This is a collection of web apps built using Shiny to illustrate and help build intuitions about some statistical concepts.

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This example demonstrates the relationship between the standard deviation of a population, the standard deviation and standard error of the mean for a sample drawn from that population, and the expected distribution of means that we would obtain if we took many samples (of the same size) from the population. It is meant to emphasize how the standard error of the mean, as calculated from the sample statistics for a single sample, corresponds to the width of the expected distribution of means (under normal assumptions).

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This example performs 1000 one-sample t tests (with different samples from the same distribution) and plots the resulting histograms of t statistics and p values. It is possible to control both the true effect size (Cohen's D) and the number of observations in a sample to show how these two parameters relate the expected distribution of scores. When the effect size is 0, the simulation shows what happens when the null hypothesis is true.

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This example demonstrates the key objective of linear regression: finding the coefficients for a linear model that minimize the squared distance from each observation to the prediction made by the model at the same value of x.

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Similar to the linear regression example, this app shows how the goal of logistic regression is to find a model (expressed in linear coefficients -- here just the intercept and a slope term) that maximizes the likelihood of the data you are fitting the model to.

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This app plots a simple linear regression and allows the user to visualize the distribution of regression estimates from bootstrap resamples of the dataset. The user can also plot a normal density with mean at y-hat and standard deviation equal to the standard error of the regression estimate at that point. The app thus draws a comparison between the bootstrap procedure, the expected sampling characteristics of the regression line, and a common way of visualizing the uncertainty of a regression.

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This app plots a basic multiple regression with two variables: x, a continuous measure, and group, a categorical measure. The app lets the user choose whether to fit a simple regression, an additive multiple regression, or an interactive multiple regression, and it shows the lm() output and a visualization for each choice. The app also lets the user control the true effect size for each component of the data to help build intuition about the visual and statistical consequences of different relationships between variables in a multiple regression.

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This app shows what happens to multiple regression results when there is considerable covariance between two contiunous predictor variables. Although the overall model fit does not change as the covariance is increased (as visualized by the regression of y onto yhat and the R squared in the model summary), the parameter estimates become unstable and the confidence intervals expand, which yields large p values even though the relationship between the predictors and the response variable does not change.

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This app is intended to provide some intuition about simple mediation models. It allows you to specify a range of causal structures by changing the strength (and direction) of the relationships between three variables. Once you have constructed a structure, you can observe the effects of manipulating the system. Finally, you can simulate data from a model with the specified structure and observe how changing the strength of the relationships influences the regression parameters and inferential statistics.