Statistics
Lecture 10
Hypothesis testing:
Non-parametric tests
•David Bartl
•Statistics
•INM/BASTA

Outline of the lecture
•About statistical hypothesis testing:
Parametric and Non-parametric tests
•Sign test for the median
•Pearson’s χ2-test for the goodness of fit
•χ2-test of independence of qualitative data items

About statistical hypothesis testing
•Parametric and Non-parametric tests
•

Parametric and Non-parametric tests
•There are two large classes of statistical tests:  parametric  and  non-parametric.
•
•The parametric tests make assumptions about the probability distributions
of the random variables that are subject to the test.  It is often assumed that
the underlying distribution is normal (Gaussian).
•
•The non-parametric tests do not make such assumptions.  The non-parametric tests can be used if
the random variables are not normally distributed.

Sign test
for the median
•Sign test for the median
•Paired sign test for
the difference of the medians
•

Sign test for the median



Sign test for the median



Sign test for the median



Sign test for the median



Sign test for the median



Sign (binomial) test for the median



Sign (binomial) test for the median
that


Sign (binomial) test for the median



Sign (z-) test for the median



Sign (z-) test for the median



Sign (z-) test for the median



Sign (z-) test for the median



Sign (z-) test for the median



Sign test for the median



Sign test for the median



Paired sign test for the difference of the medians



Paired sign test for the difference of the medians



Paired sign test for the difference of the medians



Paired sign test for the difference of the medians



χ2-test
for goodness of fit
•Pearson’s χ2-test for the goodness of fit
•

Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Pearson’s χ2-test for the goodness of fit



Example:  Tests for population proportion



Example:  Tests for population proportion



Example:  Tests for population proportion



Pearson’s χ2-test for the goodness of fit



χ2-test
of independence
of qualitative
data items
•χ2-test of independence of
qualitative data items
•

χ2-test of independence of qualitative data items



Contingency table
the observed counts of the combinations
of the categories Ai&Bj for i=1,…,r & j=1,…,s
marginal totals
marginal totals
the grand total

2 × 2 contingency table
•The 2 × 2 contingency table is popular.
•It is a contingency table with  r=2 rows  and  s=2 columns.
the observed counts of the combinations
of the categories Ai&Bj for i=1,2 & j=1,2
marginal totals
marginal totals
the grand total

χ2-test of independence of qualitative data items
•Having all the observed counts of the combinations of the categories  Ai & Bj  summarized in the
contingency table for  i=1,…,r  and for  j=1,…,s,
we ask whether the category of the data item (variable)  B depends upon
the category of the data item (variable)  A,  or whether the categories of both data items
(variables)  A  and  B  are independent of each other.
•
•Assume therefore the null hypothesis  H0:
•the categories of both data items (variables)  A  and  B  are independent
of each other

χ2-test of independence of qualitative data items
•Having all the observed counts of the combinations of the categories  Ai & Bj  summarized in the
contingency table for  i=1,…,r  and for  j=1,…,s,  assume
the null hypothesis  H0  that the categories of both data items (variables)  A  and  B  are
independent of each other.
•Now – if we choose a data unit randomly:
•What is the probability that the data item  A  of the chosen data unit is of category  Ai  for
some  i=1,…,r ?
•What is the probability that the data item  B  of the chosen data unit is of category  Bj  for
some  j=1,…,s ?

χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items



χ2-test of independence of qualitative data items