odds ratio \(= \frac{n_{11}, n_{22}}{n_{21}\cdot n_{12}}\)

concordant - in agreement

• a concordant pair is a pair of observations

  • each on two variables \((X_1,Y_1)\) and \((X_2,Y_2)\)

    • having the property that

      \({\displaystyle \operatorname {sgn} (X_{2}-X_{1})\ =\operatorname {sgn} (Y_{2}-Y_{1}),}\)

      • where \(sgn\) refers to a whether a number is positive, zero, or negative (its sign)
        • the signum function is defined as: \({\displaystyle \operatorname {sgn} x={\begin{cases}-1,&x<0\\0,&x=0\\1,&x>0\end{cases}}}\)

• a discordant pair is a pair of two variables observations such that

  \({\displaystyle \operatorname {sgn} (X_{2}-X_{1})\ =-\operatorname {sgn} (Y_{2}-Y_{1})}\)

  • if one pair contains a higher value value \(X\) then the other pair contains a higher value of \(Y\)

Somer's D

Goodman's and Kruskal's gamma

P: Concordant pair Q: Discordant pair

\(G = (P-Q)/(P+Q)\)

Kendall tau-b

Stuart tau

c

Pearson Chi-square test

Mantel-Haenszel chi-square test

significance association

• cannot assess direction at all

Spearman Correlation Statistic

Cramer's V

Practice

when a sample size decreases

• the p-value increaes
  • and the width of the CL for the odds ratio increases

there aren't upper bound and lower bound for a logit

\(logit = \log(\frac{p}{1-p}) = x\beta\)

Greenacre's method

similar to gradient descent to find the maximum likelihood

• chooses a least reduction of chi square

• hierachically clusters

• collapses levels of contigency tables

Variable Clustering

• complexity of dataset increases rapidly with increasing dimensionality

  • computation times, exploring the model, model scoring, redundancy in the datset

    Eigenvalue - variance explained by each PC for all the variables

    • column total of each PC

      • if an eigenvalue of PC is greater than specific threshold
        • then cluster is split