(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 34097, 907]*) (*NotebookOutlinePosition[ 35100, 939]*) (* CellTagsIndexPosition[ 35056, 935]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Bounding analysis applied to attributed risk", "Title"], Cell["Minh Ha-Duong, 2002, 2003", "Author"], Cell["\<\ minh.ha.duong@cmu.edu http://www.andrew.cmu.edu/user/mduong\ \>", "Address"], Cell["\<\ This notebook demonstrates a bounding analysis method to attribute \ the lung cancer risk to various environmental pollutants.\ \>", "Abstract"], Cell["\<\ This code is freely re-useable according to the GNU General Public \ License. This notebook should be distributed along with the BeliefMatrices.m package \ because it will not run without it.\ \>", "Text"], Cell["\<\ The corresponding theory is explained in: Granger-Casman NSF research grant number SES-0216897 and SRB-9521914 Ha-Duong, Casman and Granger, (2003) Bounding Analysis applied to lung cancer \ risk (submitted to ISIPTA\.b403, Lugano) Ha-Duong, Casman and Granger, (2003) Bounding analysis as a companion for \ Risk Analysis: Theory and methods (for submission to Risk Analysis)\ \>", \ "Reference"], Cell[CellGroupData[{ Cell["The Frame of reference", "Subsection"], Cell[BoxData[{ \(\(\[CapitalOmega] = {smoke, radon, asbestos, unknown};\)\), "\[IndentingNewLine]", \(\(n = Length[\[CapitalOmega]];\)\)}], "Input", CellLabel->"In[1]:="], Cell[TextData[{ "Subsets of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], " are indexed in the canonical order (canonical only in this theory and for \ Mathematica only up to version 4.1 included):\nusing binary notation from 0 \ to ", Cell[BoxData[ \(TraditionalForm\`2\^n - 1\)]], ", with 0 corresponding to the empty set and ", Cell[BoxData[ \(TraditionalForm\`2\^n - 1\)]], " to ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega]\)]], ".\nWe load the standard package to index subsets this way, with \ RankSubset and NthSubset functions. \nIt also defines the KSubsets \ function that returns all subsets with k elements." }], "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[3]:="] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Attributable risk: Upper bound is plausibility, lower bound is \ belief\ \>", "Subsection"], Cell[TextData[{ "Next we load the package implemening belief function theory for small n. \ Small means n less than 10, because the package implements Philippe Smets \ (2001) Matrix calculus for belief functions, which uses square matrices of \ size ", Cell[BoxData[ \(TraditionalForm\`2\^n - 1\)]], "." }], "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[4]:="], Cell[TextData[{ "Here is the central idea, see Ha-Duong and ... (2002) Discussion paper on \ bounding analysis:\n\nThe central variable is ", Cell[BoxData[ \(TraditionalForm\`m\)]], " , representing the distribution of exposure among all recorded cases.\n\ For example, m[smoke] = 0.50 means that within all lung cancer deaths in \ the reference year,\nfifty percent of cases were exposed to smoke and no \ other factor.\nWe adopt m as the lower bound on attributable risk, and \n\ plausibility for the bound that means the UPPER bound.\nThey are corrected by \ the background rate, which is NOT m[0], see below." }], "Text"], Cell[BoxData[{ \(\(\(bound[attributableRisk[cause_Symbol], backgroundRate_] := \(Plausibility[m, n]\)\[LeftDoubleBracket] RankBinarySubset[\[CapitalOmega], {cause}]\[RightDoubleBracket] \ \((1 - backgroundRate)\)\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(bound[attributableRisk[cause1_\ && \ cause2_], backgroundRate_] := \(Commonality[m, n]\)\[LeftDoubleBracket] RankBinarySubset[\[CapitalOmega], {cause1, cause2}]\[RightDoubleBracket] \((1 - backgroundRate)\)\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(bound[attributableRisk[cause1_\ || \ cause2_], backgroundRate_] := \(Plausibility[m, n]\)\[LeftDoubleBracket] RankBinarySubset[\[CapitalOmega], {cause1, cause2}]\[RightDoubleBracket] \((1 - backgroundRate)\)\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(lowerBound[attributableRisk[cause_Symbol], backgroundRate_] := m[RankBinarySubset[\[CapitalOmega], {cause}]] \((1 - backgroundRate)\)\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(\(lowerBound[attributableRisk[cause1_\ && \ cause2_], backgroundRate_] := m[RankBinarySubset[\[CapitalOmega], {cause1, cause2}]] \((1 - backgroundRate)\)\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(lowerBound[attributableRisk[cause1_\ || \ cause2_], backgroundRate_] := \(Belief[m, n]\)\[LeftDoubleBracket] RankBinarySubset[\[CapitalOmega], {cause1, cause2}]\[RightDoubleBracket] \((1 - backgroundRate)\)\)}], "Input", CellLabel->"In[5]:="] }, Open ]], Cell[CellGroupData[{ Cell["Coherence constraints", "Subsection"], Cell[TextData[{ "See Walley (1991) page 198.\nNote on the data structure: information about \ lower and upper probabilities shall be given as a list of pairs:\n \ boundsList = ", Cell[BoxData[ \(TraditionalForm\`{{lowerProba1, upperProba1\ }, {lowerProba2, upperProba2\ }, ... }\)]] }], "Text"], Cell[BoxData[ \(sensibleBoundQ[{lowerBound_, upperBound_}] := 0 \[LessEqual] lowerBound \[LessEqual] upperBound \[LessEqual] 1\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(avoidSureLossQ[ boundsList_] := \((Plus @@ boundsList)\)\[LeftDoubleBracket]1\[RightDoubleBracket] \ \[LessEqual] 1 \[LessEqual] \((Plus @@ boundsList)\)\[LeftDoubleBracket]2\[RightDoubleBracket]\)], \ "Input", CellLabel->"In[12]:="], Cell[BoxData[{ \(coherentQ[boundsList_, i_] := \((Plus @@ boundsList)\)\[LeftDoubleBracket]1\[RightDoubleBracket] - boundsList\[LeftDoubleBracket]i, 1\[RightDoubleBracket] + boundsList\[LeftDoubleBracket]i, 2\[RightDoubleBracket] \[LessEqual] 1 \[LessEqual] \ \((Plus @@ boundsList)\)\[LeftDoubleBracket]2\[RightDoubleBracket] - boundsList\[LeftDoubleBracket]i, 2\[RightDoubleBracket] + boundsList\[LeftDoubleBracket]i, 1\[RightDoubleBracket]\), "\n", \(coherentQ[boundsList_] := Table[coherentQ[boundsList, i], {i, 1, Length[boundsList]}]\)}], "Input",\ CellLabel->"In[13]:="], Cell[BoxData[ \(coherenceConstraints[boundsList_] := Flatten[{sensibleBoundQ /@ boundsList, \[IndentingNewLine]avoidSureLossQ[ boundsList], \[IndentingNewLine]coherentQ[ boundsList]}]\)], "Input", CellLabel->"In[15]:="] }, Open ]], Cell[CellGroupData[{ Cell["Maximum unspecificity principle", "Subsection"], Cell[TextData[{ "Constraint that no death is related to ", Cell[BoxData[ \(TraditionalForm\`num\)]], " factors or more.\nThis is a simplification. In a further study we will \ introduce the concept of interactions/independance/irrelevance to replace \ this constraint." }], "Text"], Cell[BoxData[ \(noMultipleSynergy[ num_] := \(\((m[RankBinarySubset[\[CapitalOmega], #]] \[Equal] 0)\) &\)\ /@ \((Join\ @@ \ Table[KSubsets[\[CapitalOmega], k], {k, num, n}])\)\)], "Input", CellLabel->"In[16]:="], Cell[TextData[{ "In addition we specify that ", Cell[BoxData[ \(TraditionalForm\`m\)]], " is positive and sums up to unity." }], "Text"], Cell[BoxData[ \(constraintSet[knowledge__] := Join[Flatten[{knowledge}], \[IndentingNewLine]Table[ m[k] \[GreaterEqual] 0, \ {k, 0, 2\^n - 1}], \[IndentingNewLine]{Sum[ m[k], \ {k, 0, 2\^n - 1}]\ \[Equal] \ 1\ }\[IndentingNewLine]]\)], "Input", CellLabel->"In[17]:="], Cell["Lets simplify the constraints.", "Text"], Cell[BoxData[{ \(simpleConstraints[set_] := Cases[set, m[_] \[Equal] _?NumberQ]\), "\[IndentingNewLine]", \(presolve[ set_] := \[IndentingNewLine]Join[simpleConstraints[set], set /. \((simpleConstraints[ set] /. {Equal \[Rule] \ Rule\ })\)] //. {a___, True, b___} \[Rule] {a, b}\)}], "Input", CellLabel->"In[18]:="], Cell[TextData[{ "We retain the most unspecific ", Cell[BoxData[ \(TraditionalForm\`m\)]], " compatible with available knowledge (given as a set of constraints).\n\ Unicity is NOT garanteed.\nThere is a solution to the maximization program if \ and only if the coherence constraints can be satisfied." }], "Text"], Cell[BoxData[ \(\(mostUnspecificExposure[ knowledge__] := \(ConstrainedMax[\ Unspecificity[m, n], \[IndentingNewLine]presolve[ constraintSet[knowledge]], \[IndentingNewLine]Table[ m[k], \ {k, 0, 2\^n - 1}]\[IndentingNewLine]]\)\[LeftDoubleBracket]2\ \[RightDoubleBracket];\)\)], "Input", CellLabel->"In[20]:="] }, Open ]], Cell[CellGroupData[{ Cell["Relative risk", "Subsection"], Cell[TextData[{ "The relative risk of smoking says how many times more likely you are to \ get cancer when you smoke compared to non-smokers.\nThe formula is ", Cell[BoxData[ \(ar \[Equal] \ \(p \((rr - 1)\)\)\/\(p \((rr - 1)\) + 1\)\)]], " \nwhere ", Cell[BoxData[ \(TraditionalForm\`ar\)]], " is the attributable risk, ", Cell[BoxData[ \(TraditionalForm\`rr\)]], " the relative risk and ", Cell[BoxData[ \(TraditionalForm\`p\)]], " the probability of exposure,\nsee for example Beir VI appendix C p. 229, \ or Breslow & Day Measures of Disease p. 74" }], "Text"], Cell[BoxData[ \(\(\(\ \)\(attributableFromRelativeRisk[cause_, rr_] = 1 - 1\/\(exposureProbability[cause]\ \((rr - 1)\)\ + 1\);\)\)\)], \ "Input", CellLabel->"In[21]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(relativeFromAttributableRisk[cause_, ar_] = rr /. \(Solve[ar \[Equal] attributableFromRelativeRisk[cause, rr], rr]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\)\)\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(\(\(-ar\) - exposureProbability[cause] + ar\ exposureProbability[cause]\ \)\/\(\((\(-1\) + ar)\)\ exposureProbability[cause]\)\)], "Output", CellLabel->"Out[22]="] }, Open ]], Cell[TextData[{ "It is the same to bound ", Cell[BoxData[ \(TraditionalForm\`ar\)]], " or ", Cell[BoxData[ \(TraditionalForm\`rr\)]], ", since they both increase together.\n\nHere is how to say that the \ relative risk of smoke is between 3 and 5,\nso that the inequality stays \ linear in our control variables ", Cell[BoxData[ \(TraditionalForm\`m\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bound[attributableRisk[smoke]] \[LessEqual] attributableFromRelativeRisk[smoke, 5]\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(bound[attributableRisk[smoke]] \[LessEqual] 1 - 1\/\(1 + 4\ exposureProbability[smoke]\)\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(lowerBound[attributableRisk[smoke], br] >= attributableFromRelativeRisk[smoke, 3]\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(\((1 - br)\)\ m[8] \[GreaterEqual] 1 - 1\/\(1 + 2\ exposureProbability[smoke]\)\)], "Output", CellLabel->"Out[24]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Pretty printing", "Subsection"], Cell[BoxData[{ \(toOneDigit[x_] := Round[10 x]/10. \), "\[IndentingNewLine]", \(toTwoDigits[x_] := Round[100 x]/100. \), "\[IndentingNewLine]", \(toThreeDigits[x_] := Round[1000 x]/1000. \), "\[IndentingNewLine]", \(\(Attributes[toOneDigit] = Listable;\)\), "\[IndentingNewLine]", \(\(Attributes[toTwoDigits] = Listable;\)\), "\[IndentingNewLine]", \(\(Attributes[toThreeDigits] = Listable;\)\)}], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(\(dataToDisplay := {\[IndentingNewLine]toThreeDigits[\(bound[ attributableRisk[#], backgroundRate] &\) /@ \[CapitalOmega]], \ \[IndentingNewLine]toThreeDigits[\(lowerBound[attributableRisk[#], backgroundRate] &\) /@ \[CapitalOmega]], \ \[IndentingNewLine]\(exposureProbability[#] &\)\ /@ \[CapitalOmega], \ \[IndentingNewLine]\(toTwoDigits[ relativeFromAttributableRisk[#, bound[attributableRisk[#], backgroundRate]]] &\) /@ \[CapitalOmega], \ \[IndentingNewLine]\(toTwoDigits[ relativeFromAttributableRisk[#, lowerBound[attributableRisk[#], backgroundRate]]] &\) /@ \[CapitalOmega]\ \[IndentingNewLine] (*\(,\)\(PignisticProbability[m, n]\)*) \[IndentingNewLine]};\)\)], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \(resultSummary[ rule_] := \[IndentingNewLine]ColumnForm[{TableForm[\[IndentingNewLine]\ dataToDisplay /. rule, \[IndentingNewLine]TableHeadings \[Rule] \ {{\*"\"\<\!\(attributable\&_\) %\>\"", \*"\"\<\!\(attributable\+_\) %\>\"", "\ \", \*"\"\<\!\(\(relative\\\ risk\)\&_\)\>\"", \ \*"\"\<\!\(\(relative\\\ risk\)\+_\)\>\""\ (*\(,\)\("\"\)*) }, \ \[CapitalOmega]}\[IndentingNewLine]], \[IndentingNewLine]{"\", \ Chop[m[0] /. rule]}\[IndentingNewLine]}]\)], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \(resultDetail[ rule_] := \[IndentingNewLine]rule /. \(\(Rule[m[k_], rr_]\)\(\[Rule]\)\({k, UnrankBinarySubset[k, \[CapitalOmega]], rr}\)\(\ \)\)\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \(statTable[ rule_] := \(\({Rest[BinarySubsets[\[CapitalOmega]]], Table[m[k] \((1 - backgroundRate)\), {k, 1, 2\^n - 1}], Belief[m, n] \((1 - backgroundRate)\), Commonality[m, n] \((1 - backgroundRate)\), Plausibility[m, n] \((1 - backgroundRate)\)}\)\(/.\)\(rule\)\(\ \)\)\)], "Input",\ CellLabel->"In[34]:="] }, Open ]], Cell[CellGroupData[{ Cell["Numeric real-world values", "Subsection"], Cell["\<\ The total population is 270 millions and the lung cancer deaths are \ 152 thousands. We assume statistical independance of exposure probabilities\ \>", "Text"], Cell[BoxData[{ \(\(\(backgroundFraction[backgroundRate_] := backgroundRate\ \((1 - exposureProbability[smoke])\) \((1 - exposureProbability[radon])\) \((1 - exposureProbability[asbestos])\)\ 270/ 0.152\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(exposureProbability[smoke] = 0.45;\)\), "\n", \(\(exposureProbability[radon] = 0.5;\)\), "\n", \(\(exposureProbability[asbestos] = 0.05;\)\ \), "\n", \(\(exposureProbability[unknown] = 0.05;\)\)}], "Input", CellLabel->"In[35]:="], Cell["\<\ Our information is given as upper and lower bound on the \ attributable risk.\ \>", "Text"], Cell[BoxData[ \(commonKnowledge[backgroundRate_, boundsList_] := \[IndentingNewLine]{m[0] \[Equal] \ backgroundFraction[ backgroundRate], \[IndentingNewLine]\[IndentingNewLine]bound[ attributableRisk[smoke], backgroundRate] \[Equal] boundsList\[LeftDoubleBracket]1, 2\[RightDoubleBracket], \[IndentingNewLine]lowerBound[ attributableRisk[smoke], backgroundRate] \[Equal] boundsList\[LeftDoubleBracket]1, 1\[RightDoubleBracket], \[IndentingNewLine]\ \ \[IndentingNewLine]bound[attributableRisk[radon], backgroundRate] \[Equal] \ boundsList\[LeftDoubleBracket]2, 2\[RightDoubleBracket], \[IndentingNewLine]lowerBound[ attributableRisk[radon], backgroundRate] \[Equal] boundsList\[LeftDoubleBracket]2, 1\[RightDoubleBracket], \ \[IndentingNewLine]\[IndentingNewLine]bound[attributableRisk[asbestos], backgroundRate] \[Equal] boundsList\[LeftDoubleBracket]3, 2\[RightDoubleBracket], \[IndentingNewLine]lowerBound[ attributableRisk[asbestos], backgroundRate] \[Equal] boundsList\[LeftDoubleBracket]3, 1\[RightDoubleBracket]\ , \ \[IndentingNewLine]\[IndentingNewLine]noMultipleSynergy[ 3]\[IndentingNewLine]}\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[{ \(\(expert1Bounds = {{0.7, 0.95}, {0.02, 0.21}, {0.01, 0.05}};\)\), "\[IndentingNewLine]", \(\(expert1BGRate\ = 3. /100000;\)\)}], "Input", CellLabel->"In[41]:="] }, Open ]], Cell[CellGroupData[{ Cell["Coherence check", "Subsection"], Cell["\<\ The bounds avoid sure loss if the first five lines can be True, and \ coherent if all constraints can be satisfied (neglecting the background \ rate).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(constSet = coherenceConstraints[ Append[expert1Bounds, {loX, upX}]];\)\), "\[IndentingNewLine]", \(constSet // ColumnForm\)}], "Input", CellLabel->"In[43]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {"True"}, {"True"}, {"True"}, {\(0 \[LessEqual] loX \[LessEqual] upX \[LessEqual] 1\)}, {\(\(\(0.73`\)\(\[InvisibleSpace]\)\) + loX \[LessEqual] 1 \[LessEqual] \(\(1.21`\)\(\[InvisibleSpace]\)\) + upX\)}, {\(\(\(0.98`\)\(\[InvisibleSpace]\)\) + loX \[LessEqual] 1 \[LessEqual] \(\(0.96`\)\(\[InvisibleSpace]\)\) + upX\)}, {\(\(\(0.9199999999999999`\)\(\[InvisibleSpace]\)\) + loX \[LessEqual] 1 \[LessEqual] \(\(1.02`\)\(\[InvisibleSpace]\)\) + upX\)}, {\(\(\(0.77`\)\(\[InvisibleSpace]\)\) + loX \[LessEqual] 1 \[LessEqual] \(\(1.17`\)\(\[InvisibleSpace]\)\) + upX\)}, {\(\(\(0.73`\)\(\[InvisibleSpace]\)\) + upX \[LessEqual] 1 \[LessEqual] \(\(1.21`\)\(\[InvisibleSpace]\)\) + loX\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ {True, True, True, LessEqual[ 0, loX, upX, 1], LessEqual[ Plus[ 0.72999999999999998, loX], 1, Plus[ 1.21, upX]], LessEqual[ Plus[ 0.97999999999999998, loX], 1, Plus[ 0.95999999999999996, upX]], LessEqual[ Plus[ 0.91999999999999993, loX], 1, Plus[ 1.02, upX]], LessEqual[ Plus[ 0.77000000000000002, loX], 1, Plus[ 1.1699999999999999, upX]], LessEqual[ Plus[ 0.72999999999999998, upX], 1, Plus[ 1.21, loX]]}], Editable->False]], "Output", CellLabel->"Out[44]="] }, Open ]], Cell["\<\ Here are the bounds on X so that coherence is maintained \ (neglecting the background rate).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ConstrainedMin[upX - loX, constSet, {upX, loX}]\)\[LeftDoubleBracket]2\[RightDoubleBracket]\), "\ \[IndentingNewLine]", \(\(ConstrainedMax[upX - loX, constSet, {upX, loX}]\)\[LeftDoubleBracket]2\[RightDoubleBracket]\)}], "Input", CellLabel->"In[45]:="], Cell[BoxData[ \({upX \[Rule] 0.040000000000000036`, loX \[Rule] 0.020000000000000018`}\)], "Output", CellLabel->"Out[45]="], Cell[BoxData[ \({upX \[Rule] 0.27`, loX \[Rule] 0.`}\)], "Output", CellLabel->"Out[46]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Results: Bounds on X", "Subsection"], Cell["\<\ Here are most unspecific bounds on X so that coherence is \ maintained AND 3-way synergies are disallowed AND background rate is \ non-zero.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(result1 = Append[mostUnspecificExposure[ commonKnowledge[expert1BGRate, expert1Bounds]], backgroundRate \[Rule] expert1BGRate]\)\(\n\) \)\), "\[IndentingNewLine]", \(resultSummary[result1]\), "\n", \(\(\(resultDetail[result1] // ColumnForm\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(TableForm[Transpose[statTable[result1]], TableHeadings \[Rule] {Range[ 15], {"\", \*"\"\<(1-\!\(r\_0\)) m (\!\(and\+_\))\>\"", \*"\"\ \<(1-\!\(r\_0\)) bel (\!\(or\+_\))\>\"", \*"\"\<(1-\!\(r\_0\))q \ (\!\(and\&_\))\>\"", \*"\"\<(1-\!\(r\_0\))pl (\!\(or\&_\))\>\""}}]\)}], \ "Input", CellLabel->"In[47]:="], Cell[BoxData[ \({m[0] \[Rule] 0.013921875`, m[1] \[Rule] 0.`, m[2] \[Rule] 0.01000030000900027`, m[3] \[Rule] 0.006048724117973671`, m[4] \[Rule] 0.02000060001800054`, m[5] \[Rule] 0.`, m[6] \[Rule] 0.`, m[7] \[Rule] 0.`, m[8] \[Rule] 0.7000210006300188`, m[9] \[Rule] 0.02604932413597421`, m[10] \[Rule] 0.033952475918027414`, m[11] \[Rule] 0.`, m[12] \[Rule] 0.1900057001710051`, m[13] \[Rule] 0.`, m[14] \[Rule] 0.`, m[15] \[Rule] 0.`, backgroundRate \[Rule] 0.00003`}\)], "Output", CellLabel->"Out[47]="], Cell[BoxData[ InterpretationBox[GridBox[{ { TagBox[GridBox[{ {"\<\"\"\>", "smoke", "radon", "asbestos", "unknown"}, {"\<\"\\!\\(attributable\\&_\\) %\"\>", "0.9500000000000001`", "0.21`", "0.05`", "0.032`"}, {"\<\"\\!\\(attributable\\+_\\) %\"\>", "0.7000000000000001`", "0.02`", "0.01`", "0"}, {"\<\"Exposure %\"\>", "0.45`", "0.5`", "0.05`", "0.05`"}, {"\<\"\\!\\(\\(relative\\\\ risk\\)\\&_\\)\"\>", "43.22`", "1.53`", "2.05`", "1.6600000000000001`"}, {"\<\"\\!\\(\\(relative\\\\ risk\\)\\+_\\)\"\>", "6.19`", "1.04`", "1.2`", "1.`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{"\!\(attributable\&_\) %", "\!\(attributable\+_\) %", "Exposure %", "\!\(\(relative\\ risk\)\&_\)", "\!\(\(relative\\ risk\)\+_\)"}, {smoke, radon, asbestos, unknown}}]]]}, {\({"Non attributed", 0.013921875`}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { TableForm[ {{0.95000000000000007, 0.20999999999999999, 0.050000000000000003, 0.032000000000000001}, {0.70000000000000007, 0.02, 0.01, 0}, {0.45000000000000001, 0.5, 0.050000000000000003, 0.050000000000000003}, 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