Confidence Interval for Standardized Difference Between Means by pptfiles

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									    Confidence Interval for Standardized Difference Between Means, Related Samples

       You cannot use my Conf_Interval-d2.sas program to construct a confidence interval for d
when the data are from correlated samples. With correlated samples the distributions here are
very complex, not following the noncentral t. You can construct an approximate confidence
                                     g2       2(1  r12 )
interval, g  ZccSE where SE is                         , but such a confidence interval is not very
                                   2(n  1)       n
accurate unless you have rather large sample sizes. I cannot recall where I got this formula, but
suspect it was from Kline’s Beyond Significance Testing. I probably have the source at my
office, but am stuck at home with a broken leg at the moment. I do have an SAS program that
uses this algorithm

        PS: I use “g” to symbolize the sample estimate of Cohen’s d.

        Algina & Kesselman (2003) provided a new method for computing confidence intervals
for the standardized difference between means. I shall illustrate that method here.

Run this SAS code:
options pageno=min nodate formdlim='-';
******************************************************************************;
title 'Experiment 2 of Karl''s Dissertation';
title2 'Correlated t-tests, Visits to Mus Tunnel vs Rat Tunnel, Three Nursing Groups
'; run;
data Mus; infile 'C:\D\StatData\tunnel2.dat';
input nurs $ 1-2 L1 3-5 L2 6-8 t1 9-11 t2 12-14 v_mus 15-16 v_rat 17-18;
 v_diff=v_mus - v_rat;
proc means mean stddev n skewness kurtosis t prt;
var v_mus V_rat v_diff;
run;
*****************************************************************************;
Proc Corr Cov; var V_Mus V_Rat; run;
Get this output
--------------------------------------------------------------------------------------------------

                               Experiment 2 of Karl's Dissertation                                 1
          Correlated t-tests, Visits to Mus Tunnel vs Rat Tunnel, Three Nursing Groups

                                       The MEANS Procedure

  Variable           Mean        Std Dev    N       Skewness       Kurtosis   t Value   Pr > |t|
  ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
  v_mus        22.7291667     10.4428324   48     -0.2582662     -0.3846768     15.08     <.0001
  v_rat        13.2916667     10.4656649   48      0.8021143      0.2278957      8.80     <.0001
  v_diff        9.4375000     11.6343334   48     -0.0905790     -0.6141338      5.62     <.0001
  ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
                                                                                               2


                                   The CORR Procedure


                               Covariance Matrix, DF = 47

                                            v_mus             v_rat

                       v_mus       109.0527482            41.6125887
                       v_rat        41.6125887           109.5301418



                       Pearson Correlation Coefficients, N = 48
                              Prob > |r| under H0: Rho=0

                                            v_mus          v_rat

                           v_rat           0.38075       1.00000
                                           0.0076


To obtain our estimate of the standardized difference between mean, run this SAS code:
Data G;
M1 = 22.72917 ;    SD1 = 10.44283;
M2 = 13.29167 ;    SD2 = 10.46566;
g = (m1-m2) / SQRT(.5*(sd1*sd1+sd2*sd2)); run;
Proc print; var g; run;


Obtain this result:
                                     Obs         g

                                      1        0.90274




       Using the approximation method first presented,
        .902742 2(1  .38075)
SE                           0.18567 . A 95% CI is .90274(1.96)(.18567) = [.539, 1.267].
        2( 48  1)    48


       From James Algina’s webpage, I obtained a simple SAS program for the confidence
interval. Here is it, with values from above.

*This program computes an approximate CI for the effect
size in a within-subjects design with two groups.
m2 and m1 are the means for the two groups
s1 and s2 are the standard deviations for the two groups
n1 and n2 are the sample sizes for the two groups
r is the correlation
prob is the confidence level;
data;
m1=22.7291667 ;
m2=13.2916667 ;
s1=10.4428324 ;
s2=10.4656649 ;
                                                                                                3

r= 0.38075 ;
n=48    ;
prob=.95 ;
v1=s1**2;
v2=s2**2;
s12=s1*s2*r;
se=sqrt((v1+v2-2*s12)/n);
pvar=(v1+v2)/2;
nchat=(m1-m2)/se;
es=(m1-m2)/(sqrt(pvar));
df=n-1;
ncu=TNONCT(nchat,df,(1-prob)/2);
ncl=TNONCT(nchat,df,1-(1-prob)/2);
ul=se*ncu/(sqrt(pvar));
ll=se*ncl/(sqrt(pvar));
output;
proc print;
title1 'll is the lower limit and ul is the upper limit';
title2 'of a confidence interval for the effect size';
var es ll ul ;
run;
quit;


      Here is the result:
                    ll is the lower limit and ul is the upper limit            4
                      of a confidence interval for the effect size

                            Obs      es        ll         ul

                            1     0.90274   0.53546    1.26275


       The program presented by Algina & Keselman (2003) is available at another of Algina’s
web pages . This program will compute confidence intervals for one or more standardized
contrasts between related means, with or without a Bonferroni correction, and with or without
pooling the variances across all groups. Here is code, modified to compare the two related
means from above.

* This program is used with within-subjects designs. It computes
confidence intervals for effect size estimates. To use the program one
inputs at the top of the program:
     m--a vector of means
     v--a covariance matrix in lower diagonal form, with periods for
        the upper elements
     n--the sample size
  prob--the confidence level prior to the Bonferroni adjustment
adjust--the number of contrats is a Bonferroni adjustment to the
        confidence level is requested. Otherwise adjust is set
        equal to 1
  Bird--a switch that uses the variances of all variables to calculate
        the denominator of the effect size as suggested by K. Bird
        (Bird=1). Our suggestion is to use the variance of those
        variables involved in the contrast to calculate the denominator
        of the effect size (Bird=0)
                                                               4

In addition one inputs at the bottom of the program:
     c--a vector of contrast weights
multiple contrasts can be entered. After each, type the code
run ci;
proc iml;
m={ 22.72917 13.29167};
v={109.0527482 . ,
   41.6125887 109.5301418};
do ii = 1 to nrow(v)-1;
do jj = ii+1 to nrow(v);
   v[ii,jj]=v[jj,ii];
end;
end;
n=48     ;
Bird=1;
df=n-1;
cl=.95;
adjust=1;
prob=cl;
print 'Vector of means:';
print m;
print 'Covariance matrix:';
print v;
print 'Sample size:';
print n;
print 'Confidence level before Bonferroni adjustment:';
print cl;
cl=1-(1-prob)/adjust;
print 'Confidence level with Bonferroni adjustment:';
print cl;
start CI;
pvar=0;
count=0;
if bird=1 then do;
  do mm=1 to nrow(v);
     if c[1,mm]^=0 then do;
         pvar=pvar+v[mm,mm];
            count=count+1;
     end;
  end;
end;
if bird=0 then do;
   do mm=1 to nrow(v);
       pvar=pvar+v[mm,mm];
           count=count+1;
   end;
end;
pvar=pvar/count;
es=m*c`/(sqrt(pvar));
se=sqrt(c*v*c`/n);
nchat=m*c`/se;
ncu=TNONCT(nchat,df,(1-prob)/(2*adjust));
ncl=TNONCT(nchat,df,1-(1-prob)/(2*adjust));
ll=se*ncl/(sqrt(pvar));
ul=se*ncu/(sqrt(pvar));
print 'Contrast vector';
print c;
                                                                      5

print 'Effect size:';
print es;
Print 'Estimated noncentrality parameter';
print nchat;
Print 'll is the lower limit of the CI and ul is the upper limit';
print ll ul;
finish;
c={1 -1};
run ci;
quit;


      Here is the output:


                    ll is the lower limit and ul is the upper limit
                      of a confidence interval for the effect size

                                    Vector of means:


                                           m

                                   22.72917     13.29167


                                   Covariance matrix:


                                           v

                                  109.05275 41.612589
                                  41.612589 109.53014


                                      Sample size:


                                           n

                                                48


                     Confidence level before Bonferroni adjustment:


                                           cl

                                               0.95


                      Confidence level with Bonferroni adjustment:


                                           cl

                                               0.95


                                    Contrast vector


                                           c

                                          1           -1
                                                                                                     6



                                           Effect size:

--------------------------------------------------------------------------------------------------

                         ll is the lower limit and ul is the upper limit
                           of a confidence interval for the effect size

                                                es

                                            0.9027425


                                Estimated noncentrality parameter


                                              nchat

                                             5.619997


                    ll is the lower limit of the CI and ul is the upper limit


                                              ll          ul

                                       0.5354583 1.2627535


        Notice that this produces the same CI produced by the shorter program.

        Recommended additional reading:
       Algina, J., & Keselman, H. J. (2003). Approximate confidence intervals for effect sizes.
        Educational and Psychological Measurement, 63, 537-553. DOI:
        10.1177/0013164403256358
       Grissom, R. J., & Kim, J. J. (2005). Effect sizes for research: A broad practical
        approach. Mahwah, NJ: Erlbaum. – especially pages 67 and 68
       Glass, G. V., & Hopkins, K. D. Statistical Methods in Education and Psychology (2nd
        ed.), Prentice-Hall 1984. Section 12.12: Testing the hypothesis of equal means with
        paired observations, pages 240-243. Construction of the CI is shown on page 241, with a
        numerical example on pages 242-243.

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