MATH 251, Probability and Statistics I, Fall 2001, Wed. Dec. 5, Day 39

Chi-Square (Ch. 9) wins by a nose.

Recall: A Two-way table sorts n individuals into categories (cells), where the categories are the cross-classification possibilities of two variables.  This chapter assumes each individual is counted in only one cell, and that each drops into its cell independent of what cells the others drop into. (That is, our "subjects" don't influence each other.  The experimenter may be doing a treatment to influence the subjects.).
Problem 9.6:  Does contacting people before a mail survey increase the response rate (cut nonresponse)?
                      Intervention
Response    Letter   Phone call   None  (Total)
 Yes         171      146         118    435
 No          220       68         455    743
Total        391      214         573   1178
         Looks like a difference: too much to have occurred by chance?
Note: WE controlled how many got each type of intervention.  CHANCE controlled the response within each.  Intervention is "explanatory," Response is "response"
H₀: Proportions of response in all 3 interventions (in the population) are the same.
H_a: Not all the same.

What would we expect if they were all the same?  The same proportion as if we pooled the results from all interventions..  E.g. None&Yes should be in the same proportion to None subtotal as total&yes is to grand total:  X/573 = 435/1178.  So X = 573 · 435/1178. (= 36.9% of 573 = 211.6)  X=RowTotal ·ColTotal/GrandTotal.
Likewise for all the other cells.

It's useful to compare "observed - expected" = "residual"(Cells/Residuals/Unstandardized), or the same, Standardized.  Which cells are higher than expected, which lower?

The Test statistic is the "(Pearson) Chi-square":
    For each cell, find (observed - expected) ²/expected.  Sum for all cells.
Degrees of freedom= (#rows-1)(#cols-1).  Look in table F (p. T-20): P-value is right tail.
   For this data set, chisquare = 163.413, d.f. 2.  Way off the right end of the table.
(It says "2-sided" in the SPSS output, but it is doing the right tail only of the chi-square distribution.  "2-sided" here means you don't know which way the alternative is going.)
- - - - - - - -  - - -  - -
9.17  Do alcohol use and nicotine use go together?  Data from 452 women: alcohol  and nicotine use during early pregnancy.  We control Neither variable.
H₀: Alcohol use and Nicotine use are Independent
H_a: Alcohol use and Nicotine use are Dependent.
What is Expected in Alc100&Nic16 cell?  Nic16Total/GrandTotal  x Alc100Total/GrandTotal , is proportion expected there.  Multiply by GrandTotal to get count expected.  Expected =  RowTotal ·ColTotal/GrandTotal.

This is a Different underlying model, exactly the same computations as before.
d.f.= (4-1)(3-1) = 6.  chisquare = 42.252, off the table again.

Reading: Text pp. 624-634(skim meta-analysis), Models 639-643
SPSS:  Entering data that is already in a table: You did this on Day 12 (SPSSpp. 66-72) Data>Weight, by count variable.    Note, you can import from Text files OK, but Excel files are formatted wrong.
Analyze>Descriptive Statistics>Crosstabs. Cells/Counts/Expected, + any percentages desired, Residuals. Statistics/ChiSquared.   Note, cells are arranged in alphabetical order. Mess with names to get them nice.