The analysis of all results is based on the sums over the last seven sessions of the number of stay and switch responses at each alternative, the number of reinforcers earned for staying at and for switching from each alternative and the time at each alternative (). The stay responses at an alternative included all presses after the first press during each visit to that alternative. The responses for switching from an alternative were the first press at the other alternative. The time at an alternative was the cumulative intervals from the first press at the alternative to the first press at the other alternative. The reinforcers earned for staying at an alternative were the reinforcers arranged by the schedule for staying at that alternative. The reinforcers earned for switching from an alternative were the reinforcers arranged by the schedule for switching from that alternative. All response, time and reinforcer ratios were calculated as left data divided by right data.

| **Table 2**This table shows time at each alternative (min), stay responses at each alternative and changeovers from each alternative. |

Before using the results of this experiment to evaluate the stay/switch model, the equation for describing choice among different magnitudes of reinforcers for staying at and switching from each alternative needs to be developed. Including the magnitudes of reinforcers in

Equation 3 produces,

*Mt*_{n} is the magnitude of the reinforcer for staying at alternative

*n* and

*Mw*_{n} is the magnitude the reinforcer for switching from alternative

*n*. Because the number of reinforcer earned for staying at and switching from each alternative was the same, they can be factored and cancel yielding,

Except for Rat 819’s stay-response ratios, the stay/switch model described the stay-response ratios and time ratios for all rats ( and ). The data were close to the plane. These figures show that the residuals, the distance from each data to the best-fitting plane, were small. These figures also show that the residuals appear to be randomly distributed with respect to the values of the independent variables. Finally, these figures show that the slope of the plane was not equal to zero along the ratio of the ratios of magnitudes of reinforcers, which means that the ratio of ratios of magnitudes influenced allocations of stay responses and of time; however, the slope along the ratio of the sums of magnitudes of reinforcers was not equal to zero only for the ratio of stay responses, indicating that the ratio of the sums of magnitudes influenced stay-response ratios but not time ratios.

These observations were confirmed by the results of the regressions using

Equation 5 ().

Equation 5 accounted for greater than 80% of the variance for all rats except Rat 819’s stay-response ratio. The value of

*l*, sensitivity to the ratio of the ratio of stay to switch magnitudes, was greater than zero for both the ratio of stay responses and the ratio of times for all rats. The value of

*m*, sensitivity to the ratio of the sums of stay and switch magnitudes, was greater than zero for all rats’ ratios of stay response and for time ratios for just two rats (Rats 814 and 824). Stay responding for Rats 814, 817 and 818 was biased towards the right alternative and time ratios for Rats 814, 817, 818 and 819 also were biased towards the right alternative. Rat 824’s stay response and time ratios were biased towards the left alternative.

| **Table 3**This table shows the results of least-squares linear regression using data from all conditions using the stay/switch model (Equation 5), and the generalized matching law (Equation 7). |

The generalized matching law described the response and time allocations when using the symmetrical arrangements of magnitudes of reinforcers ( and ). For each rat, the response and time ratios from the symmetrical conditions form an approximately straight line. Thus, any failure of the generalized matching law to describe the data using other arrangements of stay and switch magnitudes of reinforcers cannot result simply from using two pairs of schedules.

Before using the results of this experiment to evaluate the generalized matching law, I need to develop the equation for describing choice among different magnitudes of reinforcers for staying and switching. When applying the generalized matching law to symmetrical magnitudes of reinforcers one takes the frequency of each reinforcer type multiplied by its magnitude (e.g.,

*Mt*_{1} *

*Rt*_{1} +

*Mw*_{2} *

*Rw*_{2};

Rachlin & Baum, 1969b;

Davison & McCarthy, 1988, which produces,

To show that this equation is the correct application of the generalized matching law, consider

Equation 6 when applied to symmetrical conditions (traditional concurrent procedures). The sum of the number of stay and switch reinforcers is the same at both alternatives, so they can be factored out of the numerator and denominator and then they cancel. Because the magnitudes are equal we can substitute

*M*_{n}, the magnitude obtained at each alternative. The resulting ratio,

*M*_{1} /

*M*_{2}, is the generalized matching law applied to magnitudes of reinforcers (

Rachlin & Baum, 1969a;

Davison & McCarthy, 1988). Because the frequency of stay and switch reinforcers obtained at each alternative was the same, they cancel from the equation, leaving,

This equation was used for regressions using both symmetrical and nonsymmetrical arrangements.

Using arrangements that were not symmetrical decreased the precision of the descriptions by the generalized matching law. Considering all data points for each rat, and show for Rat 814 and 824’s response ratios and Rat 816 and 817’s time ratios, that the data points fall roughly on the same straight line as in the symmetrical conditions but there is more variability. However, for Rats 816, 817 and 818 the response allocations are systematically above and for Rat 819 it is below a line for symmetrical conditions. For Rats 814, 818, 819 and 824 the time allocations are systematically below a line for symmetrical conditions. The generalized matching law adequately described (*r*^{2} > .80) response allocations from all conditions only for Rat 817. However, it adequately described the time allocations from all conditions for all but Rat 814 (). For the results of all rats, the sensitivity to reinforcer allocation was low (< 0.60). The low slopes could result from not using a COD or from using non-independent scheduling of reinforcers. Time ratios were consistently more sensitive to reinforcers ratios than were response ratios. There was no consistent bias, although several rat’s response or time allocations were biased, usually towards the right alternative ().

The stay/switch model provided better descriptions of the rats’ response allocation data. Examining shows that, in the Necessary Condition, all 12 deviations from the best-fitting line were in the direction predicted by

Equation 5. The deviations first were towards one alternative and then the other alternative. Examining this figure also shows that, in the Sufficient Conditions, 10 of the 12 the deviations from the best-fitting line were in the direction predicted by

Equation 5. These deviations were closer to indifference. Finally, for the Unsymmetrical Conditions, 6 of the 12 deviations were in the direction predicted by

Equation 5. The deviations were closer to indifference. Combining the results from these three arrangements shows that, using a binomial test, the deviations were significantly in the direction predicted by

Equation 5 (28 of 36 conditions;

*p* < .01).

The models were also compared by plotting the obtained (stay) response and time ratios as a function of the ratios predicted by the generalized matching law and the stay/switch model. The diagonal dashed line shows the locus of perfect predictions. This figure shows that the predictions by the stay/switch model of the stay-response and time ratios are closer to the obtained values than the predictions of the generalized matching law. The difference is especially noticeable for the necessary, sufficient and unsymmetrical arrangements. Although a similar difference is evident for the time ratios, it is much smaller.

Because the stay/switch model described the stay-response and time ratios, the following analyses assessed whether changing the ratio of the magnitudes of obtained reinforcers was necessary or sufficient for changing response and time ratios. Changing the ratio of the magnitudes of obtained reinforcers was not necessary for changing the response and time ratios. For the necessary conditions, the ratio of magnitudes of reinforcers remained unchanged yet response and time ratios, for each rat, first favored one alternative and then the other alternative. For response allocations, the ranges did not overlap (triangles, ). For time allocations of Rats 814, 816, 819, and 824, the ranges did not overlap (triangles, ).

Changing the ratio of the magnitudes of obtained reinforcers was not sufficient for changing the response ratios but was sufficient for changing time ratios. That is, in the sufficient conditions, response ratios but not time ratios remained unchanged as the ratio of magnitudes of reinforcers favored one alternative and then the other alternative. For Rats 814, 816, 817, and 819 the ranges of the response ratios from the two sufficient conditions overlapped as the ratio of the magnitudes of reinforcers changed (squares, ). Changing the ratio of the magnitudes of reinforcers was sufficient for changing the time ratio for each rat (squares, ). For each rat, the ranges of time ratios did not overlap.