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between the best-fit point and the remaining points in the space are shown in Fig. 12 for the m$_{4}$ = m$_{1}$ model. The two-dimensional 90% confidence level contours for that model are shown in Fig. 13. The projections and contours for the m$_{4}$ ≫ m$_{3}$ model are shown in Figs. 14 and 15. 

As seen in these figures, θ$_{34}$ < 38 $^{◦}$(56$^{◦}$) at 90% con- fidence level for the m$_{4}$ = m$_{1}$ model. The number in parentheses represents the 90% confidence level limit ob- tained when maximal ν$_{e}$ appearance is allowed. For the m$_{4}$ ≫ m$_{3}$ model, θ$_{24}$ < 10 $^{◦}$(11$^{◦}$) and θ$_{34}$ < 38 $^{◦}$(56$^{◦}$) at the 90% confidence level. These limits indicate that any coupling between the active neutrinos and a sterile neu- trino is submaximal. Furthermore, the χ 2 values indicate that these four-flavor models fit the data no better than oscillations among only the active neutrinos. 

A straightforward method to quantify the coupling be- tween the active and sterile neutrinos is to determine the fraction of disappearing ν$_{µ}$ that transition to ν$_{s}$ . That fraction is expressed as 

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For the m$_{4}$ = m$_{1}$ model, the disappearance fraction f$_{s}$ is energy independent, as can be seen upon inserting the expressions from Eq. (12) into Eq. (15). The 90% confi- dence level limit for f$_{s}$ is determined by selecting a large number of test values of θ$_{23}$ and θ$_{34}$ from Gaussian distri- butions with mean and σ given in Table II. The value of f$_{s}$ that is larger than 90% of the test cases represents the limit. The value corresponding to the 90% confidence level is f$_{s}$ < 0 . 51 (0 . 55), with the value in parentheses indicating the value obtained for maximally-allowed ν$_{e}$ appearance in the beam. This new limit on the value of f$_{s}$ represents a reduction of 33% compared to the previ- ous MINOS result without ν$_{e}$ appearance [17]. 

For the m$_{4}$ ≫ m$_{3}$ model, inserting the expressions from Eq. (13) into (15) shows that f$_{s}$ is energy depen- dent because of the constant terms in Eq. (13). For this reason the 90% confidence level value of f$_{s}$ for this model is presented at E$_{ν}$ = 1 . 4 GeV, the energy where the ν$_{µ}$ 

