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MA181 INTRODUCTION TO STATISTICAL MODELLING

GOODNESS-OF-FIT TEST

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\item[Example - Mendel's peas]
Mendel's double intercross data for
$(Round,Yellow)\times(Wrinkled, Green)$ peas. The expected
frequencies are in the ratios 9:3:3:1 on the assumption that the
factors segregate independently.

\begin{center}
\begin{tabular}{cccccc}
\hline &$RY$&$WY$&$RG$&$WG$&Total\\ \hline Observed
frequency&315&101&108&32&556\\Expected
frequency&$312\frac{3}{4}$&$104\frac{1}{4}$&$104\frac{1}{4}$&$34\frac{3}{4}$&556\\
\hline
\end{tabular}
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$$x^2=\frac{(2\frac{1}{4})^2}{312\frac{3}{4}}+\frac{(3\frac{1}{4})^2}{104\frac{1}{4}}
+\frac{(3\frac{1}{4})^2}{104\frac{1}{4}}+\frac{(2\frac{3}{4})^2}{34\frac{3}{4}}=0.470$$

Critical regions with $\nu=3$ are $x^2>7.815$ for $\alpha=0.05$
and $x^2>11.34$ for $\alpha=0.01$. Therefore accept $H_0$: the two
genes segregate independently.

\item[Example - Pharbitis]
Double intercross data for two genes $A$ and $B$ in Pharbitis. The
expected frequencies are again in the ratios 9:3:3:1, on the
assumption that $A$ and $B$ segregate independently.

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\begin{tabular}{cccccc}
\hline &$AB$&$Ab$&$aB$&$ab$&Total\\ \hline Observed
frequency&187&35&37&31&290\\ Expected
frequency&$163\frac{3}{8}$&$54\frac{3}{8}$&$54\frac{3}{8}$&$18\frac{1}{8}$&290\\
\hline
\end{tabular}
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$$x^2=\frac{(23\frac{7}{8})^2}{163\frac{1}{8}}+\frac{(19\frac{3}{8})^2}{54\frac{3}{8}}+\frac{(17\frac{3}{8})^2}{54\frac{3}{8}}+\frac{(12\frac{7}{8})^2}{18\frac{1}{8}}=25.096$$

Critical regions with $\nu=3$

$x^2>7.815$ for $\alpha=0.05$

$x^2>11.34$ for $\alpha=0.01$

$x^2>16.27$ for $\alpha=0.001$

Reject $H_0$ and conclude (very strongly) that the genes are
linked.

\item[Estimating parameters]

\item[Example - Pharbitis revisited]
One theory suggests that the probabilities for the four cells can
be written as $(2+\theta)/4,\ (1-\theta)/4,\ (1-\theta)/4$ and
$\theta/4$ for some parameter $\theta$. The maximum likelihood
estimate of $\theta$ is $\hat{\theta}=0.4835$, which leads to the
expected frequencies given in the following.

\begin{center}
\begin{tabular}{cccccc}
\hline &$AB$&$Ab$&$aB$&$ab$&Total\\ \hline Observed
Frequency&187&35&37&31&290\\ Expected
frequency&180.054&37.446&37.446&35.054&290\\ \hline
\end{tabular}
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$$x^2=\frac{(187-180.054)^2}{180.054}+\ldots+\frac{(31-35.054)^2}{35.054}=0.902$$

Critical regions with 2 degrees of freedom are

$x^2>5.991$ for $\alpha=0.05$

$x^2>9/210$ for $\alpha=0.01$

Accept $H_0$: model given as above.

\item[Example Peas in pods]
The table below gives, in its second column, the frequency
distribution of the number $Y$£ of peas found in the pod of a
four-seeded line of pea. A total of 269 pods were inspected.

$\hat{\pi}=0.5530$

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\begin{tabular}{ccccccc}
\hline Number of peas in pod&0&1&2&3&4&Total\\ \hline Observed
frequency&16&45&100&82&26&269\\ Expected
frequency&10.74&53.15&98.62&81.33&25.15&268.99\\ \hline
\end{tabular}
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$$x^2=\frac{(16-10.74)^2}{10.74}+\ldots+\frac{(26-25.15)^2}{25.15}=3.88.$$

Critical regions with three degrees of freedom as on page 1. Do
not reject $H_0$: model given by binomial distribution.

\item[Small expected frequencies]
No expected frequency should be smaller than one and no more than
20\% should be less than five. Otherwise it is necessary to pool
cells.

\item[Example - Poisson distribution]
The number $Y$, of $\alpha$-particles emitted by a film of
Polonium in 2608 intervals of $\frac{1}{8}$ minute was given on
the Poisson distribution handout. The end of the table is as
follows:

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\begin{tabular}{|ccc|}
\hline &Frequency of Intervals&\\ $y$&Observed&Poisson, $E_y$\\
\hline 10&10&11.3\\ 11&4&4.0\\ 12&0&1.3\\ 13&1&0.4\\ 14&1&0.1\\
$\geq15$&0&0.0\\ \hline \end{tabular}
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The last four cells may be pooled to give the following complete
table.

$\hat{\mu}=3.8715$

\begin{tabular}{ccccccccccc}
\hline $y$&0&1&2&3&4&5&6&7&8&9\\ \hline
$O_y$&57&203&383&525&532&408&273&139&45&27\\
$E_y$&54.3&210.3&407.1&525.3&508.4&393.7&254.0&140.5&68.0&29.2\\
\hline
\end{tabular}

\begin{tabular}{cccc}
\hline 10&11&$\geq12$&Total\\ \hline 10&4&2&2608\\
11.3&4.0&1.8&2607.9\\ \hline
\end{tabular}


$x^2=13.0$

Critical regions with 11 degrees of freedom

$x^2>19.68$ for $\alpha=0.05$

$x^2>24.72$ for $\alpha=0.01$

$x^2>31.26$ for $\alpha=0.001$

Accept $H_0$: model is given by Poisson distribution.

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