The Blinder—Oaxaca decomposition is a statistical method that explains the difference in the means of a dependent variable between two groups by decomposing the gap into that part that is due to differences in the mean values of the independent variable within the groups, on the one hand, and group differences in the effects of the independent variable, on the other hand.

The method was introduced by sociologist and demographer Evelyn M. Kitagawa in The following three equations illustrate this decomposition. Estimate separate linear wage regressions for individuals i in groups A and B :. Then, since the average value of residuals in a linear regression is zero, we have:. The first part of the last line of 3 is the impact of between-group differences in the explanatory variables Xevaluated using the coefficients for group A.

The unexplained differential in wages for the same values of explanatory variables should not be interpreted as the amount of the difference in wages due only to discrimination. This is because other explanatory variables not included in the regression e. For example, David Card and Alan Krueger found in a paper entitled, "School Quality and Black-White Relative Earnings: A Direct Assessment" [5] that improvements in the quality of schools for Black men born in the Southern states of the United States between and increased the return to education for these men, leading to narrowing of the black-white earnings gap.

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Views Read Edit View history. Help Community portal Recent changes Upload file. Download as PDF Printable version.The consistent histories, also known as decoherent histories, approach to quantum interpretation is broadly compatible with standard quantum mechanics as found in textbooks. However, the concept of measurement by which probabilities are introduced in standard quantum theory no longer plays a fundamental role.

Instead, all quantum time dependence is probabilistic stochasticwith probabilities given by the Born rule or its extensions. In particular, quantum mechanics is local and consistent with special relativity. Classical mechanics emerges as a useful approximation to the more fundamental quantum mechanics under suitable conditions. The price to be paid for this is a set of rules for reasoning resembling, but also significantly different from, those which comprise quantum logic.

An important philosophical implication is the lack of a single universally-true state of affairs at each instant of time. However, there is a correspondence limit in which the new quantum logic becomes standard logic in the macroscopic world of everyday experience. The reader is assumed familiar with, and probably a bit confused by and frustrated with, the fundamentals of quantum mechanics found in introductory textbooks.

The histories approach extends the calculational methods found in standard textbooks and gives them a microscopic physical interpretation which the textbooks lack.

### Heckman correction

In particular, measurements are treated in the same way as all other physical processes and play no special role in the interpretation. Thus there is no measurement problem, and under appropriate conditions one can discuss the microscopic properties revealed by measurements. In particular, there is no conflict between quantum theory and relativity: superluminal influences cannot carry information or anything else for the simple reason that they do not exist.

In appropriate regimes large systems, strong decoherence the laws of classical physics provide a very good approximation, valid for all practical purposes, to the more fundamental and more exact laws of quantum mechanics.

### Causal Mediation

But at what price? The histories approach introduces a variety of concepts that go beyond textbook quantum theory, and these can be summarized under two headings. First, quantum dynamics is treated as stochastic or probabilistic. Not just when measurements are carried out, but always. Second, following Sec.

But then new logical principles are required to consistently deal with the difference between the quantum Hilbert space and a classical phase space. Since textbook quantum mechanics already contains certain rules for calculating probabilities, the first innovation of the histories approach, stochastic dynamics, is not very startling and by itself causes few conceptual difficulties.

It is the second innovation, in the domain of logic and ontology, that represents the most radical departure from classical thinking. However, the new quantum logic reduces to the old familiar classical propositional logic in the same domain where classical mechanics is a good approximation to quantum mechanics.

That is to say, the old logic is perfectly good for all practical purposes in the same domain where classical mechanics is perfectly good for all practical purposes, and a consistent, fully quantum analysis explains why this is so. The property that its energy. If infinite dimensional it must be complete—Cauchy sequences have limits. But for our purposes finite-dimensional spaces will suffice for discussing the major conceptual difficulties of quantum theory and how the histories approach resolves them.

Some examples use a harmonic oscillator with an infinite dimensional Hilbert space simply because it is relatively simple and familiar.

The ray uniquely determines and is uniquely determined by the corresponding projector. These larger subspaces also represent quantum properties and are the analogues of sets of more than one point in a classical phase space.

A quantum projector behaves in many ways like a classical indicator function, e.

Following Sec. Consider the example of a one-dimensional quantum harmonic oscillator. Despite the close analogy between classical and quantum properties there is actually a profound difference.Previouslyyou have added and simplified rational expressions, such as:. Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions.

To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. Then you write the fractions with one of the factors for each of the denominators.

Of course, you don't know what the numerators are yet, so you assign variables usually capital letters for these unknown values:. Multiply things out, and group the x -terms and the constant terms:. For the two sides to be equal, the coefficients of the two polynomials must be equal. So you "equate the coefficients" to get:. This creates a system of equations that you can solve:.

Then the original fractions were as we already know the following:. I've never seen this second method in textbooks, but it can often save you a whole lot of time over the "equate the coefficients and solve the system of equations" method that they usually teach.

If the denominator of your fraction factors into unique linear factors, then the decomposition process is fairly straightforward, as shown in the previous example. But what if the factors aren't unique or aren't linear? Stapel, Elizabeth. Accessed [Date] [Month] Study Skills Survey. Tutoring from Purplemath Find a local math tutor. Cite this article as:. Contact Us.Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity.

Use of this web site signifies your agreement to the terms and conditions. Personal Sign In. For IEEE to continue sending you helpful information on our products and services, please consent to our updated Privacy Policy. Email Address. Sign In. Multiple Reference Points-Based Decomposition for Multiobjective Feature Selection in Classification: Static and Dynamic Mechanisms Abstract: Feature selection is an important task in machine learning that has two main objectives: 1 reducing dimensionality and 2 improving learning performance.

Feature selection can be considered a multiobjective problem. However, it has its problematic characteristics, such as a highly discontinuous Pareto front, imbalance preferences, and partially conflicting objectives. These characteristics are not easy for existing evolutionary multiobjective optimization EMO algorithms. The static mechanism alleviates the dependence of the decomposition on the Pareto front shape and the effect of the discontinuity. The dynamic one is able to detect regions in which the objectives are mostly conflicting, and allocates more computational resources to the detected regions.

In comparison with other EMO algorithms on 12 different classification datasets, the proposed decomposition approach finds more diverse feature subsets with better performance in terms of hypervolume and inverted generational distance.

The dynamic mechanism successfully identifies conflicting regions and further improves the approximation quality for the Pareto fronts. Article :. Date of Publication: 30 April DOI: Need Help?The income approach to measuring gross domestic product GDP is based on the accounting reality that all expenditures in an economy should equal the total income generated by the production of all economic goods and services.

It also assumes that there are four major factors of production in an economy and that all revenues must go to one of these four sources. Therefore, by adding all of the sources of income together, a quick estimate can be made of the total productive value of economic activity over a period. Adjustments must then be made for taxes, depreciationand foreign factor payments.

There are generally two ways to calculate GDP: the expenditures approach and the income approach. Each of these approaches looks to best approximate the monetary value of all final goods and services produced in an economy over a set period of time normally one year.

The major distinction between each approach is its starting point. The expenditure approach begins with the money spent on goods and services.

Conversely, the income approach starts with the income earned wages, rents, interest, profits from the production of goods and services. It's possible to express the income approach formula to GDP as follows:. Total national income is equal to the sum of all wages plus rents plus interest and profits. Some economists illustrate the importance of GDP by comparing its ability to provide a high level picture of an economy to that of a satellite in space that can survey the weather across an entire continent.

Along with better informed policies and institutions, national accounts have contributed to a significant reduction in the severity of business cycles since the end of world War II.

However, GDP does fluctuate because of business cycles. When the economy is booming, and GDP is rising, inflationary pressures build up rapidly as labor and productive capacity near full utilization. As interest rates rise, companies cut back, and the economy slows down and companies cut costs. To break the cycle, the central bank must loosen monetary policy in order to stimulate economic growth and employment until the economy is strong again. Your Money. Personal Finance.

## How Do You Calculate GDP With the Income Approach?

Your Practice. Popular Courses. Economics Macroeconomics. Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation. Related Articles. Partner Links. Expenditure Method Definition The expenditure method is a method for determining GDP that totals consumption, investment, government spending, and net exports. Recessionary Gap Definition A recessionary gap, or contractionary gap, is where a country's real GDP is lower than it's GDP if the economy was operating at full employment.

Depression Definition An economic depression is a steep and sustained drop in economic activity featuring high unemployment and negative GDP growth. Keynesian Economics Definition Keynesian Economics is an economic theory of total spending in the economy and its effects on output and inflation developed by John Maynard Keynes.

Explaining the Wage-Price Spiral and How It Relates to Inflation A wage-price spiral is a macroeconomic theory to explain the cause-and-effect relationship between rising wages and rising prices, or inflation. Investopedia is part of the Dotdash publishing family.This page briefly compares mediation analysis from both the traditional and causal inference frameworks. An annotated resource list is provided, followed by a suggested article for a future Epi 6 project relating to causal mediation.

What is mediation? Mediation is the process through which an exposure causes disease. In the simple diagram below we examine the total effect of exposure on outcome. Researchers may hypothesize that some or all of the total effect of exposure on an outcome operates through a mediator, which is an effect of the exposure and a cause of the outcome.

When a mediator is hypothesized, the total effect can be broken into two parts: the direct and indirect effect. The direct effect is the effect of exposure on the outcome absent the mediator.

The indirect pathway is the effect of exposure on the outcome that works through the mediator.

Why care about mediation? There are many motivations for performing mediation analysis, but the overarching goal is one of causal explanation. Other more specific reasons include: increasing construct validity, strengthening evidence of the main effect hypothesis, understanding the mechanisms and active ingredients by which exposure causes disease, and evaluating and improving interventions i.

For example, if a researcher is mainly interested in eliminating mediated pathways not of interest in order to strengthen their evidence of an exposure-outcome relationship, the effect of interest is the direct effect. On the other hand, if underlying mechanisms by which exposure causes disease are of interest, the researcher may be more interested in estimating the indirect effect.

The traditional approach to mediation — what we have learned in the majority of our epidemiology and biostatistics classes — was proposed by Baron and Kenny in an early version appeared in Judd and Kenny, The four steps to identification of a mediator are summarized as:.

In epidemiology the last step is commonly utilized — that is, putting your proposed mediator in a model and assessing whether there is an appreciable reduction in magnitude of the parameter estimate comparing the adjusted estimate to the crude. One can take a more quantitative approach to mediation by obtaining an estimate of the total, direct, and indirect pathways.

In the simple figures above, the estimate of the total effect is the value of the parameter estimate for the exposure when the outcome is regressed on the exposure; the direct effect is the parameter estimate for the exposure when the outcome is regressed on the exposure and the mediator. The indirect effect can be calculated either by a product or difference method.

Of note, because estimating the indirect effect simply requires multiplying or subtracting two parameter estimates, obtaining an estimate of the statistical significance of the indirect effect is complicated and requires hand calculation or use of a macro of some kind.

**Nonnegative matrix factorisation with the beta-divergence (...) - Févotte - Workshop 3 -CEB T1 2019**

There are two main limitations of the traditional approach to estimating direct and indirect effects. First, effect decomposition — the fact that the direct and indirect effects sum to the total — using the product or difference method only works in the special case where linear regression is used for the mediator and outcome models and when there is no exposure-mediator interaction.

If interaction is present and the traditional approach is utilized, the effect estimates obtained will not be interpretable. One point worth noting about assessing mediation with binary outcomes when the outcome is common is that because of non-collapsibility of the odds ratio, the traditional approach to mediation even if there is no X-M interaction will result in a non-interpretable estimate.

Second, an often-ignored assumption of this approach is no unmeasured confounding of the M-Y path. This assumption can be violated in both observational studies as well as RCTs because while the exposure can sometimes be randomized, it is often not the case that both exposure and mediator are randomized. First, these methods allow for effect decomposition in the presence of X-M interaction by defining direct and indirect effects controlled or natural from a potential outcomes PO framework and developing estimations of these quantities that are not model specific.

Second, causal mediation clearly explicates the four main assumptions for estimating direct and indirect effects, providing clarity to the no unmeasured confounding assumptions required to perform mediation analysis. The causal mediation approach places emphasis on conducting sensitivity analyses to examine the robustness of findings to violations of these assumptions.Recent work has considerably advanced the definition, identification and estimation of controlled direct, and natural direct and indirect effects in causal mediation analysis.

Despite the various estimation methods and statistical routines being developed, a unified approach for effect estimation under different effect decomposition scenarios is still needed for epidemiologic research.

G-computation offers such unification and has been used for total effect and joint controlled direct effect estimation settings, involving different types of exposure and outcome variables. In this study, we demonstrate the utility of parametric g-computation in estimating various components of the total effect, including 1 natural direct and indirect effects, 2 standard and stochastic controlled direct effects, and 3 reference and mediated interaction effects, using Monte Carlo simulations in standard statistical software.

For each study subject, we estimated their nested potential outcomes corresponding to the mediated effects of an intervention on the exposure wherein the mediator was allowed to attain the value it would have under a possible counterfactual exposure intervention, under a pre-specified distribution of the mediator independent of any causes, or under a fixed controlled value.

A final regression of the potential outcome on the exposure intervention variable was used to compute point estimates and bootstrap was used to obtain confidence intervals. Through contrasting different potential outcomes, this analytical framework provides an intuitive way of estimating effects under the recently introduced 3- and 4-way effect decomposition.

This framework can be extended to complex multivariable and longitudinal mediation settings. Abstract Recent work has considerably advanced the definition, identification and estimation of controlled direct, and natural direct and indirect effects in causal mediation analysis.

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