question: How are statistics calculated for events influenced by multiple variables/parameters?
comment: By making a multi dimensional function, we can define some outcome by multiple inputs. For example, z = x + y, z is defined by two inputs, x and y. For probabilities involving multiple steps, or probabilities that involve previous conditions, we refer to Bayes theorem, which tells us the relationship between two statistical events.
passages: formula_1 , where formula_2 , where formula_3 is the N-dimensional sampled area in the region of support ranging from 0 to formula_4 in each dimension, formula_5 is the set of sampled points at point formula_3, and formula_7 is the reference set of data points.
   Section::::Methodology.:Empirical Methods.
  
  Methodologically, event studies imply the following: Based on an estimation window prior to the analyzed event, the method estimates what the normal stock returns of the affected firm(s) should be at the day of the event and several days prior and after the event (i.e., during the event window). Thereafter, the method deducts this 'normal returns' from the 'actual returns' to receive 'abnormal returns' attributed to the event.
   Section::::Significance of mediation.
  
  As outlined above, there are a few different options one can choose from to evaluate a mediation model.
   If sensors collect multiple data points at a given time then another method of using threshold based detection is to measure the variance of the data and only send data to a gateway node if the variance reaches a certain threshold. Events may occur in only a small portion of the data set. In this case it is important to break the data into smaller segments and take into account the variance of each individual segment. Consider an N-dimensional data set that can be divided into four equal quadrants. The sum of absolute difference can be calculated for all the samples in that section and can be used as an approximation of the variance of that quadrant. Below is what the equation would look like for one quadrant.
   Section::::Methodology.:Calculation of abnormal returns.
  
  Depending on the model chosen for the 'normal return', conducting event studies requires the researcher to implement a distinct sequence of steps. For the most common model, the 'market model', the steps are as follows:
  
  BULLET::::1. Retrieve and match time series of financial returns of the focal firm's stock and its reference index.
  
  BULLET::::2. For each event, identify the sequences of firm and market returns that need to be included in the estimation window.
   However, such analyses were not designed with epidemiological inference in mind and it may be difficult to extrapolate from standard statistics to desired epidemiological quantities.
  
  In an effort to bridge the gap between traditional evolutionary approaches and epidemiological models, several analytical methods have been developed to specifically address problems related to phylodynamics.
  
  These methods are based on coalescent theory, birth-death models, and simulation, and are used to more directly relate epidemiological parameters to observed viral sequences.
  
  Section::::Methods.:Coalescent theory and phylodynamics.
  
  Section::::Methods.:Coalescent theory and phylodynamics.:Effective population size.
   The sum of absolute difference above is one quadrant of the total space sampled and can represent the pseudo-variance of a subset of the data. By collecting all the variances for the different quadrants it is more likely to detect the occurrence of an event even if the event only affects a portion of the total region of support.
  
  Section::::Supervised detection.
   (3) Moderators:
  
  Section::::Moderated mediation.
  
  Mediation and moderation can co-occur in statistical models. It is possible to mediate moderation and moderate mediation.
   where the additional parameter ρ parameterizes the density of the unobserved component "v". Now, the different estimation methods for stock or flow sampling data are available to estimate the relevant parameters.
  
  A specific example is described by Lancaster. Assume that the conditional hazard is given by
  
  "λ(t ; x , v ) = v exp (x β) α t "
   Phylodynamic models may provide insight into epidemiological parameters that are difficult to assess through traditional surveillance means.
  
  For example, assessment of formula_1 from surveillance data requires careful control of the variation of the reporting rate and the intensity of surveillance.
  
  Inferring the demographic history of the virus population from genetic data may help to avoid these difficulties and can provide a separate avenue for inference of formula_1.
  
  Such approaches have been used to estimate formula_1 in hepatitis C virus and HIV.
   Assume the model takes the form
   Once events and event chains are defined, quantitative analysis using Monte Carlo simulation can be performed to quantify the cumulative effect of the events. Probabilities and impacts of risks assigned to activities are used as input data for Monte Carlo simulation of the project schedule. In most projects it is necessary to supplement the event based variance with uncertainties as distributions related to duration, start time, cost, and other parameters.
   Section::::In modeling.:In ANOVA.
  
  A simple setting in which interactions can arise is a two-factor experiment analyzed using Analysis of Variance (ANOVA). Suppose we have two binary factors "A" and "B". For example, these factors might indicate whether either of two treatments were administered to a patient, with the treatments applied either singly, or in combination. We can then consider the average treatment response (e.g. the symptom levels following treatment) for each patient, as a function of the treatment combination that was administered. The following table shows one possible situation:
   (2) Nonspuriousness and/or no confounds. For example, should one identify other third variables and prove that they do not alter the relationship between the independent variable and the dependent variable he/she would have a stronger argument for their mediation effect. See other 3rd variables below.
   Blood stains are an important part of forensic statistics, as the analysis of blood drop collisions may help to picture the event that had previously gone on. Commonly blood stains are an elliptical shape, because of this blood stains are usually easy to determine the blood droplets angle through the formula “"α = arcsin d/a"”. In this formula 'a' and 'd' are simply estimations of the axis of the ellipse. From these calculations, a visualization of the event causing the stains is able to be drawn, and alongside further information such as the velocity of the entity that caused such stains."
   The inductive method relies on the empirically-calculated spatial relationship between
  
  historical or known event occurrence locations and factors that make up the environment
  
  (infrastructure, socio-culture, topographic, etc.). Each event occurrence is plotted in
  
  geographic space and a quantitative relationship is defined between the event occurrence
  
  and the factors that make up the environment. The advantage of this method is that software
  
  can be developed to empirically discover – harnessing the speed of computers, which is
  
  crucial when hundreds of factors are involved – both known and unknown correlations
   Third, analysts need to generate conditional probabilities that events have on each other. Basically, this asks the question, "If event 'A' occurs, what is the new probability of event 'B' occurring?" This must be done for every possible interaction between events.
  
  Fourth, analysts must test their initial conditional probabilities to ensure that there are no mathematical errors. This is usually done by running simulations in a computer several times.
  
  Fifth, analysts can run the analysis to determine future scenarios, or determine how significant other events are to specific events.
  
  Section::::Methodology.:Futures forecasting style.:Mathematical technique.
   Physical properties and the conditions of use are also inherently variable. For example, the applied load (stress) on a mechanical part may vary. The measured strength of that part (tensile strength, etc.) may also be variable. The part will break when the stress exceeds the strength.
  
  With two normal distributions, the statistical interference may be calculated as above. (This problem is also workable for transformed units such as the log-normal distribution). With other distributions, or combinations of different distributions, a Monte Carlo method or simulation is often the most practical way to quantify the effects of statistical interference.
  
  Section::::See also.
   First, analysts must consider the number and type of events to be considered in the analysis and create an event set. Because each event will have an interaction with every other event, Gordon recommends that 10–40 events be used.
  
  Second, analysts must take the initial probability of each event into account. The probabilities of events must be taken in isolation from one another.
   BULLET::::- Zhang, Junni L.; Rubin, Donald B. (2003) "Estimation of Causal Effects via Principal Stratification When Some Outcomes are Truncated by “Death”", "Journal of Educational and Behavioral Statistics", 28: 353–368
  
  BULLET::::- Barnard, John; Frangakis, Constantine E.; Hill, Jennifer L.; Rubin, Donald B. (2003) "Principal Stratification Approach to Broken Randomized Experiments", "Journal of the American Statistical Association", 98, 299–323
  
  BULLET::::- Roy, Jason; Hogan, Joseph W.; Marcus, Bess H. (2008) "Principal stratification with predictors of compliance for randomized trials with 2 active treatments", "Biostatistics", 9 (2), 277–289.
   Section::::Hypothesis tests for interactions.
  
  Analysis of variance and regression analysis are used to test for significant interactions. 
  
  Section::::Hypothesis tests for interactions.:Example: Interaction of temperature and time in cookie baking.
  
  Is the yield of good cookies affected by the baking temperature and time in the oven? The table shows data for 8 batches of cookies.
   Event studies, however, may differ with respect to their specification of normal returns. The most common model for normal returns is the 'market model' (MacKinlay 1997). Following this model, the analysis implies to use an estimation window (typically sized 120 days) prior to the event to derive the typical relationship between the firm's stock and a reference index through a regression analysis. Based on the regression coefficients, the normal returns are then projected and used to calculate the abnormal returns. Alternative models for the normal returns include the CAPM model, or more simplistic approaches such as mean returns (see MacKinlay 1997 for an overview).
   Generally there is wide uncertainty associated with estimates of formula_25, so such aggregation should be done only with care, including sensitivity analysis. Typically, researchers examine specific elements of formula_23 rather than attempting to aggregate.
  
  The big picture result is that by collecting data on average economic sector transactions formula_16 and average sector emissions formula_18, it is possible to make quick predictions about the full supply chain emissions associated with a product of interest by representing the product as marginal changes in production from relevant sectors formula_15.
  
  Section::::Software.
   The use of descriptive and summary statistics has an extensive history and, indeed, the simple tabulation of populations and of economic data was the first way the topic of statistics appeared. More recently, a collection of summarisation techniques has been formulated under the heading of exploratory data analysis: an example of such a technique is the box plot.
   While the concept of mediation as defined within psychology is theoretically appealing, the methods used to study mediation empirically have been challenged by statisticians and epidemiologists and interpreted formally.
  
  (1) Experimental-causal-chain design
  
  An experimental-causal-chain design is used when the proposed mediator is experimentally manipulated. Such a design implies that one manipulates some controlled third variable that they have reason to believe could be the underlying mechanism of a given relationship.
  
  (2) Measurement-of-mediation design
  