#
#
######## A few Graphic Examples of Simpson's Paradox, in 100 lines or less.
#
# Failure to recognize clusters or groups within the data may lead to
# strikingly wrong conclusions about the relationships among the variables.
# http://en.wikipedia.org/wiki/Simpson%27s_paradox
#

library(car)
N <- 300
n <- 100
group <- c(rep("low", n), rep("med", n), rep("high", n))
X <- rnorm(N, 100, 15); Y <- .8*X + rnorm(N, 100, 15); cor(X,Y)
scatterplot(X,Y, smooth = FALSE)
dev.new()
scatterplot(X,Y, ellipse = FALSE, groups = group, legend.plot = TRUE)

######

graphics.off()
x1 <- rnorm(n, 50, 15); y1 <- -.6*x1 + rnorm(n, 50, 30); cor(x1,y1)
x2 <- rnorm(n, 100, 15); y2 <- -.6*x2 + rnorm(n, 100, 30); cor(x2,y2)
x3 <- rnorm(n, 150, 15); y3 <- -.6*x3 + rnorm(n, 150, 30); cor(x3,y3)
X <- c(x1, x2, x3); Y <- c(y1, y2, y3); cor(X,Y)
scatterplot(X,Y, smooth = FALSE)
dev.new()
scatterplot(X,Y, ellipse = FALSE, groups = group, legend.plot = TRUE)

######

graphics.off()
x1 <- rnorm(n, 50, 15); y1 <- -.6*x1 + rnorm(n, 50, 20); cor(x1,y1)
x2 <- rnorm(n, 100, 15); y2 <- -.6*x2 + rnorm(n, 100, 20); cor(x2,y2)
x3 <- rnorm(n, 150, 15); y3 <- -.6*x3 + rnorm(n, 150, 20); cor(x3,y3)
X <- c(x1, x2, x3); Y <- c(y1, y2, y3); cor(X,Y)
scatterplot(X,Y, smooth = FALSE)
dev.new()
scatterplot(X,Y, ellipse = FALSE, groups = group, legend.plot = TRUE)

######

graphics.off()
x1 <- rnorm(n, 50, 15); y1 <- -.6*x1 + rnorm(n,50,10); cor(x1,y1)
x2 <- rnorm(n, 100, 15); y2 <- -.6*x2 + rnorm(n,100,10); cor(x2,y2)
x3 <- rnorm(n, 150, 15); y3 <- -.6*x3 + rnorm(n,150,10); cor(x3,y3)
X <- c(x1, x2, x3); Y <- c(y1, y2, y3); cor(X,Y)
scatterplot(X,Y, smooth = FALSE)
dev.new()
scatterplot(X,Y, ellipse = FALSE, groups = group, legend.plot = TRUE)

######

graphics.off()
x1 <- rnorm(n, 50, 15); y1 <- -.6*x1 + rnorm(n,50,5); cor(x1,y1)
x2 <- rnorm(n, 100, 15); y2 <- -.6*x2 + rnorm(n,100,5); cor(x2,y2)
x3 <- rnorm(n, 150, 15); y3 <- -.6*x3 + rnorm(n,150,5); cor(x3,y3)
X <- c(x1, x2, x3); Y <- c(y1, y2, y3); cor(X,Y)
scatterplot(X,Y, smooth = FALSE)
dev.new()
scatterplot(X,Y, ellipse = FALSE, groups = group, legend.plot = TRUE)

######

graphics.off()
x1 <- rnorm(n, 50, 15); y1 <- -.6*x1 + rnorm(n,50,1); cor(x1,y1)
x2 <- rnorm(n, 100, 15); y2 <- -.6*x2 + rnorm(n,100,1); cor(x2,y2)
x3 <- rnorm(n, 150, 15); y3 <- -.6*x3 + rnorm(n,150,1); cor(x3,y3)
X <- c(x1, x2, x3); Y <- c(y1, y2, y3); cor(X,Y)
scatterplot(X,Y, smooth = FALSE)
dev.new()
scatterplot(X,Y, ellipse = FALSE, groups = group, legend.plot = TRUE)

######

graphics.off()
rm(group,N,n,x1,x2,x3,y1,y2,y3,X,Y)
detach("package:car");detach("package:MASS");detach("package:nnet")
search()
ls()

######
#
# References & Resources.
#
# Bickel, P. J., Hammel, E. A., & O'Connell. (1975). Sex bias in graduate
#    admissions: Data from Berkeley. Science, 187, 398 - 404.
# Blyth, C. R. (1972). On Simpson's Paradox and the Sure-Thing Principle.
#    Journal of the American Statistical Association, 67, 364 - 366.
# Clifford, W. H. (1982). Simpson's paradox in real life. The American
#    Statistician, 36, 46 - 48.
# Fox, J., et al., (2012). Package 'car': A companion package to An R
#    Companion to Applied Regression, (2nd Ed.), Sage, 2011.
# Kievit, R., & Epskamp. S. (2012). Package 'Simpsons': A package for
#    detecting Simpson's Paradox.
# Pearl, J. (2009). Causality: Models, reasoning, and inference (2nd ed.).
#    [Chapter 6: Simpson's paradox, confounding, and collapsibility.] New York:
#    Cambridge University Press.
# Ripley, B., Venables, B., Hornik, K., Gebhardt, A., & Firth, D. (2012).
#    Package 'MASS': Functions and datasets to support Venables and Ripley,
#    'Modern Applied Statistics with S' (4th ed.), 2002.
# Ripley, B. (2012). Package 'nnet': Software for feed-forward neural networks
#    with a single hidden layer, and for multinomial log-linear models.
# Robinson, W. S. (1950). Ecological correlations and the behavior of
#    individuals. American Sociological Review, 15, 351 - 357.
# Rucker, G., & Schumacher, M. (2008). Simpson's paradox visualized: The
#    example of the Rosiglitazone meta-analysis. BMC Medical Research
#    Methodology, 8:34. DOI: 10.1186/1471-2288-8-34
# Simpson, E. H. (1951). The interpretation of interaction in contingency
#    tables. Journal of the Royal Statistical Society (Series B:
#    Methodological), 13, 238 - 241.
#
#
#
# Updated December 11, 2012.