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

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:

• Katherine will miss class today and Monday.
• I spent until midnight last night getting today's classes semi-prepared. My 151 students claim that you'll understand that the preparation meant that I did not get your exams graded.
• Extra credit for any one Pride Week event.
• Is anyone in Symphonic band? If so, EC for Sunday's concert.
• Due Monday: Mini-Project Memos.

• Handouts: R Notes for Topic 25.
• Due: 24-6, 24-7, 24-9, 24-15, 24-20.

Overview:

• Presentation Debriefing.
• Today's Topic: Inferences from Samples for Categorical Variables, Continued.
• Some R.

Debriefing on the Presentation

• What did you see as strengths?
• What did you see as weaknesses?
• What errors did you note?
  • I apologize that correcting errors may embarrass some of you. However, I think that it's important that you hear about errors.

What Samples Tell Us About Categorical Variables, Continued

• Context: Why do we call the topics of the first few weeks descriptive statistics and the more recent topics inferential startistics?
• Topic 21: Two populations, one binary categorical variable.
• Topic 24: One population, one non-binary categorical variable.
• Topic 25: One population, two categorical variables (binary or non-binary).
  • Can you think about the values for the explanatory variable as representing different populations?
  • What might your null and alternate hypothesis be?
• Just as in topic 24, we answer questions about populations using a chi-square test.
  • That is, we compute expected values, subtract them from observed, square the difference, divide by the expected, and sum all of those values.

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