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\title{Stat 312: Lecture 27\\
Testing for Independence}
% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}
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\maketitle \thispagestyle{empty}
\subsection*{Concepts}
\begin{enumerate}
\item Suppose that each individual in a population is classified
with respect to two different {\em factors} into a two-way
contingency table of size $I \times J$.

\item The null hypothesis of interest would be testing if the two
factors are independent:
$$H_0: p_{ij} = p_{i\bullet}p_{\bullet j} \mbox{ for all } i,j,$$
where $p_{i\bullet}=\sum_{j=1}^J p_{ij}$ and $p_{\bullet
j}=\sum_{i=1}^J p_{ij}.$

 \item Under $H_0$, the expected number of element $\mathbb{E} N_{ij} =
 np_{ij} = np_{i\bullet}p_{\bullet j}$. Since the marginal
 probabilities are unknown, we estimate them by $\hat p_{i
 \bullet}=n_{i \bullet}/n$, the sample proportion for category $i$
 for factor 1 and $\hat p_{\bullet j}=n_{\bullet j}/n$, the sample proportion
 for category $j$ for factor 2. So $\mathbb{E} N_{ij} = n_{i \bullet}n_{\bullet
 j}/n$.

 \item Test statistic
 $$\chi^2 = \sum_{i,j} \frac{(n_{ij}
 -\mathbb{E}N_{ij})^2}{\mathbb{E}N_{ij}} \sim
 \chi^2_{(I-1)(J-1)}.$$
This is the test statistic that has been used in testing
homogeneity!
\end{enumerate}

\subsection*{In-class problems}
Excercise 14.41. Test if the parental use of alcohol and drugs
is independent of the student usage of marijuana for 445 college students.\\
\begin{tabular}{|c|c|c|c|}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
   parent$\backslash$student& never & occasional & regular \\
  \hline
  neither & 141 & 54 & 40 \\
  one & 68 & 44 & 51 \\
  both & 17 & 11 & 19 \\
  \hline
\end{tabular}\\
\subsection*{Self-study problems}
{\em Example 14.14.}\\
 {\em Example.} Show that the two hypotheses
$H_0: p_{ij} = p_{i\bullet}p_{\bullet j} \mbox{ for all } i,j$ and
$H_0: p_{1j} = p_{2j} = \cdots = p_{Ij} \mbox{ for all } j$ are
equivalent. Hence testing for homogeneity is equivalent to testing
independence.\\
{\bf The End of Lectures. ra...ra...ra...} Thank you all. In the next class, I will
solve some sample final exams.
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