Concordant pairs and discordant pairs refer to comparing two pairs of data points to see if they “match.” The meaning is slightly different depending on if you are finding these pairs from various coefficients (like Kendall’s Tau) or if you are performing experimental studies and clinical trials.
Concordant pairs and discordant pairs are used in Kendall’s Tau, for Goodman and Kruskal’s Gamma and in Logistic Regression. They are calculated for ordinal (ordered) variables and tell you if there is agreement (or disagreement) between scores. To calculate concordance or discordance, your data must be ordered and placed into pairs.
Note that in the first column, interviewer 1’s choices have been ordered from smallest to greatest. That way, a comparison can be made between the choices for interviewer 1 and 2. With concordant or discordant pairs, you’re basically answering the question: did the judges/raters rank the pairs in the same order? You aren’t necessarily looking for the exact same rank, but rather if one job seeker was consistently ranked higher by both interviewers.
Concordant pairs: both interviewers rank both applicants in the same order — that is, they both move in the same direction. While they aren’t the same rank (i.e. both 1st or both 2nd), each pair is ordered equally higher or equally lower. Interviewer 1 ranked F as 6th and G as 7th, while interviewer 2 ranked F as 5th and G as 8th. F and G are concordant because F was consistently ranked higher than G.
Discordant pairs: Candidates E and F are discordant because the interviewers ranked in opposite directions (one said E had a higher rank than F, while the other said F ranked higher than 6).
The term “concordant pair” is sometimes used (i.e. in case control studies) to mean a pair who are both exposed (or both not exposed) to some factor. In other words, it is based on exposure status. In a matched pairs design, a concordant pair means that the exposure status of one case is the same as a control case.
A medication’s indication can act as a confounder in observational studies, especially when the drug’s effectiveness is being assessed (Ahrens & Pigeot, 2007). Indication refers to how a particular drug is used for treatment of a certain disease. For example, aspirin is an indication for cardiovascular disease.
Confounding by indication is likely to happen when a particular medicine is linked to the outcome of interest in a study. For example, let’s say you’re observational study is looking into the effects of a new drug A on outcomes for patients with cardiovascular disease (CVD). As the study is observational, there’s no control group or experimental group, and you’re merely observing what happens. In this particular example, patients with more severe cases of CVD are more likely to be prescribed drug A, but they are also more likely to have adverse events (e.g. a stroke). Therefore, your study might conclude that drug A isn’t very effective, because it appears patients have more severe events. These misleading effects (or lack of effects) are what confounding by indication is all about.
Confounding by contraindication is, despite the similar sounding name, completely different from confounding by indication. O. Miettinen, in Epidemiological Research: Terms and Concepts, notes that it’s very rare to find confounding by contraindication in a study, while confounding by indication is quite common.
Confounding by indication is difficult to control for. One of the reasons is that the specific reason why the drug was prescribed usually isn’t recorded (Ahrens & Pigeot, 2007). The solution would be a controlled clinical trial, but they can be expensive and challenging to implement.
Confounding by indication is often confused as a type of selection bias, but it’s actually a type of confounding bias. Confounding isn’t actually a true “bias”, because bias is usually a result of data collection errors or measurement errors. Confounding by indication isn’t (despite the allusion) actually a bias; It’s just something that might result in confounding bias. On the other hand, selection bias is a specific type of bias that affects the types of people who are in your study; It removes the randomness you’re hoping to achieve. For example, the healthy worker effect results in healthier workers in your study, because people who are working are healthier than people who are unemployed or out of work due to a job-related disability.
A consistent estimate has insignificant errors (variations) as sample sizes grow larger. More specifically, the probability that those errors will vary by more than a given amount approaches zero as the sample size increases. In other words, the more data you collect, a consistent estimator will be close to the real population parameter you’re trying to measure. The sample mean and sample variance are two well-known consistent estimators.
The idea of consistency can also be applied to model selection, where you consistently select the “true” model with the associated “true” parameters. For example, a goodness of fit test can also be used as measure of consistency. One popular goodness of fit test is the chi-square test, which works on the premise that expected values for your data fit a normal distribution. And if you have data from a time-series model, data consistency can be measured with an autoregressive model. Many other measures of consistency for fitting to data to models exist. Which method you use depends on what you want your data to measure. For example, do you think your data follows a linear trend, an exponential trend, or a specific trend like the one seen in this paper, which outlines a consistent estimator for disturbance components in financial models?
The term consistent estimator is short for “consistent sequence of estimators,” an idea found in convergence in probability. The basic idea is that you repeat the estimator’s results over and over again, with steadily increasing sample sizes. Eventually — assuming that your estimator is consistent — the sequence will converge on the true population parameter. This convergence is called a limit, which is a fundamental building block of calculus.
Levinsohn, J. & MacKie-Mason, J. (1989). A simple, cons. est. for disturbance components in financial models. National Bureau of Economic Research. Technical working paper No. 80. Retrieved January 7, 2017 from http://www.nber.org/papers/t0080.pdf.
When you’re talking about a construct in relation to testing and construct validity, it has nothing to do with the way a test is designed or constructed. A construct is something that happens in the brain, like a skill, level of emotion, ability or proficiency. For example, proficiency in any language is a construct.
Construct validity is one way to test the validity of a test; it’s used in education, the social sciences, and psychology. It demonstrates that the test is actually measuring the construct it claims it’s measuring. For example, you might try to find out if an educational program increases emotional maturity in elementary school age children. Construct validity would measure if your research is actually measuring emotional maturity.
It isn’t that easy to measure construct validity–several measures are usually required to demonstrate it, including pilot studies and clinical trials. One of the reasons it’s so hard to measure is one of the very reasons it exists: in the social sciences, there’s a lot of subjectivity and most constructs have no real unit of measurement. Even those constructs that do have an acceptable measurement scale (like IQ) are open to debate.
After World War II, many efforts were made to apply statistics to construct validity, but the solutions were so complicated they couldn’t be used in real life. Experience and judgment of the researcher are the acceptable norms to testing construct validity. In some circumstances, such as in clinical trials, statistical tests like a Student’s t-test can be used to determine if there is a significant difference between pre- and post tests.
Excellent StatisticsA contour integral is what we get when we generalize what we’ve learned about taking integrals of real functions along a real line to integrals of complex functions along a contour in a two-dimensional complex plane.
It’s not quite as difficult as it sounds. To directly calculate the values of a contour integral around a given contour, all we need to do is sum the values of the “complex residues“, inside of the contour. A residue in this case is what remains when you integrate around the origin. We can also apply the Cauchy integral formula, or use an application of the residue theorem.
What is a contour in the complex plane? Think about it as a finite (fixed) number of smooth curves. We can define it more exactly as a directed curve, that is made up of a finite sequence of directed smooth curves. Each of these curves must be matched to give just one direction.
Integrating over a contour might sound intimidating, so let’s start with something a bit simpler. Suppose we want to integrate the function f(x) over the curve Γ, and suppose M ∈ â„‚1[I] defines a curve such that Γ = M(I).
That’s all well and good. But what if we want to integrate over a contour which is defined by M1,…Ml ∈ C1[I]? We could describe our contour this way:
Convergence of random variables (sometimes called stochastic convergence) is where a set of numbers settle on a particular number. It works the same way as convergence in everyday life; For example, cars on a 5-line highway might converge to one specific lane if there’s an accident closing down four of the other lanes. In the same way, a sequence of numbers (which could represent cars or anything else) can converge (mathematically, this time) on a single, specific number. Certain processes, distributions and events can result in convergence— which basically mean the values will get closer and closer together.
When Random variables converge on a single number, they may not settle exactly that number, but they come very, very close. In notation, x (xn → x) tells us that a sequence of random variables (xn) converges to the value x. This is only true if the https://www.statisticshowto.com/absolute-value-function/#absolute of the differences approaches zero as n becomes infinitely larger. In notation, that’s:
Each of these definitions is quite different from the others. However, for an infinite series of independent random variables: convergence in probability, convergence in distribution, and almost sure convergence are equivalent (Fristedt & Gray, 2013, p.272).
If you toss a coin n times, you would expect heads around 50% of the time. However, let’s say you toss the coin 10 times. You might get 7 tails and 3 heads (70%), 2 tails and 8 heads (20%), or a wide variety of other possible combinations. Eventually though, if you toss the coin enough times (say, 1,000), you’ll probably end up with about 50% tails. In other words, the percentage of heads will converge to the expected probability.
The concept of a limit is important here; in the limiting process, elements of a sequence become closer to each other as n increases. In simple terms, you can say that they converge to a single number.
Convergence in distribution (sometimes called convergence in law) is based on the distribution of random variables, rather than the individual variables themselves. It is the convergence of a sequence of cumulative distribution functions (CDF). As it’s the CDFs, and not the individual variables that converge, the variables can have different probability spaces.
In more formal terms, a sequence of random variables converges in distribution if the CDFs for that sequence converge into a single CDF. Let’s say you had a series of random variables, Xn. Each of these variables X1, X2,…Xn has a CDF FXn(x), which gives us a series of CDFs {FXn(x)}. Convergence in distribution implies that the CDFs converge to a single CDF, Fx(x) (Kapadia et. al, 2017).
Several methods are available for proving convergence in distribution. For example, Slutsky’s Theorem and the Delta Method can both help to establish convergence. Convergence of moment generating functions can prove convergence in distribution, but the converse isn’t true: lack of converging MGFs does not indicate lack of convergence in distribution. Scheffe’s Theorem is another alternative, which is stated as follows (Knight, 1999, p.126):
Let’s say that a sequence of random variables Xn has probability mass function (PMF) fn and each random variable X has a PMF f. If it’s true that fn(x) → f(x) (for all x), then this implies convergence in distribution. Similarly, suppose that Xn has cumulative distribution function (CDF) fn (n ≥ 1) and X has CDF f. If it’s true that fn(x) → f(x) (for all but a countable number of X), that also implies convergence in distribution.
Almost sure convergence (also called convergence in probability one) answers the question: given a random variable X, do the outcomes of the sequence Xn converge to the outcomes of X with a probability of 1? (Mittelhammer, 2013).
As an example of this type of convergence of random variables, let’s say an entomologist is studying feeding habits for wild house mice and records the amount of food consumed per day. The amount of food consumed will vary wildly, but we can be almost sure (quite certain) that amount will eventually become zero when the animal dies. It will almost certainly stay zero after that point. We’re “almost certain” because the animal could be revived, or appear dead for a while, or a scientist could discover the secret for eternal mouse life. In life — as in probability and statistics — nothing is certain.