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chapter10.lyx.emergency
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chapter10.lyx.emergency
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\end_header
\begin_body
\begin_layout Part*
Chapter 10: Categorical Data And Nonparametric Methods
\end_layout
\begin_layout Chapter*
10.1 Tests of Goodness-of-Fit
\end_layout
\begin_layout Standard
In some problems, we have one specific distribution in mind for the data
we will observe.
If that one distribution is not appropriate, we do not necessarily have
a parametric family of alternative distributions in mind.
In these cases and others, we can still test the null hypothesis that the
data come from the one specific distribution against the alternative hypothesis
that the data do not come from that distribution.
\end_layout
\begin_layout Standard
In Chapters 7,8,9, we have assumed that the observations that are available
to the statisticaian come from dsitributionfor which the exact form is
known, even though the values of some parameters are unknown.
In othe words, we have assumed that the observations come from a certain
parametric family of distributions, and a statistic inference must be made
about the values of the parameters defining that family.
\end_layout
\begin_layout Standard
In many of the problems to be discussed in this chapter, we shall not assume
that the the available observations come from a particular parametric family
of distributions.
Rather, we shall study inferences that can be made about the distribution
from which the observations come, without making special assumptions about
the form of that distribution.
As one example, we might assmue that the observations form a random sample
from a continuous distribution, without specifying the form of this distributio
n any further, and we might then investigate the
\series bold
possibity
\series default
that this distribution is a normal distribution.
As a third example, we might be interested in investigating the possibility
that two independent random samples actually come from the same distribution,
and we might assume only that both distributions from which the samples
are taken are continuous.
\end_layout
\begin_layout Standard
Problems in which the possible distributions of the observations are not
restricted to a specific parametric family are called
\series bold
nonparametric problems
\series default
, and the statistical methods that are appicable in such problems are called
\series bold
nonparameretric methods
\series default
.
\end_layout
\begin_layout Section*
Categorical Data
\end_layout
\begin_layout Standard
In this section and the next four sections, we shall consider statistical
problems based on data such that each observation can be classified as
belonging to one of a finite number of possible categories or types.
Observations of this type are called
\series bold
categorical data.
\series default
Since there are only a finite number of possible categories in these problems,
and since we are interested in making inferences about the probabilities
of these categories, these problems actually involve just a finite number
of parameters.
However, as we shall see, methods based on categorical data can be usefully
applied in both parametric and nonparametric problems.
\end_layout
\begin_layout Section*
The
\begin_inset Formula $\chi^{2}$
\end_inset
Test
\end_layout
\begin_layout Standard
Suppose that a larfe population consists of items of
\begin_inset Formula $k$
\end_inset
different types, and let
\begin_inset Formula $p_{i}$
\end_inset
denote the probability that an item selected at random will be of type
\begin_inset Formula $i$
\end_inset
(
\begin_inset Formula $i=1,...,k$
\end_inset
).
Example
\begin_inset Formula $10.1.2$
\end_inset
is of this type with
\begin_inset Formula $k=4$
\end_inset
.
Of course,
\begin_inset Formula $p_{i}\ge0$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
and
\begin_inset Formula $\sum_{i=1}^{k}p_{i}^{0}=1$
\end_inset
, and suppose that the following hypotheses are to be tested:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
H_{0}:p_{i}=p_{i}^{0}\text{ for }i=1,...,k,
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
H_{1}:p_{i}=p_{i}^{0}\text{ for at least one value of }i\text{ (10.1.1)}
\]
\end_inset
\end_layout
\begin_layout Standard
We shall assume that a random sample of size
\begin_inset Formula $n$
\end_inset
is to be taken from the given population.
That is,
\begin_inset Formula $n$
\end_inset
independent observations are to be taken, and there is probability
\begin_inset Formula $p_{i}$
\end_inset
that each observations will be of type
\begin_inset Formula $i$
\end_inset
\begin_inset Formula $(i=1,...,k)$
\end_inset
.
On the basis of these
\begin_inset Formula $n$
\end_inset
observations, the hypotheses (10.1.1) are to be tested.
\end_layout
\begin_layout Standard
For
\begin_inset Formula $i=1,...,k$
\end_inset
, we shall let
\begin_inset Formula $N_{i}$
\end_inset
denote the number of observations in the random sample that are of type
\begin_inset Formula $i$
\end_inset
.
Thus,
\begin_inset Formula $N_{1},..,N_{k}$
\end_inset
are nonnegative integers such that
\begin_inset Formula $\sum_{i=1}^{k}N_{i}=n$
\end_inset
.
Indeed, (
\begin_inset Formula $N_{1},...,N_{n})$
\end_inset
has the multinomial distribution (see Sec, 5.9)
\end_layout
\begin_layout Theorem*
10.1.1,
\begin_inset Formula $\chi^{2}$
\end_inset
Statistic.
The following statistic
\end_layout
\begin_layout Theorem*
\begin_inset Formula
\[
Q=\sum_{i=1}^{k}\frac{(N_{i}-np_{i}^{0})^{2}}{np_{i}^{0}}\text{ (10.1.2)}
\]
\end_inset
\end_layout
\begin_layout Theorem*
has the property that if
\begin_inset Formula $H_{0}$
\end_inset
is true and the sample size
\begin_inset Formula $n\rightarrow\infty$
\end_inset
, then
\begin_inset Formula $Q$
\end_inset
converges in distribution to the
\begin_inset Formula $\chi^{2}$
\end_inset
distribution with
\begin_inset Formula $k-1$
\end_inset
degrees of freedom.
(see Definition 6.3.1)
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Standard
Theorem 10.1.1 says that if
\begin_inset Formula $H_{0}$
\end_inset
is true and the sample size
\begin_inset Formula $n$
\end_inset
is large, the distribution of
\begin_inset Formula $Q$
\end_inset
will be approximately the
\begin_inset Formula $\chi^{2}$
\end_inset
distribution with
\begin_inset Formula $k-1$
\end_inset
degrees of freedom.
The discussion that we have presented indicates that
\begin_inset Formula $H_{0}$
\end_inset
should be rejected when
\begin_inset Formula $Q\ge c$
\end_inset
, where
\begin_inset Formula $c$
\end_inset
is an appropriate constant.
If it is desired to carry out the test at the level of significance
\begin_inset Formula $\alpha_{0}$
\end_inset
, then
\begin_inset Formula $c$
\end_inset
should be chosem to be the
\begin_inset Formula $1-\alpha_{0}$
\end_inset
quantile of
\begin_inset Formula $\chi^{2}$
\end_inset
distribution with
\begin_inset Formula $k-1$
\end_inset
degrees of freedom.
This test is called the
\begin_inset Formula $\chi^{2}$
\end_inset
\series bold
test of goodness-of-fit.
\end_layout
\begin_layout Paragraph*
Note: General form of
\begin_inset Formula $\chi^{2}$
\end_inset
test statistic.
\series medium
The form of the statistic
\begin_inset Formula $Q$
\end_inset
in (10.1.2) is common to all
\begin_inset Formula $\chi^{2}$
\end_inset
tests including those that will be introduced later in this chapter.
The form is a sum of terms, each of which is the square of different between
an observed count and an expected count divided by the expected count:
\begin_inset Formula $\sum\text{(observed-expected)}^{2}/\text{expected }$
\end_inset
.
The expected counts are computed under the assumption that the null hypothesis
is true.
\end_layout
\begin_layout Standard
Whenever the value of each expected count,
\begin_inset Formula $np_{i}^{0}\ge5$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
, and the approximation should still be satisfactory if
\begin_inset Formula $np_{i}^{0}\ge1.5$
\end_inset
for
\begin_inset Formula $i=1,...,k$
\end_inset
.
\end_layout
\begin_layout Section*
Testing Hypotheses about a Continuous Distribution
\end_layout
\begin_layout Section*
Likelihood Ratio Tests for Proportions
\end_layout
\begin_layout Chapter*
10.3.
Contingency Tables
\end_layout
\begin_layout Standard
When each observation in our sample is a bivariate discrete random vector
(a pair of discrete random variables), then there is a simple way to test
the hypothesis that the two random variables are independent.
The test is another form of
\begin_inset Formula $\chi^{2}$
\end_inset
test like the ones used earlier in this chapter.
\end_layout
\begin_layout Section*
Independence in Contingency Tables
\end_layout
\begin_layout Definition*
10.3.1 Contingency Tables.
A table in which each observation is clssified in two or more ways is called
a contingency table.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Standard
In Table 10.12, only two classifications are considered for each student,
namely, the curriculum in which he is enrolled and the candidate he prefers.
Such a table is called a two-way contingency table.
\end_layout
\begin_layout Standard
Furthermore, we shall let
\begin_inset Formula $N_{i+}$
\end_inset
denote the total number of individuals classified in the
\begin_inset Formula $i$
\end_inset
th row and
\begin_inset Formula $N_{+j}$
\end_inset
denote the total number of individuals classified in the
\begin_inset Formula $j$
\end_inset
th column.
\end_layout
\begin_layout Standard
Thus,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
N_{i+}=\sum_{j=1}^{C}N_{ij}\text{ and }N_{+j}=\sum_{i=1}^{R}N_{ij}\text{ (10.3.1)}
\]
\end_inset
\end_layout
\begin_layout Standard
Also,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{i=1}^{R}\sum_{j=1}^{C}N_{ij}=\sum_{i=1}^{R}N_{i+}=\sum_{j=1}^{C}N_{+j}=n\text{ (10.3.2)}
\]
\end_inset
\end_layout
\begin_layout Standard
On the basis of these observations, the following hypotheses are to be tested:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
H_{0}:p_{ij}=p_{i+}p_{+j}\text{ for }i=1,...,R\text{ and }j=1,...,C\text{ (10.3.3)}
\]
\end_inset
\end_layout
\begin_layout Section*
\begin_inset Formula
\[
H_{1}:\text{The hypothesis }H_{0}\text{ is not true}
\]
\end_inset
The
\begin_inset Formula $\chi^{2}$
\end_inset
Test of Independence
\end_layout
\begin_layout Standard
The
\begin_inset Formula $\chi^{2}$
\end_inset
tests described in Sec.10.2 can be applied to the problem of testing the
hypotheses (10.3.3).
Each individual in the population from which the sample is taken must belong
in one the RC cells of the contingency table.
Under the null hypothesis
\begin_inset Formula $H_{0}$
\end_inset
, the unknown probabilities
\begin_inset Formula $p_{ij}$
\end_inset
of these cells have been expressed as functions of the unknown parameters
\begin_inset Formula $p_{i+}$
\end_inset
and
\begin_inset Formula $p_{+j}$
\end_inset
.
Since
\begin_inset Formula $\sum_{i=1}^{R}p_{i+}=1$
\end_inset
and
\begin_inset Formula $\sum_{j=1}^{C}p_{+j}=1$
\end_inset
, the actual number of unknown parameters to be estimated when
\begin_inset Formula $H_{0}$
\end_inset
is true is
\begin_inset Formula $s=(R-1)+(C-1)$
\end_inset
, or
\begin_inset Formula $s=R+C-2$
\end_inset
.
\end_layout
\begin_layout Standard
For
\begin_inset Formula $i=1,..,R$
\end_inset
and
\begin_inset Formula $j=1,...,C$
\end_inset
, let
\begin_inset Formula $\hat{E_{ij}}$
\end_inset
denote the M.L.E., when
\begin_inset Formula $H_{0}$
\end_inset
is true of the expected number of observations that will be classified
in the
\begin_inset Formula $i$
\end_inset
th row and the
\begin_inset Formula $j$
\end_inset
th column of the table.
In this problem, the statistic
\begin_inset Formula $Q$
\end_inset
defined by Eq.
(10.2.4) will have the following form :
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
Q=\sum_{i=1}^{R}\sum_{j-1}^{C}\frac{(N_{ij}-\hat{E}_{ij})^{2}}{\hat{E}_{ij}}\text{ (10.3.4)}
\]
\end_inset
\end_layout
\begin_layout Standard
Furthermore, since the contingency table contains RC cells, and since
\begin_inset Formula $s=R+C-2$
\end_inset
parameters are to be estimated when
\begin_inset Formula $H_{0}$
\end_inset
is true, it follows that when
\begin_inset Formula $H_{0}$
\end_inset
is true and
\begin_inset Formula $n\rightarrow\infty$
\end_inset
, the c.d.f.
of
\begin_inset Formula $Q$
\end_inset
converges to the c.d.f.
of the
\begin_inset Formula $\chi^{2}$
\end_inset
distribution for which the number of degrees of freedom is
\begin_inset Formula $RC-1-s=(R-1)(C-1)$
\end_inset
\end_layout
\begin_layout Standard
Next, we shall consider the form of the estimator
\begin_inset Formula $\hat{E}_{ij}$
\end_inset
.
The expected number of observations in the
\begin_inset Formula $i$
\end_inset
th row and the
\begin_inset Formula $j$
\end_inset
th column is simply
\begin_inset Formula $np_{ij}$
\end_inset
.
When
\begin_inset Formula $H_{0}$
\end_inset
is true,
\begin_inset Formula $p_{ij}=p_{i+}p_{+j}$
\end_inset
.
Therefore, if
\begin_inset Formula $\hat{p}_{i+}$
\end_inset
and
\begin_inset Formula $\hat{p}_{+j}$
\end_inset
then it follows that
\begin_inset Formula $\hat{E}_{ij}=n\hat{p}_{i+}\hat{p}_{+j}$
\end_inset
.
\begin_inset Formula $\hat{p}_{i+}=N_{i+}/n$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\hat{E}_{ij}=n(\frac{N_{i+}}{n})(\frac{N_{+j}}{n})=\frac{N_{i+}N_{+j}}{n}\text{ (10.3.5)}.
\]
\end_inset
\end_layout
\begin_layout Standard
If we substitute this value of
\begin_inset Formula $\hat{E}_{ij}$
\end_inset
into Eq.
(10.3.4), we can calculate the value of
\begin_inset Formula $Q$
\end_inset
from the observed values of
\begin_inset Formula $N_{ij}.$
\end_inset
The null hypothesis
\begin_inset Formula $H_{0}$
\end_inset
should be rejected if
\begin_inset Formula $Q\geq c$
\end_inset
, where
\begin_inset Formula $c$
\end_inset
is an appropriately chosen constant.
When
\begin_inset Formula $H_{0}$
\end_inset
is true, and the sample size
\begin_inset Formula $n$
\end_inset
is large, the distribution of
\begin_inset Formula $Q$
\end_inset
will be approximately
\begin_inset Formula $\chi^{2}$
\end_inset
distribution with
\begin_inset Formula $(R-1)(C-1)$
\end_inset
degrees of freedom.
\end_layout
\begin_layout Section*
Summary
\end_layout
\begin_layout Standard
We learned how to test the null hypothesis that two discrete random variables
are independent based on a random sample of
\begin_inset Formula $n$
\end_inset
pairs.
First, form a contingency table of the counts for every pair of possible
observed values.
Then, estimate the two marginal distribution of the two random variables.
Under the null hypothesis that the random variables are independent, the
expected count for value
\begin_inset Formula $i$
\end_inset
of the first variable and value
\begin_inset Formula $j$
\end_inset
of the second variable in
\begin_inset Formula $n$
\end_inset
times the product the product of the two estimated marginal probabilites.
We then form the
\begin_inset Formula $\chi^{2}$
\end_inset
statistic
\begin_inset Formula $Q$
\end_inset
by summing (observed-expected)
\begin_inset Formula $^{2}/$
\end_inset
expected over all the cells in the contingency table.
The degrees of freedom is
\begin_inset Formula $(R-1)(C-1)$
\end_inset
, where
\begin_inset Formula $R$
\end_inset
is the number of rows in the table and
\begin_inset Formula $C$
\end_inset
is the number of columns.
\end_layout
\begin_layout Chapter*
10.4.
Tests of Homogeneity
\end_layout
\begin_layout Standard
Imagine that we select subjects from several different populations, and
that we observe a discrete random variable for each subject.
We might be interested in whether or not the distribution of that discrete
random variable is the same in each population.
There is a
\begin_inset Formula $\chi^{2}$
\end_inset
test of this hypothesis that is very similar to the
\begin_inset Formula $\chi^{2}$
\end_inset
test of independence.
\end_layout
\begin_layout Section*
Samples from Several Populations
\end_layout
\begin_layout Standard
In general, we shall consider a problem in which random samples are taken
from
\begin_inset Formula $R$
\end_inset
different populations, and each observation in each sample can be classified
as one of
\begin_inset Formula $C$
\end_inset
different types.
Thus, the data obtained from the
\begin_inset Formula $R$
\end_inset
samples,can be represented in an
\begin_inset Formula $RxC$
\end_inset
table.
For
\begin_inset Formula $i=1,...,R$
\end_inset
, and
\begin_inset Formula $j=1,..,C$
\end_inset
, we shall let
\begin_inset Formula $p_{ij}$
\end_inset
denote the probability that an observation chosen at random from the
\begin_inset Formula $i$
\end_inset
tj population will be of type
\begin_inset Formula $j$
\end_inset
.
Thus,
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{j=1}^{C}p_{ij}=1\text{ for }i=1,...,R
\]
\end_inset
\end_layout
\begin_layout Standard
The hypotheses to be tested are as follows:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
H_{0}:p_{1j}=p_{2j}=...=P_{Rj}\text{ for }j=1,...,C
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
H_{1}:\text{ The hypothesis }H_{0}\text{ is not true(10.4.1)}
\]
\end_inset
\end_layout
\begin_layout Standard
If the null hypothesis in (10.4.1) were true, then combining the
\begin_inset Formula $R$
\end_inset