System of Equations

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

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

CASE 1

eqSigma=(((-1+varJ) varK)/(2 ((1/(γ^2))+(varK/(σ^2))) (σ^4)))-((varJ varK)/(2 (σ^2)))+((varJ varK ((σT)^2))/(2 (σ^4)))-((((σT)^2) (((varJ varK)/((γ^2) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+((varJ varK (σ^2) (-(varK/(σ^2))+(γ^2) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))/((varK (γ^2)+(σ^2)) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))))/(σ^6))+(1/varM)varS ((((varK^3) ((ηT)^2))/((γ^2) ((1/(γ^2))+(varK/(σ^2))) (σ^8) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))-(((varK^2) ((ηT)^2))/((γ^2) (σ^6) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+(1/(2 (σ^8)))(varK^2) ((ηT)^2) (((varJ varK)/((γ^4) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2)))-((varK^2)/((γ^2) ((((1/(γ^2))+(varK/(σ^2))))^2) (σ^2) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+(varK/((γ^2) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2) (τ^2)))+(varK/((γ^4) ((1/(γ^2))+(varK/(σ^2))) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2) (τ^2)))))+((varK ((varJ/(γ^2))+(1/(τ^2))))/(2 (σ^4) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+(1/(2 (σ^8)))((σT)^2) ((((varJ^2) (varK^2))/((γ^4) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2)))+varJ (((varK^2)/((((1/(γ^2))+(varK/(σ^2))))^2))+((varJ (varK^4))/((γ^2) ((1/(γ^2))+(varK/(σ^2))) (σ^4) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2)))+((varK^4)/(((((1/(γ^2))+(varK/(σ^2))))^2) (σ^4) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))-((varK^3)/(((1/(γ^2))+(varK/(σ^2))) (σ^2) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))-(((varK^3) (((varJ varK)/(σ^2))+(1/(τ^2))))/((γ^2) ((1/(γ^2))+(varK/(σ^2))) (σ^2) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2)))+((varK^2)/((γ^2) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2) (τ^2))))+((varJ (varK^2))/((γ^2) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2) (τ^2))))+((varJ (varK^3) ((τT)^2))/(2 ((((1/(γ^2))+(varK/(σ^2))))^2) (σ^8)))+((varJ varK ((τT)^2))/(2 (σ^4)))-(((varJ^2) (varK^2) ((τT)^2))/((γ^2) (σ^6) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+(1/(2 (σ^8)))(varJ^2) (varK^2) (((varJ varK)/((γ^4) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2)))-((varK^2)/((γ^2) ((((1/(γ^2))+(varK/(σ^2))))^2) (σ^2) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+(varK/((γ^2) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2) (τ^2)))+(varK/((γ^4) ((1/(γ^2))+(varK/(σ^2))) (((-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))))^2) (τ^2)))) ((τT)^2)-((varJ (varK^2) ((τT)^2))/((σ^6) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))) (τ^2)))+((varR ((((varK^3) ((ηT)^2))/(2 ((((1/(γ^2))+(varK/(σ^2))))^2) (σ^8)))+((varK ((ηT)^2))/(2 (σ^4)))+((varJ (varK^3) ((τT)^2))/((γ^2) ((1/(γ^2))+(varK/(σ^2))) (σ^8) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2))))))+(((varK^2) ((τT)^2))/((σ^6) (-((varJ (varK^2))/(σ^4))+((1/(γ^2))+(varK/(σ^2))) (((varJ varK)/(σ^2))+(1/(τ^2)))) (τ^2)))-(((varK^2) (((ηT)^2)+((τT)^2)))/(((1/(γ^2))+(varK/(σ^2))) (σ^6)))))/varM);

eqTauText = ToExpression["-\\frac{1}{2}\\frac{1}{\\tau^2}+\\frac{1}{2}*((\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})*(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})-J*(\\frac{K}{\\sigma^2})^2)^{-1}*(\\frac{1}{(\\tau^2)^2})*(\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})+\\frac{1}{2}* (\\frac{1}{\\sigma^2})^2*(\\frac{1}{(\\tau^2)^2})*((\\frac{1}{\\gamma^2})^2*\\frac{1}{\\xi^2}*(J^2*K^2*\\tau^2_t+(S/M)*K^2*\\eta^2_t)+\\sigma^2_t*J*K*\\frac{1}{\\xi^2} *(\\frac{1}{\\gamma^2})^2)",TeXForm]/.{ξ -> ((K/σ^2+1/γ^2)*(J*K/σ^2+1/τ^2)-J*(K/σ^2)^2), (Subscript[τ, t]) -> τT,(Subscript[η, t]) -> ηT,(Subscript[σ, t]) -> σT};

eqGammaText = ToExpression["-\\frac{J}{2}\\frac{1}{\\gamma^2}+\\frac{1}{2}((\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})*(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})-J*(\\frac{K}{\\sigma^2})^2)^{-1}*(\\frac{1}{(\\gamma^2)^2})*(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})+\\frac{J-1}{2}*(\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^{-1}*(\\frac{1}{(\\gamma^2)^2})+\\frac{1}{2} *(\\frac{1}{\\sigma^2})^2*\\frac{1}{(\\gamma^2)^2}*((-\\frac{K}{\\sigma^2}\\frac{1}{\\xi^2} \\frac{1}{\\tau^2}-\\frac{K}{\\sigma^2}\\frac{1}{\\xi^2}* (\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})+2*(\\frac{K}{\\sigma^2})^2*((\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^2*\\xi)^{-1}+J*(\\frac{K}{\\sigma^2})^4*((\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^2*\\xi^2)^{-1})*(J^2*K^2*\\tau^2_t+(S/M)*K^2*\\eta^2_t) +
(\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^{-2}*K^2*(J*\\tau^2_t+(R/M)*\\eta^2_t)+\\sigma^2_t*(-\\frac{K}{\\sigma^2}\\frac{1}{\\xi^2} \\frac{1}{\\tau^2}*J*K+J*K*(\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^{-2} -\\frac{K}{\\sigma^2}\\frac{1}{\\xi^2}* (\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})*J*K+2*J*K*(\\frac{K}{\\sigma^2})^2*((\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^2*\\xi)^{-1}+J^2*K*(\\frac{K}{\\sigma^2})^4*((\\frac{K}{\\sigma^2}+\\frac{1}{\\gamma^2})^2*\\xi^2)^{-1}))",TeXForm]/.{ξ -> ((K/σ^2+1/γ^2)*(J*K/σ^2+1/τ^2)-J*(K/σ^2)^2), (Subscript[τ, t]) -> τT,(Subscript[η, t]) -> ηT,(Subscript[σ, t]) -> σT};

eqTau = eqTauText/.{K -> varK, J -> varJ, M -> varM, R -> varR, S -> varS};
eqGamma = eqGammaText/.{K -> varK, J -> varJ, M -> varM, R -> varR, S -> varS};

(*Root 1*)

Simplify[ExpandAll[Simplify[ExpandAll[eqSigma*σ*2]]/.{σ -> (Sqrt[((σT)^2)]),τ -> (Sqrt[(varS-varR)/(varM*(varJ^2-varJ))*((ηT)^2)+((τT)^2)]),γ -> (Sqrt[(varJ*varR-varS)/(varM*(varJ^2-varJ))*((ηT)^2)])}]]
Simplify[Simplify[ExpandAll[eqTau*τ*2]]/.{σ -> (Sqrt[((σT)^2)]),τ -> (Sqrt[(varS-varR)/(varM*(varJ^2-varJ))*((ηT)^2)+((τT)^2)]),γ -> (Sqrt[(varJ*varR-varS)/(varM*(varJ^2-varJ))*((ηT)^2)])}]
Simplify[Simplify[ExpandAll[eqGamma*γ*2]]/.{σ -> (Sqrt[((σT)^2)]),τ -> (Sqrt[(varS-varR)/(varM*(varJ^2-varJ))*((ηT)^2)+((τT)^2)]),γ -> (Sqrt[(varJ*varR-varS)/(varM*(varJ^2-varJ))*((ηT)^2)])}]

(*Root 2*)

Simplify[Simplify[ExpandAll[eqSigma*σ*2]]/.{σ -> (Sqrt[((σT)^2)]),τ -> 0,γ -> (Sqrt[varR/(varJ*varM)*((ηT)^2)+((τT)^2)])}]
Simplify[Simplify[ExpandAll[eqTau*τ*2]]/.{σ -> (Sqrt[((σT)^2)]),τ -> 0,γ -> (Sqrt[varR/(varJ*varM)*((ηT)^2)+((τT)^2)])}]
Simplify[Simplify[ExpandAll[eqGamma*γ*2]]/.{σ -> (Sqrt[((σT)^2)]),τ -> 0,γ -> (Sqrt[varR/(varJ*varM)*((ηT)^2)+((τT)^2)])}]

(*Root 3*)

Simplify[Simplify[ExpandAll[eqSigma*σ*2]]/.{σ -> (Sqrt[varK*(varJ*varR-varS)/(varJ*varM*(varJ*varK-1))*((ηT)^2)+((σT)^2)]),τ -> (Sqrt[(-varR+varK*varS)/(varJ*varM*(varJ*varK-1))*((ηT)^2)+((τT)^2)]),γ -> 0}]
Simplify[Simplify[ExpandAll[eqTau*τ*2]]/.{σ -> (Sqrt[varK*(varJ*varR-varS)/(varJ*varM*(varJ*varK-1))*((ηT)^2)+((σT)^2)]),τ -> (Sqrt[(-varR+varK*varS)/(varJ*varM*(varJ*varK-1))*((ηT)^2)+((τT)^2)]),γ -> 0}]
Simplify[Simplify[ExpandAll[eqGamma*γ*2]]/.{σ -> (Sqrt[varK*(varJ*varR-varS)/(varJ*varM*(varJ*varK-1))*((ηT)^2)+((σT)^2)]),τ -> (Sqrt[(-varR+varK*varS)/(varJ*varM*(varJ*varK-1))*((ηT)^2)+((τT)^2)]),γ -> 0}]

(*Root 4*)

Simplify[Simplify[ExpandAll[eqSigma*σ*2]]/.{σ -> (Sqrt[varR/(varJ*varM)*((ηT)^2)+((σT)^2)+((τT)^2)]),τ -> 0,γ -> 0}]
Simplify[Simplify[ExpandAll[eqTau*τ*2]]/.{σ -> (Sqrt[varR/(varJ*varM)*((ηT)^2)+((σT)^2)+((τT)^2)]),τ -> 0,γ -> 0}]
Simplify[Simplify[ExpandAll[eqGamma*γ*2]]/.{σ -> (Sqrt[varR/(varJ*varM)*((ηT)^2)+((σT)^2)+((τT)^2)]),τ -> 0,γ -> 0}]

CASE 2

Note: here, we could not write the system as a linear function of T's, so there is no R and S, just T_m for a specific sequence.  Hence, these are the set of equations for one sequence, not summed over sequences.

eqSigmaMText =ToExpression["-\\frac{J*K}{2}*\\frac{1}{\\sigma^2}-\\frac{1}{2}*((\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})*(\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})-(\\frac{T*K}{\\sigma^2})^2)^{-1}*(\\frac{-2*J*T*K^2}{(\\sigma^2)^3}-\\frac{J*K}{(\\sigma^2)^2*\\eta^2}-\\frac{K*T}{(\\sigma^2)^2*\\tau^2}+\\frac{2*K^2*T^2}{(\\sigma^2)^3})+\\frac {1}{2 *(\\sigma^2)^2}* J*K*(\\tau_t^2+\\gamma_t^2+\\sigma_t^2)
-(\\frac{1}{\\sigma^2})^3*(\\frac{1}{\\xi}*((\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})*(J^2*K^2*\\tau_t^2+J*K^2* \\gamma_t^2)+2*(\\frac{-T*K}{\\sigma^2})*(J*T*K^2*\\tau_t^2+ T*K^2*\\gamma_t^2)+(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})*(T^2*K^2*\\tau_t^2+T*K^2*\\gamma_t^2))+\\sigma^2_t* \\frac{1}{\\xi}*(2*(J-T)*K*\\frac{T*K}{\\sigma^2}+J*K*\\frac{1}{\\eta^2}+T*K*\\frac{1}{\\tau^2}))
+\\frac{1}{2}* (\\frac{1}{\\sigma^2})^4*(\\frac{1}{\\xi^2}*(((J-T)*\\frac{T^2*K^3}{(\\sigma^2)^2}+2*(J-T)*K*\\frac{T*K}{\\sigma^2}*\\frac{1}{\\eta^2}+J*K*\\frac{1}{(\\eta^2)^2})*(J^2*K^2*\\tau_t^2+J*K^2* \\gamma_t^2)+2*(T*K*\\frac{1}{\\tau^2}*\\frac{1}{\\eta^2}-(J-T)*\\frac{T^2*K^3}{(\\sigma^2)^2})*(J*T*K^2*\\tau_t^2+ T*K^2*\\gamma_t^2)+((J-T)*\\frac{J*T*K^3}{(\\sigma^2)^2}+2*(J-T)*K*\\frac{T*K}{\\sigma^2}*\\frac{1}{\\tau^2}+T*K*\\frac{1}{(\\tau^2)^2})*(T^2*K^2*\\tau_t^2+T*K^2*\\gamma_t^2))+\\sigma^2_t*\\frac{1}{\\xi^2}*(2*(J-T)^2*\\frac{T^2*K^4}{(\\sigma^2)^2}+2*(J-T)*J*K^2*\\frac{T*K}{\\sigma^2}*\\frac{1}{\\eta^2}+J^2*K^2*\\frac{1}{(\\eta^2)^2} +2*T^2*K^2*\\frac{1}{\\tau^2}*\\frac{1}{\\eta^2}+2*(J-T)*K*\\frac{T^2*K^2}{\\sigma^2}*\\frac{1}{\\tau^2}+T^2*K^2*\\frac{1}{(\\tau^2)^2}))",TeXForm]/.{ξ -> ((K*T/σ^2+1/η^2)*(J*K/σ^2+1/τ^2)-(K*T/σ^2)^2), (Subscript[τ, t]) -> τT,(Subscript[γ, t]) -> γT,(Subscript[σ, t]) -> σT};

eqTauMText =ToExpression["-\\frac{1}{2}*\\frac{1}{\\tau^2}+\\frac{1}{2}*((\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})*(\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})-(\\frac{T*K}{\\sigma^2})^2)^{-1}*(\\frac{1}{\\tau^2})^2*(\\frac{K*T}{\\sigma^2}+\\frac{1}{\\eta^2})+\\frac{1}{2}* (\\frac{1}{\\sigma^2})^2*(\\frac{1}{\\tau^2})^2*\\frac{1}{\\xi^2}*((\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})^2*(J^2*K^2*\\tau_t^2+J*K^2* \\gamma_t^2)-2*(\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})*\\frac{T*K}{\\sigma^2}*(J*T*K^2*\\tau_t^2+ T*K^2*\\gamma_t^2)+(\\frac{T*K}{\\sigma^2})^2*(T^2*K^2*\\tau_t^2+T*K^2*\\gamma_t^2)+\\sigma^2_t*(J*K*(\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})^2-(\\frac{T*K}{\\sigma^2}+\\frac{2}{\\eta^2})*\\frac{T^2*K^2}{\\sigma^2}))",TeXForm]/.{ξ -> ((K*T/σ^2+1/η^2)*(J*K/σ^2+1/τ^2)-(K*T/σ^2)^2), (Subscript[τ, t]) -> τT,(Subscript[γ, t]) -> γT,(Subscript[σ, t]) -> σT};

eqEtaMText =ToExpression["-\\frac{1}{2}*\\frac{1}{\\eta^2}+\\frac{1}{2}*((\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})*(\\frac{T*K}{\\sigma^2}+\\frac{1}{\\eta^2})-(\\frac{T*K}{\\sigma^2})^2)^{-1}*(\\frac{1}{\\eta^2})^2*(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})+\\frac{1}{2}* (\\frac{1}{\\sigma^2})^2*(\\frac{1}{\\eta^2})^2*\\frac{1}{\\xi^2}*((\\frac{T*K}{\\sigma^2})^2*(J^2*K^2*\\tau_t^2+J*K^2* \\gamma_t^2)-2*\\frac{T*K}{\\sigma^2}*(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})*(J*T*K^2*\\tau_t^2+ T*K^2*\\gamma_t^2)+(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})^2*(T^2*K^2*\\tau_t^2+T*K^2*\\gamma_t^2)+\\sigma^2_t*(\\frac{-T^2*K^2}{\\sigma^2}*(\\frac{J*K}{\\sigma^2}+\\frac{2}{\\tau^2}) +  T*K*(\\frac{J*K}{\\sigma^2}+\\frac{1}{\\tau^2})^2))",TeXForm]/.{ξ -> ((K*T/σ^2+1/η^2)*(J*K/σ^2+1/τ^2)-(K*T/σ^2)^2), (Subscript[τ, t]) -> τT,(Subscript[γ, t]) -> γT,(Subscript[σ, t]) -> σT};

eqSigmaM = eqSigmaMText/.{K -> varK, J -> varJ, T -> varT};
eqTauM = eqTauMText/.{K -> varK, J -> varJ, T -> varT};
eqEtaM = eqEtaMText/.{K -> varK, J -> varJ, T -> varT};

(*Root 3*)

Simplify[Simplify[ExpandAll[eqSigmaM*σ*2]]/.{σ->((Sqrt[varK ((γT)^2)-varJ * varK ((γT)^2)+((σT)^2)-varJ * varK ((σT)^2)])/(Sqrt[1-varJ * varK])),τ->((Sqrt[((γT)^2)-varK ((γT)^2)+((τT)^2)-varJ * varK ((τT)^2)])/(Sqrt[1-varJ * varK])),η -> 0}]
Simplify[Simplify[ExpandAll[eqTauM*τ*2]]/.{σ->((Sqrt[varK ((γT)^2)-varJ * varK ((γT)^2)+((σT)^2)-varJ * varK ((σT)^2)])/(Sqrt[1-varJ * varK])),τ->((Sqrt[((γT)^2)-varK ((γT)^2)+((τT)^2)-varJ * varK ((τT)^2)])/(Sqrt[1-varJ * varK])),η -> 0}]
Simplify[Simplify[ExpandAll[eqEtaM*η*2]]/.{σ->((Sqrt[varK ((γT)^2)-varJ * varK ((γT)^2)+((σT)^2)-varJ * varK ((σT)^2)])/(Sqrt[1-varJ * varK])),τ->((Sqrt[((γT)^2)-varK ((γT)^2)+((τT)^2)-varJ * varK ((τT)^2)])/(Sqrt[1-varJ * varK])),η -> 0}]

(*Root 4*)

Simplify[Simplify[ExpandAll[eqSigmaM*σ*2]]/.{σ->(Sqrt[((γT)^2)+((σT)^2)+((τT)^2)]),τ->0,η -> 0}]
Simplify[Simplify[ExpandAll[eqTauM*τ*2]]/.{σ->(Sqrt[((γT)^2)+((σT)^2)+((τT)^2)]),τ->0,η -> 0}]
Simplify[Simplify[ExpandAll[eqEtaM*η*2]]/.{σ->(Sqrt[((γT)^2)+((σT)^2)+((τT)^2)]),τ->0,η -> 0}]