OPTION 7

Anova Experiments with 3 or More Treatments

Ref: Abramowitz and Stegum: Handbook of Mathematical Functions, 1964. Chapter 26 - Probability Functions

This program computes the power of the F-test for a variety (listed below) of the experimental design situations where three or more groups are going to be compared and the response variable can be assumed to have an approximately normal distribution.

All calculations are performed using the approximation formulae in Chapter 26 of A&S. Most calculations are done iteratively since we've found that some of the formulae give better approximations than the others. In all cases the problem to be solved is some variation of the following:

"If an F-statistic with nu1 dof in the numerator and nu2 dof in the denominator has probability alpha from a central F-distribution, what is the corresponding probability beta (power) if we suppose that the statistic came from a non-central F-distribution with a non-centrality parameter, D, specified by the alternative hypothesis?"

Non-Centrality Parameter

The non-centrality parameter can be identified in one of two ways unless otherwise indicated.

  1. Range of group means.

    2. The user specifies the range in terms of the difference between the maximum group mean and the minimum group mean. The program assumes that the treatment means under the alternative hypothesis will be equally spaced over that range and the non-centrality parameter is computed.

  2. Individual group means.

    3. The user will be prompted for the expected mean of each group under the alternative hypothesis.

    The user is prompted for the method of calculation when a choice is available.

Types of Designs:
  1. One-way Anova

    For replications, b, and treatments, t:

    nu1 = t-1
    nu2 = (b-1)(t)
    D = b[X'X] / VAR
    X - mean corrected vector
    VAR - variance of a single observation
  2. Randomized Blocks

    For blocks, b, and treatments, t:

    nu1 = t-1
    nu2 = (b-1)(t-1)
    D = b[X'X] / VAR
    X - mean corrected vector
    VAR - variance of a single observation
  3. Latin Square

    For treatments, t, and blocks, t.

    nu1 = t-1
    nu2 = (t-1)(t-2)
    D = t[X'X] / VAR
  4. F-test

    5. The user must specify nu1, nu2 and D directly.

For each design, one of the following can be computed:

  1. Significance level
  2. Power
  3. Number of replicates or denominator dof.
  4. Non-centrality parameter or range of means.

when the user supplies the other three.

Details of Calculations
  1. Significance level

    2. A&S 26.6.26 is used to approximate the central F in terms of non-central F'. The adjusted denominator dof is rounded to the nearest integer. A&S 26.6.4 and 2 6.6.8 are used to verify iteratively that the probability is Q(F':nu1,nu2) = BETA + _ .001. The probability of the central F is then evaluated as ALPHA using A&S 26.6.4 and 26.6.8.

  2. Power

    3. First the central F is found iteratively using A&S 26.6.4 and 26.6.8 such that

    Q(F':nu1,nu2) = ALPHA + _ .001

    Then A&S 26.6.26 is used to approximate non-central F' with nu1* (rounded). Then A&S 26.6.4 and 26.6.8 are used to evaluate BETA.

  3. Number of replicates (blocks)

    4. An initial "guess" is generated such that the central F probability is ALPHA + _ .001 using A&S 26.6.4 and 26.6.8. Then A&S 26.6.26 is used to approximate non-central F' with nu1* (rounded). Next A&S 26.6.4 and 26.6.8 are used to evaluate BETA. If BETA > Power then nu2 decreased; if BETA < Power then nu2 is increased. Convergence is attained when

    Prob(nu2-1) < Power + .0005

    and Prob(nu2) > Power.

    Note: The number of replicates (blocks) per treatment can change only by integer values but this can change nu2 by more than one. Since the power associated with the selected number of replicates may be somewhat greater than the desired power, the associated power is printed. If this is much larger than the "desired power" the user is advised to run the program to compute power using one less replicate or dof for comparison. For given nu1, alpha, beta and D there may not be nu2 to satisfy the conditions. In particular, for some D's the beta will not be achieved even when nu2 is infinite; the program bounds nu2 at 150 above and 1 below.

  4. Non-centrality parameter

    5. This is computed as the non-centrality parameter for the "F-test" option. For the other options, this value is interpreted as the maximum difference (i.e. range of group means).

    The approximation proceeds as in (3). Either the range or non-centrality parameter is changed until the difference in the parameter from one iteration to the next is < .0005 or the computed power is within .0005 of the "desired power".