regression to the mean

Examples of regression to the mean in the following topics:

• The Regression Fallacy

  • Things such as golf scores, the earth's temperature, and chronic back pain fluctuate naturally and usually regress towards the mean.
  • Speed cameras are often installed after a road incurs an exceptionally high number of accidents, and this value usually falls (regression to mean) immediately afterwards.
  • The reason is that political power and occupation of territories is not primarily determined by random events, making the concept of regression to the mean inapplicable (on the large scale).
  • In essence, misapplication of regression to the mean can reduce all events to a "just so" story, without cause or effect.
  • Such misapplication takes as a premise that all events are random, as they must be for the concept of regression to the mean to be validly applied.

• Regression Toward the Mean: Estimation and Prediction

  • Regression toward the mean says that if a variable is extreme on its 1st measurement, it will tend to be closer to the average on its 2nd.
  • Historically, what is now called regression toward the mean has also been called reversion to the mean and reversion to mediocrity.
  • Thus the mean of these students would "regress" all the way back to the mean of all students who took the original test.
  • The following is a second example of regression toward the mean.
  • It is possible for changes between the measurement times to augment, offset or reverse the statistical tendency to regress toward the mean.

• Multiple Regression Models

  • Multiple regression is used to find an equation that best predicts the $Y$ variable as a linear function of the multiple $X$ variables.
  • The purpose of a multiple regression is to find an equation that best predicts the $Y$ variable as a linear function of the $X$ variables.
  • Multiple regression would give you an equation that would relate the tiger beetle density to a function of all the other variables.
  • A second use of multiple regression is to try to understand the functional relationships between the dependent and independent variables, to try to see what might be causing the variation in the dependent variable.
  • As you are doing a multiple regression, there is also a null hypothesis for each $X$ variable, meaning that adding that $X$ variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.

• Polynomial Regression

  • The goal of polynomial regression is to model a non-linear relationship between the independent and dependent variables.
  • Polynomial regression fits a nonlinear relationship between the value of $x$ and the corresponding conditional mean of $y$, denoted $E(y\ | \ x)$, and has been used to describe nonlinear phenomena such as the growth rate of tissues, the distribution of carbon isotopes in lake sediments, and the progression of disease epidemics.
  • Point-wise or simultaneous confidence bands can then be used to provide a sense of the uncertainty in the estimate of the regression function.
  • The goal of polynomial regression is to model a non-linear relationship between the independent and dependent variables (technically, between the independent variable and the conditional mean of the dependent variable).
  • This is similar to the goal of non-parametric regression, which aims to capture non-linear regression relationships.

• Introduction to Linear Regression

  • The vertical lines from the points to the regression line represent the errors of prediction.
  • That is the criterion that was used to find the line in Figure 2.
  • MX is the mean of X, MY is the mean of Y, sX is the standard deviation of X, sY is the standard deviation of Y, and r is the correlation between X and Y.
  • The formulas are the same; simply use the parameter values for means, standard deviations, and the correlation.
  • The regression equation is simpler if variables are standardized so that their means are equal to 0 and standard deviations are equal to 1, for then b = r and A = 0.

• Regression Analysis for Forecast Improvement

  • The error is a random variable with a mean of zero conditional on the explanatory variables.
  • In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.
  • Regression analysis is also used to understand which among the independent variables is related to the dependent variable, and to explore the forms of these relationships.
  • Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional.
  • The performance of regression analysis methods in practice depends on the form of the data generating process and how it relates to the regression approach being used.

• Predictions and Probabilistic Models

  • In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.
  • Regression analysis is also used to understand which among the independent variables is related to the dependent variable, and to explore the forms of these relationships.
  • This means that any extrapolation is particularly reliant on the assumptions being made about the structural form of the regression relationship.
  • A good rule of thumb when using the linear regression method is to look at the scatter plot of the data.
  • Explain how to estimate the relationship among variables using regression analysis

• Checking the Model and Assumptions

  • These assumptions are similar to those of standard linear regression models.
  • The following are the major assumptions with regard to multiple regression models:
  • Error will not be evenly distributed across the regression line.
  • In effect, residuals appear clustered and spread apart on their predicted plots for larger and smaller values for points along the linear regression line; the mean squared error for the model will be incorrect.
  • Most experts recommend that there should be at least 10 to 20 times as many observations (cases, respondents) as there are independent variables, otherwise the estimates of the regression line are probably unstable and unlikely to replicate if the study is repeated.

• A Graph of Averages

  • A graph of averages and the least-square regression line are both good ways to summarize the data in a scatterplot.
  • The regression line drawn through the points describes how the dependent variable $y$ changes with the independent variable $x$.
  • A good line of regression makes the distances from the points to the line as small as possible.
  • This line passes through the point $(\bar{x},\bar{y})$ (the means of $x$ and $y$).
  • If we needed to summarize the $y$ values whose $x$ values fall in a certain interval, the point plotted on the graph of averages would be good to use.

• Qualitative Variable Models

  • In statistics, particularly in regression analysis, a dummy variable (also known as a categorical variable, or qualitative variable) is one that takes the value 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome.
  • For example, if gender is one of the qualitative variables relevant to a regression, then the categories included under the gender variable would be female and male.
  • Analysis of variance (ANOVA) models are a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups).
  • An example with one qualitative variable might be if we wanted to run a regression to find out if the average annual salary of public school teachers differs among three geographical regions in a country .
  • Break down the method of inserting a dummy variable into a regression analysis in order to compensate for the effects of a qualitative variable.