## ----setup, include=FALSE----------------------------------------------------- knitr::opts_chunk$set(echo = TRUE) library(knitr) set.seed(12314159) ## ----libraries---------------------------------------------------------------- library(eikosograms) library(gridExtra) ## ----setup_conditional_independence_tables, eval = TRUE, echo = FALSE--------- # This function will create all the joint probabilities # and place them in the appropriate three-way table create3WayBinaryTable <- function(widths, heights) { toprow_eikos <- widths * (1 - heights) bottomrow_eikos <- widths * heights probs <- array(c(bottomrow_eikos, toprow_eikos), dim = c(2,2,2), dimnames = list(Z = c("z1", "z2"), X = c("x1", "x2"), Y = c("y1", "y2") )) as.table(probs) } # Marginal probabilities (widths) for # the two binary conditioning variates # are given by the xz values, # and the conditional probabilities (heights) # for the response variate's first value # are given by the b values. # Figure numbers here refer to those figure numbers of # the original Oldford (2003) article entitled # "Understanding probabilistic independence and its # modelling via Eikosograms and graphs." xz1 <- c(2/7, 8/35, 1/7, 12/35) b1 <- c(4/5, 1/2, 3/10, 7/10) fig5a_data <- create3WayBinaryTable(xz1, b1) b2 <- c(4/5, 1/2, 3/5, 1/5) fig5b_data <- create3WayBinaryTable(xz1, b2) b3 <- c(4/5, 1/2, 4/5, 1/2) fig5c_data <- create3WayBinaryTable(xz1, b3) b4 <- c(7/10, 7/10, 7/10, 7/10) fig6_data <- create3WayBinaryTable(xz1, b4) xz2 <- c(10/33, 4/11, 5/33, 2/11) fig7_data <- create3WayBinaryTable(xz2, b4) xz3 <- c(1/3, 1/6, 1/3, 1/6) b5 <- c(7/10, 7/10, 3/10, 3/10) fig8_data <- create3WayBinaryTable(xz2, b5) xz4 <- c(2/9, 1/9, 2/9, 4/9) b6 <- c(2/3, 2/3, 1/6, 1/6) fig9_data <- create3WayBinaryTable(xz4, b6) xz5 <- c(1/7, 1/7, 3/7, 2/7) b7 <- c(1/3, 2/3, 1/4, 1/6) fig11_data <- create3WayBinaryTable(xz5, b7) xz6 <- c(1/4, 1/4, 1/4, 1/4) b8 <- c(3/4, 1/2, 1/4, 1/4) fig12_data <- create3WayBinaryTable(xz6, b8) xz7 <- c(1/3, 1/6, 1/3, 1/6) b9 <- c(2/3, 1/2, 5/6, 1/6) fig13_data <- create3WayBinaryTable(xz7, b9) xz8 <- c(1/6, 1/3, 1/6, 1/3) b10 <- c(1/6, 2/3, 5/6, 1/3) fig14_data <- create3WayBinaryTable(xz8, b10) # Create this now for early illustration cond_indep <- fig5c_data ## ----independence_Y_given_X, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Independence of variates X and Y"---- independenceExample <- as.table(array(c(10, 40, 15, 60), dim = c(2,2), dimnames = list(X =c("x_1", "x_2"), Y = c("y_1", "y_2")))) ## ----bivariate_independence_1, eval = FALSE, echo = TRUE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%"---- # eikos("Y", "X", data = independenceExample) ## ----png_bivariate_independence_1, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Independence of variates Y and X"---- include_graphics("img/IndependenceExploration/indepYX.png") ## ----bivariate_independence_2, eval = FALSE, echo = TRUE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%"---- # eikos("X", "Y", data = independenceExample) ## ----png_bivariate_independence_2, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Independence of variates X and Y"---- include_graphics("img/IndependenceExploration/indepXY.png") ## ----conditional_independence, eval = FALSE, echo = TRUE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%"---- # eikos("Y", c("X", "Z"), data = cond_indep, xaxs = FALSE, yaxs = FALSE) ## ----png_conditional_independence, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Conditional independence of variates Y and X given Z"---- include_graphics("img/IndependenceExploration/indepYgivenXZ.png") ## ----table_creation_function , echo = TRUE, eval = FALSE---------------------- # # This function will create all the joint probabilities # # and place them in the appropriate three-way table # create3WayBinaryTable <- function(widths, heights) { # toprow_eikos <- widths * (1 - heights) # bottomrow_eikos <- widths * heights # probs <- array(c(bottomrow_eikos, toprow_eikos), # dim = c(2,2,2), # dimnames = list(Z = c("z1", "z2"), # X = c("x1", "x2"), # Y = c("y1", "y2") # )) # as.table(probs) # } ## ----some_table, echo = TRUE, eval = TRUE------------------------------------- someTable <- create3WayBinaryTable(widths = c(10/35, 8/35, 5/35, 12/35), heights = c(7/10, 7/10, 7/10, 7/10)) ## ----cond_indep_table, echo = FALSE------------------------------------------- knitr::kable(cond_indep) ## ----eikos_of_someTable, eval = FALSE, echo = TRUE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%"---- # eikos(y="Y", x = c("Z", "X"), data = someTable) ## ----png_eikos_of_someTable, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "What is the independence structure?"---- include_graphics("img/IndependenceExploration/someTableYgivenZX.png") ## ----conditional_independence_tables_display, echo = FALSE, eval = FALSE------ # # Marginal probabilities (widths) for # # the two binary conditioning variates # # are given by the xz values, # # and the conditional probabilities (heights) # # for the response variate's first value # # are given by the b values. # # Figure numbers here refer to those figure numbers of # # the original Oldford (2003) article entitled # # "Understanding probabilistic independence and its # # modelling via Eikosograms and graphs." # xz1 <- c(2/7, 8/35, 1/7, 12/35) # b1 <- c(4/5, 1/2, 3/10, 7/10) # fig5a_data <- create3WayBinaryTable(xz1, b1) # b2 <- c(4/5, 1/2, 3/5, 1/5) # fig5b_data <- create3WayBinaryTable(xz1, b2) # b3 <- c(4/5, 1/2, 4/5, 1/2) # fig5c_data <- create3WayBinaryTable(xz1, b3) # b4 <- c(7/10, 7/10, 7/10, 7/10) # fig6_data <- create3WayBinaryTable(xz1, b4) # xz2 <- c(10/33, 4/11, 5/33, 2/11) # fig7_data <- create3WayBinaryTable(xz2, b4) # xz3 <- c(1/3, 1/6, 1/3, 1/6) # b5 <- c(7/10, 7/10, 3/10, 3/10) # fig8_data <- create3WayBinaryTable(xz2, b5) # xz4 <- c(2/9, 1/9, 2/9, 4/9) # b6 <- c(2/3, 2/3, 1/6, 1/6) # fig9_data <- create3WayBinaryTable(xz4, b6) # xz5 <- c(1/7, 1/7, 3/7, 2/7) # b7 <- c(1/3, 2/3, 1/4, 1/6) # fig11_data <- create3WayBinaryTable(xz5, b7) # xz6 <- c(1/4, 1/4, 1/4, 1/4) # b8 <- c(3/4, 1/2, 1/4, 1/4) # fig12_data <- create3WayBinary Table(xz6, b8) # xz7 <- c(1/3, 1/6, 1/3, 1/6) # b9 <- c(2/3, 1/2, 5/6, 1/6) # fig13_data <- create3WayBinaryTable(xz7, b9) # xz8 <- c(1/6, 1/3, 1/6, 1/3) # b10 <- c(1/6, 2/3, 5/6, 1/3) # fig14_data <- create3WayBinaryTable(xz8, b10) # ## ----complete_independence_table, echo = TRUE--------------------------------- complete_indep <- create3WayBinaryTable(widths = c(10/33, 12/33, 5/33, 6/33), heights = c(7/10, 7/10, 7/10, 7/10)) ## ----complete_independence_code, eval = TRUE, fig.height = 0.1---------------- eikosY <- eikos(y="Y", x = c("Z", "X"), xaxs = FALSE, yaxs= FALSE, data = complete_indep, draw = FALSE) eikosX <- eikos(y="X", x = c("Z", "Y"), xaxs = FALSE, yaxs= FALSE, data = complete_indep, draw = FALSE) eikosZ <- eikos(y="Z", x = c("X", "Y"), xaxs = FALSE, yaxs= FALSE, data = complete_indep, draw = FALSE) ## ----complete_independence_arrange_3, eval = FALSE, echo = TRUE, fig.width=11, fig.height=3.5, fig.align="center", out.width="100%"---- # grid.arrange(eikosY, eikosX, eikosZ, nrow=1) ## ----png_complete_independence_arrange_3, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Complete independence"---- include_graphics("img/IndependenceExploration/completeIndep.png") ## ----conditional_on_one, echo = TRUE, fig.width=9, fig.height=0.1, fig.align="center", out.width="80%"---- eikosYX <- eikos(y = "Y", x = "X", data = complete_indep, main = "Y | X", xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosYZ <- eikos(y = "Y", x = "Z", data = complete_indep, main = "Y | Z", xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosXZ <- eikos(y = "X", x = "Z", data = complete_indep, main = "X | Z", xaxs = FALSE, yaxs= FALSE, draw = FALSE) ## ----conditional_on_one_arranged, eval = FALSE, echo = TRUE, fig.width=10, fig.height=3, fig.align="center", out.width="80%"---- # grid.arrange(eikosYX, eikosYZ, eikosXZ, nrow = 1) ## ----png_conditional_on_one_arranged, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Pairwise independence"---- include_graphics("img/IndependenceExploration/pairwise.png") ## ----grid_code, eval = TRUE, fig.height = 0.1--------------------------------- layout3and2way <- function(table) { if(length(dimnames(table))!=3) stop("Must be a three-way table") varNames <- names(dimnames(table)) zVar <- varNames[1] xVar <- varNames[2] yVar <- varNames[3] eikosY <- eikos(y = yVar, x = c(zVar, xVar), data = table, main = paste0(yVar, " | ", zVar, "&", xVar), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosX <- eikos(y = xVar, x = c(zVar, yVar), data = table, main = paste0(xVar, " | ", zVar, "&", yVar), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosZ <- eikos(y = zVar, x = c(xVar, yVar), data = table, main = paste0(zVar, " | ", xVar, "&", yVar), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosYX <- eikos(y = yVar, x = xVar, data = table, main = paste(yVar, xVar, sep =" | "), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosYZ <- eikos(y = yVar, x = zVar, data = table, main = paste(yVar, zVar, sep =" | "), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosXY <- eikos(y = xVar, x = yVar, data = table, main = paste(xVar, yVar, sep =" | "), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosXZ <- eikos(y = xVar, x = zVar, data = table, main = paste(xVar, zVar, sep =" | "), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosZY <- eikos(y = zVar, x = yVar, data = table, main = paste(zVar, yVar, sep =" | "), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) eikosZX <- eikos(y = zVar, x = xVar, data = table, main = paste(zVar, xVar, sep =" | "), xlabs = FALSE, ylabs = FALSE, xaxs = FALSE, yaxs= FALSE, draw = FALSE) layout <- rbind(c(1,1, NA, 2, 2, NA, 3, 3), rep(NA, 8), c(4, 5, NA, 6, 7, NA, 8, 9)) grid.arrange(eikosY, eikosX, eikosZ, eikosYX, eikosYZ, eikosXY, eikosXZ, eikosZY, eikosZX, layout_matrix = layout, widths = c(2,2,1,2,2,1,2,2), heights = c(2, 0.5, 1.1) ) } ## ----complete_independence_layout, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # layout3and2way(complete_indep) ## ----png_complete_independence_layout, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "Complete independence"---- include_graphics("img/IndependenceExploration/complete_indep.png") ## ----Case_2, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # table <- create3WayBinaryTable(widths = c(10/33, 12/33, 5/33, 6/33), # heights = c(7/10, 7/10, 3/10, 3/10)) # # layout3and2way(table) ## ----png_Case_2, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "One 4-flat; two 2 by 2 flats"---- include_graphics("img/IndependenceExploration/case2.png") ## ----Case_3, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # table <- create3WayBinaryTable(widths = c(2/9, 1/9, 2/9, 4/9), # heights = c(2/3, 2/3, 1/6, 1/6)) # # layout3and2way(table) ## ----png_Case_3, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "One no-flat; two 2 by 2 flats"---- include_graphics("img/IndependenceExploration/case3.png") ## ----Case_4_1, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # table <- create3WayBinaryTable(widths = c(1/7, 1/7, 3/7, 2/7), # heights = c(1/3, 2/3, 1/4, 1/6)) # # layout3and2way(table) ## ----png_Case_4_1, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "No flats; no marginal independence"---- include_graphics("img/IndependenceExploration/case41.png") ## ----Case_4_2, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # table <- create3WayBinaryTable(widths = c(1/4, 1/4, 1/4, 1/4), # heights = c(3/4, 1/2, 1/4, 1/4)) # # layout3and2way(table) ## ----png_Case_4_2, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "No flats; one marginal independence"---- include_graphics("img/IndependenceExploration/case42.png") ## ----Case_4_3, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # table <- create3WayBinaryTable(widths = c(2/6, 1/6, 2/6, 1/6), # heights = c(2/3, 1/2, 5/6, 1/6)) # # layout3and2way(table) ## ----png_Case_4_3, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "No flats; two marginal independences"---- include_graphics("img/IndependenceExploration/case43.png") ## ----Case_4_4, eval = FALSE, echo = TRUE, fig.width=11, fig.height=5.5, fig.align="center", out.width="80%"---- # table <- create3WayBinaryTable(widths = c(1/6, 2/6, 1/6, 2/6), # heights = c(1/6, 2/3, 5/6, 1/3)) # # layout3and2way(table) ## ----png_Case_4_4, eval = TRUE, echo = FALSE, fig.width=4, fig.height=3.5, fig.align="center", out.width="50%", fig.cap = "No flats; three marginal independences"---- include_graphics("img/IndependenceExploration/case44.png")