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Pullback - Revision history
2024-03-29T13:39:09Z
Revision history for this page on the wiki
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https://en.formulasearchengine.com/index.php?title=Pullback&diff=252897&oldid=prev
2.31.11.144: removed (x) from the function y(x) (strictly speaking the function is y, y(x) is the value of y at x).
2014-06-28T23:31:52Z
<p>removed (x) from the function y(x) (strictly speaking the function is y, y(x) is the value of y at x).</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:31, 29 June 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The <del style="font-weight: bold; text-decoration: none;">term '''dilution assay''' is generally used to designate a special type of [[bioassay]] in which one or more preparations (e.g. a drug) are administered to experimental units at different dose levels inducing a measurable biological response. The dose levels are prepared by dilution in a diluent that is inert in respect of the response. The experimental units can for example be cell-cultures, tissues, organs or living animals. The biological response may be quantal (e.g. positive/negative) or quantitative (e.g. growth). The goal </del>is <del style="font-weight: bold; text-decoration: none;">to relate the response to the dose, usually by [[interpolation]] techniques, and in many cases to express the potency/activity of the test preparation(s) relative to a standard of </del>known <del style="font-weight: bold; text-decoration: none;">potency/activity</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <ins style="font-weight: bold; text-decoration: none;">author </ins>is known <ins style="font-weight: bold; text-decoration: none;">as Araceli Gulledge</ins>. <ins style="font-weight: bold; text-decoration: none;">One </ins>of the <ins style="font-weight: bold; text-decoration: none;">things I love most </ins>is <ins style="font-weight: bold; text-decoration: none;">greeting card gathering but I don</ins>'<ins style="font-weight: bold; text-decoration: none;">t have </ins>the <ins style="font-weight: bold; text-decoration: none;">time lately</ins>. For <ins style="font-weight: bold; text-decoration: none;">years she's been living </ins>in <ins style="font-weight: bold; text-decoration: none;">Kansas</ins>. <ins style="font-weight: bold; text-decoration: none;">Interviewing </ins>is <ins style="font-weight: bold; text-decoration: none;">what I do </ins>for a <ins style="font-weight: bold; text-decoration: none;">residing but I strategy on altering </ins>it.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Stop </ins>by <ins style="font-weight: bold; text-decoration: none;">my blog post </ins>:: <ins style="font-weight: bold; text-decoration: none;">car warranty </ins>([http://<ins style="font-weight: bold; text-decoration: none;">Www</ins>.<ins style="font-weight: bold; text-decoration: none;">Tenx</ins>.<ins style="font-weight: bold; text-decoration: none;">com</ins>/<ins style="font-weight: bold; text-decoration: none;">UserProfile</ins>/<ins style="font-weight: bold; text-decoration: none;">tabid</ins>/<ins style="font-weight: bold; text-decoration: none;">62</ins>/<ins style="font-weight: bold; text-decoration: none;">userId</ins>/<ins style="font-weight: bold; text-decoration: none;">13166</ins>/<ins style="font-weight: bold; text-decoration: none;">Default</ins>.<ins style="font-weight: bold; text-decoration: none;">aspx similar site</ins>]<ins style="font-weight: bold; text-decoration: none;">)</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Dilution assays can be direct or indirect. In a '''direct dilution assay''' the amount </del>of <del style="font-weight: bold; text-decoration: none;">dose needed to produce a specific (fixed) response is measured, so that </del>the <del style="font-weight: bold; text-decoration: none;">dose </del>is <del style="font-weight: bold; text-decoration: none;">a stochastic variable defining the '''tolerance distribution'''. Conversely, in an '''indirect dilution assay''</del>' the <del style="font-weight: bold; text-decoration: none;">dose levels are administered at fixed dose levels, so that the response is a stochastic variable</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Statistical models==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For <del style="font-weight: bold; text-decoration: none;">a mathematical definition of a dilution assay an observation space <math>U</math> is defined and a function <math>f:U\rightarrow\mathbb{R}</math> so that the responses <math>u\</del>in <del style="font-weight: bold; text-decoration: none;">U</math> are mapped to the set of real numbers</del>. <del style="font-weight: bold; text-decoration: none;">It </del>is <del style="font-weight: bold; text-decoration: none;">now assumed that a function <math>F</math> exists which relates the dose <math>z\in[0,\infty)</math> to the response</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>f(u)=F(z)+e</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">in which <math>e</math> is an error term with expectation 0. <math>F</math> is usually assumed to be [[continuous function|continuous]] and [[monotone function|monotone]]. In situations where a standard preparation is included it is furthermore assumed that the test preparation <math>T</math> behaves like a dilution (or concentration) of the standard <math>S</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>F_{T}(z)=F_{S}(\rho z)</math>, for all <math>z</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">where <math>\rho>0</math> is the relative potency of <math>T</math>. This is the fundamental assumption of similarity of dose-response curves which is necessary </del>for a <del style="font-weight: bold; text-decoration: none;">meaningful and unambiguous definition of the relative potency. In many cases </del>it <del style="font-weight: bold; text-decoration: none;">is convenient to apply a power transformation <math>x=z^{\lambda}</math> with <math>\lambda>0</math> or a logarithmic transformation <math>x=\log (z)</math></del>. <del style="font-weight: bold; text-decoration: none;">The latter can be shown to be a limit case of <math>\lambda\downarrow0</math> so if <math>\lambda=0</math> is written for the log transformation the above equation can be redefined as</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:<math>F_{T}(x)=F_{S}(\rho^{\lambda}x)</del><<del style="font-weight: bold; text-decoration: none;">/math</del>><del style="font-weight: bold; text-decoration: none;">, for all </del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">x</math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Estimates <math>\hat F</math> of <math>F</math> are usually restricted to be member of a well-defined [[parametric family of functions]], for example the family of [[linear function]]s characterized </del>by <del style="font-weight: bold; text-decoration: none;">an intercept and a slope. Statistical techniques such as optimization by [[Maximum Likelihood]] can be used to calculate estimates of the parameters. Of notable importance in this respect is the theory of [[generalized linear models|Generalized Linear Models]] with which a wide range of dilution assays can be modelled. Estimates of <math>F</math> may describe <math>F</math> satisfactorily over the range of doses tested, but they do not necessarily have to describe <math>F</math> beyond that range. However, this does not mean that dissimilar curves can be restricted to an interval where they happen to be similar.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In practice, <math>F</math> itself is rarely of interest. More of interest is an estimate of <math>\rho</math> or an estimate of the dose that induces a specific response. These estimates involve taking ratios of statistically dependent parameter estimates. [[Fieller's theorem]] can be used to compute confidence intervals of these ratios.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Some special cases deserve particular mention because of their widespread use</del>: <del style="font-weight: bold; text-decoration: none;">If <math>F</math> is linear and <math>\lambda>0</math> this is known as a '''slope-ratio model'''. If <math>F</math> is linear and <math>\lambda=0</math> this is known as a '''parallel line model'''. Another commonly applied model is the [[probit model]] where <math>F</math> is the cumulative [[normal distribution]] function, <math>\lambda=0</math> and <math>e</math> follows a [[binomial distribution]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Example</del>: <del style="font-weight: bold; text-decoration: none;">Microbiological assay of antibiotics==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[image:DilutionAssay.png|Parallel line assay]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">An [[antibiotic]] standard (shown in red) and test preparation </del>(<del style="font-weight: bold; text-decoration: none;">shown in blue) are applied at three dose levels to sensitive [[microorganism]]s on a layer of [[agar]] in [[petri dish]]es. The stronger the dose the larger the zone of inhibition of growth of the microorganisms. The biological response <math>u</math> is in this case the zone of inhibition and the diameter of this zone <math>f(u)</math> can be used as the measurable response. The doses <math>z</math> are transformed to logarithms <math>x=\log (z)</math> and the method of least squares is used to fit two parallel lines to the data. The horizontal distance <math>\log (\hat\rho)</math> between the two lines (shown in green) serves as an estimate of the potency <math>\rho</math> of the test preparation relative to the standard.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Software==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The major statistical software packages do not cover dilution assays although a statistician should not have difficulties to write suitable scripts or macros to that end. Several special purpose software packages for dilution assays exist.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==References==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Finney, D.J. (1971). Probit Analysis, 3rd Ed. Cambridge University Press, Cambridge. ISBN 0-521-08041-X</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Finney, D.J. (1978). Statistical Method in Biological Assay, 3rd Ed. Griffin, London. ISBN 0-02-844640-2</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Govindarajulu, Z. (2001). Statistical Techniques in Bioassay, 2nd revised and enlarged edition, Karger, New York. ISBN 3-8055-7119-4</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==External links==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''Software for dilution assays:'''</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*</del>[http://<del style="font-weight: bold; text-decoration: none;">www</del>.<del style="font-weight: bold; text-decoration: none;">bioassay</del>.<del style="font-weight: bold; text-decoration: none;">de PLA]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[http:</del>//<del style="font-weight: bold; text-decoration: none;">combistats.edqm.eu CombiStats]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[http:</del>//<del style="font-weight: bold; text-decoration: none;">www.unistat.com Unistat]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[http:</del>//<del style="font-weight: bold; text-decoration: none;">www</del>.<del style="font-weight: bold; text-decoration: none;">cambridgesoft.com/software/details/?ds=12&dsv=85 BioAssay]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
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2.31.11.144
https://en.formulasearchengine.com/index.php?title=Pullback&diff=16293&oldid=prev
en>Alborzagros at 14:46, 8 June 2013
2013-06-08T14:46:06Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:46, 8 June 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Her title </del>is <del style="font-weight: bold; text-decoration: none;">Felicidad Ahmad</del>. <del style="font-weight: bold; text-decoration: none;">She functions as </del>a <del style="font-weight: bold; text-decoration: none;">monetary officer </del>and <del style="font-weight: bold; text-decoration: none;">she will not alter </del>it <del style="font-weight: bold; text-decoration: none;">anytime quickly</del>. <del style="font-weight: bold; text-decoration: none;">Alabama has usually been his home</del>. The <del style="font-weight: bold; text-decoration: none;">favorite pastime </del>for <del style="font-weight: bold; text-decoration: none;">my children </del>and <del style="font-weight: bold; text-decoration: none;">me </del>is <del style="font-weight: bold; text-decoration: none;">dancing </del>and <del style="font-weight: bold; text-decoration: none;">now I</del>'<del style="font-weight: bold; text-decoration: none;">m trying </del>to <del style="font-weight: bold; text-decoration: none;">earn cash with it</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Here </del>is <del style="font-weight: bold; text-decoration: none;">my page </del>:: [http://<del style="font-weight: bold; text-decoration: none;">Vozdemujer</del>.<del style="font-weight: bold; text-decoration: none;">net</del>/<del style="font-weight: bold; text-decoration: none;">index</del>.<del style="font-weight: bold; text-decoration: none;">php</del>?<del style="font-weight: bold; text-decoration: none;">mod</del>=<del style="font-weight: bold; text-decoration: none;">users</del>&<del style="font-weight: bold; text-decoration: none;">action</del>=<del style="font-weight: bold; text-decoration: none;">view&id=5527 extended auto warranty</del>]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The term '''dilution assay''' is generally used to designate a special type of [[bioassay]] in which one or more preparations (e.g. a drug) are administered to experimental units at different dose levels inducing a measurable biological response. The dose levels are prepared by dilution in a diluent that is inert in respect of the response. The experimental units can for example be cell-cultures, tissues, organs or living animals. The biological response may be quantal (e.g. positive/negative) or quantitative (e.g. growth). The goal is to relate the response to the dose, usually by [[interpolation]] techniques, and in many cases to express the potency/activity of the test preparation(s) relative to a standard of known potency/activity.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Dilution assays can be direct or indirect. In a '''direct dilution assay''' the amount of dose needed to produce a specific (fixed) response is measured, so that the dose is a stochastic variable defining the '''tolerance distribution'''. Conversely, in an '''indirect dilution assay''' the dose levels are administered at fixed dose levels, so that the response </ins>is <ins style="font-weight: bold; text-decoration: none;">a stochastic variable.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Statistical models==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For a mathematical definition of a dilution assay an observation space <math>U</math> is defined and a function <math>f:U\rightarrow\mathbb{R}</math> so that the responses <math>u\in U</math> are mapped to the set of real numbers</ins>. <ins style="font-weight: bold; text-decoration: none;">It is now assumed that </ins>a <ins style="font-weight: bold; text-decoration: none;">function <math>F</math> exists which relates the dose <math>z\in[0,\infty)</math> to the response</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>f(u)=F(z)+e</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">in which <math>e</math> is an error term with expectation 0. <math>F</math> is usually assumed to be [[continuous function|continuous]] </ins>and <ins style="font-weight: bold; text-decoration: none;">[[monotone function|monotone]]. In situations where a standard preparation is included </ins>it <ins style="font-weight: bold; text-decoration: none;">is furthermore assumed that the test preparation <math>T</math> behaves like a dilution (or concentration) of the standard <math>S</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>F_{T}(z)=F_{S}(\rho z)</math>, for all <math>z</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where <math>\rho>0</math> is the relative potency of <math>T</math>. This is the fundamental assumption of similarity of dose-response curves which is necessary for a meaningful and unambiguous definition of the relative potency</ins>. <ins style="font-weight: bold; text-decoration: none;">In many cases it is convenient to apply a power transformation <math>x=z^{\lambda}</math> with <math>\lambda>0</math> or a logarithmic transformation <math>x=\log (z)</math></ins>. The <ins style="font-weight: bold; text-decoration: none;">latter can be shown to be a limit case of <math>\lambda\downarrow0</math> so if <math>\lambda=0</math> is written for the log transformation the above equation can be redefined as</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>F_{T}(x)=F_{S}(\rho^{\lambda}x)</math>, </ins>for <ins style="font-weight: bold; text-decoration: none;">all <math>x</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Estimates <math>\hat F</math> of <math>F</math> are usually restricted to be member of a well-defined [[parametric family of functions]], for example the family of [[linear function]]s characterized by an intercept and a slope. Statistical techniques such as optimization by [[Maximum Likelihood]] can be used to calculate estimates of the parameters. Of notable importance in this respect is the theory of [[generalized linear models|Generalized Linear Models]] with which a wide range of dilution assays can be modelled. Estimates of <math>F</math> may describe <math>F</math> satisfactorily over the range of doses tested, but they do not necessarily have to describe <math>F</math> beyond that range. However, this does not mean that dissimilar curves can be restricted to an interval where they happen to be similar.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In practice, <math>F</math> itself is rarely of interest. More of interest is an estimate of <math>\rho</math> or an estimate of the dose that induces a specific response. These estimates involve taking ratios of statistically dependent parameter estimates. [[Fieller's theorem]] can be used to compute confidence intervals of these ratios.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Some special cases deserve particular mention because of their widespread use: If <math>F</math> is linear </ins>and <ins style="font-weight: bold; text-decoration: none;"><math>\lambda>0</math> this </ins>is <ins style="font-weight: bold; text-decoration: none;">known as a '''slope-ratio model'''. If <math>F</math> is linear </ins>and <ins style="font-weight: bold; text-decoration: none;"><math>\lambda=0</math> this is known as a '''parallel line model''</ins>'<ins style="font-weight: bold; text-decoration: none;">. Another commonly applied model is the [[probit model]] where <math>F</math> is the cumulative [[normal distribution]] function, <math>\lambda=0</math> and <math>e</math> follows a [[binomial distribution]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Example: Microbiological assay of antibiotics==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[image:DilutionAssay.png|Parallel line assay]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">An [[antibiotic]] standard (shown in red) and test preparation (shown in blue) are applied at three dose levels </ins>to <ins style="font-weight: bold; text-decoration: none;">sensitive [[microorganism]]s on a layer of [[agar]] in [[petri dish]]es. The stronger the dose the larger the zone of inhibition of growth of the microorganisms. The biological response <math>u</math> is in this case the zone of inhibition and the diameter of this zone <math>f(u)</math> can be used as the measurable response</ins>. <ins style="font-weight: bold; text-decoration: none;">The doses </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">z</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">are transformed to logarithms <math>x=\log (z)</math> and the method of least squares </ins>is <ins style="font-weight: bold; text-decoration: none;"> used to fit two parallel lines to the data. The horizontal distance <math>\log (\hat\rho)</math> between the two lines (shown in green) serves as an estimate of the potency <math>\rho</math> of the test preparation relative to the standard.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Software==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The major statistical software packages do not cover dilution assays although a statistician should not have difficulties to write suitable scripts or macros to that end. Several special purpose software packages for dilution assays exist.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Finney, D.J. (1971). Probit Analysis, 3rd Ed. Cambridge University Press, Cambridge. ISBN 0-521-08041-X</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Finney, D.J. (1978). Statistical Method in Biological Assay, 3rd Ed. Griffin, London. ISBN 0-02-844640-2</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Govindarajulu, Z. (2001). Statistical Techniques in Bioassay, 2nd revised and enlarged edition, Karger, New York. ISBN 3-8055-7119-4</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==External links==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''Software for dilution assays</ins>:<ins style="font-weight: bold; text-decoration: none;">'''</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[http</ins>:<ins style="font-weight: bold; text-decoration: none;">//www.bioassay.de PLA]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*</ins>[http://<ins style="font-weight: bold; text-decoration: none;">combistats</ins>.<ins style="font-weight: bold; text-decoration: none;">edqm.eu CombiStats]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[http:/</ins>/<ins style="font-weight: bold; text-decoration: none;">www.unistat</ins>.<ins style="font-weight: bold; text-decoration: none;">com Unistat]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[http://www.cambridgesoft.com/software/details/</ins>?<ins style="font-weight: bold; text-decoration: none;">ds</ins>=<ins style="font-weight: bold; text-decoration: none;">12</ins>&<ins style="font-weight: bold; text-decoration: none;">dsv</ins>=<ins style="font-weight: bold; text-decoration: none;">85 BioAssay]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Pharmacology]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Biostatistics]</ins>]</div></td></tr>
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en>Alborzagros
https://en.formulasearchengine.com/index.php?title=Pullback&diff=252896&oldid=prev
en>Michael Hardy at 19:29, 11 August 2011
2011-08-11T19:29:18Z
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en>Michael Hardy