(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 84608, 2197] NotebookOptionsPosition[ 79119, 2115] NotebookOutlinePosition[ 79452, 2130] CellTagsIndexPosition[ 79409, 2127] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["System of Equations", "Title", CellChangeTimes->{{3.807304116062188*^9, 3.807304126043613*^9}, { 3.84211185428517*^9, 3.842111860627666*^9}},ExpressionUUID->"249660d2-3298-4bc1-987e-\ b80b051d1ec6"], Cell["\<\ Supplemental file for Model misspecification in stepped wedge trials: Random \ effects for time or treatment Emily C. Voldal, Fan Xia, Avi Kenny, Patrick J. Heagerty, and James P. Hughes\ \ \>", "Subtitle", CellChangeTimes->{{3.807304131263259*^9, 3.807304149598194*^9}, { 3.822615708974361*^9, 3.822615743653187*^9}, {3.8240750696207438`*^9, 3.8240751675849123`*^9}, {3.842111866885234*^9, 3.842111880855781*^9}},ExpressionUUID->"6c1035df-565b-45e0-8b0e-\ 2faf522e1111"], Cell["\<\ This document contains the systems of equations for both the time-fitted \ random treatment case (Case 1) and the treatment-fitted random time case \ (Case 2), and demonstrates the closed-form solutions when available Notation: R stands for the sum of T_m across sequences; S stands for the sum \ of T_m^2 across sequences. To avoid confusion, any single-letter variables \ (e.g. K) will have an extra prefix (e.g. varK). Other notation should be \ consistent with the main paper. We demonstrate here that the closed-form roots listed in the paper exist; \ based on results for specific designs and on simulations, it doesn't appear \ that there are additional roots but we certainly haven't proved that. For updates and any corrections, see \ https://github.com/voldal/SWT_model_misspecification\ \>", "Text", CellChangeTimes->{{3.807304153134701*^9, 3.807304188664811*^9}, { 3.807306339601892*^9, 3.807306398690393*^9}, {3.807306463772435*^9, 3.8073064903156967`*^9}, {3.807310882595138*^9, 3.807311024537902*^9}, { 3.8074722247201977`*^9, 3.8074722528357286`*^9}, {3.8074785225857067`*^9, 3.807478569598668*^9}, {3.822615797036356*^9, 3.822615819238717*^9}, { 3.8226158602667513`*^9, 3.822615865023712*^9}, {3.8421118908218813`*^9, 3.8421119136190157`*^9}, {3.848191479241337*^9, 3.848191499091031*^9}, { 3.848191553904133*^9, 3.848191554607647*^9}},ExpressionUUID->"f018248a-2847-4967-9f89-\ 2a513f956a2b"], Cell[CellGroupData[{ Cell["CASE 1", "Subsection", CellChangeTimes->{{3.8073042838720083`*^9, 3.807304287126093*^9}, { 3.84211192292332*^9, 3.842111928869272*^9}},ExpressionUUID->"4d5e59c5-d1f7-4276-9f13-\ 26b19daeb1a0"], Cell[BoxData[ RowBox[{ RowBox[{"eqSigma", "=", RowBox[{ FractionBox[ 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