Statistical Analysis for Education and Psychology Researchers

pooledstandard

deviation—52

where and are the sample variances, and n 1 and n 2 are the respective sample sizes.
For worked examples and what to do when sample sizes are substantively different see
Lipsey (1990).
As the population effect size increases, the t-ratio increases as does the likelihood of
attaining statistical significance. Put simply, the power of a statistical test is related to the
effect size, the larger the effect the more probable is statistical significance and the
greater is the statistical power.
To summarize, the following design attributes increase statistical power:

• larger sample sizes;


• homogeneous populations (low variability in population measures of interest);


• larger alpha Type I errors (problem of finding a difference that does not actually exist in
  the population);


• larger effect sizes.

Estimating Sample Size and/or Power for a Design

To compute a sample size for an investigation using charts that depict statistical power
for various values of effect size, alpha, and sample size, the general procedure is to enter
the power charts with any of the three parameters, say effect size, alpha and power, and
the fourth parameter the corresponding sample size, can be determined. Alternatively,
you could enter the power charts with a sample size and determine the statistical power of
a test. The reader is referred to Lipsey (1990) for power charts and illustrated examples of
how to use them.
In this section, rather than refer to charts and tables, three SAS programmes are
presented for sample size and power estimation calculations. These SAS programmes use
the SAS functions PROBIT and PROBNORM to generate appropriate values and thereby
avoid the necessity to use tables and charts. The programmes are suitable for four
common study designs: binomial data two independent groups; binomial data (paired
groups cross-over design); normally distributed data independent groups design; and
normally distributed data paired groups cross-over design (or simply paired/related
groups). These programmes are presented in Appendix A3.
For all three programmes either sample size or power can be estimated provided three
other parameters are entered. For example, to calculate sample size for a normally
distributed response variable in an independent groups design the following three
parameters would need to be specified (in the programme): Power (power), Type I error
(alpha), difference between the two means that is to be detected (diff), pooled within
group standard deviation (sd), and −9 for the fourth parameter, sample size. When −9 is
entered this missing parameter is estimated by the programme. If we wanted to estimate
power for a given sample size, then −9 would be entered for power and the given sample
size would be substituted for the −9 in the above example. There can only be one missing
parameter for each sample size or power calculation.

Statistical analysis for education and psychology researchers 134