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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.

## Definition

In matrix notation a mixed model can be represented as

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

where

• 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

## Estimation

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 : R3â†’ 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.

## Background

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

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Marine Biology Marine Biology

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