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From Wikipedia

Mixed model

A mixed model is a statistical model containing both fixed effects and random effects, that is mixed effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. They are particularly useful in settings where repeated measurements are made on the same statistical units, or where measurements are made on clusters of related statistical units.

History and current status

Ronald Fisher introduced random effects models to study the correlations of trait values between relatives. In the 1950s, Charles Roy Henderson provided best linear unbiased estimates (BLUE) of fixed effects and best linear unbiased predictions (BLUP) of random effects. Subsequently, mixed modeling has become a major area of statistical research, including work on computation of maximum likelihood estimates, non-linear mixed effect models, missing data in mixed effects models, and Bayesian estimation of mixed effects models. Mixed models are applied in many disciplines where multiple correlated measurements are made on each unit of interest. They are prominently used in research involving human and animal subjects in fields ranging from genetics to marketing, and have also been used in industrial statistics.


In matrix notation a mixed model can be represented as

\ y = X \beta + Zu + \epsilon\,\!


  • y is a vector of observations, with mean E(y)=X\beta
  • \beta is a vector of fixed effects
  • u is a vector of independent and identically-distributed (IID) random effects with mean E(u)=0 and variance-covariance matrix \operatorname{var}(u)=G
  • \epsilon is a vector of IID random error terms with mean E(\epsilon)=0 and variance \operatorname{var}(\epsilon)=R
  • X and Z are matrices of regressors relating the observations y to \beta and u


Henderson's "mixed model equations" (MME) are:

\begin{pmatrix} X'R^{-1}X & X'R^{-1}Z \\ Z'R^{-1}X & Z'R^{-1}Z + G^{-1}

\end{pmatrix}\begin{pmatrix} \tilde{\beta} \\ \tilde{u} \end{pmatrix}=\begin{pmatrix} X'R^{-1}y \\ Z'R^{-1}y \end{pmatrix}

The solutions to the MME, \textstyle\tilde{\beta} and \textstyle\tilde{u} are best linear unbiased estimates and predictors for \beta and u, respectively.

Mixed models require somewhat sophisticated computing algorithms to fit. Solutions to the MME are obtained by methods similar to those used for linear least squares. For complicated models and large datasets, iterative methods may be needed.

More recently, methods for maximum likelihood estimation of mixed models have become more widely used than least-squares based methods.

Mathematical model

Note: The term model has a different meaning inmodel theory, a branch of mathematical logic. An artifact which is used to illustrate a mathematical idea may also be called a mathematical model, and this usage is the reverse of the sense explained below.

A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written modeling). Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively.

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.

Examples of mathematical models

  • Population Growth. A simple (though approximate) model of population growth is theMalthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
  • Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function V : R3→ R and the trajectory is a solution of the differential equation
m \frac{d^2}{dt^2} x(t) = - \operatorname{grad} \left( V \right) (x(t)).
Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.
  • Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which is used to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes an optimization problem, that is:
\max U(x_1,x_2,\ldots, x_n)
subject to:
\sum_{i=1}^n p_i x_i \leq M.
x_{i} \geq 0 \; \; \; \forall i \in \{1, 2, \ldots, n \}
This model has been used in general equilibrium theory, particularly to show existence and Pareto efficiency of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.

Modelling requires selecting and identifying relevant aspects of a situation in the real world.


Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The values of the variables can be practically anything; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.

Building blocks


From Encyclopedia

Marine Biology Marine Biology

Sources Origins of Modern Marine Biology. During the last quarter of the nineteenth century, several leading American biologists became interested in establishing a marine station capable of promoting and sustaining advanced research and instruction in marine biology along the lines of successful and influential European biological stations, such as Anton Dorrn’s marine-biology station in Naples (in which four American universities officially participated) or Henri Lacaze-Duthier’s laboratory at Banyuls-sur-Mer, France. Louis Agassiz, a Swiss-born zoologist, (1807-1873) mounted the first such endeavor at Penikese Island, Massachusetts, in 1873, and during the same period his students Alpheus Hyatt (1838-1902) and Alpheus Packard (1839-1905) established summer programs at the Massachusetts seaports of Annisquam and Salem, respectively. These programs were all short-lived. The Woods Hole Laboratory. The most influential early research station was the U.S. Fish Commission laboratory, at Woods Hole, Massachusetts, directed by Spencer Fullerton Baird between 1871 and 1887. Created to survey and study the offshore fish populations and to manage a hatchery, this laboratory contributed to scientific knowledge mainly in the classification of fish species and marine ecology. The Marine Biology Laboratory. Hoping to expand the scope of his laboratory in concert with a coalition of research-oriented universities, Baird drew up a plan that was the basis for the establishment of the Marine Biological Laboratory (MBL) at Woods Hole in 1888, the year after Baird’s death. Under its first director, Charles O. Whitman (1842-1910), who ran the laboratory until 1908, the MBL established a summer program with instruction in invertebrate zoology in 1888, adding marine botany in 1890, general physiology in 1892, and embryology in 1893. The courses were designed to address problems of marine biology but also employ the specific advantages offered by marine biology in order to push ahead the research perspectives of biology generally. The general-physiology course taught at Woods Hole was the first of its kind in the world. Everywhere else physiology was limited to the study of mammals. At Woods Hole in 1899-1900 German-born physiologist Jacques Loeb (1859-1924), conducted his famous experiments on artificial parthenogenesis of sea-urchin eggs, causing unfertilized eggs to develop into new organisms. Embryology courses at Woods Hole took advantage of the specific characteristics of marine eggs, many of which (including those of the sea urchin) have no outer shell and are transparent. A Pioneering Institution. By giving the study of marine biology the broadest possible focus, the MBL became a leader in experimental biology. Until universities developed their own research facilities, institutions such as the MBL functioned as centers for scientific research and discovery. Following Dorrn’s model, the MBL allocated benches in its laboratory to participating universities, which sent their most promising students to Woods Hole. In addition to Loeb, the earliest researchers at the MBL included geneticists T. H. Morgan (1866-1945), who won a Nobel Prize in Physiology and Medicine in 1933, and E. B. Wilson (1856-1939)—who, like Morgan, had earlier worked in the U.S. Fish Commission laboratory—as well as embryologist Frank R. Lillie (1870-1947) and geneticist Nettie Stevens (1861-1912). By the time he delivered the Lowell Lectures on Light Waves and Their Uses at Harvard University in 1899, Albert A. Michelson had come to believe that the age of great scientific discoveries had come to an end and that the physicist’s future role would be in the realm of refining previously established laws through precise measurement: Six years after Michelson spoke these words Albert Einstein published his special theory of relativity, overturning all of Michelson’s assumptions about the nature of the universe. Source: Albert A. Michelson, Light Waves and Their Uses (Chicago: University of Chicago Press, 1903). Dean C. Allard, “The Fish Commission Laboratory and Its Influence on the Founding of the Marine Biological Laboratory,� Journal of the History of Biology, 23 (1990): 251-270; Frank R. Lillie, The Woods Hole Marine Biological Laboratory (Chicago: University of Chicago Press, 1944).

From Yahoo Answers

Question:I'm making a 72(progress report grade) in Biology Pre-Ap but I really want to bring it up to a B+ or preferably an A. I have worked so hard in that class but the problem is the test he gives us. We are now doing an animal kingdom notebook project and we are allowed to use our notes, but when it is test time, it's not easy looking through the whole notebook secton in less than 50 mins on a 40-50 question test. Please, what are some advice to help me prepare better on my test. I write my notes in a sub-divided format( I forgot what the format is called) so they are very organized. I don't know, I don't think I study enough but I really want to do well on my next test and this six weels report card. Please help(Tips!!, are welcome) ---------------- -10 points-best answer -5 stars -added contact(if I already added you, I'll answer 1 or 2 of your questions) p.s I have one week because after this week is final exams and I would like some tips to prepare for the biology exam... thx

Answers:It is good you can use your notes, but don't use them as a crutch. I home schooled my daughter and she always did well on the animal kingdom because she loves animals. Math went in one ear and out the other. To this day, she has problems with math. It is not that she can't do it, it is her attitude toward it. My point, hit the books and fall in love with the subject. I know it sounds hard, but you are seeing the animals a seperate. In the larger picture, they are a part of us. Find out how. My husband and I went to 'The Bodies Exhibit' when we were in Vegas. He didn't want to go at first, but then he changed his thinking about it and enjoyed it. It had a bunch of bodies sliced in different ways so you can see inside the human body, in case you never heard of it. You would be surprised how some parts of our inner bodies look like different fish and animals. It was amazing. Anyway, attitude is key. Once you make the decision to really know the subject, you will be more aware of thing you wouldn't have noticed before. Just learning it for a test will only get you so far. Learning to have an understanding and really seeing the subject will give you a greater understanding and then you will be able to mark references that will take you quicker to areas with facts that will be usable on your test. Use those post it tabs to help you move quicker. I hope I helped you. Good luck.

Question:I'm currently in my 3rd year and am a double major in genetics and math. So far for math, I have been taking the courses that are required of both pure and applied math majors (real analysis, abstract algebra, numerical analysis, etc) so I haven't yet decided whether I want to do pure math or applied math. I was originally only planning to be a genetics major, but after working a year in a biochemistry lab I found I couldn't stomach working with bodily fluids and chemicals as a career after college. I still find genetics to be very exciting, when I am sitting in front of a book. I know what mathematical biology means from Wiki, but what I wish to know is whether a mathematical biology lab similar to that of a biochemistry lab (with mice gore and stuff) or is it a more sterile environment with only computers and machines here and there? Otherwise I might just do either pure or applied math in econ or actuarial science.

Answers:I think you might find an area like bio-informatics interesting. YOu have the perfect background for it; chemistry, genetics, both pure and applied math. Another relatively new area that would make use of a lot of your skills is proteomics, the large scale study of proteins which is often done via computer modeling and simulations, rather than with pipetttes, titrations and the accompanying ickiness. Talk with your advisors or faculty in bio, math or computer science and see what your univ has to offer. Or, look for bioinformatics programs across the country. Believe me, with your background, there are a lot of things you can do. Good luck.

Question:im working on a 3-d model of a plant cell using household items and i dont know wht type of items i should use and its due on thursday i think and i almost got a few things but its really confusing heres a link to a picture of a plant cell http://www.plyojump.com/courses/biology/images/cell_plant.jpg http://www.lclark.edu/~seavey/bio210/images%20for%20210/plant_cell_diagram.gif i can use food for this

Answers:can u use food?

Question:1. Give 2 ways in which a person can get each type of immunity? 2. Describe the functions of insulin and glucogon in the body 3. Is mammal reproduction Internal or External fertilization 4. are fish ecothermy or endothermy 5. what are the major traits of arthropods? 6. What are the host(s) of pen worms and hook worms 7. Explain how jellyfish follows alternation of generation in its life cycle. 8. What characteristic is an advantage to ferns over mosses making them able to grow to a larger size in port of ther life cycle? 9. Give the general traits of ferns. 10. Name the classes of fungi and give at least one example of each class. 11. Their are four groups of animal like protists. name the four groups and tell how each moves and gets food. 12. Distinguish between the lysogenic and lytic cycles of a bacteriophage. 13. Is bacteria and obligate aerobic bacteria: Prokaryotic or eukaryotic, name its traits or food and energy sources, and habitat(s)

Answers:im only in 8th grade! u can call a friend to help though

From Youtube

Composites Modeler for SolidWorks :Composites Modeler for SolidWorks allows users to define complex ply layup models of high performance composite structures in SolidWorks . Draping simulation is supported to verify ply producibility, create manufacturing data and calculate resulting fiber orientations on the surface. Flat patterns can be exported to nesting and cutting systems to minimize material waste. The ply layup can be transferred efficiently to finite element codes like Abaqus, Nastran and Ansys via Simulayt's Layup file pipeline.

Davson-Danielli Model & Singer-Nicolson Fluid Mosaic Model :Davson-Danielli Model & Singer-Nicoloson Fluid Mosaic Model Video Summary Life occurs in an aqueous environment. Proteins are essential to life because they help maintain solute homeostasis. The phospholipid bi-layer, which is found in cell membranes, is one way a phospholipid arranges itself. The phosphate head, which is hydrophilic, moves toward the water. The hydrocarbon tail is hydrophobic. The Davson-Danielli model is an idea proposed on how cells regulate their environment. The proteins are on the outside of the bi-layer membrane. However, membranes need to allow for solute particles to pass back and forth. When proteins are on the outside, it blocks the solutes from passing through, preventing homeostasis. Therefore, we have to reject this model. In the Singer-Nicholson Fluid Mosaic Model, the integral proteins are embedded in the bi-layer. They have the ability to open up and let particles move back and forth. This model differs from the Davson-Danielli Model because it recognizes that biological systems are open systems that interact with their environments. This is a much more dynamic entity. This model allows for homeostasis, or a balance. The embedded proteins act as gatekeepers. They transport with the concentration gradient. Passive transport and facilitated transport move from a region of high solute concentration to a region of low solute concentration. Neither of these processes require energy because they are working with the concentration gradient. The ...