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**wik_chick88** two clinical psychiatrists assessed five patients for depression, one using the Beck depression inventory and tyhe other using the Hamilton rating scale for depression. the scores recorded for each scale are given in the following table:

$\displaystyle

\begin{array}{c|ccccc}Scale&1&2&3&4&5\\\hline Beck&20&11&13&22&37\\Hamilton&22&14&10&17&31\\\hli ne\end{array}

$

(a) are the two depression scales related? a regression analysis in R gave the following output:

$\displaystyle

\begin{array}{ccc}\ &Estimate\ Std.&Error\\(Intercept)&3.7070&3.8433\\Beck&0.7327 &0.1704\end{array}

$

Residual standard error: 3.498 on 3 degrees of freedom

Based on this analysis, perform a hypothesis test for association between the Hamilton and Beck scales

(b) Kendall's $\displaystyle \tau $ is a nonparametric measure of correlation based on concordant and discordant pairs. two points $\displaystyle (x_{1},y_{1}) $ and $\displaystyle (x_{2},y_{2})$ are concordant if the signs of $\displaystyle x_{2} - x_{1}$ and $\displaystyle y_{2} - y_{1}$ match; otherwise they are discordant. for example, in this data, the pair (20, 22) and (11, 14) are concordant while the pair (20, 22) and (22, 17) are discordant.

given the numbers of concordant and discordant pairs, c and d, respectively, Kendall's $\displaystyle \tau$ is defined by

$\displaystyle \tau = \frac{c - d}{\frac{1}{2}n(n-1)}$

the difference divided by the total number of pairs. calculate the value of $\displaystyle \tau $ for the relationship between the Hamilton and Beck scales in the data above.

(c) based on the value from (b), use this approximation to perform a hypothesis test for association between the Hamilton and Beck scales based on Kendall's $\displaystyle \tau$.