Introductory concepts - Practical Biomedical Signal Analysis Using MATLAB - page 16

is needed. We assume
that the data are random variables and represent a sample taken from some popu-
lation characterized by properties which are in accordance with the null hypothesis.
Statistic is a function of a random variable. The main thing that must be known about
the statistic is the probability with which it takes different values for the random vari-
ables conforming to the null hypothesis.
If for the given data the computed statistics is S x

To actually perform a test, a function called statistic S ( x )

= ( )

then the p value returned
by the test is the probability of observing statistics with values equal or more extreme
than S x .Ifthe p value is high, then we assume that the data conform to the null
hypothesis. But if the probability of observing such a value of the statistic is low
then we can doubt that the data agree with the null hypothesis. Consequently we
reject that hypothesis and accept the alternative one. The critical level of probability
used to make the decision is called the significance level α. It expresses how low the
p value must be to doubt the null hypothesis.

S x

1.5.2 Types of tests
To select the type of test we need to answer the question: do we know the proba-
bility distribution from which the data were sampled?

Ye s , we know or can assume, or can transform, e.g., by Box-Cox transform [Box

and Cox, 1964], the data to one of the known probability distributions. In this
case we select appropriate classical parametric tests based on the normal, t , F ,
χ 2 or some other known statistics. In MATLAB Statistics Toolbox many such
tests are available e.g., ttest , ttest2 , anova , chi2gof . To test the normal-
ity assumption we can use Lilliefors' test [Lilliefors, 1967], implemented as
lillietest , or use a qualitative graphical test implemented as normplot .

No, we do not know the probability distribution. In this case we have two possibili-
ties:

• Use a classial non-parametric test e.g.:

Wilcoxon rank sum test — tests, if two independent samples come
from identical continuous distributions with equal medians, against
the alternative that they do not have equal medians. In MATLAB
Statistics Toolbox it is implemented as ranksum .
Wilcoxon signed rank test — One-sample or paired-sample Wilcoxon
signed rank test. It tests, if a sample comes from a continuous distri-
bution symmetric about a specified median, against the alternative
that it does not have that median. In MATLAB Statistics Toolbox it
is implemented as signrank .
Sign test — One-sample or paired-sample sign test. It tests, if a sam-
ple comes from an arbitrary continuous distribution with a specified
median, against the alternative that it does not have that median. In
MATLAB Statistics Toolbox it is implemented as signtest .