Real Statistics Release 8.5

Tests for Homogeneity of Variances

The following new worksheet functions have been added which test for homogeneity of variances in addition to the existing Levene’s test (LEVENE) and the Fligner-Killeen test (FKTEST).

BARTLETT(R1) = p-value of Bartlett’s test

CONTEST(R1) = p-value of the Conover Squared Ranks test

Bradley-Terry Model

The objective of the Bradley-Terry model is to determine the ranking of a number of variables representing competitors, based on the results of pairwise competitions. The following new worksheet array functions have been added to support such models. Here R1 is an n × n square array or cell range that contains the number of wins/losses between n competitors and R2 is a column array with n rows containing the rankings of n competitors.

BT_MODEL(R1, prec, iter): returns a column array with the rankings of the n competitors based on the data in R1. prec = a small positive number (default .0000001); iterations stop when this level of precision is reached with a maximum of iter iterations (default 100).

Pairwise_Sample(R1): returns an n × n array containing the probability of a win for each pair based on the data in R1.

Pairwise_Pop(R2): returns an n × n square array containing the probability of a win for each pair based on the rankings in the column array R2.

Divergence

The following new worksheet functions have been added that provide a measure of diversity based on column arrays or cell ranges R1 and R2 with the same number of rows.

KL_DIVERGE(R1, R2) = the Kullback-Leibler Divergence for R1 || R2

JS_DIVERGE(R1, R2) = the Jenson-Shannon Divergence for R1 || R2

In addition, the new release also supports the divergence measurement used for credit scoring. This takes the form of a new CS_DIVERGE worksheet function that uses any of the following formats:

CS_DIVERGE(R1): an array R1 that contains three columns: the first column contains data values and the second and third columns contain frequencies (non-negative integers) for the two samples

CS_DIVERGE(R1): an array R1 that contains two columns consisting of the data for two samples

CS_DIVERGE(R1, R2): column arrays R1 and R2 that contain the data for two samples

CS_DIVERGE(R1, R2, R3): column arrays R1, R2, and R3: R1 contains data values and R2 and R3 contain frequencies for the two samples

Tukey’s Additivity Test

In the case of two-factor ANOVA without replications, there is no interaction term (it is subsumed in the error term), and so there is no straightforward way to determine whether the interaction effect is significant. Instead, we can use Tukey’s Additivity Test to determine whether the interaction between the factors is indeed significant.

The following new worksheet function supports this test.

ADDITIVITY_ANOVA(R1) = p-value of Turkey’s Additivity Test for the data in R1 in Excel format.

If the data is in standard (stacked) format, then you can use the array formula =ADDITIVITY_ANOVA(StdAnova2(R1)).

Dummy (Tag) Code Enhancement

The existing TAGCODE array function has been enhanced to optionally use the first value in a column array as the reference code. Previously only the last value could serve as the reference code. The new form of this worksheet function is as follows:

TAGCODE(R1, code, last): returns an array with k-1 columns where k = the number of unique values in the column array R1. If code = TRUE (default) then 0, 1 coding is used; otherwise, 0, 1, -1 coding is used. If last = TRUE (default) the last value in R1 is used as the reference category; otherwise, the first value is used as the reference category.

Note that if R1 contains the values A, B, A, C, B, B (in that order), then the first value is A, while the last value is C (and not B) since it is the last value to make its (first) appearance.

The Extract Columns from a Data Range data analysis tool (on the Desc tab) has also been revised to allow you to choose either the first or last element as the reference.

List of Worksheet Names

The following array function has been added.

SheetNames(): returns a column array with a list of all the sheet names for the current workbook

Fisher Exact Test Enhancement

A new option has been added to the FISHER_MIDP function for a 2 × 2 matrix both of whose opposite diagonal elements are equal.

Bug Fixes

• Fixed an error when choosing the Residuals and Cook’s D option from the Linear Regression data analysis tool. Previously the Studentized Residuals chart inadvertently plotted the standardized residuals instead of the studentized residuals. This has now been corrected.
• Fixed an error in the calculation of Cohen’s h effect size reported by the Two-sample Proportion Test data analysis tool.
• In some languages, TRUE and FALSE were not translated properly by Real Statistics. This should be corrected in this release.