Bayesian Simultaneous Intervals for Small Areas: An Application to Variation in Maps Erik Barry Erhardt, Balgobin Nandram, Jai Won Choi International Journal of Statistics and Probability Vol 1, No 2, pp. 229–243 Received: September 19, 2012 Accepted: October 24, 2012 Online: October 29, 2012 http://www.ccsenet.org/journal/index.php/ijsp/article/view/20714 doi:10.5539/ijsp.v1n2p229 Abstract Bayesian inference about small areas is of considerable current interest, and simultaneous intervals for the parameters for the areas are needed because these parameters are correlated. This is not usually pursued because with many areas the problem becomes difficult. We describe a method for finding simultaneous credible intervals for a relatively large number of parameters, each corresponding to a single area. Our method is model based, it uses a hierarchical Bayesian model, and it starts with either the 100(1-alpha)% (e.g., alpha=0.05 for 95%) credible interval or highest posterior density (HPD) interval for each area. As in the construction of the HPD interval, our method is the result of the solution of two simultaneous equations, an equation that accounts for the probability content, 100(1-alpha)% of all the intervals combined, and an equation that contains an optimality condition like the “equal ordinates” condition in the HPD interval. We compare our method with one based on a nonparametric method, which as expected under a parametric model, does not perform as well as ours, but is a good competitor. We illustrate our method and compare it with the nonparametric method using an example on disease mapping which utilizes a standard Poisson regression model.
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The UNM Department of Mathematics and Statistics funded my travel request to go to 2012 WNAR – Graybill June 17-20, 2012 at Colorado State University – Fort Collins, Colorado. I intend to participate with a talk on my trophic-level modeling in the “Biostatistics and systems biology” session, meet and discuss research with those working in similar areas, and look for opportunities for cross-collaboration for students at these other western universities.
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A morphometric analysis of Actaea racemosa L. (Ranunculaceae) Z. Gardner, L. Lueck, E.B. Erhardt, L.E. Craker Journal of Medicinally Active Plants scholarworks.umass.edu/jmap/vol1/iss2/3/ Abstract Actaea racemosa L. (syn. Cimicifuga racemosa [L.] Nutt.), Ranunculaceae, commonly known as black cohosh, is an herbaceous, perennial, medicinal plant native to the deciduous woodlands of eastern North America. Historical texts and current sales data indicate the continued popularity of this plant as an herbal remedy for over 175 years. Much of the present supply of A. racemosa is harvested from the wild. Diversity within and between populations of the species has not been well characterized. The purpose of this study was to assess the morphological variation of A. racemosa and identify patterns of variation at the population and species levels. A total of twentysix populations representative of a significant portion of the natural range of the species were surveyed and plant material was collected for the morphological analysis of 37 leaflet, flower, and whole plant characteristics. In total, 511 leaflet samples and 83 flower samples were examined. Several of the populations surveyed had sets of relatively unique characteristics (large leaflet measurements, tall leaves and flowers, and a large number of stamen) and Tukey-Kramer multiple comparisons revealed significant differences between specific populations for 20 different characteristics. However, no unique phenotype was found. Considerable morphological plasticity was noted in the apices of the staminodia. Cluster analyses showed that the morphological variation within populations is not smaller than between population and that this variation in not influenced by their geographic distribution.
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This article was printed in the Daily Lobo on 11/10/2011. Presenting information in a way that clearly answers interesting questions is challenging. Every plot has an implicit question (hypothesis) that it helps you answer. Therefore, it is important to align a visual display of information with the intended interesting question(s). Collaboration or consultation with a statistician can clarify interesting questions and lead to answers through appropriate data analysis (visit UNM’s free statistics consulting clinic, www.stat.unm.edu/~clinic). Suicide was the topic of the front cover story in the Daily Lobo on Thurs, Nov 3rd. With the story, two pie charts displayed average annual proportions of “successful” and “unsuccessful” suicides by method in NM. The “successful” pie chart answers this statement of conditional probability (their implied question): “given a successful suicide, what percentage used certain methods?” A question I consider more interesting reverses the conditioning (my question): “given an attempted suicide with a certain method, what percentage were successful?” Furthermore, I want to know the overall frequency and percentage of each method attempted. How can we present the information in a way that simultaneously answers these questions? The Suicide Prevention Resource Center (SPRC.org) maintains national and state suicide fact sheets, last updated September 2008, describing “deaths by suicide, estimated hospitalized attempts, and data on medical costs, work loss costs, gender, race/ethnicity, age, and method of suicide.” The pie charts in Thursday’s Daily Lobo were reproductions of those found on the NM fact sheet. From their NM summaries, below is the SPRC table for estimated mean frequencies by method for “successful” and “unsuccessful” suicides.

Method	Successful	Unsuccessful	Total
Cut/Pierce	4	229	233
Firearms	191	16	207
Poisoning	60	1097	1157
Suffocation	73	23	96
Other/Unspecified	13	91	104
Total	341	1456	1797

Their question and pie charts (below) consider percentages down columns. When the data are reduced to row percentages for “successful” and “unsuccessful” attempts separately, you lose the relative frequency of attempts. The percentage of firearms “successes” (56%), for example, depends on all the other “successful” attempts. Because proportions for “successful” and “unsuccessful” attempts are separate, you can’t learn about how successful firearm attempts are. [caption id="" align="alignnone" width="549" caption="Original pie charts of proportions of method conditional on attempt "success", which doesn't ask/answer the interesting/relavant question."][/caption] It is critical to consider the temporal process: a person first chooses a method, then makes an attempt, and is either “successful” or not. The data display and questions should follow these temporal steps. The pie chart displays ignore this process. My question and plot (below) considers the temporal process of attempting suicide, considering percentages across rows, including row total information. First, the relative use of various methods is clear; almost two-thirds of attempts are by poisoning, and firearm and cut/pierce are each just above one in ten. However, though attempts by firearms (12%) and cut/pierce (13%) are relatively rare, the “success” rates are extremely different (92% versus 2%)! The plot has been sorted by the numbers of “successes” to emphasize the relative risk of the methods in terms of lives, information which is lost in the pie charts. Also, the area of each box is relative to the frequency in each box. The Agora Crisis Center (505-277-3013, 9am-midnight, every day) plays a critical role in our community, and our education as individuals around these issues can save someone. Using statistics and visualization to tell and understand the important story in the data can lead to improvements in strategies and resource allocation for treatment and prevention. [caption id="" align="alignnone" width="368" caption="Improved visualization has relative use of methods across the horizontal and proportion of successes along the vertical. Area is proportional to people."][/caption] R code follows to produce plot above (with modest post-production necessary).

# following example from http://learnr.wordpress.com/2009/03/29/ggplot2_marimekko_mosaic_chart/
setwd("F:\\Dropbox\\UNM\\seminar\\DailyLobo\\20111103_SuicideStatistics")

################################################################################
df <- data.frame(
          M = c(
            "Cut/Pierce",       # "C/P",
            "Firearms",         # "F",
            "Poisoning",        # "P",
            "Suffocation",      # "S",
            "Other/Unspecified")# "O/U")       # Method
        , S = c(4,191,60,73,13)     # Successful
        , U = c(229,16,1097,23,91)  # Unsuccessful
      )

df$T  <- df$S+df$U;                 # Total across method
#df$pS <- df$S/sum(df$S);            # prop S
#df$pU <- df$U/sum(df$U);            # prop U
df$Successful <- 100*round(df$S/df$T, digits = 2);     # prop S by M
df$Unsuccessful <- 100*round(df$U/df$T, digits = 2);   # prop S by M
df$pT  <- 100*round(df$T/sum(df$T), digits = 2);       # prop total
df <- df[order(-df$S),];  # sort so largest method is first

# proportions on x-axis
df$xmax <- cumsum(df$pT);
df$xmin <- df$xmax - df$pT;

#Data looks like this before the long-format conversion:
df

library(ggplot2)
dfm <- melt(df, id = c("M", "T", "pT", "S", "U", "xmin", "xmax"))
dfm

#Now we need to determine how the columns are stacked and where to position the text labels.

#Calculate ymin and ymax:
dfm1 <- ddply(dfm , .(M), transform, ymax = cumsum(value))
dfm1 <- ddply(dfm1, .(M), transform, ymin = ymax - value)
n <- dim(dfm1)[1];
dfm1$F <- as.vector(t(matrix(c(dfm1$S[seq(1,n-1,by=2)],dfm1$U[seq(2,n,by=2)]),ncol=2)))

# Positioning of text:
dfm1$xtext <- with(dfm1, xmin + (xmax - xmin)/2)
dfm1$ytext <- with(dfm1, ymin + (ymax - ymin)/2)

# Finally, we are ready to start the plotting process:
p <- ggplot(dfm1, aes(ymin = ymin, ymax = ymax, xmin = xmin, xmax = xmax, fill = variable))

# Use grey border to distinguish between the segments:
p1 <- p + geom_rect(colour = I("grey"))

# The explanation of different fill colours will be included in the text label of Segment A using the ifelse function.
p2 <- p1 + geom_text(aes(x = xtext, y = ytext,
      label = ifelse(M == df$M[1],
                     paste(variable, "\n", value, "%\n", F, sep = ""),
                     paste(value, "%\n", F, sep = "")))
      , size = 3.5)

# The maximum y-axes value is 100 (as in 100%), and to add the segment description above each column I manually specify the text position.
p3 <- p2 + geom_text(aes(x = xtext, y = 107, label = paste(M,"\n",pT,"%\n",T,sep="")), size = 4)
#p3

# Some last-minute changes to the default formatting: remove axis labels, legend and gridlines.
p4 <- p3 + theme_bw() + labs(x = "Percent Attempts by Method", y = "Percent Success by Method",
     fill = NULL) + opts(legend.position = "none",
     panel.grid.major = theme_line(colour = NA),
     panel.grid.minor = theme_line(colour = NA))
p4

pdf("fromR.pdf")
p4
dev.off()

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