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Fig. 6. Loss in predictability (proportion of variance accounted for) of the method of least-squares from as- suming the matching relation (slope a = 1.0) to be true. See text for further explanation.

Figure 6 Loss in predictability (proportion of variance accounted for) of the method of least-squares from as- suming the matching relation (slope a = 1.0) to be true. See text for further explanation.

Related Figures (7)

Fig. 1. Appearance of overmatching and undermatch- ing when relative responding and reinforcement are represented as proportions. Different shapes arise from different values of a in Equation 2.
Fig. 1. Appearance of overmatching and undermatch- ing when relative responding and reinforcement are represented as proportions. Different shapes arise from different values of a in Equation 2.
Fig. 2. Proportion of variance accounted for by least- squares regression using Equation | plotted against the fitted slope a. Horizontal lines indicate 80% and 100% of variance accounted for. Note logarithmic x-axis; see text for further explanation. The distributions of Figure 3 show varia- tion to both sides of a single mode. They are Figure 2 indicates that Equation | generally described the data well. The majority of points lie above .9. Out of 103 sets, less than 80% of the variance was accounted for in only eleven sets; less than 70% in only three. The fits to the time ratios and response ratios were equally good. The averages for time and re- sponses were 90.5% and 90.3%. Figure 2 shows also that there was no systematic relation be- tween goodness of fit and slope.
Fig. 2. Proportion of variance accounted for by least- squares regression using Equation | plotted against the fitted slope a. Horizontal lines indicate 80% and 100% of variance accounted for. Note logarithmic x-axis; see text for further explanation. The distributions of Figure 3 show varia- tion to both sides of a single mode. They are Figure 2 indicates that Equation | generally described the data well. The majority of points lie above .9. Out of 103 sets, less than 80% of the variance was accounted for in only eleven sets; less than 70% in only three. The fits to the time ratios and response ratios were equally good. The averages for time and re- sponses were 90.5% and 90.3%. Figure 2 shows also that there was no systematic relation be- tween goodness of fit and slope.
MATCHING, UNDERMATCHING, AND OVERMATCHING Fig. 3. Frequency distributions of the slope a obtained by fitting Equation 1 to the various sets of data. Verti- cal lines show the location of 1.0, the value required by the matching relation. The two distributions sepa- rate studies that measured responses (bottom) and time (top).
MATCHING, UNDERMATCHING, AND OVERMATCHING Fig. 3. Frequency distributions of the slope a obtained by fitting Equation 1 to the various sets of data. Verti- cal lines show the location of 1.0, the value required by the matching relation. The two distributions sepa- rate studies that measured responses (bottom) and time (top).
A nonparametric test was used to assess sta- tistical significance (Hollander & Wolfe, 1973, pp. 200-204). For this, a nonparametric proce- dure was used to fit a regression line (column vii in Table 1; Hollander & Wolfe, 1973, pp. 205-216). The number of data (column ix) was sometimes less than for the least-square method (column iv) because ordinates with identical abscissae were averaged into a single value. The probability that the slope could equal 1.0 (column x) appears in Figure 5 plotted against the nonparametric-estimated slope. Apart from four or five outliers, the points conform to a roughly symmetrical curve fall- ing away to both sides of a peak at 1.0. A hori- zontal line indicates the level of probability equal to .05. Points lie above this line for slopes from about .60 to 1.56. No points lie below the line in the range between approxi- mately .90 and 1.20. According to this test, then, slopes between .9 and 1.2 approximate 1.0. Outside this range, unsystematic variation presumably determines whether the slope should be considered distinct from 1.0. Although Figures 2 and 5 suggest that the Fig. 5. Probability of a true slope of 1.0 (nonpara- metric test of Hollander & Wolfe, 1973, pp. 201-204) versus the slope estimated by a nonparametric fitting technique (Hollander & Wolfe, 1973, pp. 205-206). The graph includes 103 points. The horizontal line shows probability equal to .05.
A nonparametric test was used to assess sta- tistical significance (Hollander & Wolfe, 1973, pp. 200-204). For this, a nonparametric proce- dure was used to fit a regression line (column vii in Table 1; Hollander & Wolfe, 1973, pp. 205-216). The number of data (column ix) was sometimes less than for the least-square method (column iv) because ordinates with identical abscissae were averaged into a single value. The probability that the slope could equal 1.0 (column x) appears in Figure 5 plotted against the nonparametric-estimated slope. Apart from four or five outliers, the points conform to a roughly symmetrical curve fall- ing away to both sides of a peak at 1.0. A hori- zontal line indicates the level of probability equal to .05. Points lie above this line for slopes from about .60 to 1.56. No points lie below the line in the range between approxi- mately .90 and 1.20. According to this test, then, slopes between .9 and 1.2 approximate 1.0. Outside this range, unsystematic variation presumably determines whether the slope should be considered distinct from 1.0. Although Figures 2 and 5 suggest that the Fig. 5. Probability of a true slope of 1.0 (nonpara- metric test of Hollander & Wolfe, 1973, pp. 201-204) versus the slope estimated by a nonparametric fitting technique (Hollander & Wolfe, 1973, pp. 205-206). The graph includes 103 points. The horizontal line shows probability equal to .05.
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