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Quasi-Newton methods are based on Newton's method to find the stationary point of a function, where the gradient is 0. Newton's method assumes that the function can be locally approximated as a quadratic in the region around the optimum, and uses the first and second derivatives to find the stationary point.

In higher dimensions, Newton's method uses the gradient and the Hessian matrix of second derivatives of the function to be minimized. In quasi-Newton methods the Hessian matrix does not need to be computed. The Hessian is updated by analyzing successive gradient vectors instead. Quasi-Newton methods are a generalization of the secant method to find the root of the first derivative for multidimensional problems.

In multiple dimensions the secant equation is under-determined , and quasi-Newton methods differ in how they constrain the solution, typically by adding a simple low-rank update to the current estimate of the Hessian.

The first quasi-Newton algorithm was proposed by William C. Davidon , a physicist working at Argonne National Laboratory. He developed the first quasi-Newton algorithm in The SR1 formula does not guarantee the update matrix to maintain positive-definiteness and can be used for indefinite problems.

The Broyden's method does not require the update matrix to be symmetric and is used to find the root of a general system of equations rather than the gradient by updating the Jacobian rather than the Hessian. Newton's method, and its derivatives such as interior point methods , require the Hessian to be inverted, which is typically implemented by solving a system of linear equations and is often quite costly.

The various quasi-Newton methods differ in their choice of the solution to the secant equation in one dimension, all the variants are equivalent. This is indeed the case for the class of quasi-Newton methods based on least-change updates. Owing to their success, there are implementations of quasi-Newton methods in almost all programming languages. The NAG Library contains several routines [4] for minimizing or maximizing a function [5] which use quasi-Newton algorithms.

In the SciPy extension to Python , the scipy. Quasi-experimental studies can use both preintervention and postintervention measurements as well as nonrandomly selected control groups.

Using this basic definition, it is evident that many published studies in medical informatics utilize the quasi-experimental design. Although the randomized controlled trial is generally considered to have the highest level of credibility with regard to assessing causality, in medical informatics, researchers often choose not to randomize the intervention for one or more reasons: Each of these reasons is discussed below. Ethical considerations typically will not allow random withholding of an intervention with known efficacy.

Thus, if the efficacy of an intervention has not been established, a randomized controlled trial is the design of choice to determine efficacy. But if the intervention under study incorporates an accepted, well-established therapeutic intervention, or if the intervention has either questionable efficacy or safety based on previously conducted studies, then the ethical issues of randomizing patients are sometimes raised.

In the area of medical informatics, it is often believed prior to an implementation that an informatics intervention will likely be beneficial and thus medical informaticians and hospital administrators are often reluctant to randomize medical informatics interventions. In addition, there is often pressure to implement the intervention quickly because of its believed efficacy, thus not allowing researchers sufficient time to plan a randomized trial.

For medical informatics interventions, it is often difficult to randomize the intervention to individual patients or to individual informatics users. So while this randomization is technically possible, it is underused and thus compromises the eventual strength of concluding that an informatics intervention resulted in an outcome.

For example, randomly allowing only half of medical residents to use pharmacy order-entry software at a tertiary care hospital is a scenario that hospital administrators and informatics users may not agree to for numerous reasons. Similarly, informatics interventions often cannot be randomized to individual locations.

Using the pharmacy order-entry system example, it may be difficult to randomize use of the system to only certain locations in a hospital or portions of certain locations. For example, if the pharmacy order-entry system involves an educational component, then people may apply the knowledge learned to nonintervention wards, thereby potentially masking the true effect of the intervention.

When a design using randomized locations is employed successfully, the locations may be different in other respects confounding variables , and this further complicates the analysis and interpretation. In situations where it is known that only a small sample size will be available to test the efficacy of an intervention, randomization may not be a viable option.

Randomization is beneficial because on average it tends to evenly distribute both known and unknown confounding variables between the intervention and control group. However, when the sample size is small, randomization may not adequately accomplish this balance. Thus, alternative design and analytical methods are often used in place of randomization when only small sample sizes are available. The lack of random assignment is the major weakness of the quasi-experimental study design.

Associations identified in quasi-experiments meet one important requirement of causality since the intervention precedes the measurement of the outcome. Another requirement is that the outcome can be demonstrated to vary statistically with the intervention. Unfortunately, statistical association does not imply causality, especially if the study is poorly designed. Thus, in many quasi-experiments, one is most often left with the question: These rival hypotheses, or alternative explanations, arise from principles of epidemiologic study design.

Internal validity is defined as the degree to which observed changes in outcomes can be correctly inferred to be caused by an exposure or an intervention.

Each of these latter two principles is discussed in turn. Adapted from Shadish et al. An inability to sufficiently control for important confounding variables arises from the lack of randomization. A variable is a confounding variable if it is associated with the exposure of interest and is also associated with the outcome of interest; the confounding variable leads to a situation where a causal association between a given exposure and an outcome is observed as a result of the influence of the confounding variable.

For example, in a study aiming to demonstrate that the introduction of a pharmacy order-entry system led to lower pharmacy costs, there are a number of important potential confounding variables e.

In a multivariable regression, the first confounding variable could be addressed with severity of illness measures, but the second confounding variable would be difficult if not nearly impossible to measure and control. In addition, potential confounding variables that are unmeasured or immeasurable cannot be controlled for in nonrandomized quasi-experimental study designs and can only be properly controlled by the randomization process in randomized controlled trials.

To get the true effect of the intervention of interest, we need to control for the confounding variable. Another important threat to establishing causality is regression to the mean. The phenomenon was first described in by Francis Galton who measured the adult height of children and their parents. He noted that when the average height of the parents was greater than the mean of the population, the children tended to be shorter than their parents, and conversely, when the average height of the parents was shorter than the population mean, the children tended to be taller than their parents.

In medical informatics, what often triggers the development and implementation of an intervention is a rise in the rate above the mean or norm. For example, increasing pharmacy costs and adverse events may prompt hospital informatics personnel to design and implement pharmacy order-entry systems. If this rise in costs or adverse events is really just an extreme observation that is still within the normal range of the hospital's pharmaceutical costs i.

However, often informatics personnel and hospital administrators cannot wait passively for this decline to occur. Therefore, hospital personnel often implement one or more interventions, and if a decline in the rate occurs, they may mistakenly conclude that the decline is causally related to the intervention.

In fact, an alternative explanation for the finding could be regression to the mean. In the social sciences literature, quasi-experimental studies are divided into four study design groups 4 , There is a relative hierarchy within these categories of study designs, with category D studies being sounder than categories C, B, or A in terms of establishing causality. Thus, if feasible from a design and implementation point of view, investigators should aim to design studies that fall in to the higher rated categories.

In our review, we determined that most medical informatics quasi-experiments could be characterized by 11 of 17 designs, with six study designs in category A, one in category B, three designs in category C, and one design in category D because the other study designs were not used or feasible in the medical informatics literature.

Time moves from left to right. For example, there may be instances where an A6 design established stronger causality than a B1 design. Here, X is the intervention and O is the outcome variable this notation is continued throughout the article. In this study design, an intervention X is implemented and a posttest observation O1 is taken. For example, X could be the introduction of a pharmacy order-entry intervention and O1 could be the pharmacy costs following the intervention.

This design is the weakest of the quasi-experimental designs that are discussed in this article. Without any pretest observations or a control group, there are multiple threats to internal validity. Unfortunately, this study design is often used in medical informatics when new software is introduced since it may be difficult to have pretest measurements due to time, technical, or cost constraints.

This is a commonly used study design. A single pretest measurement is taken O1 , an intervention X is implemented, and a posttest measurement is taken O2. For example, O1 could be pharmacy costs prior to the intervention, X could be the introduction of a pharmacy order-entry system, and O2 could be the pharmacy costs following the intervention. Including a pretest provides some information about what the pharmacy costs would have been had the intervention not occurred.

The advantage of this study design over A2 is that adding a second pretest prior to the intervention helps provide evidence that can be used to refute the phenomenon of regression to the mean and confounding as alternative explanations for any observed association between the intervention and the posttest outcome.

Similarly, extending this study design by increasing the number of measurements postintervention could also help to provide evidence against confounding and regression to the mean as alternate explanations for observed associations.

This design involves the inclusion of a nonequivalent dependent variable b in addition to the primary dependent variable a. Variables a and b should assess similar constructs; that is, the two measures should be affected by similar factors and confounding variables except for the effect of the intervention. Variable a is expected to change because of the intervention X, whereas variable b is not.

Taking our example, variable a could be pharmacy costs and variable b could be the length of stay of patients. If our informatics intervention is aimed at decreasing pharmacy costs, we would expect to observe a decrease in pharmacy costs but not in the average length of stay of patients. However, a number of important confounding variables, such as severity of illness and knowledge of software users, might affect both outcome measures.

Thus, if the average length of stay did not change following the intervention but pharmacy costs did, then the data are more convincing than if just pharmacy costs were measured.

This design adds a third posttest measurement O3 to the one-group pretest-posttest design and then removes the intervention before a final measure O4 is made. The advantage of this design is that it allows one to test hypotheses about the outcome in the presence of the intervention and in the absence of the intervention.

Thus, if one predicts a decrease in the outcome between O1 and O2 after implementation of the intervention , then one would predict an increase in the outcome between O3 and O4 after removal of the intervention. One caveat is that if the intervention is thought to have persistent effects, then O4 needs to be measured after these effects are likely to have disappeared. For example, a study would be more convincing if it demonstrated that pharmacy costs decreased after pharmacy order-entry system introduction O2 and O3 less than O1 and that when the order-entry system was removed or disabled, the costs increased O4 greater than O2 and O3 and closer to O1.

In addition, there are often ethical issues in this design in terms of removing an intervention that may be providing benefit. The advantage of this design is that it demonstrates reproducibility of the association between the intervention and the outcome.

For example, the association is more likely to be causal if one demonstrates that a pharmacy order-entry system results in decreased pharmacy costs when it is first introduced and again when it is reintroduced following an interruption of the intervention.

As for design A5, the assumption must be made that the effect of the intervention is transient, which is most often applicable to medical informatics interventions. Because in this design, subjects may serve as their own controls, this may yield greater statistical efficiency with fewer numbers of subjects. Posttest-Only Design with Nonequivalent Groups: An intervention X is implemented for one group and compared to a second group. The use of a comparison group helps prevent certain threats to validity including the ability to statistically adjust for confounding variables.

Because in this study design, the two groups may not be equivalent assignment to the groups is not by randomization , confounding may exist. For example, suppose that a pharmacy order-entry intervention was instituted in the medical intensive care unit MICU and not the surgical intensive care unit SICU. The absence of a pretest makes it difficult to know whether a change has occurred in the MICU. Also, the absence of pretest measurements comparing the SICU to the MICU makes it difficult to know whether differences in O1 and O2 are due to the intervention or due to other differences in the two units confounding variables.

The reader should note that with all the studies in this category, the intervention is not randomized. The control groups chosen are comparison groups. Obtaining pretest measurements on both the intervention and control groups allows one to assess the initial comparability of the groups. The assumption is that if the intervention and the control groups are similar at the pretest, the smaller the likelihood there is of important confounding variables differing between the two groups.

The use of both a pretest and a comparison group makes it easier to avoid certain threats to validity. However, because the two groups are nonequivalent assignment to the groups is not by randomization , selection bias may exist.

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Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. A systematic review of comparisons of effect sizes derived from randomised and non-randomised studies.

Yet little has been written about the benefits and limitations of the quasi-experimental approach as applied to informatics studies.