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Purpose: The purpose of this study was to evaluate the impact of using Imagine Math Facts on student mathematics proficiency. Methods: This retrospective study compared the performance of Grade 1-5 students who used or did not use the Imagine Math Facts program on a standardized assessment of mathematics proficiency. Statistical matching was utilized to account for baseline differences and generate comparable study groups. Comparisons were conducted globally and across individual grades. Results: Early elementary students who used Imagine Math Facts scored 6.7 points higher on a standardized mathematics assessment than similar non-user students. More specifically, Grade 2 users of Imagine Math Facts achieved statistically higher scores on the standardized mathematics assessment than students who did not use the program. Implications: This study provides evidence of the effectiveness of Imagine Math Facts on mathematics proficiency for early elementary students. [This report was prepared by Imagine Learning.]

We study the effects of missingness on the estimation of population parameters. Moving beyond restrictive missing completely at random (MCAR) assumptions, we first formulate a missing data analogue of Huber's arbitrary $\epsilon$-contamination model. For mean estimation with respect to squared Euclidean error loss, we show that the minimax quantiles decompose as a sum of the corresponding minimax quantiles under a heterogeneous, MCAR assumption, and a robust error term, depending on $\epsilon$, that reflects the additional error incurred by departure from MCAR. We next introduce natural classes of realisable $\epsilon$-contamination models, where an MCAR version of a base distribution $P$ is contaminated by an arbitrary missing not at random (MNAR) version of $P$. These classes are rich enough to capture various notions of biased sampling and sensitivity conditions, yet we show that they enjoy improved minimax performance relative to our earlier arbitrary contamination classes for both parametric and nonparametric classes of base distributions. For instance, with a univariate Gaussian base distribution, consistent mean estimation over realisable $\epsilon$-contamination classes is possible even when $\epsilon$ and the proportion of missingness converge (slowly) to 1. Finally, we extend our results to the setting of departures from missing at random (MAR) in normal linear regression with a realisable missing response., Comment: 86 pages, 6 figures

We study the efficient estimation of a class of mean functionals in settings where a complete multivariate dataset is complemented by additional datasets recording subsets of the variables of interest. These datasets are allowed to have a general, in particular non-monotonic, structure. Our main contribution is to characterise the asymptotic minimal mean squared error for these problems and to introduce an estimator whose risk approximately matches this lower bound. We show that the efficient rescaled variance can be expressed as the minimal value of a quadratic optimisation problem over a function space, thus establishing a fundamental link between these estimation problems and the theory of generalised ANOVA decompositions. Our estimation procedure uses iterated nonparametric regression to mimic an approximate influence function derived through gradient descent. We prove that this estimator is approximately normally distributed, provide an estimator of its variance and thus develop confidence intervals of asymptotically minimal width. Finally we study a more direct estimator, which can be seen as a U-statistic with a data-dependent kernel, showing that it is also efficient under stronger regularity conditions., Comment: 69 pages

Most of the literature on differential privacy considers the item-level case where each user has a single observation, but a growing field of interest is that of user-level privacy where each of the $n$ users holds $T$ observations and wishes to maintain the privacy of their entire collection. In this paper, we derive a general minimax lower bound, which shows that, for locally private user-level estimation problems, the risk cannot, in general, be made to vanish for a fixed number of users even when each user holds an arbitrarily large number of observations. We then derive matching, up to logarithmic factors, lower and upper bounds for univariate and multidimensional mean estimation, sparse mean estimation and non-parametric density estimation. In particular, with other model parameters held fixed, we observe phase transition phenomena in the minimax rates as $T$ the number of observations each user holds varies. In the case of (non-sparse) mean estimation and density estimation, we see that, for $T$ below a phase transition boundary, the rate is the same as having $nT$ users in the item-level setting. Different behaviour is however observed in the case of $s$-sparse $d$-dimensional mean estimation, wherein consistent estimation is impossible when $d$ exceeds the number of observations in the item-level setting, but is possible in the user-level setting when $T \gtrsim s \log (d)$, up to logarithmic factors. This may be of independent interest for applications as an example of a high-dimensional problem that is feasible under local privacy constraints., Comment: 57 pages, 1 figure, 4 tables

(Purpose) This study aimed to evaluate the efficacy of Imagine Math by comparing the performance of program users and non-users on the NWEA MAP Growth Math assessment. (Methods) Propensity score matching was utilized to compare statistically similar southeastern U.S. charter school students who used or did not use the Imagine Math program during the 2020-2021 school year. Imagine Math users and non-users were matched based on baseline performance and demographics and compared based on NWEA MAP Growth Math assessment performance. (Results) Use of Imagine Math was found to generate a positive and statistically significant impact on students' mathematics performance. Imagine Math users scored an average of 2.00 points higher on the Spring 2021 NWEA MAP Growth Math assessment than statistically similar non-user students. Grade-level models revealed statistically positive effects for Imagine Math students in Kindergarten through 8th grade with all findings being statistically significant except for Kindergarten. (Implications) Results show that students who used Imagine Math scored higher on the NWEA MAP Growth Math assessment than similar comparison students. Thus, this study provides evidence that the use of Imagine Math supports students' mathematics achievement. [This report was prepared by Imagine Learning.]

We study the problem of testing whether the missing values of a potentially high-dimensional dataset are Missing Completely at Random (MCAR). We relax the problem of testing MCAR to the problem of testing the compatibility of a sequence of covariance matrices, motivated by the fact that this procedure is feasible when the dimension grows with the sample size. Tests of compatibility can be used to test the feasibility of positive semi-definite matrix completion problems with noisy observations, and thus our results may be of independent interest. Our first contributions are to define a natural measure of the incompatibility of a sequence of correlation matrices, which can be characterised as the optimal value of a Semi-definite Programming (SDP) problem, and to establish a key duality result allowing its practical computation and interpretation. By studying the concentration properties of the natural plug-in estimator of this measure, we introduce novel hypothesis tests that we prove have power against all distributions with incompatible covariance matrices. The choice of critical values for our tests rely on a new concentration inequality for the Pearson sample correlation matrix, which may be of interest more widely. By considering key examples of missingness structures, we demonstrate that our procedures are minimax rate optimal in certain cases. We further validate our methodology with numerical simulations that provide evidence of validity and power, even when data are heavy tailed., Comment: 88 pages, 15 figures

We introduce a new nonparametric framework for classification problems in the presence of missing data. The key aspect of our framework is that the regression function decomposes into an anova-type sum of orthogonal functions, of which some (or even many) may be zero. Working under a general missingness setting, which allows features to be missing not at random, our main goal is to derive the minimax rate for the excess risk in this problem. In addition to the decomposition property, the rate depends on parameters that control the tail behaviour of the marginal feature distributions, the smoothness of the regression function and a margin condition. The ambient data dimension does not appear in the minimax rate, which can therefore be faster than in the classical nonparametric setting. We further propose a new method, called the Hard-thresholding Anova Missing data (HAM) classifier, based on a careful combination of a k-nearest neighbour algorithm and a thresholding step. The HAM classifier attains the minimax rate up to polylogarithmic factors and numerical experiments further illustrate its utility., Comment: 73 pages, 6 figures

Although there is substantial literature on identifying structural changes for continuous spatio-temporal processes, the same is not true for categorical spatio-temporal data. This work bridges that gap and proposes a novel spatio-temporal model to identify changepoints in ordered categorical data. The model leverages an additive mean structure with separable Gaussian space-time processes for the latent variable. Our proposed methodology can detect significant changes in the mean structure as well as in the spatio-temporal covariance structures. We implement the model through a Bayesian framework that gives a computational edge over conventional approaches. From an application perspective, our approach's capability to handle ordinal categorical data provides an added advantage in real applications. This is illustrated using county-wise COVID-19 data (converted to categories according to CDC guidelines) from the state of New York in the USA. Our model identifies three changepoints in the transmission levels of COVID-19, which are indeed aligned with the ``waves'' due to specific variants encountered during the pandemic. The findings also provide interesting insights into the effects of vaccination and the extent of spatial and temporal dependence in different phases of the pandemic.

In the last two decades, considerable research has been devoted to a phenomenon known as spatial confounding. Spatial confounding is thought to occur when there is multicollinearity between a covariate and the random effect in a spatial regression model. This multicollinearity is considered highly problematic when the inferential goal is estimating regression coefficients and various methodologies have been proposed to attempt to alleviate it. Recently, it has become apparent that many of these methodologies are flawed, yet the field continues to expand. In this paper, we offer a novel perspective of synthesizing the work in the field of spatial confounding. We propose that at least two distinct phenomena are currently conflated with the term spatial confounding. We refer to these as the ``analysis model'' and the ``data generation'' types of spatial confounding. We show that these two issues can lead to contradicting conclusions about whether spatial confounding exists and whether methods to alleviate it will improve inference. Our results also illustrate that in most cases, traditional spatial linear mixed models do help to improve inference on regression coefficients. Drawing on the insights gained, we offer a path forward for research in spatial confounding., Comment: 38 pages main text; 8 figures; code available upon request

Given a set of incomplete observations, we study the nonparametric problem of testing whether data are Missing Completely At Random (MCAR). Our first contribution is to characterise precisely the set of alternatives that can be distinguished from the MCAR null hypothesis. This reveals interesting and novel links to the theory of Fr\'echet classes (in particular, compatible distributions) and linear programming, that allow us to propose MCAR tests that are consistent against all detectable alternatives. We define an incompatibility index as a natural measure of ease of detectability, establish its key properties, and show how it can be computed exactly in some cases and bounded in others. Moreover, we prove that our tests can attain the minimax separation rate according to this measure, up to logarithmic factors. Our methodology does not require any complete cases to be effective, and is available in the R package MCARtest., Comment: 66 pages, 4 figures

Network data are ubiquitous in our daily life, containing rich but often sensitive information. In this paper, we expand the current static analysis of privatised networks to a dynamic framework by considering a sequence of networks with potential change points. We investigate the fundamental limits in consistently localising change points under both node and edge privacy constraints, demonstrating interesting phase transition in terms of the signal-to-noise ratio condition, accompanied by polynomial-time algorithms. The private signal-to-noise ratio conditions quantify the costs of the privacy for change point localisation problems and exhibit a different scaling in the sparsity parameter compared to the non-private counterparts. Our algorithms are shown to be optimal under the edge LDP constraint up to log factors. Under node LDP constraint, a gap exists between our upper bound and lower bound and we leave it as an interesting open problem., Comment: Accepted at NeurIPS 2022. This version includes interactive mechanisms in Sections 2 and 3

Invited discussion for Biometrika of 'Multivariate Fisher's independence test for multivariate dependence' by Gorsky and Ma (2022)., Comment: 4 pages

It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. In this paper, we start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study four concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem, a nonparametric density estimation problem and a univariate median estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.

We address the problem of goodness-of-fit testing for H\"older continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of $f_0$ including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.

We study online change point detection problems under the constraint of local differential privacy (LDP) where, in particular, the statistician does not have access to the raw data. As a concrete problem, we study a multivariate nonparametric regression problem. At each time point $t$, the raw data are assumed to be of the form $(X_t, Y_t)$, where $X_t$ is a $d$-dimensional feature vector and $Y_t$ is a response variable. Our primary aim is to detect changes in the regression function $m_t(x)=\mathbb{E}(Y_t |X_t=x)$ as soon as the change occurs. We provide algorithms which respect the LDP constraint, which control the false alarm probability, and which detect changes with a minimal (minimax rate-optimal) delay. To quantify the cost of privacy, we also present the optimal rate in the benchmark, non-private setting. These non-private results are also new to the literature and thus are interesting \emph{per se}. In addition, we study the univariate mean online change point detection problem, under privacy constraints. This serves as the blueprint of studying more complicated private change point detection problems.

Urbanization of an area is known to increase the temperature of the surrounding area. This phenomenon -- a so-called urban heat island (UHI) -- occurs at a local level over a period of time and has lasting impacts for historical data analysis. We propose a methodology to examine if long-term changes in temperature increases and decreases across time exist (and to what extent) at the local level for a given set of temperature readings at various locations. Specifically, we propose a Bayesian change point model for spatio-temporally dependent data where we select the number of change points at each location using a "forwards" selection process using deviance information criteria (DIC). We then fit the selected model and examine the linear slopes across time to quantify changes in long-term temperature behavior. We show the utility of this model and method using a synthetic data set and temperature measurements from eight stations in Utah consisting of daily temperature data for 60 years., Comment: 24 pages, 5 figures, 5 tables

We present the $U$-Statistic Permutation (USP) test of independence in the context of discrete data displayed in a contingency table. Either Pearson's chi-squared test of independence, or the $G$-test, are typically used for this task, but we argue that these tests have serious deficiencies, both in terms of their inability to control the size of the test, and their power properties. By contrast, the USP test is guaranteed to control the size of the test at the nominal level for all sample sizes, has no issues with small (or zero) cell counts, and is able to detect distributions that violate independence in only a minimal way. The test statistic is derived from a $U$-statistic estimator of a natural population measure of dependence, and we prove that this is the unique minimum variance unbiased estimator of this population quantity. The practical utility of the USP test is demonstrated on both simulated data, where its power can be dramatically greater than those of Pearson's test and the $G$-test, and on real data. The USP test is implemented in the R package USP., Comment: 27 pages, 7 figures

In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data $(X_1,Y_1),\ldots,(X_n,Y_n)$, taking values in $\mathbb{R}^d \times \mathbb{R}$, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector $X_i$ and to the value of its response variable $Y_i$. Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data., Comment: 25 pages

We find separation rates for testing multinomial or more general discrete distributions under the constraint of local differential privacy. We construct efficient randomized algorithms and test procedures, in both the case where only non-interactive privacy mechanisms are allowed and also in the case where all sequentially interactive privacy mechanisms are allowed. The separation rates are faster in the latter case. We prove general information theoretical bounds that allow us to establish the optimality of our algorithms among all pairs of privacy mechanisms and test procedures, in most usual cases. Considered examples include testing uniform, polynomially and exponentially decreasing distributions., Comment: 31 pages

We study the problem of independence testing given independent and identically distributed pairs taking values in a $\sigma$-finite, separable measure space. Defining a natural measure of dependence $D(f)$ as the squared $L^2$-distance between a joint density $f$ and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form $\{f: D(f) \geq \rho^2 \}$. We therefore restrict attention to alternatives that impose additional Sobolev-type smoothness constraints, and define a permutation test based on a basis expansion and a $U$-statistic estimator of $D(f)$ that we prove is minimax optimal in terms of its separation rates in many instances. Finally, for the case of a Fourier basis on $[0,1]^2$, we provide an approximation to the power function that offers several additional insights. Our methodology is implemented in the R package USP., Comment: 58 pages, 4 figures

We consider the binary classification problem in a setup that preserves the privacy of the original sample. We provide a privacy mechanism that is locally differentially private and then construct a classifier based on the private sample that is universally consistent in Euclidean spaces. Under stronger assumptions, we establish the minimax rates of convergence of the excess risk and see that they are slower than in the case when the original sample is available., Comment: 12 pages

We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide generality, a weighted nearest neighbour estimator is efficient, in the sense of achieving the local asymptotic minimax lower bound. Moreover, we also prove a corresponding central limit theorem, which facilitates the construction of asymptotically valid confidence intervals for the functional, having asymptotically minimal width. One interesting consequence of our results is the discovery that, for certain functionals, the worst-case performance of our estimator may improve on that of the natural `oracle' estimator, which is given access to the values of the unknown densities at the observations., Comment: 76 pages

We propose a general new method, the conditional permutation test, for testing the conditional independence of variables $X$ and $Y$ given a potentially high-dimensional random vector $Z$ that may contain confounding factors. The proposed test permutes entries of $X$ non-uniformly, so as to respect the existing dependence between $X$ and $Z$ and thus account for the presence of these confounders. Like the conditional randomization test of Cand\`es et al. (2018), our test relies on the availability of an approximation to the distribution of $X \mid Z$. While Cand\`es et al. (2018)'s test uses this estimate to draw new $X$ values, for our test we use this approximation to design an appropriate non-uniform distribution on permutations of the $X$ values already seen in the true data. We provide an efficient Markov Chain Monte Carlo sampler for the implementation of our method, and establish bounds on the Type I error in terms of the error in the approximation of the conditional distribution of $X\mid Z$, finding that, for the worst case test statistic, the inflation in Type I error of the conditional permutation test is no larger than that of the conditional randomization test. We validate these theoretical results with experiments on simulated data and on the Capital Bikeshare data set., Comment: 31 pages, 4 figures

We propose a test of independence of two multivariate random vectors, given a sample from the underlying population. Our approach, which we call MINT, is based on the estimation of mutual information, whose decomposition into joint and marginal entropies facilitates the use of recently-developed efficient entropy estimators derived from nearest neighbour distances. The proposed critical values, which may be obtained from simulation (in the case where one marginal is known) or resampling, guarantee that the test has nominal size, and we provide local power analyses, uniformly over classes of densities whose mutual information satisfies a lower bound. Our ideas may be extended to provide a new goodness-of-fit tests of normal linear models based on assessing the independence of our vector of covariates and an appropriately-defined notion of an error vector. The theory is supported by numerical studies on both simulated and real data., Comment: 46 pages, 2 figures

We derive a new asymptotic expansion for the global excess risk of a local-$k$-nearest neighbour classifier, where the choice of $k$ may depend upon the test point. This expansion elucidates conditions under which the dominant contribution to the excess risk comes from the decision boundary of the optimal Bayes classifier, but we also show that if these conditions are not satisfied, then the dominant contribution may arise from the tails of the marginal distribution of the features. Moreover, we prove that, provided the $d$-dimensional marginal distribution of the features has a finite $\rho$th moment for some $\rho > 4$ (as well as other regularity conditions), a local choice of $k$ can yield a rate of convergence of the excess risk of $O(n^{-4/(d+4)})$, where $n$ is the sample size, whereas for the standard $k$-nearest neighbour classifier, our theory would require $d \geq 5$ and $\rho > 4d/(d-4)$ finite moments to achieve this rate. These results motivate a new $k$-nearest neighbour classifier for semi-supervised learning problems, where the unlabelled data are used to obtain an estimate of the marginal feature density, and fewer neighbours are used for classification when this density estimate is small. Our worst-case rates are complemented by a minimax lower bound, which reveals that the local, semi-supervised $k$-nearest neighbour classifier attains the minimax optimal rate over our classes for the excess risk, up to a subpolynomial factor in $n$. These theoretical improvements over the standard $k$-nearest neighbour classifier are also illustrated through a simulation study., Comment: 60 pages

The Rapid Carbon Assessment (RaCA) project was conducted by the US Department of Agriculture's National Resources Conservation Service between 2010-2012 in order to provide contemporaneous measurements of soil organic carbon (SOC) across the US. Despite the broad extent of the RaCA data collection effort, direct observations of SOC are not available at the high spatial resolution needed for studying carbon storage in soil and its implications for important problems in climate science and agriculture. As a result, there is a need for predicting SOC at spatial locations not included as part of the RaCA project. In this paper, we compare spatial prediction of SOC using a subset of the RaCA data for a variety of statistical methods. We investigate the performance of methods with off-the-shelf software available (both stationary and nonstationary) as well as a novel nonstationary approach based on partitioning relevant spatially-varying covariate processes. Our new method addresses open questions regarding (1) how to partition the spatial domain for segmentation-based nonstationary methods, (2) incorporating partially observed covariates into a spatial model, and (3) accounting for uncertainty in the partitioning. In applying the various statistical methods we find that there are minimal differences in out-of-sample criteria for this particular data set, however, there are major differences in maps of uncertainty in SOC predictions. We argue that the spatially-varying measures of prediction uncertainty produced by our new approach are valuable to decision makers, as they can be used to better benchmark mechanistic models, identify target areas for soil restoration projects, and inform carbon sequestration projects.

The Open Aid Malawi initiative has collected an unprecedented database that identifies as much location-specific information as possible for each of over 2500 individual foreign aid donations to Malawi since 2003. Ensuring efficient use and distribution of that aid is important to donors and to Malawi citizens. However, because of individual donor goals and difficulty in tracking donor coordination, determining presence or absence of efficient aid allocation is difficult. We compare several Bayesian spatial generalized linear mixed models to relate aid allocation to various economic indicators within seven donation sectors. We find that the spatial gamma regression model best predicts current aid allocation. Using this model, first we use inferences on coefficients to examine whether or not there is evidence of efficient aid allocation within each sector. Second, we use this model to determine a more efficient aid allocation scenario and compare this scenario to the current allocation to provide insight for future aid donations.

Many statistical procedures, including goodness-of-fit tests and methods for independent component analysis, rely critically on the estimation of the entropy of a distribution. In this paper, we seek entropy estimators that are efficient and achieve the local asymptotic minimax lower bound with respect to squared error loss. To this end, we study weighted averages of the estimators originally proposed by Kozachenko and Leonenko (1987), based on the $k$-nearest neighbour distances of a sample of $n$ independent and identically distributed random vectors in $\mathbb{R}^d$. A careful choice of weights enables us to obtain an efficient estimator in arbitrary dimensions, given sufficient smoothness, while the original unweighted estimator is typically only efficient when $d \leq 3$. In addition to the new estimator proposed and theoretical understanding provided, our results facilitate the construction of asymptotically valid confidence intervals for the entropy of asymptotically minimal width., Comment: 69 pages, 0 figures

In analyses of spatially-referenced data, researchers often have one of two goals: to quantify relationships between a response variable and covariates while accounting for residual spatial dependence or to predict the value of a response variable at unobserved locations. In this second case, when the response variable is categorical, prediction can be viewed as a classification problem. Many classification methods either ignore response-variable/covariate relationships and rely only on spatially proximate observations for classification, or they ignore spatial dependence and use only the covariates for classification. The Bayesian spatial generalized linear (mixed) model offers a tool to accommodate both spatial and covariate sources of information in classification problems. In this paper, we formally define spatial classification rules based on these models. We also take a close look at two of these models that have been proposed in the literature, namely the probit versions of the spatial generalized linear model (SGLM) and the Bayesian spatial generalized linear mixed model (SGLMM). We describe the implications of the seemingly slight differences between these models for spatial classification and explore the issue of robustness to model misspecification through a simulation study. We also provide an overview of alternatives to the SGLM/SGLMM-based classifiers and illustrate the various methods using satellite-derived land cover data from Southeast Asia.