Pullback - Revision history

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

removed (x) from the function y(x) (strictly speaking the function is y, y(x) is the value of y at x).

← Older revision		Revision as of 01:31, 29 June 2014
Line 1:		Line 1:
	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~~.		The author is known as Araceli Gulledge. One of the things I love most is greeting card gathering but I don't have the time lately. For years she's been living in Kansas. Interviewing is what I do for a residing but I strategy on altering it.<br><br>Stop by my blog post :: car warranty ([http://Www.Tenx.com/UserProfile/tabid/62/userId/13166/Default.aspx similar site])

	~~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 is a stochastic variable~~.

	~~==Statistical models==~~
	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~~. It is ~~now assumed that a function <math>F</math> exists which relates the dose <math>z\in[0,\infty)</math> to the response~~
	~~:<math>f(u)=F(z)+e</math>~~
	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>
	~~:<math>F_{T}(z)=F_{S}(\rho z)</math>, for all <math>z</math>~~
	~~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. 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>~~. ~~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~~
	~~:<math>F_{T}(x)=F_{S}(\rho^{\lambda}x)~~<~~/math~~>~~, for all~~ <~~math~~>~~x</math>.~~

	~~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.

	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.

	~~Some special cases deserve particular mention because of their widespread use~~: 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]].

	~~==Example~~: ~~Microbiological assay of antibiotics==~~

	~~[[image:DilutionAssay.png\|Parallel line assay]]~~

	~~An [[antibiotic]] standard (shown in red) and test preparation~~ (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.

	~~==Software==~~

	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.

	~~==References==~~

	* Finney, D.J. (1971). Probit Analysis, 3rd Ed. Cambridge University Press, Cambridge. ISBN 0-521-08041-X
	* Finney, D.J. (1978). Statistical Method in Biological Assay, 3rd Ed. Griffin, London. ISBN 0-02-844640-2
	* Govindarajulu, Z. (2001). Statistical Techniques in Bioassay, 2nd revised and enlarged edition, Karger, New York. ISBN 3-8055-7119-4

	~~==External links==~~

	~~'''Software for dilution assays:'''~~
	*[http://~~www~~.~~bioassay~~.~~de PLA]~~
	*[http://~~combistats.edqm.eu CombiStats]~~
	*[http://~~www.unistat.com Unistat]~~
	*[http://~~www~~.~~cambridgesoft.com/software/details/?ds=12&dsv=85 BioAssay]~~

	~~[[Category:Pharmacology]]~~
	~~[[Category:Biostatistics]~~]

en>Alborzagros at 14:46, 8 June 2013

2013-06-08T14:46:06Z

← Older revision		Revision as of 16:46, 8 June 2013
Line 1:		Line 1:
	~~Her title~~ is ~~Felicidad Ahmad~~. ~~She functions as~~ a ~~monetary officer~~ and ~~she will not alter~~ it ~~anytime quickly~~. ~~Alabama has usually been his home~~. The ~~favorite pastime~~ for ~~my children~~ and me is ~~dancing~~ and ~~now I~~'~~m trying~~ to ~~earn cash with it~~.<br><br>~~Here~~ is ~~my page~~ :: [http://~~Vozdemujer~~.~~net~~/~~index~~.~~php~~?~~mod~~=~~users~~&~~action~~=~~view&id=5527 extended auto warranty~~]		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.

			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 is a stochastic variable.

			==Statistical models==
			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. It is now assumed that a function <math>F</math> exists which relates the dose <math>z\in[0,\infty)</math> to the response
			:<math>f(u)=F(z)+e</math>
			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>
			:<math>F_{T}(z)=F_{S}(\rho z)</math>, for all <math>z</math>
			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. 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>. 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
			:<math>F_{T}(x)=F_{S}(\rho^{\lambda}x)</math>, for all <math>x</math>.

			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.

			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.

			Some special cases deserve particular mention because of their widespread use: 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]].

			==Example: Microbiological assay of antibiotics==

			[[image:DilutionAssay.png\|Parallel line assay]]

			An [[antibiotic]] standard (shown in red) and test preparation (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.

			==Software==

			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.

			==References==

			* Finney, D.J. (1971). Probit Analysis, 3rd Ed. Cambridge University Press, Cambridge. ISBN 0-521-08041-X
			* Finney, D.J. (1978). Statistical Method in Biological Assay, 3rd Ed. Griffin, London. ISBN 0-02-844640-2
			* Govindarajulu, Z. (2001). Statistical Techniques in Bioassay, 2nd revised and enlarged edition, Karger, New York. ISBN 3-8055-7119-4

			==External links==

			'''Software for dilution assays:'''
			*[http://www.bioassay.de PLA]
			*[http://combistats.edqm.eu CombiStats]
			*[http://www.unistat.com Unistat]
			*[http://www.cambridgesoft.com/software/details/?ds=12&dsv=85 BioAssay]

			[[Category:Pharmacology]]
			[[Category:Biostatistics]]

en>Michael Hardy at 19:29, 11 August 2011

2011-08-11T19:29:18Z

New page

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