PART IIa – Essentials--Analysis of Enumerative Data

Enumerate: to count, usually after classification has been performed

Enumerative data: data obtained by classifying and counting occurrences

Multinomial experiment--like the binomial experiment, except each trial has more than two outcomes

n identical trials; k possible outcomes on each trial

Independence--the outcome of one trial does not affect the outcome of any other trial

Constant probabilities for each outcome from trial to trial

p1, p2, p3, . . ., pk are the probabilities of the various outcomes

Cell counts (number of times each outcome occurs) are the variables to be

analyzed

Chi-square (χ2) distribution: continuous, positively skewed

One-dimensional chi-square test--“goodness of fit” tests

Ho: that a population conforms to some expected distribution.

A cell consists of an expectation (E) and an observation (O).

Expected values (E) are derived from Ho.

The number of cells is denoted by k.

Calculated chi-square (test statistic, χ2c) for a cell is the squared deviation

(E-O)2divided by E. The χ2c is the total of all the cells.

Degrees of freedom (df): the number of cells minus one (k-1)

(d.f. = k – 3 when the normal distribution is used.)

(d.f. = k – 2 when the Poisson distribution is used.)

Ho is rejected if χ2c  χ2t , also if p ≤ 

If Ho is rejected, additional information should be reported as to the nature of thedeviation from the expected distribution.

Often used to test for normal distributions.

For the sample size to be sufficient, the expected number (e) in each cell

should equal or exceed 5.

Two-dimensional chi-square test

H0: in the population the row variable and column variable are independent.

Ha: in the population the row variable and column variable are dependent.

Contingency (dependency) table contains a matrix of cells

A cell consists of an expectation (E) and an observation (O).

Expected values (E) are derived from H0 using the multiplication rule for

intersections of independent events: P(A B) = P(A) * P(B).

Calculated chi-square for a cell is (E-O)2 / E (same as above).

Ho is rejected if χ2c  χ2t , also if p ≤ 

Degrees of freedom: number of rows minus one, times number of columns

minus one; (r-1)(c-1) where r and c are the numbers of rows and columns

If H0 is rejected, additional information should be reported as to the nature of the dependencies.

For the sample size to be sufficient, the expected number (e) in each cell

should equal or exceed 5.

Terminology--explain each of the following:

enumerative data, multinomial experiment, binomial experiment, identical trials, independence, one-dimensional or one-way chi-square test, “goodness-of-fit” test, two-dimensional or two-way chi-square test, dependency, contingency table, multiplication rule for intersections of independent events.

Skills and Procedures

  • given appropriate data, conduct a one-way chi-square test and interpret the results
  • given appropriate data, conduct a two-way chi-square test and interpret the results

Concepts

  • describe what is meant by “goodness-of-fit”
  • explain how expected values are determined in a one-way chi-square test
  • explain how the concept of “deviation” applies in chi-square test computations
  • explain how expected values are determined in a two-way chi-square test
  • describe the application of the “multiplication rule for independent events” in two-way chi-square analysis