Analyzes a one-way classification model.

The type double function is imsls_d_anova_oneway

The p-value for the F-statistic.

Function imsls_f_anova_oneway performs an analysis of variance of responses from a oneway classification design. The model is

where the observed value yij constitutes the j-th response in the i-th group, μi denotes the population mean for the i-th group, and the ɛij arguments are errors that are identically and independently distributed normal with mean 0 and variance σ2. Function imsls_f_anova_oneway requires the yij observed responses as input into a single vector y with responses in each group occupying contiguous locations. The analysis of variance table is computed along with the group sample means and standard deviations. A discussion of formulas and interpretations for the one-way analysis of variance problem appears in most elementary statistics texts, e.g., Snedecor and Cochran (1967, Chapter 10).

Function imsls_f_anova_oneway computes simultaneous confidence intervals on all

pairwise comparisons of k means μ1 μ2, …, μk in the one-way analysis of variance model. Any of several methods can be chosen. A good review of these methods is given by Stoline (1981). The methods are also discussed in many elementary statistics texts, e.g., Kirk (1982, pp. 114−127).

Let s2 be the estimated variance of a single observation. Let v be the degrees of freedom associated with s2. Let

The methods are summarized as follows:

Tukey method: The Tukey method gives the narrowest simultaneous confidence intervals for all pairwise differences of means μi − μj in balanced (n1 = n2 = … = nk = n) one-way designs. The method is exact and uses the Studentized range distribution. The formula for the difference μi − μj is given by

where q1−a;k,v is the (1 − α) 100 percentage point of the Studentized range distribution with parameters k and v.

Tukey-Kramer method: The Tukey-Kramer method is an approximate extension of the Tukey method for the unbalanced case. (The method simplifies to the Tukey method for the balanced case.) The method always produces confidence intervals narrower than the Dunn-Šidák and Bonferroni methods. Hayter (1984) proved that the method is conservative, i.e., the method guarantees a confidence coverage of at least (1 − α) 100. Hayter’s proof gave further support to earlier recommendations for its use (Stoline 1981). (Methods that are currently better are restricted to special cases and only offer improvement in severely unbalanced cases; see, for example, Spurrier and Isham 1985.) The formula for the difference μi − μj is given by the following:

Dunn-Šidák method: The Dunn-Šidák method is a conservative method. The method gives wider intervals than the Tukey-Kramer method. (For large v and small α and k, the difference is only slight.) The method is slightly better than the Bonferroni method and is based on an improved Bonferroni (multiplicative) inequality (Miller 1980, pp. 101, 254−255). The method uses the t distribution (see function imsls_f_t_inverse_cdf, Chapter 11, Probability Distribution Functions and Inverses). The formula for the difference μi − μj is given by

where tf ;v is the 100f percentage point of the t distribution with ν degrees of freedom.

Bonferroni method: The Bonferroni method is a conservative method based on the Bonferroni (additive) inequality (Miller, p. 8). The method uses the t distribution. The formula for the difference μi − μj is given by the following:

Scheffé method: The Scheffé method is an overly conservative method for simultaneous confidence intervals on pairwise difference of means. The method is applicable for simultaneous confidence intervals on all contrasts, i.e., all linear combinations

where the following is true:

This method can be recommended here only if a large number of confidence intervals on contrasts in addition to the pairwise differences of means are to be constructed. The method uses the F distribution (see function imsls_f_F_inverse_cdf, Chapter 11, Probability Distribution Functions and Inverses). The formula for the difference μi − μj is given by

where F1−a; ( k−1),v is the (1 − α) 100 percentage point of the F distribution with k − 1 and ν degrees of freedom.

One-at-a-Time t method (Fisher’s LSD): The One-at-a-Time t method is appropriate for constructing a single confidence interval. The confidence percentage input is appropriate for one interval at a time. The method has been used widely in conjunction with the overall test of the null hypothesis μ1 = μ2 = … = μk by the use of the F statistic. Fisher’s LSD (least significant difference) test is a two-stage test that proceeds to make pairwise comparisons of means only if the overall F test is significant. Milliken and Johnson (1984, p. 31) recommend LSD comparisons after a significant F only if the number of comparisons is small and the comparisons were planned prior to the analysis. If many unplanned comparisons are made, they recommend Scheffé’s method. If the F test is insignificant, a few planned comparisons for differences in means can still be performed by using either Tukey, Tukey-Kramer, Dunn-Šidák,or Bonferroni methods. Because the F test is insignificant, Scheffé’s method does not yield any significant differences. The formula for the difference μi − μj is given by the following:

This example computes a one-way analysis of variance for data discussed by Searle (1971, Table 5.1, pp. 165−179). The responses are plant weights for six plants of three different types—three normal, two off-types, and one aberrant. The responses are given by type of plant in the following table:

The data used in this example is the same as that used in the initial example. Here, the anova_table is printed.

Simultaneous confidence intervals are generated for the following measurements of cold-cranking power for five models of automobile batteries. Nelson (1989, pp. 232−241) provided the data and approach.

The Tukey method is chosen for the analysis of pairwise comparisons, with a confidence level of 99 percent. The means and their confidence limits are output.