NCP	OPOWER
1	1	1	1.23	31	1.59	.1218	.0609	.0500	16.4848	1.5908	.3381
2	2	1	.87	28	1.02	.3157	.1579	.0500	15.0000	1.0215	.1668
3	3	-1	-.91	28	-.93	.3597	.1798	.0500	14.9333	.9313	.1466
4	4	1	1.73	48	2.76	.0082	.0041	.0500	24.8400	2.7588	.7713
5	5	1	1.04	19	.79	.4386	.2193	.0500	10.4762	.7912	.1168
6	6	-1	-2.25	22	-2.27	.0334	.0167	.0500	12.0000	2.2689	.5829
7	7	-1	-1.32	65	-2.21	.0304	.0152	.0500	33.4925	2.2138	.5875
8	8	1	1.25	38	1.28	.2086	.1043	.0500	20.0000	1.2792	.2386
9	9	1	1.13	34	1.40	.1707	.0854	.0500	18.0000	1.3994	.2747
10	10	-1	-.12	33	-.13	.8944	.4472	.0500	17.4857	.1337	.0519
11	11	1	2.47	27	2.40	.0236	.0118	.0500	14.4828	2.3995	.6382
12	12	1	2.09	28	2.37	.0246	.0123	.0500	15.0000	2.3750	.6304
13	13	1	3.07	40	4.17	.0002	.0001	.0500	21.0000	4.1723	.9826
14	14	-1	-2.11	37	-2.71	.0101	.0051	.0500	19.4872	2.7113	.7519
15	15	-1	-2.22	23	-2.95	.0072	.0036	.0500	12.4800	2.9498	.8064
16	16	1	3.17	34	4.90	.0000	.0000	.0500	18.0000	4.9021	.9974
17	17	1	.	38	4.74	.0000	.0000	.0500	20.0000	4.7400	.9961
18	18	-1	.	27	-3.17	.0038	.0019	.0500	14.4828	3.1700	.8633
19	19	1	.	54	5.46	.0000	.0000	.0500	28.0000	5.4600	.9997
20	20	-1	.	49	-2.27	.0276	.0138	.0500	25.4902	2.2700	.6047

Table 4 - Measures of effect size and nonoverlap
	Study	Direct	D	D_H	R	R2	BESD_LO	BESD_UP	U1	U2	U3
1	1	1	.5541	.5406	.2747	.0755	.3626	.6374	35.8	60.9	71.0
2	2	1	.3730	.3629	.1896	.0359	.4052	.5948	25.8	57.4	64.5
3	3	-1	-.3408	-.3316	-.1733	.0300	.5867	.4133	23.8	43.2	36.7
4	4	1	.7828	.7705	.3699	.1369	.3150	.6850	46.7	65.2	78.3
5	5	1	.3457	.3319	.1786	.0319	.4107	.5893	24.1	56.9	63.5
6	6	-1	-.9263	-.8943	-.4355	.1896	.7177	.2823	52.6	32.2	17.7
7	7	-1	-.5410	-.5347	-.2648	.0701	.6324	.3676	35.1	39.3	29.4
8	8	1	.4045	.3965	.2032	.0413	.3984	.6016	27.6	58.0	65.7
9	9	1	.4665	.4561	.2334	.0545	.3833	.6167	31.1	59.2	68.0
10	10	-1	-.0452	-.0442	-.0233	.0005	.5116	.4884	3.5	49.1	48.2
11	11	1	.8917	.8667	.4192	.1758	.2904	.7096	51.2	67.2	81.4
12	12	1	.8672	.8438	.4095	.1677	.2953	.7047	50.2	66.8	80.7
13	13	1	1.2876	1.2633	.5507	.3032	.2247	.7753	64.9	74.0	90.1
14	14	-1	-.8686	-.8509	-.4071	.1657	.7036	.2964	50.3	33.2	19.3
15	15	-1	-1.1809	-1.1419	-.5239	.2745	.7620	.2380	61.6	27.7	11.9
16	16	1	1.6340	1.5977	.6435	.4141	.1782	.8218	73.9	79.3	94.9
17	17	1	1.4989	1.4691	.6096	.3716	.1952	.8048	70.7	77.3	93.3
18	18	-1	-1.1780	-1.1450	-.5208	.2712	.7604	.2396	61.5	27.8	11.9
19	19	1	1.4592	1.4389	.5964	.3557	.2018	.7982	69.7	76.7	92.8
20	20	-1	-.6358	-.6261	-.3085	.0952	.6542	.3458	39.9	37.5	26.2

Table 5 - Non weighted effect size Descriptive statistics: Cohen's d and Hedges' correction
	D	D_H
N	20	20
Minimum	-1.1809	-1.1450
Maximum	1.6340	1.5977
Mean	.242433	.238464
Std. Error of Mean	.2042005	.1995043
Std. Deviation	.9132124	.8922102

Table 6 - Fixed effects model: Weighted average effect size, z test, and confidence interval
	NSTUDIES	EF_SIZE	VARIANCE	SE	Z	P_TWO	P_ONE	CL	CI_LOWER	CI_UPPER
1	20	.2469	.00591	.07689	3.2110	.0013	.0007	95	.0962	.3976

Table 7 - Chi-square test for homogeneity of effect size
	Q	K	P_CHISQ
1	121.2757	19	.0000

Table 8 - Random variance component

Mean

V0

.63979

Table 9 - Random effects model: Weighted average effect size, z test, and confidence interval
	NSTUDIES	EF_SIZE	VARIANCE	SE	Z	P_TWO	P_ONE	CL	CI_LOWER	CI_UPPER
1	20	.2450	.03833	.19578	1.2515	.2107	.1054	95	-.1387	.6288

****************************************************************************

** Note **

**

** Beginning in line:

**

** COMPUTE W=1/V.

**

** with effect sizes (D) and variances (V) from original sources, this

** syntax was tested with data reported in Lipsey and Wilson (2001, p. 130,

** Table 7.1) and Shadish and Haddock (1994, p. 267, Table 18.2).

**

** Imputations procedures and Individual T Tests were tested in SPSS,

** comparing the results with outputs obtained from raw data examples.

**

** Power calculations are the same given by SamplePower (Borenstein et al.,

** 2001) and measures of effect size and nonoverlap were tested with

** tabulated values and examples given by Cohen (1988) and Rosenthal et al.

** (2000).

**

** Feel free to use and modify this syntax as you wish. In case you want to

** refer it, the proper form is:

**

** Alferes, V. R. (2003). Meta-analysis: Fixed and random effects models

** [SPSS Syntax File]. Retrieved [Date], from [URL]

****************************************************************************

** References **

**

** Borenstein, M., Rothstein, H., & Cohen, J. (2001). SamplePower 2.0

** [Computer Manual]. Chicago: SPSS Inc.

** Cohen, J. (1988). Statistical power analysis for the behavioral

** sciences (2nd ed.). Hillsdale, NJ: Lawrence Erbaum.

** Cortina, J. M., & Nouri, H. (2000). Effect sizes for ANOVA designs.

** Thousand Oaks, CA: Sage.

** Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis.

** Orlando, FL: Academic Press.

** Hunter, J. E., & Schmidt, F. L. (1990). Methods of meta-analysis:

** Correcting error and bias in research findings. Newbury Park, CA:

** Sage.

** Lipsey, M. W., & Wilson, D. B. (2001). Pratical meta-analysis. Thousand

** Oaks, CA: Sage.

** Rosenthal, R., Rosnow, R. L, & Rubin, D. B. (2000). Contrasts and

** effect sizes in behavioral research: A correlational approach.

** Cambridge, UK: Cambridge University Press.

** Shadish, W. R., & Haddock, C. K. (1994). Combining estimates of effect

** size. In H. Cooper and L. V. Hedges (Eds.), The handbook of research

** synthesis (pp. 261-281). New York: Russell Sage Foundation.

***************************************************************************.

restore.