The James-Stein estimator's dominance over maximum likelihood in mean square error has been called a paradox because maximum likelihood is known to be superior in many other respects. One response, due to Efron, is to question maximum likelihood. Another is to question MSE. We pursue the second and compare MSE with Λ-information (Vos and Wu, 2025) as criteria for assessing estimators. The comparison rests on two distinctions: between point estimators and generalized estimators – functions of the sample and parameter jointly, with the score as archetype – as inferential objects, and between pointwise and family-aware assessment criteria. An elementary lemma shows that no pointwise criterion, MSE or any other risk built from a loss function, admits a uniformly optimal estimator; Λ-information, which is family-aware and parameter-invariant, is uniformly maximized by the score. A point estimator is assessed through the generalized estimators it induces, and under the score map its Λ-efficiency is the fraction of Fisher information the statistic retains, placing the criterion in Fisher's information-loss tradition. On unbiased estimators, Λ-efficiency coincides with variance-based efficiency. Returning to James-Stein, the paradox dissolves: maximum likelihood is fully efficient because it is sufficient, while the James-Stein statistic is exactly two-to-one in the sample, and the information it destroys – computed exactly – is concentrated precisely where its MSE advantage is greatest. MSE retains its proper domain under genuine squared-error loss.

During fertilization, mammalian sperm undergo a winnowing selection process that reduces the candidate pool of potential fertilizers from ~106-1011 cells to 101-102 cells (depending on the species). Classical sperm competition theory addresses the positive or 'stabilizing' selection acting on sperm phenotypes within populations of organisms but does not strictly address the developmental consequences of sperm traits among individual organisms that are under purifying selection during fertilization. It is the latter that is of utmost concern for improving assisted reproductive technologies (ART) because 'low fitness' sperm may be inadvertently used for fertilization during interventions that rely heavily on artificial sperm selection, such as intracytoplasmic sperm injection (ICSI). Importantly, some form of sperm selection is used in nearly all forms of ART (e.g., differential centrifugation, swim-up, or hyaluronan binding assays, etc.). To date, there is no unifying quantitative framework (i.e., theory of sperm selection) that synthesizes causal mechanisms of selection with observed natural variation in individual sperm traits. In this report, we reframe the physiological function of sperm as a collective diffusive search process and develop multi-scale computational models to explore the causal dynamics that constrain sperm 'fitness' during fertilization. Several experimentally useful concepts are developed, including a probabilistic measure of sperm 'fitness' as well as an information theoretic measure of the magnitude of sperm selection, each of which are assessed under systematic increases in microenvironmental selective pressure acting on sperm motility patterns.

This paper develops profile score confidence intervals (i.e., z-standard intervals) by inverting orthogonalized score estimators for comparing two independent binomial proportions in rate difference, rate ratio, or odds ratio and utilizes a generalized estimation framework of Vos and Wu (Inf. Geom. 8: 99–123, 2025) to evaluate different confidence interval methods. The orthogonalized score estimators dominate other generalized estimators in Λ-efficiency for distinguishing parameter values around the truth, so the z-standard intervals become more efficient and acquire coverage closer to the nominal level than other types of confidence intervals. In addition, the degrees of freedom and small-sample corrections (applied to the profile nuisance parameter estimates) are expected to improve the coverage of the z-standard intervals and help maintain them above the nominal level. Computational algorithms are developed to find the z-standard intervals using R’s polyroot function. Numerical studies are conducted to compare both coverage and endpoints of different types of confidence intervals.

This paper extends the idea of a generalized estimator for a scalar parameter (Vos in Inform Geom 7:151–170, 2022) to multi-dimensional parameters both with and without nuisance parameters. The title reflects the fact that generalized estimators provide more than simply another method to find point estimators, and that the methods to assess generalized estimators differ from those for point estimators. By generalized estimation we mean the use of generalized estimators together with an extended definition of information to assess their inferential properties. We show that Fisher information provides an upper bound for the information utilized by an estimator and that the score attains this bound. This optimality result provides theoretical justification for likelihood-based inference, effectively narrowing the search for optimal estimators when the score is computationally feasible.

The permutation inference distribution (PID) introduced in Wu and Vos (2019), The Journal of Nonparametric Statistics 31(3), 722-742) is extended to the joint PID for multiple regression. The joint PID is used for both hypothesis testing and the construction of confidence regions and its computational burden is similar to that of conducting a single hypothesis test. Asymptotic normality results show that PID confidence regions are asymptotically ellipsoidal and exact. In finite samples, the PID confidence regions can be used to check the fidelity of normal approximations. In some cases, normal theory-based confidence regions may be adjusted to approach nominal confidence errors. Simulation studies and real data applications are used to evaluate inferences obtained from the PID.

Learning science is a social enterprise that involves students communicating ideas, observations, and findings. Navigating talk between students about scientific concepts and practices is a complex task for teachers. Traditionally, science educators have used a method called microteaching (teaching to peers) as a context for practicing teaching. In this study, science educators created practice sessions called rehearsals, designed for elementary teacher candidates (ETCs) to participate in deliberate teaching episodes using discourse skills followed by instructor feedback. This NSF-supported work explores the use of rehearsals within virtual simulations software called Mursion (R) (developed as TeachLivE (TM)) as compared to traditional rehearsals by using treatment and comparison groups. This study found that ETCs in both contexts increased in their use of various talk moves between rehearsal timepoints. Compared to the control group, the treatment group was able to address students' ideas and understanding more frequently in the rehearsal settings as well as the classroom setting (practicum). This finding indicates that ETCs felt comfortable utilizing student avatars when practicing specific teaching skills and were able to transfer some of these skills into the classroom setting. Creating valuable teaching experiences in teacher preparation is essential and this study explores the possibility of using a new innovative context as a space for ETCs to practice those skills.

The distinction between the two uses of p-values described by Professor Greenland is related to two distinct interpretations of frequentist probability—that is, probability used to describe a random event. I will illustrate with a simple example. In the North Carolina Pick-4 lottery, 10 ping pong balls labeled with distinct digits from I 9 = 0 , 1 , … , 9 $$ {I}_9=\left\{0,1,\dots, 9\right\} $$ are mixed in a clear container and opening a door allows a single ball to be selected. Prior to opening the door, blown air mixes the balls making equally likely selection of each ball plausible. This is repeated with three identical containers to obtain the remaining three digits. If a winning ticket is defined as one where the sum of the four digits exceeds 28, the state can charge $5 for a ticket with a $100 prize and expect a profit. There are 330 of 1 0 4 $$ 1{0}^4 $$ possible outcomes where the sum exceeds 28 so the expected value is 0 . 033 × $ 100 = $ 3 . 30 $$ 0.033\times \$100=\$3.30 $$ . This calculation requires no repeated sampling but it is natural for the state to interpret this value in the long run. For an individual ticket holder, all that is required is that each ball is given an equal chance to be selected for the drawing associated with his ticket. The ticket holder does not need to imagine a long sequence of draws just as a cancer patient does not need to consider a long sequence of 5-year periods to understand a 30% 5-year survival. Using terminology from Vos and Holbert (2022), the scope for the ticket holder is specific while that of the state is generic. The uniform distribution on 4-tuples I 9 4 = I 9 × I 9 × I 9 × I 9 $$ {I}_9^4={I}_9\times {I}_9\times {I}_9\times {I}_9 $$ provides a model for repeated draws of the Pick-4 lottery, that is, of the data generation process. For most inference applications, the distribution of an unknown population can be modeled rather than the process that generated the data. We modify this example to consider inference. We are told the sum of a single lottery draw and we are to infer whether the draw came from the NC lottery or lottery A that also has four containers but each contains 8 balls with labels from I 7 = 0 , 1 , … , 7 $$ {I}_7=\left\{0,1,\dots, 7\right\} $$ . The sum of the digits is 29 but no other information is given. A reduction-to-contradiction argument establishes that the result came from the NC lottery. Premise: lottery A produced our data; every possible sum from lottery A belongs to the set 0 , 1 , … , 28 $$ \left\{0,1,\dots, 28\right\} $$ ; 29 is not in this set; conclusion: the contradiction means it is impossible that the premise is true. The deductive argument used for a sum of 29 does not work if the sum is 28. Logical certainty is no longer possible but sums of 28 or less still provide evidence, to varying degrees, regarding which lottery was used. A reduction-to-incredibility argument modeled on the above deduction can be used. Premise: lottery A produced the sum of 28; of the 8 4 $$ {8}^4 $$ possible 4-tuples only one produces a sum as large as 28; each 4-tuple had an equal chance of being selected; the probability of a sum of 28 is 1 / 8 4 < 0 . 00025 $$ 1/{8}^4<0.00025 $$ ; conclusion: the unlikely observation makes it doubtful that the premise is true. An important distinction from the deductive argument is the second step regarding all possible outcomes being equally likely. Without this we can say 28 is in the upper 0.025 percentile of the sampling distribution of 4-tuples ordered by their sum, but we cannot say the probability is less than 0.00025. In contrast, the deductive argument is valid even if the balls are hand-picked; randomization plays no role. In the conclusion of the inductive argument, the word “unlikely” refers to the stochastic probability of obtaining a sum of 28 while “doubtful” describes a degree-of-belief regarding the lottery that was used. While these are related quantities—observations that are less likely to have occurred would create greater doubt—failure to understand these as distinct can lead to confusion, especially when the numeric value of the stochastic probability, 1 / 8 4 $$ 1/{8}^4 $$ , is used to assign a numeric measure of one's doubt in the absence of any other information regarding the two lotteries. The p-value, 1 / 8 4 $$ 1/{8}^4 $$ , is obtained from a measurable function and so, by definition, is a random variable. All p-values are measurable functions and so all p-values are random variables. However, the adjective random describes only one use for this measurable function, namely to model a random process. Random variables also provide distributions that are relevant to the inference question. Although randomization plays no role in the definition of these distributions, their relevance to inference does depend on how the observed sample was obtained from the population. As a random process, 1 / 8 4 $$ 1/{8}^4 $$ is the limiting relative frequency of draws from lottery A that result in a sum of 28. Using the random process interpretation means we have to create a hypothetical process by imagining repeated draws from lottery A when, in fact, the actual sample may have come from the NC lottery. That is, the hypothetical samples do not come from the population, as the actual sample did, but from a model for the population. This distinction between population and model for the population is especially important when the model is infinite. A more realistic example is inference for a dichotomous attribute of a population, say, high blood pressure (BP). The population distribution is the ordered pair of relative frequencies associated with the two attributes, ( 1 − p pop , p pop ) $$ \left(1-{p}_{\mathrm{pop}},{p}_{\mathrm{pop}}\right) $$ where p pop $$ {p}_{\mathrm{pop}} $$ is the unknown proportion with high BP. The Bernoulli family of distributions, ( 1 − p , p ) , 0 < p < 1 $$ \left\{\left(1-p,p\right),0

Patients whose cancer was found during an Emergency Department (ED) visit often present at later stages when survival outcomes are worse. Limited research has characterized the survival experience of cancer patients who receive their diagnosis through the ED versus those who do not. A retrospective cohort study identified all patients presenting to the ED between 2014 and 2015 in a rural, regional hospital system with a visit or resulting admission associated with an oncologic ICD-9 code. The chart was abstracted to determine a new cancer diagnosis versus an existing case. Cox proportional hazards (HR) estimated survival time. Patient and cancer characteristics were compared between those who were newly diagnosed through the ED and patients who were not. Thirty-nine percent of patients in our sample received their new cancer diagnosis as a result of an ED visit. The median survival was lower in cancer cases diagnosed through the ED (13 vs. 39 months, P < .001), men (20 vs. 32 months, P < .001), and patients aged ≥ 65 (22 months vs. 32 months, P < .001). Factors associated with lower survival were having a type of cancer location other than breast (HR = 1.96; P < .001), followed by being newly diagnosed with cancer through the ED (HR = 1.71; P < .001), and stage IV at diagnosis (HR = 1.70; P < .001). Patients who received a new cancer diagnosis through the ED and required subsequent hospitalization had shorter overall survival and presented with advanced disease. Future research should address socioeconomic factors that may influence these patterns of cancer presentation.

Frequentist inference typically is described in terms of hypothetical repeated sampling but there are advantages to an interpretation that uses a single random sample. Contemporary examples are given that indicate probabilities for random phenomena are interpreted as classical probabilities, and this interpretation of equally likely chance outcomes is applied to statistical inference using urn models. These are used to address Bayesian criticisms of frequentist methods. Recent descriptions of p -values, confidence intervals, and power are viewed through the lens of classical probability based on a single random sample from the population.

Point estimators may not exist, need not be unique, and their distributions are not parameter invariant. Generalized estimators provide distributions that are parameter invariant, unique, and exist when point estimates do not. Comparing point estimators using variance is less useful when estimators are biased. A squared slope Λ is defined that can be used to compare both point and generalized estimators and is unaffected by bias. Fisher information I and variance are fundamentally different quantities: the latter is defined at a distribution that need not belong to a family, while the former cannot be defined without a family of distributions, M. Fisher information and Λ are similar quantities as both are defined on the tangent bundle TM and I provides an upper bound, Λ≤ I , that holds for all sample sizes—asymptotics are not required. Comparing estimators using Λ rather than variance supports Fisher’s claim that I provides a bound even in small samples. Λ -efficiency is defined that extends the efficiency of unbiased estimators based on variance. While defined by the slope, Λ -efficiency is simply ρ ^2 , the square of the correlation between estimator and score function.

Minimizing metabolic energy expenditure (MEE) plays an important role in increasing mobility in people with locomotor disabilities, as movements that require high energy lead to less activity. Rehabilitation programs and devices use MEE to determine how effective they are, but using indirect calorimetry is limiting due to time delays and non-real-world conditions. Electromyography (EMG) offers insight into how muscles activate; thus, the purpose of this study was to develop a real-time MEE feedback system through the utilization of EMG signals. Participants completed five walking conditions at different stride frequencies (preferred, +/- 15%, +/- 30%), while breath-by-breath gas exchange, ground reaction forces and EMG signals were collected. The live EMG signal was numerically integrated and separated into strides, then scaled by a cost of force (COF) coefficient. MEE had the expected quadratic relationship seen in previous literature (R2 = 0.967), along with COF data (R2 = 0.701). The EMG method stabilized between 75.1% - 133.1%, which is not within a close range (90% - 110%) of MEE; thus, future studies must investigate other mathematical methods. Our results indicate a qualitative association between MEE and EMG activity, which could be used to increase mobility and quality of life for populations with disability.

The aim of this article is to assess the safety and efficacy of cyberknife-based Stereotactic Body Radiotherapy (SBRT) among older patients (at least 70 years-old) whose medical conditions preclude surgery for their early stage Non-Small Cell Lung Cancer (NSCLC). A retrospective review of older patients with a median age of 82 (range: 70 to 88 year-old) who had cyberknife-based SBRT for early stage biopsy proven NSCLC in a community hospital was performed. Morbidity, complications, and local control were assessed. Among 33 older patients who had SBRT for early stage NSCLC, no patient developed grade 3 to 4 complications at a median follow-up of 19 months (range: 5 to 32 months). Only two patients (6%) developed local recurrences. Cyberknife-based SBRT is safe and effective for local control in older patients with early stage biopsy proven NSCLC whose medical conditions preclude surgery.

Objectives: To assess the effectiveness of prophylactic percutaneous endoscopic gastrostomy (PEG) tube feedings in locally advanced head and neck cancer patients undergoing image-guided radiotherapy (IGRT) and concurrent chemotherapy.

Atherosclerosis develops over a long period of time and often begins in childhood. The goal of this study was to make a cross-sectional assessment of the pattern of cardiovascular disease risk factors among Australian vegetarian (n = 49) and nonvegetarian (n = 639) 14- to 17-year-old participants from New South Wales, Australia. Vegetarians had statistically significant lower mean total (4.05 vs 4.4 mmol/L;P < .001) and low-density lipoprotein (LDL) cholesterol (2.18 vs 2.55 mmol/L; P < .001) and lower incidence of abnormal total and LDL cholesterol (31.1% vs 46.2%, P = .036, having total cholesterol ≥4.4 mmol/L and 13.3% vs 29.6%, P = .021, having LDL cholesterol ≥2.84 mmol/L). Vegetarians had a higher diastolic BP (72.0 vs 69.7 mm Hg; P = .038). No statistically significant difference was found in other risk factors including high-density lipoprotein cholesterol (P = .83), triglycerides (P = .601), systolic blood pressure (P = .727), body mass index (P = .159), plasma glucose (P = .09), C-reactive protein (P = .527), or homocysteine (P = .45). The prevalence rate with 3 or more risk factors was 12.2% among vegetarians and 13.9% among nonvegetarians (P = .156). The high percentage of abnormal total cholesterol in both diet groups and, in addition, LDL cholesterol in nonvegetarians is a cause of concern and underlines the need for lifestyle change.

BACKGROUND:This study examined the relationship of medical mistrust using the Group-based Medical Mistrust Scale (GBMMS), and Papanicolaou testing behaviors among rural Black and White women.METHODS:Utilizing a convenience sample, a cross-sectional study was performed. Inclusion criteria included self-identification as a non-Hispanic Black or White woman, at least 21 years of age, and a resident of one of the selected counties in the region. Analyses conducted were two-sample t-tests, Fisher's exact tests, Spearman's rho, and logistical regression.RESULTS:Among 338 women, four GBMMS items had statistically significant outcomes using multiple significance tests; significance remained when adjusting for demographic variables. Analyses indicated that Whites were dissatisfied with the health care system to a greater extent than Blacks.CONCLUSIONS:The impact of medical mistrust should be explored beyond individuals of a racial/ethnic minority group. Future directions include the development of a community-informed screening intervention to foster adherence among diverse rural populations.

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For linear regression and related models, a permutation inference distribution (PID) is introduced. Like the confidence distribution in the Bayesian/Fiducial/Frequentist inference framework, the PID allows the construction of both confidence intervals and p-values. For two-sample problems and pairwise comparisons in ANOVA models, a fast Fourier transformation method can be used to find the exact PID. In general, however, random permutations are required except for small samples where all permutations can be generated. Simulation studies and real data applications are used to evaluate inferences obtained from the PID. PID methods are close to standard parametric methods when the errors are iid and normal. For skewed and heavy tailed errors, PID methods are superior to bootstrap and standard parametric methods.