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\title{Stat 312: Lecture 23\\
Categorical Data}
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\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\section*{Concepts}
\begin{enumerate}
\item Multinomial data. There are $k$ outcomes (categories): $C_1,
C_2, \cdots, C_k$. The probability of sample $X_i$  belonging to
category $C_j$ is $p_j$, i.e.
$$P(X_i \in C_1)=p_1, \cdots, P(X_i \in C_k)=p_k.$$

\item The null hypothesis of interest is
$$H_0: p_1=c_1, \cdots, p_k=c_k$$
for given constants $c_1,\cdots,c_k$ ($c_1 + \cdots + c_k=1$).
$$H_1: p_j \neq c_j \mbox{ for some } j.$$

\item Let $x_1,\cdots,x_n$ be the observations. Suppose that there
are $N_j$ items in category $C_j$. Note that $N_1 + \cdots +
N_k=n$. We assume $N_1,\cdots N_j$ to be distributed as {\em
multinomial} with parameter $p_1,\cdots,p_k$. The {\em expected}
number of elements in category $C_j$ is $\mathbb{E} N_j = np_j$.

\item {\em Chi-
squared goodness-of-fit test.} The test procedure
involes measuring the relative discrepancy between the expectation
and observation using
$$\chi^2 =\Big(\frac{N_1 -\mathbb{E}N_1}{\mathbb{E}N_1}\Big)^2 + \cdots
\Big(\frac{N_k -\mathbb{E}N_k}{\mathbb{E}N_k}\Big)^2.$$ Assuming
$np_j \geq 5$ for every $j$, $$\chi^2 \sim \chi_{k-1}^2.$$

\end{enumerate}

\section*{In-class problems}
Example 1. There are four blood types (ABO system). It is believed
that 34, 15, 23 and 28 $\%$ of people have blood type A, B, AB and
O respectively. The blood samples of 100 students were collected:
A(12), B(56), AB(2), O(30). Test if the collected blood sample
contradicts the previous belief.

\section*{Self-study problems}
Read p. 616-619. Example 14.1.



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