On the statistical properties of operant settings and their contribution to the evaluation of sensitivity to reinforcement

Abstract

When using the matching law in applied settings, a recurring problem is to assess when subjects adjust their responses as a function of their associated reinforcers. Specifically, the main concern is to determine whether subjects’ behavior are sensitive to reinforcement or not. Many researchers have followed (explicitly or implicitly) the criterion that 50% of explained variance is deemed acceptable to consider the subject sensitive. However, it is neither theoretically nor empirically grounded. This article presents a null hypothesis statistical test to assess whether an organism’s behavior is sensitive to reinforcement as quantitatively expressed by the matching law. We first introduce the motivation as to why such a test is warranted, formally described the basis of the model used to compute the null hypothesis and then show some of its advantages. We conclude the article with a hypothetical example.

Keywords

Matching law sensitivity to reinforcement ratio correlation

1. Introduction

A problem when using the matching law (Herrnstein, 1961), especially in the applied literature, is to assess when subjects’ behavior are sensitive to reinforcement (when they adjust their responses as a function of associated reinforcers)¹. The purpose of the current article is to propose a null hypothesis statistical test (NHST) to assess whether an organism’s behavior is, in all likelihood, sensitive to reinforcement as quantitatively expressed by the matching law. Specifically, it tests whether the matching relation lies solely on a pure random process or whether some behavioral processes are required to explain the data. First, we define some preliminary matters such as the matching law and sensitivity to reinforcement. Later, we explain the motives warranting the test and the basis of our model. Finally, we end the article with a hypothetical example.

2. The matching law

The matching law states that the ratio of two response rates is correlated to the ratio of their respective reinforcers rate (Herrnstein, 1961). This relation is quantitatively formalized by the equation

\frac{B_{i}}{B_{e}} = \frac{R_{i}}{R_{e}}

where Bs refer to response rate, Rs to reinforcer rate, and the indices (i and e) distinguish between at least two options. This equation is generally referred to as the strict matching law (SML). To account for systematic deviation found in organisms’ response allocation (Baum, 1979; Davison & McCarthy, 1988; McDowell, 2013), the generalized matching law (GML; Baum, 1974) has been proposed. It is represented by the equation

\frac{B_{i}}{B_{e}} = c {(\frac{R_{i}}{R_{e}})}^{a}

where a refers to sensitivity to reinforcement (i.e. the degree to which organisms adjust their behavior according to reinforcer ratio) and c refers to the response rate independent of the reinforcer rates. The GML is often presented in a logarithm form.

\log (\frac{B_{i}}{B_{e}}) = a \log (\frac{R_{i}}{R_{e}}) + \log c

In log form, the relation is transformed into a linear equation, where a becomes the slope and log c the intercept. They are more easily distinguishable in a graph than their curvilinear counterpart (see Baum, 1974; Caron, 2017).

3. Sensitivity to reinforcement

It is of interest to evaluate a subject’s sensitivity to reinforcement because it provides information on how subjects will adjust their behavior according to the modification in reinforcer rates (Forget & Rivard, 2010; McDowell, 1981; Myerson & Hale, 1984; Noll, 1995). For example, Rivard, Forget, Kerr, and Bégin (2014) investigated social sensitivity of preschoolers with autism spectrum disorder (ASD) in an early behavioral intervention setting. A challenging problem here is to separate the broad social deficit, one of the main symptoms of ASD (American Psychiatric Association, 2013), into specific quantitative measures. The authors relied on the GML to assess how much children changed their social behaviors in response to changes in reinforcement rates. The sensitivity parameter provided quantitative information on how children would adjust their behavior according to their social environment.

The GML alone does not inform if the subject is sensitive to reinforcement. A sensitivity value (a) tending toward zero or largely exceeding unity is insufficient. The explained variance (or the correlation) also has to be considered. For example, sensitivity can be low and the correlation significant, or sensitivity can be high but the correlation non-significant, which for practitioners, in both cases, are not straightforward to interpret. Authors adopting an applied standpoint such as Rivard et al. (2014) and others, accept that explained variances over 50% are good evidences in favor to sensitivity to reinforcement. This convention is well captured by Reed’s statement as well as being referred by Rivard et al. (2014) that

In applied studies using naturally occurring matching relations (i.e. not instances in which the researcher programs rates of reinforcement), [variance accounted for] is typically deemed acceptable if the metric is greater than 50%. (Reed, 2009, p. 874)

The main concern here is that the criterion is not theoretically orientated. Still, the criterion is a testimony that, since few experimental studies found explained variances lower than 50%, then it must be somewhat of a lower bound in applied studies also. In the experimental literature, the matching law explains so much variance in behavioral data, over 80% in many procedures and with many species (Baum, 1979; Davison & McCarthy, 1988; McDowell, 2013), that the traditional null hypothesis would obviously be rejected. At the very least, the criterion illustrates that researchers expect high explained variances when they used the GML and that using NHST with a null hypothesis where the population parameter equals zero increases what would be considered false positives. A surprising fact for non-behavior analysts is that NHST is not explicitly recommended by Reed’s tutorial on correlation (and other similar works such as Dallery & Soto, 2013). In statistical textbooks, the correlation procedure, which the matching law is, would imperatively include NHST. Yet, there is actually no other method than Reed’s suggestion to evaluate sensitivity to reinforcement.

We do not argue that NHST with population correlation parameter equaling zero is the appropriate way to assess sensitivity to reinforcement. In fact, as we pointed out, the matching relation will always be significant because the correlations are always large. However, to ensure that subjects adjust their behavior according to reinforcers, enough evidence has to favor the presence of a behavioral process (any kind of processes leading organisms’ behavior to follow the matching law, see Caron, 2017; Herrnstein, 1997). We argue that the obtained matching relation should be tested to see if it significantly deviates from pure randomness. Subject’s behavior should be purely random if there is no behavioral process (null hypothesis) or show trends otherwise.

The question now is what random behavior looks like. A possible approach is that components (numerators and denominators, being B_i, B_e, R_i, and R_e) of the matching equation are correlated together. This concern on the matching law was studied by Caron (2015) who evaluated the influence of conditional distributions on the matching law by simulating a feedback system similar to reinforcement schedules. The author tested the relation that, in most operant settings, the expected occurrence of reinforcers is conditional to the occurrence of a response and found that this could explain 47% of the variance. This effect could reach 63% with the addition of another dependency between responses like a maximum of behavior by session or, in other terms, that response rates are reciprocal. Specifically, high correlations between ratios were due to the conditional dependencies between response rates and reinforcer rates rather than a behavioral process trying to balance ratios.

It has been long acknowledged by statisticians (Pearson, 1897; Yule, 1910) that correlations between two ratios will be inflated if components (B_i, B_e, R_i, R_e) are correlated. Worse, correlations between ratios can often be larger than the simple correlation between their components. In other terms, the hypothesis test of a ratio correlation is biased if the correlations between components are not taken into account. To resolve this issue, Pearson (1897) proposed an equation to account for correlation between components. If we can formally derive the spuriousness held in the correlations between components, we would then obtain a good approximation for a null hypothesis for testing the matching law.

4. The new NHST for the matching law

If there is no correlation between components, then the NHST that the population parameter is zero is legitimate. However, this is inappropriate when there are correlations between components because they will bias the correlation between ratios. Pearson (1897) proposed an approximate equation to evaluate the expected correlation between ratios when numerators and denominators are correlated together, which is the following equation.

r_{\frac{A}{B}, \frac{C}{D}} = \frac{r_{AC} V_{A} V_{C} - r_{AD} V_{A} V_{D} - r_{BC} V_{B} V_{C} + r_{BD} V_{B} V_{D}}{(\sqrt{V_{A}^{2} + V_{B}^{2} - 2 r_{AB} V_{A} V_{B}}) (\sqrt{V_{C}^{2} + V_{D}^{2} - 2 r_{CD} V_{C} V_{D}})}

where rs represent correlations, Vs represent the relative variance (standard deviation divided by mean), and indices correspond to the component. To simplify notation we used A = B₁, B = B_e, C = R_i, D = R_e either for the SML or the GML.² For the remaining of the current article, this notation will be kept.

The purpose of computing $r_{A / B, C / D}$ is that, if we can define a priori the correlations r_AC, r_AD, r_BC, r_BD, r_AB, and r_CD, as well as means and variances of each component to compute the relative variance, then it would be possible to assess a null hypothesis to compare to the empirical correlation. If the observed correlation between ratios is significantly different from $r_{A / B, C / D}$ , then the null hypothesis that the matching relation is due solely to the correlation between behavior and reinforcer in a given schedules of reinforcement (i.e. correlation between components) can be rejected. As such, the conclusion that the subject is sensitive to the ratio of reinforcers would be warranted. To define a priori the correlations, the means and the variances, a model of concurrent reinforcement schedules describing the distributions of response rates and reinforcer rates, and their conditional relations is needed.

5. The model

In the current section, a model involving three assumptions is developed based on the inherent properties of concurrent reinforcement schedules. They describe how behavior and reinforcers are generated, the conditional relation between behaviors, and between behaviors and reinforcers. While this may come forth as too restrictive or inaccurate, it relies on our knowledge about the operant setting. For the moment, an approximation of the quantitative relation can be deemed sufficient. As the understanding of the relation between responses and reinforcers within reinforcement schedules increase, the model will need to undergo revision.

5.1. Assumption 1: how responses and reinforcers are generated

At first, it must be acknowledged that the occurrence of behavior and reinforcer has to follow a statistical distribution (a description of the frequency each outcome will occur). We recommend generating responses and reinforcers with a Bernoulli process because it is a stochastic process involving binary random variables and discrete values. The number of emitted responses and contingent reinforcers are sum of Bernoulli process (a binomial distribution). The binomial distribution has the interesting properties of having two parameters (p, the probability of occurrence, and n, the sample size). It is noteworthy that any other discrete distributions such as a Poisson or a uniform distribution could be used. The binomial was preferred over the uniform distribution because the probabilities of occurrence can be manipulated, and the Poisson distribution because it converges toward the binomial distribution as the sample size increases.

5.2. Assumption 2: the conditional relation between responses

A second assumption, suggested by Caron (2015), is that the correlation between components A and B (i.e. B_i and B_e) is negatively perfect. Since there are only two possible responses which are mutually exclusive, knowing one determines the other. This is especially true is most applied settings where an interval sample recording is used to observe behavior. Let n be the number of behaviors observable in a session and $θ$ the probability of occurrence of A, then the frequency of A for the session will be $A ~ binom (θ, n)$ and $B = n - A$ . This assumption corresponds to an operant procedure ending when a given time has elapsed or a given amount of reinforcers has been distributed (Caron, 2015). No organism can emit an infinite number of responses in a finite time length. It is worth noting that this assumption can be altered by proposing any other correlation between −1 and 1 as long as this likely reflects the operant setting. A correlation of zero would imply no correlation between behaviors (like Caron (2014) simulated).

5.3. Assumption 3: the conditional relations between responses and reinforcers

Finally, the last assumption of the model is that the occurrence of reinforcer is conditional to the occurrence of behavior. This is similar to the constraint that the reinforcer rate is always lower than the response rate (Caron, 2015; McDowell & Ansari, 2005). For example, the organism has to emit at least one response to receive one reinforcer. This is true in ratio schedules of reinforcement as well as in interval schedules. The effect of this last assumption is minimized in experimental settings but can be quite detrimental in applied settings where reinforcer rates are not experimentally controlled (see, St Peter et al., 2005). This is also accounted by the mean probabilities of reinforcement where it is quite high in ratio schedules and lower in interval schedules. Formally, the assumption of the model is that the expected number of reinforcers is conditional to the expected number of response emitted given a probability of reinforcement p_j, such that $C ~ binom (p_{1}, A)$ and $D ~ binom (p_{2}, B)$ , where j represents the option: j = {1,2}. Thus, the mean and variance of C and D directly depend on A and B. This is a generalization of the constraint that the reinforcer rate is always lower than the response rate because probabilities p_j can be varied.

6. Means, variances, and covariance

From the assumptions, we can compute the mean, the variance, and the covariance between components which are needed to calculate $r_{A / B, C / D}$ . The variance of A is $σ_{A}^{2} = n θ (1 - θ)$ , the mean of A is $E (A) = n θ$ , the conditional variance of C as $σ_{C}^{2} = n θ p_{1} (1 - θ p_{1})$ , and the conditional mean of C is $E (C) = n θ p_{1}$ (see Casella & Berger, 2002, for details). This is the same for B and D, respectively. We can also compute the covariance of A and C (as well as B and D) as $cov (A, C) = θ p_{1} - θ^{2} p_{1}$ . Then the relative variance is computed as $V_{x} = σ_{x} / σ_{x} E (x)$ (where x refers to a component). Once these elements are gathered, the estimation of $r_{A / B, C / D}$ is possible.

7. The hypothesis test

To complete the hypothesis test, $r_{A / B, C / D}$ has to be computed and used as a null hypothesis to compare statistically to the empirical correlation found with the GML. The NHST will give the likelihood of data given the assumptions of the model. The resulting p-value will inform on the statistical likelihood of sensitivity of reinforcement. If the outcome is significant, then the null hypothesis that the data are well accounted by a model involving no learning can be rejected.

Details to compute NHST with an alternative value are found in many statistical textbooks such as Howell (2012). We have to note that testing the correlation is the same as testing the sensitivity parameter. In the bivariate case, if there is a correlation between both variables, then there must be a slope relating both. Testing either will lead to the same outcome. For the sake of simplicity, the current description will rely on correlation. Most statistical programs will test for a null hypothesis that the correlation is not different from zero. To use an alternate null hypothesis, such as $r_{A / B, C / D}$ , the computation is

z = \frac{atanh (r) - atanh (r_{\frac{A}{B}, \frac{C}{D}})}{\sqrt{\frac{1}{n - 3}}}

where r is the empirical correlation and n the sample size. The inverse hyperbolic tangent (atanh) is used to account for the non-normality of the sampling distribution of r when the null hypothesis is non-zero. This equation yields a z-value that can be compared to chosen alpha (generally α = .05, which yields a critical z-value of 1.64 in a unidirectional hypothesis test). When the z-value exceeds the threshold, the null hypothesis is rejected. A behavioral process, such as sensitivity to reinforcement, is more likely to take place. The null model alone does not account for enough of the variance, and another phenomenon is probably at play. This provides evidence to applied researchers that the subjects are sensitive to reinforcement before conducting their intervention.

8. Advantages

$r_{A / B, C / D}$ is a convenient null hypothesis because it takes into account the response rates and the reinforcer rates. Using a binomial distribution enables to vary the probability parameter compared to a discrete uniform distribution where the probabilities of the sampling distribution is fixed to .50 (e.g. Caron’s (2014, 2015) simulations). This also resolves the problem, as St Peter et al. (2005) pointed out, that

In many cases, attention [the consequence] occurred at a very high rate and duration throughout the observation period. However, because only attention following problem behavior was used in the matching analysis, spurious matching was obtained because the increased response rates resulted in higher rates of contact with attention. (p. 441)

Quantitatively speaking, when the expected occurrence of reinforcers is high and since the occurrence of reinforcers is conditional to the occurrence of a response, spurious correlations are more likely to be found. The correlation $r_{A / B, C / D}$ is sensitive to response rates and reinforcer rates. If this last rate is high, the correlation between components is also higher and the null hypothesis increases. Alternatively, if the reinforcer rate is low, the null hypothesis becomes also lower.

9. An illustration

To illustrate the hypothesis test, we present an example, which is inspired by the procedure of Rivard et al. (2014) and Caron, Forget, and Rivard (in press). Hypothetically, a subject was placed in an operant setting where it had to choose between two options and was rewarded accordingly. The reinforcer rates ratios were not manipulated experimentally. An interval-sampling procedure, that is, separating a session into many equal intervals in which a single instance of behavior can be recorded, was used to sample subject’s responses and their associated reinforcers. A session lasted 20 instances of recording, which was the maximum number of behavior the subject could emit. Finally, 20 sessions were recorded.

Figure 1 shows the response rates ratios and the reinforcer rates ratios of a hypothetical subject. The response allocation of the subject is well described by the GML, r(18) = .86 with 74% of explained variance. The subject shows an undermatching of .58 and no bias (.00) toward an option or the other. This subject’s matching relation is very similar to others found in the applied literature such as Rivard et al.’s (2014) or St Peter et al.’s (2005) studies.

To prepare the hypothetical test, subject’s probabilities of emitting response 1 and 2 as well as their respective probability to be reinforced have to be computed. This information can be calculated from the original data. Later, according to the operant setting, a hypothetical inter-response correlation has to be chosen between −1 and 1 (the R implementation chosen by default is −1, see Appendix 1). According to the hypothetical example, the probabilities of emitting a behavior to option 1 or 2 were of .56 and .44, respectively. Also, the reinforcement probabilities of these options were .44 and .29, respectively. By proceeding with the computation of reinforced behavior probabilities (p_j), that is, by dividing the current estimate of reinforcer probability by the probability to emit the behavior, we see that the occurrences of reinforcers were high (.78 and .67, respectively). Applied researcher could be worried that the rate is so high that it could inflate the correlation found by the GML.

Once all the data are gathered, we can compute $r_{A / B, C / D}$ from the probabilities of each component. Appendix 1 shows the code in R to implement the evaluation. It gives the hypothetical correlation, if the first four arguments are given. It gives the hypothetical correlation as well as the hypothesis test if the empirical correlation (r.emp) is added. Table 1 summarizes the calculation for the subject’s behavior. Section 3.4 and 3.5 already presented most of the equations used in Table 1. Be reminded that correlations are the covariance between components divided by the product of their standard deviations. The two most interesting data in Table 1 are the $r_{A / B, C / D}$ and the z-value which were both, respectively, .85 and .22. The expected correlation between response rates ratios and reinforcer rates ratios was high, but it could stem from the already large correlations between components. As a reference point, consider reinforcement probabilities both of .40 (instead of .78 and .67) for option 1 and 2 (see Appendix 2 for several examples of combination). With these values, $r_{A / B, C / D}$ would be .63 rather than .85 which illustrates the substantial influence of component correlations, the sensitivity of $r_{A / B, C / D}$ to different responses and reinforcers rates, and that $r_{A / B, C / D}$ accounts for high correlation between components by being more restrictive. The z-value obtained for the results was of −.69 which is below the traditional unidirectional test of 1.64 (for α = .05). This indicates that the data provide weak evidence for a behavioral process outside of the proposed model. Applied researchers should reconsider the legitimacy of the matching relation. Other procedure should be implemented to further analyze subject’s behavior, such as experimental control of the probabilities of reinforcement. Interestingly, the herein hypothetical example is quite similar to subject’s behavior in Caron et al. (in press), and other related works, where the GML seems to describe adequately behavior. As St Peter et al. (2005) suggested, this may be only because of the high probabilities of reinforcement, a problem addressed by the proposed null model. Certainly, previous works on the GML in natural settings should be re-evaluated in regards of the current study.

Table 1.

Summary of the hypothetical example.

	Notation	Components
		A	B	C	D
Mean	$E (x)$	11.03	8.70	8.75	5.85
Probability	p_j or θ	0.56	0.44	0.44	0.29
Variance	σ_x	4.92	4.92	4.92	4.14
Relative variance	V_x	0.20	0.25	0.25	0.35
Correlation	Corr(x,y)	−1.00			B
		0.73	−0.73		C
		−0.88	0.88	−0.64	D
NHST	$r_{A / B, C / D}$	0.90
	z-value	−0.69

NHST: null hypothesis statistical test.

10. Conclusion

A null hypothesis for statistical testing of the GML, and especially of sensitivity to reinforcement, has been proposed. Although the test is not necessary for experimental studies in which reinforcers ratio is controlled (though it can be interesting), it is convenient and informative for applied researchers who want to investigate the GML in uncontrolled environments.

Compared to Reed’s criterion, the current hypothesis testing is statistically grounded and flexible rather than being a mere rule of thumb. $r_{A / B, C / D}$ can be computed a priori and has a theoretical interpretation in terms of correlations between components. It is what is expected if only responses and reinforcers were generated by their probability distribution without any behavioral process. Moreover, it accounts for different response and reinforcer rates as well as the increase in correlation due to high reinforcer rates. It answers a concern raised by applied researchers who were worried that high occurrence of reinforcer (reinforcer rates highly correlated to response rates) could spuriously inflate the explained variance of the GML. $r_{A / B, C / D}$ automatically accounts for high or low correlations between components.

The model for implementing the null hypothesis entails some limits as it is based on a hypothetical model of reinforcement schedules. Notwithstanding this limitation, the current study requires molecular studies on feedback functions inherent to reinforcement schedules and on the moment-by-moment generating processes of responses and reinforcers. By investigating thoroughly the processes by which they occur, we will deepen our understanding in the mechanisms generating the matching law, but more importantly the behavior of organism.

Footnotes

Appendix

Appendix 2

The following table contains every $r_{A / B, C / D}$ for reinforcement probabilities p_j from .10 to .90. Only θ of .10–.50 are given, since .60–.90 is an approximate (at the second decimal) reflection of these values.

θ	p ₁	p ₂
		.10	.20	.30	.40	.50	.60	.70	.80	.90
.10	.10	.31	.33	.34	.35	.35	.35	.36	.36	.36
	.20	.41	.44	.46	.47	.47	.48	.48	.48	.48
	.30	.48	.53	.55	.56	.56	.57	.57	.58	.58
	.40	.54	.59	.62	.63	.64	.65	.65	.66	.66
	.50	.59	.65	.68	.70	.71	.71	.72	.73	.73
	.60	.62	.70	.73	.75	.76	.77	.78	.79	.79
	.70	.66	.74	.78	.80	.82	.83	.84	.84	.85
	.80	.68	.77	.82	.85	.86	.88	.89	.89	.90
	.90	.71	.81	.86	.89	.91	.92	.93	.94	.95

.20	.10	.31	.34	.36	.36	.37	.37	.37	.38	.38
	.20	.40	.45	.47	.48	.49	.49	.50	.50	.50
	.30	.46	.52	.55	.56	.58	.58	.59	.60	.60
	.40	.50	.57	.61	.63	.65	.66	.67	.67	.68
	.50	.53	.62	.66	.69	.71	.72	.73	.74	.75
	.60	.56	.65	.70	.74	.76	.77	.79	.80	.80
	.70	.58	.68	.74	.78	.80	.82	.84	.85	.86
	.80	.59	.71	.77	.81	.84	.86	.88	.89	.91
	.90	.61	.73	.80	.85	.88	.90	.92	.94	.95

.30	.10	.32	.35	.37	.38	.39	.39	.39	.40	.40
	.20	.39	.45	.48	.49	.50	.51	.52	.52	.53
	.30	.44	.51	.55	.57	.59	.60	.61	.62	.62
	.40	.47	.55	.60	.63	.65	.67	.68	.69	.70
	.50	.49	.59	.64	.68	.71	.73	.74	.75	.76
	.60	.51	.61	.68	.72	.75	.77	.79	.81	.82
	.70	.52	.64	.71	.75	.79	.82	.84	.85	.87
	.80	.53	.65	.73	.78	.82	.85	.88	.89	.91
	.90	.54	.67	.75	.81	.85	.88	.91	.93	.95

.40	.10	.32	.36	.38	.40	.41	.41	.42	.42	.42
	.20	.38	.45	.48	.51	.52	.53	.54	.55	.55
	.30	.42	.50	.55	.58	.60	.62	.63	.64	.65
	.40	.44	.54	.59	.63	.66	.68	.70	.71	.72
	.50	.46	.56	.63	.67	.71	.73	.75	.77	.78
	.60	.47	.58	.66	.71	.75	.77	.80	.82	.83
	.70	.48	.60	.68	.73	.78	.81	.84	.86	.88
	.80	.48	.61	.70	.76	.80	.84	.87	.89	.92
	.90	.49	.62	.71	.78	.83	.87	.90	.93	.95

.50	.10	.32	.37	.40	.42	.43	.44	.44	.45	.45
	.20	.37	.45	.49	.52	.54	.56	.57	.58	.59
	.30	.40	.49	.55	.59	.61	.64	.65	.67	.68
	.40	.42	.52	.59	.63	.67	.69	.72	.73	.75
	.50	.43	.54	.61	.67	.71	.74	.76	.79	.80
	.60	.44	.56	.64	.69	.74	.77	.80	.83	.85
	.70	.44	.57	.65	.72	.76	.80	.84	.86	.89
	.80	.45	.58	.67	.73	.79	.83	.86	.89	.92
	.90	.45	.59	.68	.75	.80	.85	.89	.92	.95

Acknowledgements

I thank Anne-Josée Piazza for her comments on an earlier draft.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author’s works are currently subsidized by the Fonds de recherche du Québec—Société et culture.

Notes

About the Author

Pier-Olivier Caron is completing his doctorate in psychology at the University of Quebec at Montreal. He is now doing a postdoctoral fellowship at the Canada Research Chair in Interpersonal Traumas and Resilience. His main interests are in quantitative models from the experimental analysis of behavior, decision-making from an individual and social perspective, multivariate data analyses, statistical modeling and computation.

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θ	p ₁	p ₂
		.10	.20	.30	.40	.50	.60	.70	.80	.90
.10	.10	.31	.33	.34	.35	.35	.35	.36	.36	.36
	.20	.41	.44	.46	.47	.47	.48	.48	.48	.48
	.30	.48	.53	.55	.56	.56	.57	.57	.58	.58
	.40	.54	.59	.62	.63	.64	.65	.65	.66	.66
	.50	.59	.65	.68	.70	.71	.71	.72	.73	.73
	.60	.62	.70	.73	.75	.76	.77	.78	.79	.79
	.70	.66	.74	.78	.80	.82	.83	.84	.84	.85
	.80	.68	.77	.82	.85	.86	.88	.89	.89	.90
	.90	.71	.81	.86	.89	.91	.92	.93	.94	.95

.20	.10	.31	.34	.36	.36	.37	.37	.37	.38	.38
	.20	.40	.45	.47	.48	.49	.49	.50	.50	.50
	.30	.46	.52	.55	.56	.58	.58	.59	.60	.60
	.40	.50	.57	.61	.63	.65	.66	.67	.67	.68
	.50	.53	.62	.66	.69	.71	.72	.73	.74	.75
	.60	.56	.65	.70	.74	.76	.77	.79	.80	.80
	.70	.58	.68	.74	.78	.80	.82	.84	.85	.86
	.80	.59	.71	.77	.81	.84	.86	.88	.89	.91
	.90	.61	.73	.80	.85	.88	.90	.92	.94	.95

.30	.10	.32	.35	.37	.38	.39	.39	.39	.40	.40
	.20	.39	.45	.48	.49	.50	.51	.52	.52	.53
	.30	.44	.51	.55	.57	.59	.60	.61	.62	.62
	.40	.47	.55	.60	.63	.65	.67	.68	.69	.70
	.50	.49	.59	.64	.68	.71	.73	.74	.75	.76
	.60	.51	.61	.68	.72	.75	.77	.79	.81	.82
	.70	.52	.64	.71	.75	.79	.82	.84	.85	.87
	.80	.53	.65	.73	.78	.82	.85	.88	.89	.91
	.90	.54	.67	.75	.81	.85	.88	.91	.93	.95

.40	.10	.32	.36	.38	.40	.41	.41	.42	.42	.42
	.20	.38	.45	.48	.51	.52	.53	.54	.55	.55
	.30	.42	.50	.55	.58	.60	.62	.63	.64	.65
	.40	.44	.54	.59	.63	.66	.68	.70	.71	.72
	.50	.46	.56	.63	.67	.71	.73	.75	.77	.78
	.60	.47	.58	.66	.71	.75	.77	.80	.82	.83
	.70	.48	.60	.68	.73	.78	.81	.84	.86	.88
	.80	.48	.61	.70	.76	.80	.84	.87	.89	.92
	.90	.49	.62	.71	.78	.83	.87	.90	.93	.95

.50	.10	.32	.37	.40	.42	.43	.44	.44	.45	.45
	.20	.37	.45	.49	.52	.54	.56	.57	.58	.59
	.30	.40	.49	.55	.59	.61	.64	.65	.67	.68
	.40	.42	.52	.59	.63	.67	.69	.72	.73	.75
	.50	.43	.54	.61	.67	.71	.74	.76	.79	.80
	.60	.44	.56	.64	.69	.74	.77	.80	.83	.85
	.70	.44	.57	.65	.72	.76	.80	.84	.86	.89
	.80	.45	.58	.67	.73	.79	.83	.86	.89	.92
	.90	.45	.59	.68	.75	.80	.85	.89	.92	.95

θ	p ₁	p ₂
		.10	.20	.30	.40	.50	.60	.70	.80	.90
.10	.10	.31	.33	.34	.35	.35	.35	.36	.36	.36
	.20	.41	.44	.46	.47	.47	.48	.48	.48	.48
	.30	.48	.53	.55	.56	.56	.57	.57	.58	.58
	.40	.54	.59	.62	.63	.64	.65	.65	.66	.66
	.50	.59	.65	.68	.70	.71	.71	.72	.73	.73
	.60	.62	.70	.73	.75	.76	.77	.78	.79	.79
	.70	.66	.74	.78	.80	.82	.83	.84	.84	.85
	.80	.68	.77	.82	.85	.86	.88	.89	.89	.90
	.90	.71	.81	.86	.89	.91	.92	.93	.94	.95

.20	.10	.31	.34	.36	.36	.37	.37	.37	.38	.38
	.20	.40	.45	.47	.48	.49	.49	.50	.50	.50
	.30	.46	.52	.55	.56	.58	.58	.59	.60	.60
	.40	.50	.57	.61	.63	.65	.66	.67	.67	.68
	.50	.53	.62	.66	.69	.71	.72	.73	.74	.75
	.60	.56	.65	.70	.74	.76	.77	.79	.80	.80
	.70	.58	.68	.74	.78	.80	.82	.84	.85	.86
	.80	.59	.71	.77	.81	.84	.86	.88	.89	.91
	.90	.61	.73	.80	.85	.88	.90	.92	.94	.95

.30	.10	.32	.35	.37	.38	.39	.39	.39	.40	.40
	.20	.39	.45	.48	.49	.50	.51	.52	.52	.53
	.30	.44	.51	.55	.57	.59	.60	.61	.62	.62
	.40	.47	.55	.60	.63	.65	.67	.68	.69	.70
	.50	.49	.59	.64	.68	.71	.73	.74	.75	.76
	.60	.51	.61	.68	.72	.75	.77	.79	.81	.82
	.70	.52	.64	.71	.75	.79	.82	.84	.85	.87
	.80	.53	.65	.73	.78	.82	.85	.88	.89	.91
	.90	.54	.67	.75	.81	.85	.88	.91	.93	.95

.40	.10	.32	.36	.38	.40	.41	.41	.42	.42	.42
	.20	.38	.45	.48	.51	.52	.53	.54	.55	.55
	.30	.42	.50	.55	.58	.60	.62	.63	.64	.65
	.40	.44	.54	.59	.63	.66	.68	.70	.71	.72
	.50	.46	.56	.63	.67	.71	.73	.75	.77	.78
	.60	.47	.58	.66	.71	.75	.77	.80	.82	.83
	.70	.48	.60	.68	.73	.78	.81	.84	.86	.88
	.80	.48	.61	.70	.76	.80	.84	.87	.89	.92
	.90	.49	.62	.71	.78	.83	.87	.90	.93	.95

.50	.10	.32	.37	.40	.42	.43	.44	.44	.45	.45
	.20	.37	.45	.49	.52	.54	.56	.57	.58	.59
	.30	.40	.49	.55	.59	.61	.64	.65	.67	.68
	.40	.42	.52	.59	.63	.67	.69	.72	.73	.75
	.50	.43	.54	.61	.67	.71	.74	.76	.79	.80
	.60	.44	.56	.64	.69	.74	.77	.80	.83	.85
	.70	.44	.57	.65	.72	.76	.80	.84	.86	.89
	.80	.45	.58	.67	.73	.79	.83	.86	.89	.92
	.90	.45	.59	.68	.75	.80	.85	.89	.92	.95

θ	p ₁	p ₂
		.10	.20	.30	.40	.50	.60	.70	.80	.90
.10	.10	.31	.33	.34	.35	.35	.35	.36	.36	.36
	.20	.41	.44	.46	.47	.47	.48	.48	.48	.48
	.30	.48	.53	.55	.56	.56	.57	.57	.58	.58
	.40	.54	.59	.62	.63	.64	.65	.65	.66	.66
	.50	.59	.65	.68	.70	.71	.71	.72	.73	.73
	.60	.62	.70	.73	.75	.76	.77	.78	.79	.79
	.70	.66	.74	.78	.80	.82	.83	.84	.84	.85
	.80	.68	.77	.82	.85	.86	.88	.89	.89	.90
	.90	.71	.81	.86	.89	.91	.92	.93	.94	.95

.20	.10	.31	.34	.36	.36	.37	.37	.37	.38	.38
	.20	.40	.45	.47	.48	.49	.49	.50	.50	.50
	.30	.46	.52	.55	.56	.58	.58	.59	.60	.60
	.40	.50	.57	.61	.63	.65	.66	.67	.67	.68
	.50	.53	.62	.66	.69	.71	.72	.73	.74	.75
	.60	.56	.65	.70	.74	.76	.77	.79	.80	.80
	.70	.58	.68	.74	.78	.80	.82	.84	.85	.86
	.80	.59	.71	.77	.81	.84	.86	.88	.89	.91
	.90	.61	.73	.80	.85	.88	.90	.92	.94	.95

.30	.10	.32	.35	.37	.38	.39	.39	.39	.40	.40
	.20	.39	.45	.48	.49	.50	.51	.52	.52	.53
	.30	.44	.51	.55	.57	.59	.60	.61	.62	.62
	.40	.47	.55	.60	.63	.65	.67	.68	.69	.70
	.50	.49	.59	.64	.68	.71	.73	.74	.75	.76
	.60	.51	.61	.68	.72	.75	.77	.79	.81	.82
	.70	.52	.64	.71	.75	.79	.82	.84	.85	.87
	.80	.53	.65	.73	.78	.82	.85	.88	.89	.91
	.90	.54	.67	.75	.81	.85	.88	.91	.93	.95

.40	.10	.32	.36	.38	.40	.41	.41	.42	.42	.42
	.20	.38	.45	.48	.51	.52	.53	.54	.55	.55
	.30	.42	.50	.55	.58	.60	.62	.63	.64	.65
	.40	.44	.54	.59	.63	.66	.68	.70	.71	.72
	.50	.46	.56	.63	.67	.71	.73	.75	.77	.78
	.60	.47	.58	.66	.71	.75	.77	.80	.82	.83
	.70	.48	.60	.68	.73	.78	.81	.84	.86	.88
	.80	.48	.61	.70	.76	.80	.84	.87	.89	.92
	.90	.49	.62	.71	.78	.83	.87	.90	.93	.95

.50	.10	.32	.37	.40	.42	.43	.44	.44	.45	.45
	.20	.37	.45	.49	.52	.54	.56	.57	.58	.59
	.30	.40	.49	.55	.59	.61	.64	.65	.67	.68
	.40	.42	.52	.59	.63	.67	.69	.72	.73	.75
	.50	.43	.54	.61	.67	.71	.74	.76	.79	.80
	.60	.44	.56	.64	.69	.74	.77	.80	.83	.85
	.70	.44	.57	.65	.72	.76	.80	.84	.86	.89
	.80	.45	.58	.67	.73	.79	.83	.86	.89	.92
	.90	.45	.59	.68	.75	.80	.85	.89	.92	.95