Hydroperoxyalkylperoxy species (OOQOOH) are important intermediates that are generated during the auto ignition of transport fuels. A key reaction of hydroperoxyalkylperoxy radicals is a [1,5] hydrogen shift, for which kinetics data are experimentally unavailable. Here we study two typical OOQOOH reactions and compare their kinetics to one another and to a previous study to learn the effect of structural variations of the alkyl group on the competition between alternative [1,5] hydrogen shifts of hydroperoxyalkylperoxy species. We use electronic structure calculations to determine previously missing thermochemical data, and we use variational transition state theory with multidimensional tunneling, multiple structures, torsional potential anharmonicity, and high-frequency anharmonicity to obtain more accurate rate constants than the ones that can be computed by conventional single-structure harmonic transition state theory and than the empirically estimated rate constants that are currently used in combustion modeling. The calculated temperature range is 298-1500 K. The roles of various factors in determining the rates are elucidated, and we find an especially strong effect of multiple structure anharmonicity due to torsions. Thus, even though there is some cancellation between the anharmonicity of the reactant and the anharmonicity of the transition state, and even though the reactants are very similar in structure, differing only by a methyl group, the effect of multiple structure anharmonicity has a large effect on the relative rates, as large as a factor of 17 at room temperature and as large as a factor of 7 at 1500 K. This has broad implications for the estimation of reaction rates in many subfields of chemistry, including combustion chemistry and atmospheric chemistry, where rates of reaction of complex molecules are usually estimated without explicit consideration of this fundamental entropic effect. In addition, the pressure-dependence of the rate constants is modeled by system-specific quantum Rice-Ramsperger-Kassel theory for a reversible isomerization.