Tests of association
- Chi-squared test
  - Contingency table
  - Chi-squared test
- Other tests

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: $chi^2 = sum{}{}{{(O - E)^2}/E}$

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”:

> Period <- Date
> Period[(Date>650)&(Date<=1200)] <- 1
> Period[(Date>100)&(Date<=650)] <- 2
> Period[(Date<=100)] <- 3

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

> 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

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

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

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:

> 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

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

Tests of association

Chi-squared test

Contingency table

Chi-squared test

Other tests

Cramer's V

Guttman's lambda

Kendall's tau-b

Kendall's tau-c