14,539 research outputs found
Rediscovering a little known fact about the t-test and the F-test: Algebraic, Geometric, Distributional and Graphical Considerations
We discuss the role that the null hypothesis should play in the construction
of a test statistic used to make a decision about that hypothesis. To construct
the test statistic for a point null hypothesis about a binomial proportion, a
common recommendation is to act as if the null hypothesis is true. We argue
that, on the surface, the one-sample t-test of a point null hypothesis about a
Gaussian population mean does not appear to follow the recommendation. We show
how simple algebraic manipulations of the usual t-statistic lead to an
equivalent test procedure consistent with the recommendation. We provide
geometric intuition regarding this equivalence and we consider extensions to
testing nested hypotheses in Gaussian linear models. We discuss an application
to graphical residual diagnostics where the form of the test statistic makes a
practical difference. By examining the formulation of the test statistic from
multiple perspectives in this familiar example, we provide simple, concrete
illustrations of some important issues that can guide the formulation of
effective solutions to more complex statistical problems.Comment: 22 pages, 5 figure
Indispensable monomials of toric ideals and Markov bases
Extending the notion of indispensable binomials of a toric ideal, we define
indispensable monomials of a toric ideal and establish some of their
properties. They are useful for searching indispensable binomials of a toric
ideal and for proving the existence or non-existence of a unique minimal system
of binomials generators of a toric ideal. Some examples of indispensable
monomials from statistical models for contingency tables are given.Comment: 20 pages, 5 figure
Sample size for comparing negative binomial rates in noninferiority and equivalence trials with unequal follow-up times
We derive the sample size formulae for comparing two negative binomial rates
based on both the relative and absolute rate difference metrics in
noninferiority and equivalence trials with unequal follow-up times, and
establish an approximate relationship between the sample sizes required for the
treatment comparison based on the two treatment effect metrics. The proposed
method allows the dispersion parameter to vary by treatment groups. The
accuracy of these methods is assessed by simulations. It is demonstrated that
ignoring the between-subject variation in the follow-up time by setting the
follow-up time for all individuals to be the mean follow-up time may greatly
underestimate the required size, resulting in underpowered studies. Methods are
provided for back-calculating the dispersion parameter based on the published
summary results
Comparison of Thematic Maps Using Symbolic Entropy
Comparison of thematic maps is an important task in a number of disciplines. Map comparison has traditionally been conducted using cell-by-cell agreement indicators, such as the Kappa measure. More recently, other methods have been proposed that take into account not only spatially coincident cells in two maps, but also their surroundings or the spatial structure of their differences. The objective of this paper is to propose a framework for map comparison that considers 1) the patterns of spatial association in two maps, in other words, the map elements in their surroundings; 2) the equivalence of those patterns; and 3) the independence of patterns between maps. Two new statistics for the spatial analysis of qualitative data are introduced. These statistics are based on the symbolic entropy of the maps, and function as measures of map compositional equivalence and independence. As well, all inferential elements to conduct hypothesis testing are developed. The framework is illustrated using real and synthetic maps. Key word: Thematic maps, map comparison, qualitative variables, spatial association, symbolic entropy, hypothesis tests
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