====== Tests of association ======


===== Chi-squared test =====

The Chi-squared test is a common test of association between nominal or ordinal data. More powerful tests exist (see below), but the X-squared is by far the most simple one.

The statistics value is calculated as:
<m>chi^2 = sum{}{}{{(O - E)^2}/E}</m>

Where **O** means //Observed values// and **E** means //Expected values//. See the book for a detailed explanation.

==== Contingency table ====


Produce a contingency table of ''Mat'' by ''Period'', a new variable made from ''Date'' to have three categories, and calculate chi-squared.

Creating the ''Period'' variable from ''Date'' has already been covered in “[[Transforming variables]]”:
<code C>
> Period <- Date
> Period[(Date>650)&(Date<=1200)] <- 1
> Period[(Date>100)&(Date<=650)] <- 2
> Period[(Date<=100)] <- 3
</code>

You should probably tell R these values aren't numbers but categories. Notice the difference:

<code C>
> summary(Period)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1.00    1.00    1.50    1.55    2.00    3.00 
> Period <- factor(Period)
> Mat <- factor(Mat)
> summary(Period)
 1  2  3 
20 18  2 
</code>

This hasn't really effect on the following operations, but it helps you keeping a clean working environment.

<code C>
> table(Mat,Period)
    Period
Mat   1  2  3
   1 20  0  0
   2  0 18  2

</code>

See http://finzi.psych.upenn.edu/R/Rhelp02a/archive/2847.html for another method using ''xtabs()''.

==== Chi-squared test ====

We are now ready to perform the Chi-squared test:

<code C>
> crosstab <- table(Mat,Period)
> xtabs() # similar to table, but different results
> chisq.test(crosstab)

	Pearson`s Chi-squared test

data:  table(Mat, Period) 
X-squared = 40, df = 2, p-value = 2.061e-09

Warning message:
In chisq.test(table(Mat, Period)) :
  Chi-squared approximation may be incorrect

</code>

This result is OK, but has some differences from the one you would get doing all the operations by hand:
  * the //''p-value''// is not a fixed one (because you're not using tables), but rather a floating point number, expressed in scientific notation. It is very low however.
  * there's a warning about a possible approximation of the //''χ-squared''// value

===== Other tests =====

Other tests of association mentioned in //Digging Numbers// don't seem so widely used, and this is probably the reason why they are not part of the standard R distribution.

==== Cramer's V ====

This test is included in the ''[[http://cran.r-project.org/web/packages/cramer/index.html|cramer]]'' contributed package 

==== Guttman's lambda ====

==== Kendall's tau-b ====

This test is included in the contributed package ''[[http://cran.r-project.org/web/packages/Kendall/index.html|Kendall]]''. See 

==== Kendall's tau-c ====

Kendall's tau-c is not included in any package, but it can be defined as a custom function. See https://stat.ethz.ch/pipermail/r-help/2006-September/112806.html





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[[Start]] · [[Data description]] · [[Transforming variables]] · [[Tables]] · [[Pictorial displays]] · [[Measures of position and variability]] · [[Sampling]] · [[Tests of difference]] · [[Tests of distribution]] · [[Correlation]] · **Tests of association**