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PubMed1993

[Probability in medicine].

The term "probability", apparently vague, is defined as the ratio of favourable and all cases, e. g. 1/4. Sometimes is expressed as "odds", in the above example 1:3. Mathematical (a priori) probability is calculated according to the following consideration: during accidental tossing of an intact coin the probability of head is 1/2. Statistical (a posteriori) probability is assessed experimentally. If randomization is not respected, the result is doubtful. We call it bias. Probabilities are multiplied (added) if one and (or) another phenomenon should occur. Phenomena which occur or do not occur are frequently governed by binomial division. According to the law of large numbers the a posteriori probability will become the closer to the a priori probability, the larger the number of experiments. Forecasts based on probability are made possible by interferential statistics. In this respect they differ from mere statistical description. Thus optimal decision is achieved under conditions of uncertainty, typical for medicine. This applies to diagnosis as well as treatment. In this way a cut-off value can be selected for a given diagnostic test to achieve an optimal combination of the resulting sensitivity and specificity and the probability of disease (health) in case of a positive (negative) finding can be expressed. The results of statistical testing (e. g. the effectiveness of treatment) is as a rule presented by means of the probability value p, as compared with error alpha which should be assessed in advance. It is, however, necessary to respect also an alternative hypothesis. The latter is unfortunately formulated only rarely.

Probability↗
PubMed1998

Choice maintained by cocaine or food in monkeys: effects of varying probability of reinforcement.

Several theoretical models of choice, including matching, maximizing and probability matching, have evolved from the experimental analysis of behavior. However, these models are based largely on results of studies involving non-drug reinforcers. The extent to which the choice to self-administer a drug of abuse conforms to these models is largely unknown. The present study was designed to examine the effects of varying probability of reinforcement of two options on choice by rhesus monkeys maintained under a discrete-trials choice procedure. One group of monkeys (n=5) was prepared with chronic indwelling venous catheters and responding was maintained by i.v. injections of cocaine (0.05 or 0.2 mg/kg per injection). In a second group, behavioral conditions were identical, except that responding was maintained by the delivery of food pellets. Two choice options were available to each group. Each option was maintained by the identical dose of cocaine or amount of food, but with varying probabilities of delivery. The percentage of trials on which monkeys chose option 1 increased with its probability of reinforcement for cocaine (0.05 or 0.2 mg/kg per injection) and this effect did not change with dose of cocaine. When probabilities of reinforcement for option 1/option 2 were 0.75/0.25, choice was exclusive for the higher probability option. Results were identical when behavior was maintained by food. The present experiment supports the conclusion that exclusive choice of the alternative with the higher probability of reinforcement can be extended to cocaine as a reinforcer. The present results also extend the known conditions under which drug-maintained responding is similar to food-maintained responding to situations involving options with different probabilities of reinforcement, i.e., probability choice. Probability matching was not observed, arguing that this model does not predict drug or food choice.

Animals↗
PubMed2000

Subsequent investigation and management of patients with intermediate-category and -probability ventilation-perfusion scintigraphy.

The authors wished to determine the proportion of patients with intermediate-category and intermediate-probability ventilation-perfusion scintigraphy (IVQS) who proceed to further imaging for investigation of thromboembolism, to identify the defining clinical parameters and to determine the proportion of patients who have a definite imaging diagnosis of thromboembolism prior to discharge from hospital on anticoagulation therapy. One hundred and twelve VQS studies performed at the Flinders Medical Centre over a 9-month period were reported as having intermediate category and probability for pulmonary embolism. Medical case notes were available for review in 99 of these patients and from these the pretest clinical probability, subsequent patient progress and treatment were recorded. Eight cases were excluded because they were already receiving anticoagulation therapy. In the remaining 91 patients the pretest clinical probability was considered to be low in 25; intermediate in 30; and high in 36 cases. In total, 51.6% (n = 47) of these patients (8% (n = 2) with low, 66% (n = 20) with intermediate, and 69.4% (n = 25) with high pretest probability) proceeded to CT pulmonary angiography (CTPA) and/or lower limb duplex Doppler ultrasound (DUS) evaluation. Of the patients with IVQS results, 30.7% (n = 28) were evaluated with CTPA. No patient with a low, all patients with a high and 46% of patients with an intermediate pretest probability initially received anticoagulation therapy. This was discontinued in three patients with high and in 12 patients with intermediate clinical probability prior to discharge from hospital. Overall, 40% of patients discharged on anticoagulation therapy (including 39% of those with a high pretest probability) had a positive imaging diagnosis of thromboembolism The results suggest that, although the majority of patients with intermediate-to-high pretest probability and IVQS proceed to further imaging investigation, CTPA is relatively underused in this group. Most patients with a high pretest clinical probability receive anticoagulation therapy irrespective of imaging findings, and less than half of all patients discharged from hospital on anticoagulation therapy have a positive imaging diagnosis of thromboembolism.

Angiography↗
PubMed1984

Bayesian probability analysis: a prospective demonstration of its clinical utility in diagnosing coronary disease.

One hundred fifty-four patients referred for coronary arteriography were prospectively studied with stress electrocardiography, stress thallium scintigraphy, cine fluoroscopy (for coronary calcifications), and coronary angiography. Pretest probabilities of coronary disease were determined based on age, sex, and type of chest pain. These and pooled literature values for the conditional probabilities of test results based on disease state were used in Bayes' theorem to calculate posttest probabilities of disease. The results of the three noninvasive tests were compared for statistical independence, a necessary condition for their simultaneous use in Bayes' theorem. The test results were found to demonstrate pairwise independence in patients with and those without disease. Some dependencies that were observed between the test results and the clinical variables of age and sex were not sufficient to invalidate application of the theorem. Sixty-eight of the study patients had at least one major coronary artery obstruction of greater than 50%. When these patients were divided into low-, intermediate-, and high-probability subgroups according to their pretest probabilities, noninvasive test results analyzed by Bayesian probability analysis appropriately advanced 17 of them by at least one probability subgroup while only seven were moved backward. Of the 76 patients without disease, 34 were appropriately moved into a lower probability subgroup while 10 were incorrectly moved up. We conclude that posttest probabilities calculated from Bayes' theorem more accurately classified patients with and without disease than did pretest probabilities, thus demonstrating the utility of the theorem in this application.

Angiography↗
PubMed2005

A simple method for determining the probability a new vertebral fracture is present in postmenopausal women with osteoporosis.

The vertebral fracture status of women with osteoporosis has strong prognostic implications that may influence clinical decisions. We developed a simple method for estimating the probability that a new vertebral fracture has occurred in postmenopausal women with osteoporosis. Data was from the placebo groups of the Fracture Prevention Trial (median observation =21 months) and the MORE Trial at 2 years. A logistic regression analysis identified prior vertebral fracture (yes/no), new or worsening back pain (yes/no), and height loss (> or =2 cm, yes/no) as significant predictors for the presence of a new vertebral fracture. The actual probability of a new vertebral fracture in patients without these predictors, over the median observation period of 23 months, was 2.1%. Presence of back pain increased this probability fourfold; prior vertebral fracture increased this probability threefold, and height loss > or =2 cm increased this probability threefold. The predicted probabilities of a new vertebral fracture being present for each subgroup representing each of the eight possible combinations of back pain, prior vertebral fracture, and height loss were highly correlated with both the multivariate logistic regression-derived probabilities (r=0.98, p <0.001) and with the actual probabilities (r=0.99, p <0.001). The validity of this simple method was confirmed in patients from the MORE trial at both 2 years and 3 years, and in the Fracture Prevention Trial alone. This simple method provides clinicians with an estimate of the probability that a new vertebral fracture has occurred in postmenopausal women with osteoporosis.

Adult↗
PubMed2003

Minimal entropy probability paths between genome families.

We develop a metric for probability distributions with applications to biological sequence analysis. Our distance metric is obtained by minimizing a functional defined on the class of paths over probability measures on N categories. The underlying mathematical theory is connected to a constrained problem in the calculus of variations. The solution presented is a numerical solution, which approximates the true solution in a set of cases called rich paths where none of the components of the path is zero. The functional to be minimized is motivated by entropy considerations, reflecting the idea that nature might efficiently carry out mutations of genome sequences in such a way that the increase in entropy involved in transformation is as small as possible. We characterize sequences by frequency profiles or probability vectors, in the case of DNA where N is 4 and the components of the probability vector are the frequency of occurrence of each of the bases A, C, G and T. Given two probability vectors a and b, we define a distance function based as the infimum of path integrals of the entropy function H( p) over all admissible paths p(t), 0 < or = t< or =1, with p(t) a probability vector such that p(0)=a and p(1)=b. If the probability paths p(t) are parameterized as y(s) in terms of arc length s and the optimal path is smooth with arc length L, then smooth and "rich" optimal probability paths may be numerically estimated by a hybrid method of iterating Newton's method on solutions of a two point boundary value problem, with unknown distance L between the abscissas, for the Euler-Lagrange equations resulting from a multiplier rule for the constrained optimization problem together with linear regression to improve the arc length estimate L. Matlab code for these numerical methods is provided which works only for "rich" optimal probability vectors. These methods motivate a definition of an elementary distance function which is easier and faster to calculate, works on non-rich vectors, does not involve variational theory and does not involve differential equations, but is a better approximation of the minimal entropy path distance than the distance //b-a//(2). We compute minimal entropy distance matrices for examples of DNA myostatin genes and amino-acid sequences across several species. Output tree dendograms for our minimal entropy metric are compared with dendograms based on BLAST and BLAST identity scores.

Algorithms↗
PubMed2005

The effect of numerical distance and stimulus probability on ERP components elicited by numerical incongruencies in mental addition.

In two experiments, we investigated the effect of the numerical distance of incongruous results from correct results and stimulus probability on the N270/N400 event-related brain potential components. In Experiment 1, 12 subjects saw two one-digit addends and a possible solution and signaled if the proposed result (falling in the range of 3-17) was true or false. Incorrect results could deviate by +/-2 or by +/-9 from the correct answer. The probability of correct results was 50%. Twelve subjects carried out a similar task in Experiment 2 without giving behavioral responses. The probability of incorrect results was 20%, 50% or 80% in different conditions. Both raw potentials and incorrect minus correct difference potentials were analyzed. A fronto-central N3 and a centro-parietal dN3 (incorrect-correct difference) were present for incongruous results in both experiments. The amplitude of the dN3 was not sensitive to numerical distance, but the latency of the dN3 was longer when numerical distance was larger. The overall amplitude of the N3 and of the dN3 was not sensitive to the probability manipulation. However, there was a parietally localized effect of probability on N3 amplitude. The dN3 in mental addition is most probably identical to the arithmetic N400 effect reported earlier in mental multiplication. The distance effect in latency may be a correlate of the discrimination of correct vs. incorrect results. A parietally localized probability effect (right greater than left) was found in the N3 amplitude. The dN3 was insensitive to the probability manipulation. In accord with its insensitivity to stimulus probability, the dN3 seems to be more related to the N400 than to the N2b. Posterior attentional processes sensitive to the allocation of attentional resources may have contributed to the topography of the dN3. The N3 is more related to the detection of expectation violation, while the P3 reflects the ease of identifying stimulus categories.

Adolescent↗
PubMed1991

Probability of appendicitis before and after observation.

STUDY OBJECTIVE: To examine patients with abdominal pain for changes in probability of appendicitis during observation. STUDY DESIGN: Retrospective cohort study. SETTING: University-affiliated community hospital. METHODS: 252 patients with abdominal pain who were examined underwent short-term (10.4 hours) observation (95% confidence interval [CI], 8.7, 12.1) before the decision to operate during a one-year period. Alvarado's scoring system and a probability-of-diagnosis nomogram were used to assign scores and estimate probability of appendicitis. MEASUREMENTS AND RESULTS: In the study group, mean score of patients with appendicitis increased after observation from 6.8 (95% CI, 6.2, 7.4) to 7.8 (95% CI, 7.3, 8.3), corresponding to a change in probability of appendicitis from 50% to 65%. Mean score of patients without appendicitis decreased from 3.8 (95% CI, 3.5, 4.1) to 1.6 (95% CI, 1.58, 1.62), corresponding to a change in probability from 35% to 22%. The difference between mean scores for patients with and without appendicitis increased from 2.6 (95% CI, 2.0, 3.2) to 6.2 (95% CI, 6.15, 6.25) during observation. The study group initially had intermediate probability of appendicitis (score, 4.35; 95% CI, 4.04, 4.66) compared with high probability for patients who went directly to surgery after their initial evaluation (63 patients; score, 7.59; 95% CI, 7.05, 8.73) and low probability for patients with abdominal pain who were sent home after their initial evaluation without observation or surgery (2,097 patients; score, 1.87; 95% CI, 1.48, 2.26). CONCLUSION: In this group of patients with intermediate initial probability of appendicitis, observation improved the ability to distinguish patients with from those without appendicitis.

Abdominal Pain↗
PubMed2003

Does assessment of pretest probability of disease improve the utility of echocardiography in suspected endocarditis in children?

OBJECTIVE: To compare the yield rate (YR) of echocardiography when evaluating children with suspected infectious endocarditis (IE) in both the actual clinical setting and in the hypothetic setting where strict clinical criteria are applied. Study design Medical records of 101 children undergoing echocardiography for suspected IE were reviewed. Echocardiograms with positive findings were identified and the actual diagnostic YR was calculated. With the use of clinical criteria proposed by von Reyn (VR), the probability of IE was retrospectively classified as (1) rejected, (2) possible, or (3) probable. Theoretic YR of echocardiography was calculated for each classification. RESULTS: The actual YR of echocardiography was 12% (12/101). The YR of echocardiography by VR class was 0% in rejected, 20% in possible, and 80% in probable cases (chi(2) = 55.1, P <.0001). Echocardiography did not change the probability of IE in any patient classified as rejected, but allowed reassignment of disease probability in a significant proportion of patients with possible or probable IE. CONCLUSIONS: The YR of echocardiography was significant when clinical probability of IE was intermediate-to-high, and low, with marginal clinical utility, when clinical probability was low. Strict pretest assessment of disease probability may lead to more effective utilization of echocardiography in this population.

Bacteria↗
PubMed2005

Extinction probabilities and times to extinction for populations of tsetse flies Glossina spp. (Diptera: Glossinidae) subjected to various control measures.

A stochastic branching process was used to derive equations for the mean and variance of the probability of, and time to, extinction in tsetse populations. If the remnant population is a single inseminated female, the extinction probability increases linearly with adult mortality and is always certain if this mortality >3.5% per day even for zero pupal mortality. If the latter mortality is 4% per day, certain extinction is only avoided if adult mortality <1.5% per day. For remnant female populations >1, the extinction probability increases in a non-linear manner with adult mortality. Extinction is still certain for adult mortality >3.5% per day but, when the remnant population is >16, extinction is highly unlikely for adult mortality <2.5% per day if all females are inseminated. Extinction probability increases with increasing probability of sterile mating in much the same way as it does with increasing adult mortality. Extinction is assured if the probability of insemination can be reduced to 0.1. The required reduction decreases with increasing adult mortality. For adult mortality = 6-8% per day, the time to extinction increases only by one generation per order of magnitude increase in the starting population. Time to extinction is less sensitive to changes in the pupal than in the adult mortality. Reductions in the probability of insemination only become important when adult mortality is small; if the adult mortality is 8% per day, reducing the insemination probability from 1 to 0.1 only reduces the expected time to extinction by two generations. Conversely, increases in adult mortality produce important reductions in the required time even when the probability of insemination is 0.1. The practical, economic implication for the sterile insect technique is that the low-tech methods used to suppress tsetse populations should not be halted when the release of sterile males is initiated. The sterile insect technique should only be contemplated when it has been demonstrated that the low-tech methods have failed to effect eradication. The theory is shown to be in good accord with the observed results of tsetse control campaigns involving the use of odour-baited targets in Zimbabwe and the sterile insect technique on Unguja Island, Zanzibar.

Animals↗
PubMed2003

Comparing bootstrap and posterior probability values in the four-taxon case.

Assessment of the reliability of a given phylogenetic hypothesis is an important step in phylogenetic analysis. Historically, the nonparametric bootstrap procedure has been the most frequently used method for assessing the support for specific phylogenetic relationships. The recent employment of Bayesian methods for phylogenetic inference problems has resulted in clade support being expressed in terms of posterior probabilities. We used simulated data and the four-taxon case to explore the relationship between nonparametric bootstrap values (as inferred by maximum likelihood) and posterior probabilities (as inferred by Bayesian analysis). The results suggest a complex association between the two measures. Three general regions of tree space can be identified: (1) the neutral zone, where differences between mean bootstrap and mean posterior probability values are not significant, (2) near the two-branch corner, and (3) deep in the two-branch corner. In the last two regions, significant differences occur between mean bootstrap and mean posterior probability values. Whether bootstrap or posterior probability values are higher depends on the data in support of alternative topologies. Examination of star topologies revealed that both bootstrap and posterior probability values differ significantly from theoretical expectations; in particular, there are more posterior probability values in the range 0.85-1 than expected by theory. Therefore, our results corroborate the findings of others that posterior probability values are excessively high. Our results also suggest that extrapolations from single topology branch-length studies are unlikely to provide any general conclusions regarding the relationship between bootstrap and posterior probability values.

Computer Simulation↗
PubMed2001

Distribution of the probability of survival is a strategic issue for randomized trials in critically ill patients.

BACKGROUND: Many randomized clinical trials in trauma have failed to demonstrate a significant improvement in survival rate. Using a trauma patient database, we simulated what could happen in a trial designed to improve survival rate in this setting. METHODS: The predicted probability of survival was assessed using the TRISS methodology in 350 severely injured trauma patients. Using this probability of survival, the authors simulated the effects of a drug that may increase the probability of survival by 10-50% and calculated the number of patients to be included in a triad, assuming alpha = 0.05 and beta = 0.10 by using the percentage of survivors or the individual probability of survival. Other distributions (Gaussian, J shape, uniform) of the probability of survival were also simulated and tested. RESULTS: The distribution of the probability of survival was bimodal with two peaks (< 0.10 and > 0.90). There were major discrepancies between the number of patients to be included when considering the percentage of survivors or the individual value of the probability of survival: 63,202 versus 2,848 if the drug increases the probability of survival by 20%. This discrepancy also occurred in other types of distribution (uniform, J shape) but to a lesser degree, whereas it was very limited in a Gaussian distribution. CONCLUSIONS: The bimodal distribution of the probability of survival in trauma patients has major consequences on hypothesis testing, leading to overestimation of the power. This statistical pitfall may also occur in other critically ill patients.

Adolescent↗
PubMed1995

Ventilation/perfusion lung scan probability category distributions in university and community hospitals.

The criteria used in the Prospective Investigation of Pulmonary Embolism Diagnosis (PIOPED) study for the interpretation of ventilation/perfusion scans are widely used and the probability of pulmonary embolism is determined from these criteria. The prevalence of pulmonary embolism in the PIOPED study was 33%. To investigate the similarity of patient populations who have ventilation/perfusion scans at one of the medical centers that participated in the PIOPED study and a small community hospital, the authors evaluated the probability category distributions of lung scans at the two institutions. They retrospectively interpreted 54 and 49 ventilation/perfusion lung scans selected from January, 1991, to June, 1992, at Duke University Medical Center and at Central Carolina Hospital, respectively. Studies were interpreted according to the PIOPED criteria. The percentage of studies assigned to each category at Duke University Medical Center and Central Carolina Hospital were 17% and 27% normal or very low probability, 31% and 59% low probability, 39% and 10% intermediate probability, and 13% and 4% high probability, respectively. The different distribution of probability categories between university and community hospitals suggests that the prevalence of disease may also be different. The post-test probability of pulmonary embolism is related to the prevalence of disease and the sensitivity and specificity of the ventilation/perfusion scan. Because these variables may differ in community hospital settings, the post-test probability of pulmonary embolism as determined by data from the PIOPED study should only be used in institutions with similar populations. Clinical management based upon the results of the PIOPED study may not be applicable to patients who have ventilation/perfusion scans performed in a community hospital.

Hospitals, Community↗
PubMed1994

Certainty and uncertainty in science: the subjectivistic concept of probability in physiology and medicine.

Most physiological scientists have restricted understanding of probability as relative frequency in a large collection (for example, of atoms). Most appropriate for the relatively circumscribed problems of the physical sciences, this understanding of probability as a physical property has conveyed the widespread impression that the "proper" statistical "method" can eliminate uncertainty by determining the "correct" frequency or frequency distribution. However, many relatively recent developments in the theory of probability and decision making deny such exalted statistical ability. Proponents of Bayes's subjectivist theory, for example, assert that probability is "degree of belief," a more tentative idea than relative frequency or physical probability, even though degree of belief assessment may utilize frequency information. In the subjectivist view, probability and statistics are means of expressing a consistent opinion (a probability) to handle uncertainty but never means to eliminate it. In the physiological sciences the contrast between the two views is critical, because problems dealt with are generally more complex than those of physics, requiring judgments and decisions. We illustrate this in testing the efficacy of penicillin by showing how the physical probability method of "hypothesis testing" may contribute to the erroneous idea that science consists of "verified truths" or "conclusive evidence" and how this impression is avoided in subjectivist probability analysis.

Bayes Theorem↗
PubMed1993

Sodium-lithium countertransport genotype and the probability of hypertension in adults.

The objective of the present study was to determine whether information about a biometrically inferred single gene with large effects on erythrocyte sodium-lithium countertransport is useful in predicting the probability of having hypertension. We used multivariate logistic regression to model the relationship between the probability of having hypertension and predictor traits in a sample of 382 unrelated adult women and 347 unrelated adult men from Rochester, Minn. First, we identified a set of demographic, biochemical, and physiological predictors. Second, we analyzed whether the relationship between the probability of having hypertension and the identified predictor traits was heterogeneous between the biometrically inferred single locus genotypes with large effects on sodium-lithium countertransport level. Third, if there was no heterogeneity, we assessed whether sodium-lithium countertransport genotypes made an additional contribution to predicting the probability of having hypertension after other predictors were considered. In women, the predictors of the probability of having hypertension were age, plasma apolipoprotein CIII, body mass index, and an interaction term involving age and body mass index. The relationship between the probability of having hypertension and the identified predictors was not heterogeneous between sodium-lithium countertransport genotypes, and genotype did not contribute to the prediction of the probability of having hypertension after the identified predictors were considered. In men, predictors of the probability of having hypertension were age, plasma levels of high-density lipoprotein cholesterol, apolipoproteins AI and CII, sodium-lithium countertransport level, and sodium-lithium countertransport genotype. The relationship between the probability of having hypertension and sodium-lithium countertransport level and age were heterogeneous between biometrically inferred sodium-lithium countertransport genotypes.(ABSTRACT TRUNCATED AT 250 WORDS)

Aging↗
PubMed1985

Probability and delay of reinforcement as factors in discrete-trial choice.

Pigeons chose between two alternatives that differed in the probability of reinforcement and the delay to reinforcement. A peck on the red key always produced a delay of 5 s and then a possible reinforcer. The probability of reinforcement for responding on this key varied from .05 to 1.0 in different conditions. A response on the green key produced a delay of adjustable duration and then a possible reinforcer, with the probability of reinforcement ranging from .25 to 1.0 in different conditions. The green-key delay was increased or decreased many times per session, depending on a subject's previous choices. The purpose of these adjustments was to estimate an indifference point, or a delay that resulted in a subject's choosing each alternative about equally often. In conditions where the probability of reinforcement was five times higher on the green key, the green-key delay averaged about 12 s at the indifference point. In conditions where the probability of reinforcement was twice as high on the green key, the green-key delay at the indifference point was about 8 s with high probabilities and about 6 s with low probabilities. An analysis based on these results and those from studies on delay of reinforcement suggests that pigeons' choices are relatively insensitive to variations in the probability of reinforcement between .2 and 1.0, but quite sensitive to variations in probability between .2 and 0.

Animals↗
PubMed1990

Choice behavior in transition: development of preference for the higher probability of reinforcement.

Ten acquisition curves were obtained from each of 4 pigeons in a two-choice discrete-trial procedure. In each of these 10 conditions, the two response keys initially had equal probabilities of reinforcement, and subjects' choice responses were about equally divided between the two keys. Then the reinforcement probabilities were changed so that one key had a higher probability of reinforcement (the left key in half of the conditions and the right key in the other half), and in nearly every case the subjects developed a preference for this key. The rate of acquisition of preference for this key was faster when the ratio of the two reinforcement probabilities was higher. For instance, acquisition of preference was faster in conditions with reinforcement probabilities of .12 and .02 than in conditions with reinforcement probabilities of .40 and .30, even though the pairs of probabilities differed by .10 in both cases. These results were used to evaluate the predictions of some theories of transitional behavior in choice situations. A trial-by-trial analysis of individual responses and reinforcers suggested that reinforcement had both short-term and long-term effects on choice. The short-term effect was an increased probability of returning to the same key on the one or two trials following a reinforcer. The long-term effect was a gradual increase in the proportion of responses on the key with the higher probability of reinforcement, an increase that usually continued for several hundred trials.

Animals↗
PubMed2004

Combining probability values from independent permutation tests: a discrete analog of Fisher's classical method.

Permutation tests are based on all possible arrangements of observed data sets. Consequently, such tests yield exact probability values obtained from discrete probability distributions. An exact nondirectional method to combine independent probability values that obey discrete probability distributions is introduced. The exact method is the discrete analog to Fisher's classical method for combining probability values from independent continuous probability distributions. If the combination of probability values includes even one probability value that obeys a sparse discrete probability distribution, then Fisher's classical method may be grossly inadequate.

Humans↗