HEDGES G CONFIDENCE LIMITS

HEDGES G CONFIDENCE LIMITS

Analysis Command

Generates a confidence interval for the Hedge's g statistic.

The Hedge's g statistic is used to measure the effect size for the difference between two means. The formula is
\( g = \frac{\bar{y}_{1} - \bar{y}_{2}} {s_{p}} \)

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

\( s_{p} = \sqrt{\frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}} {(n_{1} - 1) + (n_{2} - 1)}} \)

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. It is typically used in the context where one of the samples is a control sample. That is, we are interested in the effect size relative to a control sample.

The confidence interval for the Hedge's g statistic is

\( g \pm \Phi^{-1}(1 - (\alpha/2)) g_{se} \)

where

\( \Phi^{-1} \)	=	the percent point function of the normal distribution
\( g_{se} \)	=	the standard error of the g statistic
	=	\( \sqrt{\frac{n_1 + n_2}{n_1 n_2} + \frac{g^2}{2(n_1 + n_2)}} \)

HEDGES G CONFIDENCE LIMITS <y1> <y2>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

HEDGES G CONFIDENCE LIMITS Y1 Y2
HEDGES G CONFIDENCE LIMITS Y1 Y2 SUBSET TAG = 1

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
\( s_{p} = \sqrt{\frac{s_{1}^{2} + s_{2}^{2}} {2}} \)

That is, Cohen's d does not weight the standard deviations based on the sample sizes.

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

0.2	=>	small effect
0.5	=>	medium effect
0.8	=>	large effect

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.

In addition to the HEDGES G CONFIDENCE LIMIT command, the following commands can also be used:
LET ALPHA = 0.05


LET A = HEDGES G Y1 Y2
LET A = HEDGES G LOWER CONFIDENCE LIMIT Y1 Y2
LET A = HEDGES G UPPER CONFIDENCE LIMIT Y1 Y2
LET A = HEDGES G STANDARD ERROR Y1 Y2


In addition to the above LET commands, built-in statistics are supported for 20+ different commands (enter HELP STATISTICS for details).

None

CONFIDENCE INTERVAL is a synonym for CONFIDENCE LIMITS

HEDGES G	=	Compute Hedge's g statistic.
T TEST	=	Perform a two sample t-test.
QUANTILE QUANTILE PLOT	=	Generate a quantile-quantile plot.

Hedges and Olkin (1985), "Statistical Methods for Meta-Analysis", New York: Academic Press.

Durlak (2009), "How to Select, Calculate, and Interpret Effect Sizes", Journal of Pediatric Psychology, Vol. 34, No. 9, pp. 917-928.

Hedges (1981), "Distribution Theory for Glass's Estimator of Effect Size and Related Estimators", Journal of Educational Statistics, Vol. 6, No. 2, pp. 107-128.

Cohen (1977), "Statistical Power Analysis for the Behavioral Sciences", Routledge.

Glass (1976), "Primary, Secondary, and Meta-Analysis of Research", Educational Researcher, Vol. 5, pp. 3-8.

Confirmatory Data Analysis

2018/08

 
SKIP 25
READ AUTO83B.DAT Y1 Y2
DELETE Y2 SUBSET Y2 < 0
SET WRITE DECIMALS 5
.
LET HG    = HEDGES G Y1 Y2
LET HGSE  = HEDGES G STANDARD ERROR Y1 Y2
LET HGLCL = HEDGES G LOWER CONFIDENCE LIMIT Y1 Y2
LET HGUCL = HEDGES G UPPER CONFIDENCE LIMIT Y1 Y2
.
PRINT HG HGSE HGLCL HGUCL
HEDGES G CONFIDENCE LIMITS Y1 Y2
    

The following output is returned

 PARAMETERS AND CONSTANTS--

    HG      --       -1.62968
    HGSE    --        0.14395
    HGLCL   --       -1.91183
    HGUCL   --       -1.34754
  
             Confidence Limits for the Hedges G Statistic
  
 Response Variable 1: Y1
 Response Variable 2: Y2
  
  
 Summary Statistics for Variable 1:
 Number of Observations:                  0
 Sample Mean:                             20.14458
 Sample Standard Deviation:               6.41470
  
 Summary Statistics for Variable 2:
 Number of Observations:                  0
 Sample Mean:                             30.48101
 Sample Standard Deviation:               6.10771
  
 Pooled Standard Deviation                6.34260
 Hedges G:                                -1.62968
 Bias Corrected Hedges G:                 -1.62593
 Standard Error for Hedges G:             0.14395
  
  
  
 -----------------------------------------------------------------
   Confidence       Z      Z-Value X          Lower          Upper
    Value (%)   Value         StdErr          Limit          Limit
 -----------------------------------------------------------------
       50.000   0.674        0.09710       -1.72678       -1.53259
       75.000   1.150        0.16560       -1.79528       -1.46409
       80.000   1.282        0.18449       -1.81417       -1.44520
       90.000   1.645        0.23678       -1.86647       -1.39290
       95.000   1.960        0.28215       -1.91183       -1.34754
       99.000   2.576        0.37080       -2.00049       -1.25888
       99.900   3.291        0.47369       -2.10337       -1.15600