# Package SHELF. February 5, 2016

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1 Type Package Package SHELF February 5, 2016 Title Tools to Support the Sheffield Elicitation Framework (SHELF) Version Date Author Jeremy Oakley Maintainer Jeremy Oakley Implements various methods for eliciting a probability distribution for a single parameter from an expert or a group of experts. The expert provides a small number of probability or quantile judgements, corresponding to points on his or her cumulative distribution function. A range of parametric distributions can then be fitted and displayed, with feedback provided in the form of additional quantiles. A graphical interface for the roulette elicitation method is also provided. For multiple experts, a weighted linear pool can be calculated. Also includes functions for eliciting beliefs about population distributions. License GPL-2 GPL-3 Depends R (>= 3.2.2) RoxygenNote Suggests testthat Imports ggplot2, grid, shiny, stats, graphics URL BugReports NeedsCompilation no Repository CRAN Date/Publication :15:23 R topics documented: SHELF-package cdffeedback cdfplot

2 2 SHELF-package elicit feedback fitdist fitprecision pdfplots plinearpool plotfit roulette Index 17 SHELF-package Tools to Support the Sheffield Elicitation Framework Implements various methods for eliciting a probability distribution for a single parameter from an expert or a group of experts. The expert provides a small number of probability or quantile judgements, corresponding to points on his or her cumulative distribution function. A range of parametric distributions can then be fitted and displayed, with feedback provided in the form of additional quantiles. A graphical interface for the roulette elicitation method is also provided. For multiple experts, a weighted linear pool can be calculated. Package: SHELF Type: Package Version: Date: License: GPL-2 GPL-3 Author(s) Jeremy Oakley References The SHELF homepage ## 1) Elicit judgements from two experts individually # Expert 1 states P(X<30)=0.25, P(X<40)=0.5, P(X<50)=0.75 # Expert 2 states P(X<20)=0.25, P(X<25)=0.5, P(X<35)=0.75 # Both experts state 0<X<100. ## 2) Fit distributions to each expert s judgements v <- matrix(c(30, 40, 50, 20, 25, 35), 3, 2) p <- c(0.25, 0.5, 0.75)

3 cdffeedback 3 myfit <- fitdist(vals = v, probs = p, lower = 0, upper = 100) ## 3) Plot the fitted distributions, including a linear pool plotfit(myfit, lp=t) ## 4) Now elicit a single consensus distribution from the two experts # Suppose they agree P(X<25)=0.25, P(X<30)=0.5, P(X<40)=0.75 v <-c(25, 30, 40) p <-c(0.25, 0.5, 0.75) myfit <- fitdist(vals = v, probs = p, lower = 0, upper = 100) ## 5) Plot the fitted density, and report some feedback, such as the # fitted 5th and 95th percentiles plotfit(myfit, ql=0.05, qu=0.95) feedback(myfit, quantiles=c(0.05,0.95)) ## Can also use interactive plotting v <- matrix(c(30, 40, 50, 20, 25, 35), 3, 2) p <- c(0.25, 0.5, 0.75) myfit <- fitdist(vals = v, probs = p, lower = 0, upper = 100) # plot each distribution plotfit(myfit, int = TRUE) ## plot the distribution for one expert only plotfit(myfit, int= TRUE, ex=1) ## Enter judgements in interactive mode elicit() ## Enter judgements using the roulette method roulette(lo=0, Up=100, nbins=10, gridheight=10) cdffeedback Feedback for the elicited distribution of the population CDF Report the median and 100(1-alpha)% credible interval for point on the population CDF cdffeedback(medianfit, precisionfit, quantiles = c(0.05, 0.95), vals = NA, alpha = 0.05, median.dist = "best", precision.dist = "gamma", n.rep = 10000)

4 4 cdffeedback medianfit precisionfit quantiles vals Details alpha median.dist The output of a fitdist command following elicitation of the expert s beliefs about the population median. The output of a fitprecision command following elicitation of the expert s beliefs about the population precision. A vector of quantiles q 1,..., q n required for feedback A vector of population values x 1,..., x n required for feedback The size of the 100(1-alpha)% credible interval The fitted distribution for the population median. Can be one of "normal", "lognormal" or "best", where "best" will select the best fitting out of normal and lognormal. precision.dist The fitted distribution for the population precision. Can either be "gamma" or "lognormal". n.rep The number of randomly sampled CDFs used to estimated the median and credible interval. Denote the uncertain population CDF by P (X x µ, σ 2 ), where µ is the uncertain population median and σ ( 2) is the uncertain population precision. Feedback can be reported in the form of the median and 100(1-alpha)% credible interval for (a) an uncertain probability P (X x µ, σ 2 ), where x is a specified population value and (b) an uncertain quantile x q defined by P (X x q µ, σ 2 ) = q, where q is a specified population probability. Value Fitted median and 100(1-alpha)% credible interval for population quantiles and probabilities. \$quantiles \$probs Each row gives the fitted median and 100(1-alpha)% credible interval for each uncertain population quantile specified in quantiles: the fitted median and 100(1-alpha)% credible interval for the value of x qi where P (X x qi µ, σ 2 ) = q i. Each row gives the fitted median and 100(1-alpha)% credible interval for each uncertain population probability specified in probs: the fitted median and 100(1- alpha)% credible interval for the value of P (X x i µ, σ 2 ). prfit <- fitprecision(interval = c(60, 70), propvals = c(0.2, 0.4), trans = "log") medianfit <- fitdist(vals = c(50, 60, 70), probs = c(0.05, 0.5, 0.95), lower = 0) cdffeedback(medianfit, prfit, quantiles = c(0.01, 0.99), vals = c(65, 75), alpha = 0.05, n.rep = 10000)

5 cdfplot 5 cdfplot Plot distribution of CDF Plot the elicited pointwise median and credible interval for an uncertain population CDF cdfplot(medianfit, precisionfit, lower = NA, upper = NA, ql = 0.025, qu = 0.975, median.dist = "best", precision.dist = "gamma", n.rep = 10000, n.x = 100, fontsize = 18) medianfit precisionfit lower upper ql qu median.dist The output of a fitdist command following elicitation of the expert s beliefs about the population median. The output of a fitdist command following elicitation of the expert s beliefs about the population precision. lower limit on the x-axis for plotting. upper limit on the x-axis for plotting. lower quantile for the plotted pointwise credible interval. upper quantile for the plotted pointwise credible interval. The fitted distribution for the population median. Can be one of "normal", "lognormal" or "best", where "best" will select the best fitting out of normal and lognormal. precision.dist The fitted distribution for the population precision. Can either be "gamma" or "lognormal". n.rep n.x fontsize The number of randomly sampled CDFs used to estimated the median and credible interval. The number of points on the x-axis at which the CDF is evaluated. Font size used in the plots. prfit <- fitprecision(interval = c(60, 70), propvals = c(0.2, 0.4), trans = "log") medianfit <- fitdist(vals = c(50, 60, 70), probs = c(0.05, 0.5, 0.95), lower = 0) cdfplot(medianfit, prfit)

6 6 feedback elicit Elicit judgements and fit distributions interactively Opens up a web browser (using the shiny package), from which you can specify judgements, fit distributions and plot the fitted density functions with additional feedback. elicit() Details Parameter limits determine which distributions can be fitted. Non-negative lower limits are needed for the gamma, lognormal and log-t distributions, and both limits must be finite for to fit a beta distribution. If a histogram is fitted without specifying finite limits, endpoints are chosen based on fitting a normal distribution. As an example, if the elicited judgements are P(X<15)=0.25, P(X<20)=0.5) and P(X<40)=0.75, specify the parameter values as 15,20,40 and the cumulative probabilities as 0.25,0.5,0.75. Press Esc in the R console window to exit the elicitation session. Author(s) Jeremy Oakley elicit() feedback Report quantiles and probabilities from the fitted probability distributions Having fitted appropriate distributions to one or more expert s judgements individually using the fitdist command, use this command to get quantiles and probabilities from the fitted distributions

7 feedback 7 feedback(fit, quantiles = NA, values = NA, dist = "best", ex = NA, sf = 3) fit quantiles values dist ex sf The output of a fitdist command. A vector of desired quantiles for feedback. If this argument is left out, the default is to use the same quantiles that were elicited from the experts. A vector of desired probabilities; desired values of a for reporting back fitted values of P(X<a). If this argument is left out, the default is to use the same values provided by the experts. If fit contains judgements from multiple experts, dist is distribution to be used for calculating probabilities and quantiles. Options are "normal", "t", "gamma", "lognormal", "logt", "beta", or "best". The default option, "best", uses the best fitting distribution for each expert. If fit contains judgements from multiple experts, specifying a value for ex will select a single expert for feedback. Note that for a single expert, feedback is given for all suitable types of distribution, but for multiple experts, feedback is given for one type of distribution only. The number of significant figures to be displayed in the output. Value fitted.quantiles Fitted quantiles for each expert fitted.probs distributions Fitted probabilities for each expert The distribution used to calculate fitted probabilities/quantiles for each expert, if feedback is given for multiple experts. Author(s) Jeremy Oakley # Two experts # Expert 1 states P(X<30)=0.25, P(X<40)=0.5, P(X<50)=0.75 # Expert 2 states P(X<20)=0.25, P(X<25)=0.5, P(X<35)=0.75 # Both experts state 0<X<100. v <- matrix(c(30, 40, 50, 20, 25, 35), 3, 2) p <- c(0.25, 0.5, 0.75) myfit <- fitdist(vals = v, probs = p, lower = 0, upper = 100) feedback(myfit)

8 8 fitdist # Feedback P(X<60) and the tertiles feedback(myfit, values=60, quantiles=c(0.33,0.66)) # Compare fitted tertiles for different distributions, expert 2 only feedback(myfit, quantiles=c(0.33,0.66), ex=2) fitdist Fit distributions to elicited probabilities Takes elicited probabilities as inputs, and fits parametric distributions using least squares on the cumulative distribution function. If separate judgements from multiple experts are specified, the function will fit one set of distributions per expert. fitdist(vals, probs, lower = -Inf, upper = Inf, weights = 1, tdf = 3) vals probs lower upper weights tdf A vector of elicited values for one expert, or a matrix of elicited values for multiple experts (one column per expert). Note that the an elicited judgement about X should be of the form P(X<= vals[i,j]) = probs[i,j] A vector of elicited probabilies for one expert, or a matrix of elicited values for multiple experts (one column per expert). A single vector can be used if the probabilities are the same for each expert. For each expert, the smallest elicited probability must be less than 0.4, and the largest elicited probability must be greater than 0.6. A single lower limit for the uncertain quantity X, or a vector of different lower limits for each expert. Specifying a lower limit will allow the fitting of distributions bounded below. A single upper limit for the uncertain quantity X, or a vector of different lower limits for each expert. Specifying both a lower limit and an upper limit will allow the fitting of a Beta distribution. A vector or matrix of weights corresponding to vals if weighted least squares is to be used in the parameter fitting. The number of degrees of freedom to be used when fitting a t-distribution.

9 fitdist 9 Value Normal Student.t Gamma Log.normal Log.Student.t Beta ssq best.fitting vals probs limits Parameters of the fitted normal distributions. Parameters of the fitted t distributions. Note that (X - location) / scale has a standard t distribution. The degrees of freedom is not fitted; it is specified as an argument to fitdist. Parameters of the fitted gamma distributions. Note that E(X) = shape / rate. Parameters of the fitted log normal distributions: the mean and standard deviation of log X. Parameters of the fitted log student t distributions. Note that (log(x) - location) / scale has a standard t distribution. The degrees of freedom is not fitted; it is specified as an argument to fitdist. Parameters of the fitted beta distributions. X is scaled to the interval [0,1] via Y = (X - lower)/(upper - lower), and E(Y) = shape1 / (shape1 + shape2). Sum of squared errors for each fitted distribution and expert. Each error is the different between an elicited cumulative probability and the corresponding fitted cumulative probability. The best fitting distribution for each expert, determined by the smallest sum of squared errors. The elicited values used to fit the distributions. The elicited probabilities used to fit the distributions. The lower and upper limits specified by each expert (+/- Inf if not specified). Note The least squares parameter values are found numerically using the optim command. Starting values for the distribution parameters are chosen based on a simple normal approximation: linear interpolation is used to estimate the 0.4, 0.5 and 0.6 quantiles, and starting parameter values are chosen by setting E(X) equal to the 0.5th quantile, and Var(X) = (0.6 quantile quantile)^2 / Note that the arguments lower and upper are not included as elicited values on the cumulative distribution function. To include a judgement such as P(X<=a)=0, the values a and 0 must be included in vals and probs respectively. Author(s) Jeremy Oakley # One expert, with elicited probabilities # P(X<20)=0.25, P(X<30)=0.5, P(X<50)=0.75 # and X>0. v <- c(20,30,50) p <- c(0.25,0.5,0.75) fitdist(vals=v, probs=p, lower=0)

10 10 fitprecision # Now add a second expert, with elicited probabilities # P(X<55)=0.25, P(X<60=0.5), P(X<70)=0.75 v <- matrix(c(20,30,50,55,60,70),3,2) p <- c(0.25,0.5,0.75) fitdist(vals=v, probs=p, lower=0) # Two experts, different elicited quantiles and limits. # Expert 1: P(X<50)=0.25, P(X<60=0.5), P(X<65)=0.75, and provides bounds 10<X<100 # Expert 2: P(X<40)=0.33, P(X<50=0.5), P(X<60)=0.66, and provides bounds 0<X v <- matrix(c(50,60,65,40,50,60),3,2) p <- matrix(c(.25,.5,.75,.33,.5,.66),3,2) l <- c(10,0) u <- c(100, Inf) fitdist(vals=v, probs=p, lower=l, upper=u) fitprecision Fit a distribution to judgements about a population precision Takes elicited probabilities about proportion of a population lying in a specfied interval as inputs, converts the judgements into probability judgements about the population precision, and fits gamma and lognormal distributions to these judgements using the fitdist function. fitprecision(interval, propvals, propprobs = c(0.05, 0.95), trans = "identity", pplot = TRUE, fontsize = 18) interval A vector specifying the endpoints of an interval [k 1, k 2 ]. propvals A vector specifying two values θ 1, θ 2 for the proportion. propprobs A vector specifying two probabilities p 1, p 2. trans pplot fontsize A string variable taking the value "identity", "log" or "logit" corresponding to whether the population distribution is normal, lognormal or logit-normal respectively. Plot the population distributions with median set at k 1 and precision fixed at the two elicited quantiles implied by propvals and propprobs. Font size used in the plots.

11 pdfplots 11 Details The expert provides a pair of probability judgements P (θ < θ 1 ) = p 1, and P (θ < θ 2 ) = p 2, where θ is the proportion of the population that lies in the interval [k 1, k 2 ]. The judgements are made conditional on the population median equalling k 1. Note that, unlike the fitdist command, a best fitting distribution is not reported, as the distributions are fitted to two elicited probabilities only. Value Gamma Parameters of the fitted gamma distribution. Note that E(precision) = shape / rate. Log.normal Parameters of the fitted log normal distribution: the mean and standard deviation of log precision. vals The elicited values θ 1, θ 2 probs The elicited probabilities p 1, p 2 limits transform The lower and upper limits specified by each expert (+/- Inf if not specified). Transformation used for a normal population distribution. fitprecision(interval=c(60, 70), propvals=c(0.2, 0.4), trans = "log") pdfplots Plot fitted population pdfs Plot fitted population pdfs at combinations of two different values of the population mean and variance. pdfplots(medianfit, precisionfit, alpha = 0.05, tails = 0.05, lower = NA, upper = NA, n.x = 100, d = "best", fontsize = 18)

12 12 pdfplots medianfit precisionfit alpha tails lower upper n.x d fontsize The output of a fitdist command following elicitation of the expert s beliefs about the population median. The output of a fitdist command following elicitation of the expert s beliefs about the population precision. Value between 0 and 1 to determine choice of means and variances used in plots Value between 0 and 1 to determine the tail area shown in the pdf plots lower limit on the x-axis for plotting. upper limit on the x-axis for plotting. The number of points on the x-axis at which the pdf is plotted. The fitted distribution for the population median. Can be one of "normal", "lognormal" or "best", where "best" will select the best fitting out of normal and lognormal. Font size used in the plots. Details Four pdfs are plotted, using each combination of the alpha/2 and 1-alpha/2 quantiles of the fitted distributions for the population median and standard deviation Value A plot and a list, containing mu sigma The two population mean values used in the plots. The two population standard deviation values used in the plots. References multiplot function obtained from on_one_page_(ggplot2)/ prfit <- fitprecision(interval = c(60, 70), propvals = c(0.2, 0.4), trans = "log") medianfit <- fitdist(vals = c(50, 60, 70), probs = c(0.05, 0.5, 0.95), lower = 0) pdfplots(medianfit, prfit, alpha = 0.01)

13 plinearpool 13 plinearpool Calculate fitted probabilities or quantiles from a (weighted) linear pool Calculates a linear pool given a set of elicited judgements in a fit object. Then calculates required probabilities or quantiles from the pooled cumulative distribution function. plinearpool(fit, x, d = "best", w = 1) qlinearpool(fit, q, d = "best", w = 1) fit x d w q The output of a fitdist command. A vector of required cumulative probabilities P(X<=x) The distribution fitted to each expert s probabilities. This must either be the same distribution for each expert, or the best fitting distribution for each expert. Options are "normal", "t", "gamma", "lognormal", "logt","beta", "best". A vector of weights to be used in the weighted linear pool. A vector of required quantiles Details Value Quantiles are calculate by first calculating the pooled cumulative distribution function at 100 points, and then using linear interpolation to invert the CDF. A probability or quantile, calculate from a (weighted) linear pool (arithmetic mean) of the experts individual fitted probability. Author(s) Jeremy Oakley # Expert 1 states P(X<30)=0.25, P(X<40)=0.5, P(X<50)=0.75 # Expert 2 states P(X<20)=0.25, P(X<25)=0.5, P(X<35)=0.75 # Both experts state 0<X<100. v <- matrix(c(30, 40, 50, 20, 25, 35), 3, 2) p <- c(0.25, 0.5, 0.75)

14 14 plotfit myfit <- fitdist(vals = v, probs = p, lower = 0, upper = 100) plinearpool(myfit, x=c(20, 50, 80)) qlinearpool(myfit, q=c(0.05, 0.5, 0.95)) # give more weight to first expert plinearpool(myfit, x=c(20, 50, 80), w=c(0.7, 0.3)) # force the use of gamma distributions for each expert qlinearpool(myfit, q=c(0.05, 0.5, 0.95), d="gamma") plotfit Plot the fitted density function for one or more experts Plots the fitted density function for one or more experts. Can also plot a fitted linear pool if more than one expert. If plotting the density function of one expert, or the linear pool only, can also indicated desired lower and upper fitted quantiles. plotfit(fit, d = "best", int = FALSE, xl = -Inf, xu = Inf, ql = NA, qu = NA, lp = FALSE, ex = NA, sf = 3, ind = TRUE, lpw = 1) fit The output of a fitdist command. d The distribution fitted to each expert s probabilities. Options are "normal", "t", "gamma", "lognormal", "logt","beta", "hist" (for a histogram fit), and "best" (for best fitting) int xl xu ql Set int = TRUE to use interactive plotting (using the shiny package). If plotting for a single expert, the argument d is ignored, as distributions can be chosen within the display. If plotting for multiple experts, feedback quantiles are not displayed, and the argument lp is ignored, as the option to show a linear pool can be chosen within the display. The lower limit for the x-axis. The default is the quantile of the fitted distribution (or the quantile of a fitted normal distribution, if a histogram fit is chosen). The upper limit for the x-axis. The default is the quantile of the fitted distribution (or the quantile of a fitted normal distribution, if a histogram fit is chosen). A lower quantile to be indicated on the density function plot. Only displayed when plotting the density function for a single expert.

15 plotfit 15 qu lp ex sf ind lpw An upper quantile to be indicated on the density function plot. Only displayed when plotting the density function for a single expert. For multiple experts, set lp=true to plot a linear pool. If judgements have been elicited from multiple experts, but a density plot for one expert only is required, the expert to be used in the plot. The number of significant figures to be displayed for the parameter values. If plotting a linear pool, set ind=false to suppress plotting of the individual density functions. A vector of weights to be used in linear pool, if unequal weighting is desired. Author(s) Jeremy Oakley # Two experts # Expert 1 states P(X<30)=0.25, P(X<40)=0.5, P(X<50)=0.75 # Expert 2 states P(X<20)=0.25, P(X<25)=0.5, P(X<35)=0.75 # Both experts state 0<X<100. v <- matrix(c(30, 40, 50, 20, 25, 35), 3, 2) p <- c(0.25, 0.5, 0.75) myfit <- fitdist(vals = v, probs = p, lower = 0, upper = 100) # Plot both fitted densities, using the best fitted distribution plotfit(myfit) # Plot a fitted beta distribution for expert 2, and show 5th and 95th percentiles plotfit(myfit, d = "beta", ql = 0.05, qu = 0.95, ex = 2) # Use interactive plotting for for expert 2, and show 5th and 95th percentiles plotfit(myfit, int = T, ex = 2) # Plot a linear pool, giving double weight to expert 1 plotfit(myfit, lp = T, lpw = c(2,1)) # Use interactive plotting, giving double weight to expert 1, if a linear pool is displayed plotfit(myfit, int = T, lpw = c(2,1)) # Plot a linear pool, giving double weight to expert 1, # show 5th and 95th percentiles, supress plotting of individual distributions, # and force use of Beta distributions plotfit(myfit, d = "beta", lp = T, lpw = c(2,1), ql = 0.05, qu = 0.95, ind=false )

16 16 roulette roulette Elicit one set of probabilities using the roulette method. Produces a graphics window with the roulette grid. The user clicks in the window to allocate chips to bins. The elicited probability inside each bin is the proportion of chips in each bin. roulette(lower = 0, upper = 100, gridheight = 10, nbins = 10, round.end = F) lower upper gridheight nbins round.end The lower limit on the x-axis of the roulette grid. The upper limit on the x-axis of the roulette grid. The maximum number of chips that can be allocated to a single bin. The number of equally sized bins drawn between Lo and Up. If set to T, empty bins and the uppermost non-empty bin will be ignored. For example, with 20 chips in total, if the uppermost non-empty bin is [70,80] and contains 1 chip, setting round.end = F will result in an elicited probability P(X>80)=0, but setting round.end = T will remove this judgement, instead only having P(X>70)=0.05. Value A list, with outputs v p upper limits of each bin. cumulative probabilities for each upper bin limit. Author(s) Jeremy Oakley x <- roulette() # Then allocate chips to bins # To fit distributions and see the results myfit <- fitdist(vals = x\$v, probs = x\$p) plotfit(myfit)

17 Index cdffeedback, 3 cdfplot, 5 elicit, 6 feedback, 6 fitdist, 4, 6, 8, 10, 11 fitprecision, 4, 10 pdfplots, 11 plinearpool, 13 plotfit, 14 qlinearpool (plinearpool), 13 roulette, 16 SHELF (SHELF-package), 2 SHELF-package, 2 17

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