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How to Calculate Sample Size &
Power Analysis Information
for Dissertation Students & Researchers
Power Analysis Basics
To review, power is defined as the probability
that a statistical test will reject the null hypothesis or the ability of a
statistical test to detect an effect. Power is equal to 1-b (beta). In order to
reject the null hypothesis (which states that there is no relationship between
the variables of interest), power should be at least .80. In general, the larger
your sample size, the greater the power, but sometimes having too many
subjects can decrease your power. A power analysis provides information for
determining the minimum number of subjects you need to collect in order to make your study worthwhile.
All quantitative studies should conduct a power analysis to ensure that certain
conditions are met to correctly reject the null hypothesis. Before conducting a
power analysis, you must know which statistical test to choose to analyze your data. If you do not know what statistical test might be best or
most powerful for your study, we can help you decide before conducting a power analysis. To conduct a power analysis, you must
also have a sense of the significance level you want to use in your statistical
analysis. A common choice in research is alpha=.05. You should also have a sense
of how much power is required to detect an effect when conducting a power
analysis. This is expressed as power=1 - ß, where ß is the probability of a Type
II error. Power=0.80 is common, but one could also choose Power=.95 or.90.
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When conducting a power analysis, you must also choose the
appropriate effect size; not choosing the appropriate effect size can result in
a sample size that is too large or too small. Keep in mind that the smaller the
effect, the more difficult it will be to detect. Choose effect sizes based on
what is typically observed in your field or use Cohen’s estimates of small,
medium and large effect sizes. Finally, its good to have a sense of the sample
size you are capable of handling (in terms of time and resources) when
conducting a power analysis. A larger sample size is likely to increase your
ability to detect a significant difference or effect, but you may not have the
resources required for collecting a large sample. Based on the power analysis, you can
determine whether the study is worth pursuing.
Many students think that there is a simple formula for a power analysis, but
power analysis varies by research situation. If you are unsure of how to conduct a power analysis on your own,
contact a consultant to assist you in this very important step in the research
process.
An Example of a Power Analysis
You should always perform a power analysis before you begin your data
collection, ideally when you are designing your study. This is referred to as an
a priori power analysis which is done before you conduct your study. When doing
this type of power analysis you need to know the alpha, the power you would like
to achieve (.e.g., .80) and the effect size (small, medium or large). You also
want to have a general sense of your desired sample size. Suppose you want to
compare the effects of two educational interventions administered to two
different classrooms in your study. In this case you would do an a priori power
analysis for a two-tailed t-test (means). You might also choose a large effect
size (d = .80), and want alpha to equal .05 (power = .95). If you used these
numbers in your analysis you would see that you would need about 84 participants
to address your research question appropriately. Request Power Analysis Help
Today
A post-hoc power analyses is done
after you have completed your research. When doing a post hoc power analysis you
need to know the alpha, the power you would like to achieve (.e.g., .80) and the
effect size (small, medium or large). In this example suppose you conducted a
study that examined the effect of two educational interventions on two separate
classrooms but did not find a significant difference between the two groups on
your dependent variable. Imagine that your total sample only consisted of 36
students (22 in classroom A and 14 in classroom B). In this situation you could
do a post-hoc power analysis for a t- test, with a medium effect size (d = .50),
and an alpha of .05 for a two-tailed test. Based on these numbers a power
analysis could tell you how many participants you needed to detect the effect
you were seeking.
Power Analysis for Regression
To perform a power analysis for
regression, you need to know the effect size, and the number of predictors you
would like to include (or included in your model). Here an example of a power
analysis for simultaneous multiple regression. The first model will test whether
classical music predicts female adolescents’ levels of calmness, and whether
this relationship is mediated by the musical rhythm. The second model will test
whether classical music predicts female adolescents’ level of attachment, and
whether this relationship is mediated by musical rhythm. Given the number of
predictors here (2), the α for the test of these model will be set at .05. To
achieve power of .95 and a medium effect size (f 2=.15), a sample size of at
least 107 is required to detect a significant model (F (2,104) =3.0837). Power
analysis for regression can tell you the exact sample size you need based on
your research question. Request Power
Analysis Help Today
Power Analysis for ANOVA
Computing required sample size for
experiments to be analyzed by ANOVA is pretty complicated. Power analysis for
ANOVA will depend on the number of effects. The example we present here is for a
single factor design. Suppose you are conducting a study where you want to
compare 10 groups of occupations on your independent variable and you expect a
"medium" effect size (f = .25) based on prior research. You may also know that
you want 95% power (alpha=.05). Given these numbers you would need a total
sample of 390 people or 39 people in each of the 10 groups to detect the effect
you are looking for. Power analysis for regression can tell you the exact sample
size you need based on your research question.
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Power Analysis for t-tests
Suppose you have two groups that
you want to compare on a continuous variable (Group 1 vs. Group 2 on age in
years). You can use a power analysis to determine the sample size needed to
obtain a t statistic equal to or larger than a critical value with an alpha =
.05. Suppose you know that you are looking for a medium effect (d=.5) and 90%
power. Given these numbers you would need a total sample of 172 people for your
study. In another example, suppose you need to do a one sample t-test to
compare pre and post test means on the outcome variable with an absolute mean
difference of 0.5 on the variable of interest. To achieve a medium effect size
(d=.06), a total sample of 122 analyzable subjects will provide 95% power to
detect a difference at the 0.05 significance level. The critical t value needed
to detect a difference in this case is t (120) = 1.657. Power analysis a t-test
can tell you the exact sample size you need based on your research question.
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