Analysis of Enumerative Data

ANALYSIS OF ENUMERATIVE DATA

"Enumerate" -- to count.

This type of data is usually generated by a process of observing, classifying, and counting.

MULTINOMIAL EXPERIMENT

Same as a binomial experiment, except there are more than two outcomes for each trial.

* n identical trials,

* k possible outcomes on each trial,

* trials must be independent (the outcome of one trial must not affect any other trial),

* probability of each of the k possible outcomes must be constant from trial to trial,

* of interest is the number of times each of the k possible outcomes occurs.

ONE-WAY (ONE-DIMENSIONAL) CHI-SQUARE (Χ²) TEST

H0: in the population, the distribution of occurrences conforms to some expected

distribution (such as the uniform distribution or the normal distribution).

Each of the k possible outcomes corresponds to a "cell." Cells may be arranged in a one-row or a one-column table.

Each cell contains an observed number of outcomes (o), and an expected number of outcomes (e).

The expected number (e) is derived from Ho.

The observed number (o) is the result of the experiment.

A chi-square (χ²) value is computed for each cell.

χ² for each cell = ( e - o )² / e.

The total χ² for all cells is the test statistic, the calculated-χ² (χ²c).

Ho is rejected if χ²c ³ χ²t and if p £ a.

χ²t is based on α and d.f. (d.f. = number of cells - 1).

(d.f. = number of cells – 3 when the normal distribution is used.)

(d.f. = number of cells – 2 when the Poisson distribution is used.)

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

This test is often used to test for the presence of a normally-distributed population.

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

equal or exceed 5.

TWO-WAY (TWO DIMENSIONAL) CHI-SQUARE (Χ²) TEST -- CONTINGENCY TABLE

"Contingency" means dependency.

A two-way contingency table contains rows and columns, representing the two variables under study. The number of rows = r ; the number of columns = c.

H0: in the population, there is independence between the row variable and the

column variable.

Ha: in the population, there is dependence between the row variable and the

column variable.

Each cell contains an observed number of outcomes (o), and an expected number of outcomes (e).

The expected number (e) is derived from Ho, using the multiplicative rule for independent events: If A and B are independent, then P(A Ç B) = P(A) * P(B).

The observed number (o) is the result of the experiment, obtained by counting.