Introduction to Statistics (MAT/SST 115.03 2008S)

Class 32: Topic 25: Inference for Two-Way Tables

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This outline is also available in PDF.

Held: Friday, 18 April 2008

Summary: We consider yet another way in which we use samples to explore populations. We continue to emphasize categorical variables, this time considering pairs of categorical variables. We look at a more general version of the chi-square test from the previous topic.

Notes:

Overview:

Debriefing on the Presentation

What Samples Tell Us About Categorical Variables, Continued

Doing Chi-Square Computations in R

So, how do we do a chi-squre computation in R? It ends up being fairly straightforward.

We start by making a table or data frame from our data.

> SC = fys[,c(1,3)]
> table(SC)
   CHOICE
SEX   1   2   3   4
  1   4  11  34 108
  2  15   7  35 127
> SCframe = data.frame(row.names=c("Below Third","Third","Second","First"),
+   Male=SCtable[1,],
+   Female=SCtable[2,])
> SCframe
            Male Female
Below Third    4     15
Third         11      7
Second        34     35
First        108    127

We can simple apply the chisq.test procedure to this table to get the important values.

> chisq.test(SCframe)
	Pearson's Chi-squared test
data:  SCframe 
X-squared = 6.7122, df = 3, p-value = 0.08166

However, we will often want to look more carefuly at the differences between observed and expected values. The computation of the expected values is a strange formula that I don't expect you to understand. (I do expect that you could do an individual computation by hand, but this does all of them at once.)

> SCexpected = rowSums(SCframe) %o% colSums(SCframe)/sum(SCframe)
> SCexpected
                 Male    Female
Below Third   8.74780  10.25220
Third         8.28739   9.71261
Second       31.76833  37.23167
First       108.19648 126.80352

We can now compare directly.

> SCframe - SCexpected
                  Male     Female
Below Third -4.7478006  4.7478006
Third        2.7126100 -2.7126100
Second       2.2316716 -2.2316716
First       -0.1964809  0.1964809

As importantly, we can compute the deviations.

> (SCframe-SCexpected)^2/SCexpected
                    Male       Female
Below Third 2.5768317632 2.1987097110
Third       0.8878854292 0.7575978934
Second      0.1567711671 0.1337667023
First       0.0003568024 0.0003044455

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Samuel A. Rebelsky, rebelsky@grinnell.edu

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