%This is written for easy conversion into a LaTeX document. I can convert it into normal text if this proves to be inconvenient. \documentclass[12pt]{article} \usepackage{amsmath} \begin{document} \title{Tests of Equivalence for Microarrays} %everybody in renn lab in same order as other paper: \author{Jones, A., Machado, H.E., Langdon, Q., Natale, S., Lee, T., Winterman, A., and Renn S.C.P.} \date{} \maketitle \begin{abstract} A test of equivalence is motivated and described in detail. Use of tests of equivalence for micro-array experiments is advocated. Examples are given which compare gene lists found equivalent by a test of equivalence against gene lists with no significant evidence of differential expression. \end{abstract} \section{ Introduction} The significance tests commonly employed by biologists in analysis of micro-arrays (citations of examples?), such as two sided t- tests, are designed to find evidence of differential expression at some p- value, $p = 0.05$, for example (Smyth 2004). If the p-value for a given feature is above $0.05$, it is commonly said that the replicates under comparison exhibit no difference in expression. This is an inaccurate, an example of the logical fallacy: argumentum ad ignorantiam. The test makes no statement about the genes above the $p = 0.05$ threshold level. It has found no evidence of differential expression, but that lack of evidence is not the same as evidence of equivalence (Altman 2005). A different test is required to test for equivalence. As those genes which are expressed equivalently across hybridizations are of interest, tests of equivalence should be a part of any micro-array analystÕs statistical tool kit. Tests of equivalence are not a new statistical idea. Most commonly used by manufacturers of generic drugs to avoid expensive, full-blown clinical trials, tests of equivalence have found widespread use in the industry since the 1960Õs, following a Federal Drug Admistration decision (and similar rulings by similar bodies elsewhere) stating that new drugs could be approved based on statistical equivalence to existing, approved drugs (Wellek 2003). For this reason, even a casual search turns up ample examples of the implementation of such tests in a clinical setting (citations needed). In the micro-array context, use of tests of equivalence is sparse. In an exhaustive survey only one paper, Eijgelaar 2010, was found which employed tests of equivalence in experiments using micro-array analysis --(This will be reworked with more accurate information once IÕve looked a little more)--. In addition to the test of equivalence, Eijgelaar carried out complicated statistical analyses involving permutation tests and false discovery rates. It is the intention of this paper to elaborate on tests of equivalence in the micro-array setting and thereby make them accessible to the average micro-array analyst. Micro-arrays present a difficult setting for such a test. It is possible to formulate a naive test such as the one currently presented below, however making one which maximizes \section{Methods} \subsection{The Test} This section lacks references, but can mostly be supported by Wellek A test of equivalence compares a measurement, $d$, of the difference between two groups of data against a pre-specified error term, $\epsilon$, which represents the acceptable level of error in the experiment. Epsilon should be determined through biological or biochemical criterion having to do with the nature and quality of the arrays and the experimental design. If the difference ($d$) is less in absolute value than the error ($\epsilon$), the two groups are indistinguishable and hence equivalent. In this sense, epsilon specifies the resolution of the experiment. The null hypothesis in the test is: \[1]begin{equation} \begin{aligned} & H_0 : |d| \geq \epsilon \\ \mbox{vs.} H_a : |d| \leq \epsilon \end{aligned} \end{equation} The difference, $|d| - \epsilon$, gives the largest possible confidence interval for the true value of |d| which is still completely contained in the error bars. The confidence level - the probability that the interval contains the true value of d - of this maximal interval, leads directly to a p-value. Suppose $|d|$ has standard error (or is it deviation?) sigma, and that $D$ is a random variable with the same distribution as that which produced $|d|$. Then the significance level, $\alpha$, for the confidence interval can be computed: \begin{equation} \alpha = Pr \bigg( D \leq [2]\frac{|d| - \epsilon}{\sigma} \bigg), \end{equation} meaning the probability that |d| is actually greater than this alpha epsilon is less than alpha. If using the R environment for statistical computing, the above command \subsection{[3]Measuring Epsilon} One source of difficulty is specifying exactly what is meant by the term Òequivalence,Ó that is, specifying the difference $\epsilon$ below which we consider two groups indistinguishable. We would like epsilon to be the threshold below which actual variation is indistinguishable from experimental error. If there are self-hybridizations, we can measure the sample standard error, $s$ about the mean, in these hybridization and set epsilon equal to a proportion (i.e. $s / 2$) of this value. A non-zero mean may be expected because the two different dyes used do not necessarily hybridize equally to the array. Any difference in hybridization falling inside this epsilon cannot be distinguished from a self hybridization and is hence equivalent in a very clear sense. The next best thing is to compare members of the same phenotype, or even better, tissues from the same individual from the same organ. To measure epsilon... The test of Equivalence described will be first described itself and then shown to be a compound hypothesis test: a double one sided t - test. The hope is that the average biologist with limited mathematical training will be able to understand and employ the test. Supplemental material available at http://wherever_this_will_be_available.com will include scripts written for the R environment for statistical computing (cite) which carry out the test. \section{Results} Compare Tof EQ genelist to genes which are not signifficantly differentiall expressed \section{discussion} Other tests of equivalence, reiterate reasons for use, improvements. This test does not make any attempt to maximize power - that is the probability of rejecting a false null hypotheses. Power maximizing tests modify the size of the confidence interval. Ours is rather naive in its assumptions. \end{document} A few resources: Requested on ILLiad http://www.informaworld.com/smpp/content~db=all~content=a920110117 @article{ title = ÒEvaluation of a Statistical Equivalence Test Applied to Microarray DataÓ, author = Ò Qiu J (Qiu, Jing), Cui XQ (Cui, Xiangqin)Ó, journal = ÒJOURNAL OF BIOPHARMACEUTICAL STATISTICSÓ, volume = Ò20Ó, issue = 2 , pages = 240-266 , year = 2010 , } http://library-catalog.reed.edu/search/i?SEARCH=0148-7299 This stuff is written in LaTeX code. The \geq and \leq stand for greater than or equal to and less than or equal to respectively - andywinterman \frac{top}{bottom} stands for a fraction top/bottom, but written so the fraction bar is horizontal.- andywinterman Still needs to be fleshed out. - andywinterman