
HEDGES GName:
BIAS CORRECTED HEDGES G (LET) COHENS D (LET) GLASS G (LET)
with \( \bar{y}_{1} \), \( \bar{y}_{2} \), and \( s_{p} \) denoting the mean of sample 1, the mean of sample 2, and the pooled standard deviation, respectively. The formula for the pooled standard deviation is
with \( s_{1} \) and \( n_{1} \) denoting the standard deviation and number of observations for sample 1, respectively, and \( s_{2} \) and \( n_{2} \) denoting the standard deviation and number of observations for sample 2, respectively. The Hedge's g statistic expresses the difference of the means in units of the pooled standard deviation. For small samples, the following bias correction is recommended
where \( n = n_{1} + n_{2} \). This bias correction is typically recommended when n < 50. NOTE:
The term being approximated is
where
with \( \Gamma \) denoting the Gamma function. Running a comparison indicated that Hedge's original approximation is more accurate than that given by Durlak. The 2018/08 version of Dataplot modified the bias correction to use the original J function with Gamma functions for \( n_1 + n_2 \le 40 \) and to use Hedge's original approximation otherwise. Hedge's g is similar to the Cohen's d statistic and the Glass g statistic. The difference is what is used for the estimate of the pooled standard deviation. The Hedge's g uses a sample size weighted pooled standard deviation while Cohen's d uses
These statistic are typically used to compare an experimental sample to a control sample. The Glass g statistic uses the standard deviation of the control sample rather than the pooled standard deviation. His argument for this is that experimental samples with very different standard deviations can result in significant differences in the g statistic for equivalent differences in the mean. So the Glass g statistic measures the difference in means in units of the control sample standard deviation. Hedge's g, Cohen's d, and Glass's g are interpreted in the same way. Cohen recommended the following rule of thumb
However, Cohen did suggest caution for this rule of thumb as the meaning of small, medium and large may vary depending on the context of a particular study. The Hedge's g statistic is generally preferred to Cohen's d statistic. It has better small sample properties and has better properties when the sample sizes are signigicantly different. For large samples where \( n_{1} \) and \( n_{2} \) are similar, the two statistics should be almost the same. The Glass g statistic may be preferred when the standard deviations are quite different. In many cases, there are multiple experimental groups being compared to the control. This could be either a separate control sample for each experiment (e.g., we are comparing effect sizes from different experiments) or a common control (e.g., different laboratories are measuring identical material and are being compared to a reference measurement). In these cases, you can compute the Hedge's g (or Glass's g or Cohen's d) for each experiment relative to its control group. You can then obtain an "overall" value of the statistic by averaging these individual statistics.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Hedge's g statistic is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Hedge's g statistic without the bias correction.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed bias corrected Hedge's g statistic is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes the Hedge's g statistic with the bias correction.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Glass's g statistic is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Glass's g statistic. The <y2> variable will be treated as the "control" sample. That is, the standard deviation of <y2> will be used as the pooled standard deviation.
<SUBSET/EXCEPT/FOR qualification> where <y1> is the first response variable; <y2> is the second response variable; <par> is a parameter where the computed Cohen's d statistic is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes Cohen's d statistic.
LET A = BIAS CORRECTED HEDGES G Y1 Y2 LET A = BIAS CORRECTED HEDGES G Y1 Y2 SUBSET Y1 > 0 LET A = GLASS G Y1 Y2 LET A = COHENS D Y1 Y2
Cohen (1977), "Statistical Power Analysis for the Behavioral Sciences", Routledge. Glass (1976), "Primary, Secondary, and MetaAnalysis of Research", Educational Researcher, Vol. 5, pp. 38. Durlak (2009), "How to Select, Calculate, and Interpret Effect Sizes", Journal of Pediatric Psychology, Vol. 34, No. 9, pp. 917928. Hedges and Olkin (1985), "Statistical Methods for MetaAnalysis", New York: Academic Press.
SKIP 25 READ IRIS.DAT Y1 TO Y4 X . LET A = HEDGES G Y1 Y2 TABULATE HEDGES G Y1 Y2 X . LABEL CASE ASIS TITLE CASE ASIS Y1LABEL Hedge's G Statistic X1LABEL Group TITLE Hedge's G Statistic for IRIS.DAT X1TIC MARK OFFSET 0.5 0.5 TIC MARK OFFSET UNITS DATA CHAR X LINE BLANK HEDGES G PLOT Y1 Y2 X . Y1LABEL Hedge's G Statistic X1LABEL Group TITLE Bootstrap of Hedge's G Statistic for IRIS.DAT CHAR X ALL LINE BLANK ALL BOOTSTRAP SAMPLES 1000 BOOTSTRAP HEDGES G PLOT Y1 Y2 XThe following output is generated Cross Tabulate HEDGES G (Response Variables: Y1 Y2 )  X  HEDGES G  0.1000000E+01  0.4311260E+01 0.2000000E+01  0.7412040E+01 0.3000000E+01  0.7168413E+01 Bootstrap Analysis for the HEDGES G Response Variable One: Y1 Response Variable Two: Y2 Group ID Variable One (X ): 1.000000 Number of Bootstrap Samples: 1000 Number of Observations: 50 Mean of Bootstrap Samples: 4.389604 Standard Deviation of Bootstrap Samples: 0.4288846 Median of Bootstrap Samples: 4.335823 MAD of Bootstrap Samples: 0.2615915 Minimum of Bootstrap Samples: 3.342993 Maximum of Bootstrap Samples: 6.389001 Percent Points of the Bootstrap Samples  Percent Point Value  0.1 = 3.343002 0.5 = 3.511410 1.0 = 3.546756 2.5 = 3.687836 5.0 = 3.798257 10.0 = 3.910058 20.0 = 4.037920 50.0 = 4.335823 80.0 = 4.705708 90.0 = 4.947669 95.0 = 5.174219 97.5 = 5.409715 99.0 = 5.681841 99.5 = 5.823756 99.9 = 6.388623 Percentile Confidence Interval for Statistic  Confidence Lower Upper Coefficient Limit Limit  50.00 4.087805 4.619201 75.00 3.934208 4.869211 90.00 3.798257 5.174219 95.00 3.687836 5.409715 99.00 3.511410 5.823756 99.90 3.342993 6.389001  Response Variable One: Y1 Response Variable Two: Y2 Group ID Variable One (X ): 2.000000 Number of Bootstrap Samples: 1000 Number of Observations: 50 Mean of Bootstrap Samples: 7.537408 Standard Deviation of Bootstrap Samples: 0.5465833 Median of Bootstrap Samples: 7.499111 MAD of Bootstrap Samples: 0.3508403 Minimum of Bootstrap Samples: 6.092766 Maximum of Bootstrap Samples: 9.751977 Percent Points of the Bootstrap Samples  Percent Point Value  0.1 = 6.092945 0.5 = 6.287894 1.0 = 6.397113 2.5 = 6.541644 5.0 = 6.668351 10.0 = 6.842498 20.0 = 7.102805 50.0 = 7.499111 80.0 = 7.976231 90.0 = 8.257037 95.0 = 8.490531 97.5 = 8.726730 99.0 = 8.898412 99.5 = 9.116481 99.9 = 9.751736 Percentile Confidence Interval for Statistic  Confidence Lower Upper Coefficient Limit Limit  50.00 7.171687 7.894311 75.00 6.920084 8.177537 90.00 6.668351 8.490531 95.00 6.541644 8.726730 99.00 6.287894 9.116481 99.90 6.092766 9.751977  Response Variable One: Y1 Response Variable Two: Y2 Group ID Variable One (X ): 3.000000 Number of Bootstrap Samples: 1000 Number of Observations: 50 Mean of Bootstrap Samples: 7.338842 Standard Deviation of Bootstrap Samples: 0.6960418 Median of Bootstrap Samples: 7.310299 MAD of Bootstrap Samples: 0.4584347 Minimum of Bootstrap Samples: 5.688356 Maximum of Bootstrap Samples: 11.27284 Percent Points of the Bootstrap Samples  Percent Point Value  0.1 = 5.688367 0.5 = 5.747268 1.0 = 5.929775 2.5 = 6.122410 5.0 = 6.271287 10.0 = 6.488546 20.0 = 6.749534 50.0 = 7.310299 80.0 = 7.873042 90.0 = 8.208357 95.0 = 8.487544 97.5 = 8.823580 99.0 = 9.252583 99.5 = 9.609171 99.9 = 11.27161 Percentile Confidence Interval for Statistic  Confidence Lower Upper Coefficient Limit Limit  50.00 6.862967 7.771134 75.00 6.543979 8.090032 90.00 6.271287 8.487544 95.00 6.122410 8.823580 99.00 5.747268 9.609171 99.90 5.688356 11.27284  Response Variable One: Y1 Response Variable Two: Y2 Group ID Variable One (All Data): Number of Bootstrap Samples: 1000 Number of Observations: 150 Mean of Bootstrap Samples: 6.421951 Standard Deviation of Bootstrap Samples: 1.547468 Median of Bootstrap Samples: 7.007179 MAD of Bootstrap Samples: 0.8567694 Minimum of Bootstrap Samples: 3.342993 Maximum of Bootstrap Samples: 11.27284 Percent Points of the Bootstrap Samples  Percent Point Value  0.1 = 3.409331 0.5 = 3.625963 1.0 = 3.741478 2.5 = 3.852497 5.0 = 3.968823 10.0 = 4.131668 20.0 = 4.447991 50.0 = 7.007179 80.0 = 7.739732 90.0 = 8.062719 95.0 = 8.351421 97.5 = 8.614502 99.0 = 8.894499 99.5 = 9.203490 99.9 = 9.751902 Percentile Confidence Interval for Statistic  Confidence Lower Upper Coefficient Limit Limit  50.00 4.617406 7.610766 75.00 4.210556 7.962787 90.00 3.968823 8.351421 95.00 3.852497 8.614502 99.00 3.625963 9.203490 99.90 3.347339 10.65474   
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Date created: 07/26/2017 