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Forecasts of the probability of a large earthquake occurring on a fault during a specific time interval assume that a probability distribution describes the interevent times between large earthquakes. However, current models have features that we consider unrealistic. In these models, earthquake probabilities remain constant or even decrease after the expected mean recurrence interval, implying that additional accumulated strain does not make an earthquake more likely. Moreover, these models assume that large earthquakes release all accumulated strain, despite evidence for partial strain release in earthquake histories showing clusters and gaps. As an alternative, we derive the necessary equations to calculate earthquake probabilities using the long-term fault memory (LTFM) model. By accounting for partial strain release, LTFM incorporates the specific timing of past earthquakes, which commonly used probability models cannot do, so it can forecast gaps and clusters. We apply LTFM to the southern San Andreas fault as an example and show how LTFM can produce better forecasts when clusters and gaps are present. LTFM better forecasts the exceptionally short interevent time before the 1857 Fort Tejon earthquake. Although LTFM is more complex than existing models, it is more powerful because (unlike current models) it incorporates fundamental aspects of the strain accumulation and release processes causing earthquakes.

The federal statistical system is experiencing competing pressures for change. On the one hand, for confidentiality reasons, much socially valuable data currently held by federal agencies is either not made available to researchers at all or only made available under onerous conditions. On the other hand, agencies which release public databases face new challenges in protecting the privacy of the subjects in those databases, which leads them to consider releasing fewer data or masking the data in ways that will reduce their accuracy. In this essay, we argue that the discussion has not given proper consideration to the reduced social benefits of data availability and their usability relative to the value of increased levels of privacy protection. A more balanced benefit–cost framework should be used to assess these trade-offs. We express concerns both with synthetic data methods for disclosure limitation, which will reduce the types of research that can be reliably conducted in unknown ways, and with differential privacy criteria that use what we argue is an inappropriate measure of disclosure risk. We recommend that the measure of disclosure risk used to assess all disclosure protection methods focus on what we believe is the risk that individuals should care about, that more study of the impact of differential privacy criteria and synthetic data methods on data usability for research be conducted before either is put into widespread use, and that more research be conducted on alternative methods of disclosure risk reduction that better balance benefits and costs.

Catalogs of moment tensors form the foundation for a wide variety of seismological studies. However, assessing uncertainties in the moment tensors and the quantities derived from them is difficult. To gain insight, we compare 5000 moment tensors in the U.S. Geological Survey (USGS) and the Global Centroid Moment Tensor (Global CMT) Project catalogs for November 2015–December 2020 and use the differences to illustrate the uncertainties. The differences are typically an order of magnitude larger than the reported errors, suggesting that the errors substantially underestimate the uncertainty. The catalogs are generally consistent, with intriguing differences. Global CMT generally reports larger scalar moments than USGS, with the difference decreasing with magnitude. This difference is larger than and of the opposite sign from what is expected due to the different definitions of the scalar moment. Instead, the differences are intrinsic to the tensors, presumably in part due to different phases used in the inversions. The differences in double-couple components of source mechanisms and the fault angles derived from them decrease with magnitude. Non-double-couple (NDC) components decrease somewhat with magnitude. These components are moderately correlated between catalogs, with correlations stronger for larger earthquakes. Hence, small earthquakes often show large NDC components, but many have large uncertainties and are likely to be artifacts of the inversion. Conversely, larger earthquakes are less likely to have large NDC components, but these components are typically robust between catalogs. If so, these can indicate either true deviation from a double couple or source complexity. The differences between catalogs in scalar moment, source geometry, or NDC fraction of individual earthquakes are essentially uncorrelated, suggesting that the differences reflect the inversion rather than the source process. Despite the differences in moment tensors, the location and depth of the centroids are consistent between catalogs. Our results apply to earthquakes after 2012, before which many moment tensors were common to both catalogs.

Earthquake stress drop, the stress change on a fault due to an earthquake, is important for seismic hazard analysis because it controls the level of high-frequency ground motions that damage structures. Numerous studies report that stress drops vary by tectonic environment, providing insight into a region’s seismic hazard. Here, we show that teleseismic stress-drop estimates have large uncertainties that make it challenging to distinguish differences between the stress drops of different earthquakes. We compared stress drops for ∼900 earthquakes derived from two independent studies using teleseismic data and found practically zero correlation. Estimates for the same earthquake can differ by orders of magnitude. Therefore, reported stress-drop differences between earthquakes may not reflect true differences. As a result of these larger uncertainties, some tectonic environment stress-drop patterns that appear in one study do not appear in the other analysis of the same earthquakes. These large uncertainties in teleseismic estimates might lead to erroneous inferences about earthquake hazards. In many applications, it may be more appropriate to assume that earthquakes in different regions have approximately the same average stress drop.

Seismic intensity data based on first-hand accounts of shaking give valuable insight into historical and early instrumental earthquakes. Comparing an observed intensity distribution to intensity-prediction models based on modern calibration events allows the magnitude to be estimated for many historic earthquakes. Magnitude estimates can also potentially be refined for earthquakes for which limited instrumental data are available. However, the complicated nature of macroseismic data and the methods used to collect and interpret the data introduce significant uncertainties. In this paper, we illustrate these challenges and possible solutions using the 1952 Kern County, California, earthquake as a case study. Published estimates of its magnitude vary from MW 7.2–7.5, making it possibly the second largest in California during the 20th century. We considered over 1100 first-hand reports of shaking, supplemented with other data, and inferred the magnitude in several ways using intensity prediction equations, yielding a preferred intensity magnitude MI 7.2 ± 0.2, where the uncertainty reflects our judgement. The revised intensity distribution reveals stronger shaking on the hanging wall, south of the surface expression of the White Wolf fault, than on the footwall. Characterizing the magnitude and shaking distribution of this early instrumental earthquake can help improve estimation of the seismic hazard of the region. Such reinterpreted intensities for historic earthquakes, combined with U.S. Geological Survey (USGS) Did You Feel It? data for more recent events, can be used to produce a uniform shaking dataset with which earthquake hazard map performance can be assessed.

The question of whether to carry out a quinquennial Census is faced by national statistical offices in increasingly many countries, including Canada, Nigeria, Ireland, Australia, and South Africa. We describe uses and limitations of cost-benefit analysis in this decision problem in the case of the 2016 Census of South Africa. The government of South Africa needed to decide whether to conduct a 2016 Census or to rely on increasingly inaccurate postcensal estimates accounting for births, deaths, and migration since the previous (2011) Census. The cost-benefit analysis compared predicted costs of the 2016 Census to the benefits of improved allocation of intergovernmental revenue, which was considered by the government to be a critical use of the 2016 Census, although not the only important benefit. Without the 2016 Census, allocations would be based on population estimates. Accuracy of the postcensal estimates was estimated from the performance of past estimates, and the hypothetical expected reduction in errors in allocation due to the 2016 Census was estimated. A loss function was introduced to quantify the improvement in allocation. With this evidence, the government was able to decide not to conduct the 2016 Census, but instead to improve data and capacity for producing post-censal estimates.

Long records often show large earthquakes occurring in supercycles, sequences of temporal clusters of seismicity, cumulative displacement, and cumulative strain release separated by less active intervals. Supercycles and associated earthquake clusters are only partly characterized via the traditionally used aperiodicity, which measures the extent that a sequence differs from perfectly periodic. Supercycles are not well described by commonly used models of earthquake recurrence. In the Poisson model, the probability of a large earthquake is constant with time, so the fault has no memory. In a seismic cycle/renewal model, the probability is quasi-periodic, dropping to zero after a large earthquake, then increasing with time, so the probability of a large earthquake depends only on the time since the past one, and the fault has only “short-term memory.” We describe supercycles with a Long-Term Fault Memory (LTFM) model, where the probability of a large earthquake reflects the accumulated strain rather than elapsed time. The probability increases with accumulated strain (and time) until an earthquake happens, after which it decreases, but not necessarily to zero. Hence, the probability of an earthquake can depend on the earthquake history over multiple prior cycles. We use LTFM to simulate paleoseismic records from plate boundaries and intraplate areas. Simulations suggest that over timescales corresponding to the duration of paleoseismic records, the distribution of earthquake recurrence times can appear strongly periodic, weakly periodic, Poissonian, or bursty. Thus, a given paleoseismic window may not capture long-term trends in seismicity. This effect is significant for earthquake hazard assessment because whether an earthquake history is assumed to contain clusters can be more important than the probability density function chosen to describe the recurrence times. In such cases, probability estimates of the next earthquake will depend crucially on whether the cluster is treated as ongoing or over.

Following the 2011 magnitude 9.1 Tohoku earthquake, Geller (2011) argued that “all of Japan is at risk from earthquakes, and the present state of seismological science does not allow us to reliably differentiate the risk level in particular geographic areas,” so a map showing uniform hazard would be preferable to the existing map. We explore this by comparing how well a 510‐yr‐long record of earthquake shaking in Japan is described by the Japanese national‐hazard (JNH) maps, uniform maps, and randomized maps. Surprisingly, as measured by the metric implicit in the JNH maps (i.e., a metric that requires only a specific fraction of the sites during the chosen time interval to exceed the predicted ground motion), both uniform and randomized maps do better compared with the actual maps. However, using the squared misfit between maximum observed shaking and the predicted shaking as a metric, the JNH maps do better compared with the uniform or randomized maps. These results indicate that (1) the JNH maps are not performing as well as expected, (2) identification of the factors controlling map performance is complicated, and (3) learning more about how maps perform and why would be valuable in making more effective policy.

Recent large earthquakes that caused great damage in areas predicted to be relatively safe, illustrate the importance of criteria that assess how well earthquake hazard maps used to develop codes for earthquake‐resistant construction are actually performing. At present, there is no agreed‐upon way of assessing how well a map performed and thus determining whether one map performed better than another. The fractional site exceedance metric implicit in current maps, that during the chosen time interval the predicted ground motion will be exceeded only at a specific fraction of the sites, is useful but permits maps to be nominally successful even if they significantly underpredict or overpredict shaking, or permits them to be nominally unsuccessful but do well in terms of predicting shaking. We explore some possible metrics that better measure the effects of overprediction and underprediction and can be weighted to reflect the two differently and to reflect differences in populations and property at risk. Although no single metric alone fully characterizes map behavior, using several metrics can provide useful insight for comparing and improving hazard maps. For example, both probabilistic and deterministic hazard maps for Italy dramatically overpredict the recorded shaking in a 2200‐yr‐long historical intensity catalog, illustrating problems in the data (most likely), models, or both.

Recent large earthquakes that caused ground shaking larger than anticipated have generated interest in how to improve earthquake hazard mapping. Issues under discussion include how to evaluate maps’ performance, how to assess their uncertainties, how to make better maps, and how to best use maps given their limitations. An important question is what to do after an earthquake yielding shaking larger than anticipated. Hazard mappers have two choices. One is to regard the high shaking as a low‐probability event allowed by the map, which used estimates of the probability of future earthquakes and the resulting shaking to predict the maximum shaking expected with a certain probability over a given time (Hanks et al. , 2012; Frankel, 2013). The usual choice, however, is to accept that high shaking was not simply a low‐probability event consistent with the map, and revise the map to show increased hazard in the heavily shaken area (Fig. 1). Figure 1. (top) Japanese seismic‐hazard maps before and after the 2011 Tohoku earthquake. The predicted hazard has been increased both along the east coast, where the 2011 earthquake occurred, and on the west coast. (http://www.j-shis.bosai.go.jp/map/?lang=en; last accessed December 2014.) (bottom) Comparison of successive Italian hazard maps (Stein et al. , 2013). The 1999 map was updated to reflect the 2002 Molise earthquake, and the 2006 map will likely be updated after the 2012 Emilia earthquake. Whether and how much to revise a map is complicated, because a new map that better describes the past may or may not better predict the future. For example, increasing the predicted hazard after an earthquake on a fault will make better predictions if the average recurrence time is short compared to the map’s time window but will overpredict future shaking if the average recurrence time is much longer than the map’s time window. …

Current earthquake hazard maps for Nepal predict substantial variations in hazard within the nation, with noticeable differences between maps. We thus suggest that given present knowledge, all of Nepal may be better regarded as equally hazardous and perhaps vulnerable to much larger earthquakes than those currently known because of their long recurrence times. This proposal is based on the limitations of the historical earthquake record, the recognized deficit in seismic moment release, and GPS data showing a similar level of coupling along the arc. Support for using smoother maps can be had from analysis for Japan, which is also located on and parallel to a subduction boundary, showing that in some ways the hazard maps may be overparameterized, in that including too high a level of detail may lower the maps’ ability to predict shaking. Treating Nepal’s hazard as uniform and developing mitigation strategies accordingly may help reduce damage in future earthquakes.

Summary Latent class models are increasingly used to assess the accuracy of medical diagnostic tests and other classifications when no gold standard is available and the true state is unknown. When the latent class is treated as the true class, the latent class models provide measures of components of accuracy including specificity and sensitivity and their complements, type I and type II error rates. The error rates according to the latent class model differ from the true error rates, however, and empirical comparisons with a gold standard suggest the true error rates often are larger. We investigate conditions under which the true type I and type II error rates are larger than those provided by the latent class models. Results from Uebersax (1988, Psychological Bulletin 104, 405–416) are extended to accommodate random effects and covariates affecting the responses. The results are important for interpreting the results of latent class analyses. An error decomposition is presented that incorporates an error component from invalidity of the latent class model.

Average accuracy of jury verdicts for a set of cases can be studied empirically and systematically even when the correct verdict cannot be known. The key is to obtain a second rating of the verdict, for example, the judge’s, as in the recent study of criminal cases in the United States by the National Center for State Courts (NCSC). That study, like the famous Kalven-Zeisel study, showed only modest judge-jury agreement. Simple estimates of jury accuracy can be developed from the judge-jury agreement rate; the judge’s verdict is not taken as the gold standard. Although the estimates of accuracy are subject to error, under plausible conditions they tend to overestimate the average accuracy of jury verdicts. The jury verdict was estimated to be accurate in no more than 87 percent of the NCSC cases (which, however, should not be regarded as a representative sample with respect to jury accuracy). More refined estimates, including false conviction and false acquittal rates, are developed with models using stronger assumptions. For example, the conditional probability that the jury incorrectly convicts given that the defendant truly was not guilty (a “Type I error”) was estimated at 0.25, with an estimated standard error (s.e.) of 0.07, the conditional probability that a jury incorrectly acquits given that the defendant truly was guilty (“Type II error”) was estimated at 0.14 (s.e. 0.03), and the difference was estimated at 0.12 (s.e. 0.08). The estimated number of defendants in the NCSC cases who truly are not guilty but are convicted does seem to be smaller than the number who truly are guilty but are acquitted. The conditional probability of a wrongful conviction, given that the defendant was convicted, is estimated at 0.10 (s.e. 0.03).

In cellular biology, node-and-edge graph or “network” data collection often uses bait-prey technologies such as co-immunoprecipitation (CoIP). Bait-prey technologies assay relationships or “interactions” between protein pairs, with CoIP specifically measuring protein complex co-membership. Analyses of CoIP data frequently focus on estimating protein complex membership. Due to budgetary and other constraints, exhaustive assay of the entire network using CoIP is not always possible. We describe a stratified sampling scheme to select baits for CoIP experiments when protein complex estimation is the main goal. Expanding upon the classic framework in which nodes represent proteins and edges represent pairwise interactions, we define generalized nodes as sets of adjacent nodes with identical adjacency outside the set and use these as strata from which to select the next set of baits. Strata are redefined at each round of sampling to incorporate accumulating data. This scheme maintains user-specified quality thresholds for protein complex estimates and, relative to simple random sampling, leads to a marked increase in the number of correctly estimated complexes at each round of sampling. The R package seqSample contains all source code and is available at

We explore whether less detailed probabilistic hazard maps might perform better by assessing how smoothing Japan's national earthquake hazard maps affects their fit to a 510-year record of shaking. As measured by the fractional exceedance metric implicit in such probabilistic hazard maps, simple smoothing over progressively larger areas improves the maps' performance such that in the limit a uniform map performs best. However, using the squared misfit between maximum observed shaking and that predicted as a metric, map performance improves up to a ~75-150 km smoothing window, and then decreases with further smoothing. This result suggests that the probabilistic hazard models and the resulting maps may be over-parameterized, in that including too high a level of detail to describe past and future earthquakes may lower the maps' ability to predict future shaking.

Networks are a popular tool for representing elements in a system and their interconnectedness. Many observed networks can be viewed as only samples of some true underlying network. Such is frequently the case, for example, in the monitoring and study of massive, online social networks. We study the problem of how to estimate the degree distribution - an object of fundamental interest - of a true underlying network from its sampled network. In particular, we show that this problem can be formulated as an inverse problem. Playing a key role in this formulation is a matrix relating the expectation of our sampled degree distribution to the true underlying degree distribution. Under many network sampling designs, this matrix can be defined entirely in terms of the design and is found to be ill-conditioned. As a result, our inverse problem frequently is ill-posed. Accordingly, we offer a constrained, penalized weighted least-squares approach to solving this problem. A Monte Carlo variant of Stein's unbiased risk estimation (SURE) is used to select the penalization parameter. We explore the behavior of our resulting estimator of network degree distribution in simulation, using a variety of combinations of network models and sampling regimes. In addition, we demonstrate the ability of our method to accurately reconstruct the degree distributions of various sub-communities within online social networks corresponding to Friendster, Orkut and LiveJournal. Overall, our results show that the true degree distributions from both homogeneous and inhomogeneous networks can be recovered with substantially greater accuracy than reflected in the empirical degree distribution resulting from the original sampling., Published at http://dx.doi.org/10.1214/14-AOAS800 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The sensitivity of a benefit-cost analysis of a data program to incorrect specifications of the cost function C and benefit function B is an important consideration for producers and consumers of the analyses. We consider the effects of misspecification of B or C on the optimal data expenditure and optimal data quality. Geometric and analytical results show when the analysis will be sensitive and when order of magnitude estimates for costs and benefits will suffice in practice. A benefit-cost analysis of the 1970 census illustrates these results. We also show why misspecification of benefits typically has a larger effect than comparable misspecification of costs. If the benefits and the costs are both large but their difference small, the optimal expenditure for data (or optimal data quality) will be highly sensitive to changes in specifications of costs, benefits, data use, etc. In other cases, the sensitivity of the prescribed optimal expenditure to scalemisspecification of costs or benefits depends on the sharpness of the bend of the graph of B versus C near the optimum. Although the prescribed optimal expenditure is the same when B is misspecified as when C is comparably misspecified, the levels of data quality attained can differ. When B is misspecified, the data quality attained decreases if benefits are understated. When C is misspecified, the data quality attained usually decreases if costs are overstated, but for some unusual decision problems it increases and in rare, ‘self-correcting’ problems, it remains unchanged.

In July 1991 the Census Bureau recommended to its parent agency, the Department of Commerce, that the 1990 census be adjusted for undercount. The Secretary of Commerce decided not to adjust, however. Those decisions relied at least partly on the Census Bureau's analyses of the accuracy of the census and of the proposed undercount adjustments based on the Post-Enumeration Survey (PES). Error distributions for the nation, states, and smaller geographic units were estimated with extensions of methods applied to test censuses. To summarize and assess the relative importance of errors in different units, the Census Bureau used aggregate loss functions. This article describes the total error analysis and loss function analysis of the Census Bureau. In its decision not to adjust the census, the Department of Commerce cited different criteria than aggregate loss functions. Those criteria are identified and discussed.

This article considers surveys where one can observe, after the sample is selected, that each member of the sample belongs to one or more aggregations. The population of aggregations is of interest, and we consider the probability that a given aggregation contains at least one sample member. An expression for the probability is derived as a function of population parameters, many of which can be known only if additional, costly data collection is undertaken. A variety of model-based estimators of those parameters is discussed, and their relative advantages and disadvantages are noted. The estimators are applied to a particular case, the National Educational Longitudinal Study of 1988 (NELS:88), in which the aggregations are tenth-grade schools. Evaluations of the estimators are presented. In NELS:88, a sample of eighth-grade schools and eighth-grade students in those schools was surveyed and 2 years later, when most of the students were in tenth grade, the students were resurveyed. If the tenth-grade school sample (i.e., the set of tenth-grade schools enrolling one or more sampled students) is a probabiliy sample, then we can make inferences to the population of tenth-grade schools. A requirement for a sample to be a probability sample is that the selection probabilities be nonzero for all units and be known for the selected units. This article discusses how those probabilities can be estimated from complete and incomplete data. Various estimators based on incomplete data are derived and empirically evaluated with data from a special test sample of schools and data from the Houston Independent School District.

The amount worth spending to collect and analyze data depends on the uses of the data. We consider the determination of the optimal expenditure in some simple situations when only some aspects of the data use are known. In particular, we consider uses where an action is taken if a statistic exceeds a threshold τ, which is unknown at the time of data planning and collection (though its value will not depend on the value of the statistic). We analyze the sensitivity of the optimal expenditure to the a priori uncertainty in τ. For many decision problems, the optimal expenditure decreases with increasing uncertainty about τ, but we identify some uses for which the optimal expenditure increases with increasing uncertainty about τ. We also study the effects of other forms of ambiguity. Uncertainty about whether the data will be used typically decreases the optimal expenditure, as does uncertainty about the preferences represented in the decision making.

Official forecasts of mortality depend on assumptions about target values for the future rates of decline in mortality rates. Smooth functions connect the jump‐off (base‐year) mortality to the level implied by the targets. Three alternative sets of targets are assumed, leading to high, middle, and low forecasts. We show that this process can be closely modeled using simple linear statistical models. These explicit models allow us to analyze the error structure of the forecasts. We show that the current assumption of perfect correlation between errors in different ages, at different forecast years, and for different causes of death, is erroneous. An alternative correlation structure is suggested, and we show how its parameters can be estimated from the past data. The effect of the level of aggregation on the accuracy of mortality forecasts is considered. It is not clear whether or not age‐ and cause‐specific analyses have been more accurate in the past than analyses based on age‐specific mortality alone wo...

Verdicts in criminal trials can err by convicting an innocent person or acquitting a guilty defendant. Recent work shows how the rates and relative numbers of both types of errors can be measured statistically both for judge verdicts and for jury verdicts (Spencer 2007, Gastwirth and Sinclair 1998). The key is to have multiple ratings of the cases, e.g., also record the judge's belief of the correct verdict in a jury trial or an expert observer's belief of the correct verdict in a non-jury trial. Analysis of a non-random sample of state court cases in 2000-01 from four jurisdictions studied by the National Center for State Courts (Hannaford-Agor et al 2003) yields estimates that approximately 17% of the jury verdicts were incorrect: 7% were wrongful convictions and 10% were wrongful acquittals; it was estimated that 12% of the judges' verdicts were incorrect: 10% of judges' verdicts wrongful convictions and 1% wrongful acquittals. Those numbers are subject to a number of limitations, not least the non-representative nature of the sample of cases; this paper reviews those limitations not only for the analysis of the NCSC data but, more importantly, with an eye toward future studies.

"We describe a methodology for estimating the accuracy of dual systems estimates (DSE's) of population, census estimates of population, and estimates of undercount in the census. The DSE's are based on the census and a post-enumeration survey (PES). We apply the methodology to the 1988 dress rehearsal census of St. Louis and east-central Missouri and we discuss its applicability to the 1990 [U.S.] census and PES. The methodology is based on decompositions of the total (or net) error into components, such as sampling error, matching error, and other nonsampling errors. Limited information about the accuracy of certain components of error, notably failure of assumptions in the 'capture-recapture' model, but others as well, lead us to offer tentative estimates of the errors of the census, DSE, and undercount estimates for 1988. Improved estimates are anticipated for 1990." Comments are included by Eugene P. Ericksen and Joseph B. Kadane (pp. 855-7) and Kenneth W. Wachter and Terence P. Speed (pp. 858-61), as well as a rejoinder by Mulry and Spencer (pp. 861-3).

The Office of the Actuary, U.S. Social Security Administration, produces alternative forecasts of mortality to reflect uncertainty about the future. Appropriate probabilistic interpretations of the intervals have not been provided, however, because explicit stochastic models are not used. In this article we identify the components and assumptions of the official forecasts and approximate them by stochastic parametric models. We estimate parameters of the models from past data, derive statistical intervals for the forecasts, and compare them with the official high-low intervals. We use the models to evaluate the forecasts rather than to develop different predictions of the future. Analysis of data from 1972 to 1985 shows that the official intervals for mortality forecasts for males or females aged 45–70 have approximately a 95% chance of including the true mortality rate in any year. For other ages the chances are much less than 95%. The Office of the Actuary's interval estimates of age- and cause...

PART I: USES OF STATISTICS FOR DESCRIPTION PART II DATA COLLECTION PART III USES OF STATISTICS FOR POLICY ANALYSIS

Two problems that arise in apportioning the U.S. House of Representatives are: (a) fractional numbers of representatives cannot be allocated, so states receive different per capita representation, and (b) the state population sizes are known only with error. Both problems are addressed in a unified way with decision theory. Although the method currently in use, equal proportions, has poor properties when the populations are assumed perfectly known, it performs surprisingly well in the presence of modest errors in the data. The converse is true for the quota method. Previously developed qualitative notions of bias in apportionment methods are extended to provide a quantitative definition of bias. The new definition accounts both for bias in the apportionment method and for bias arising from imperfect population measurements. Illustrative estimates of the bias against states with large black populations are developed.

"This report contains recommendations reflecting the views of the ASA Technical Panel on the Census Undercount concerning [U.S.] Census Bureau procedures and plans in the fall of 1982." The report consists of technical statistical advice on measurement and estimation problems. A response by Barbara A. Bailar, reflecting the views of the U.S. Census Bureau, is also included (pp. 257-60).

Commonly used techniques for estimating the effect of data error on fund allocations are biased because they fail to account for numerous sources of data error and for misspecification of the allocation formula itself. Alternative techniquesfor estimating the effect of data error are presented and applied to estimate the effect of census undercount on general revenue sharing allocations.

Statistical data are expanding in use and importance in our society. Good data are expensive, and the increasing demand must compete with other needs for scarce resources. A recommended way to cope with this problem involves rational setting of statistical priorities. The need for benefit-cost analysis of statistical programs has risen sharply with the mushrooming uses of data in determining the allocations of funds. Thus, a recent study found that statistical data were used in determining the distribution of $122 billion - one-fifth of the federal budget - under 150 domestic assistance programs in fiscal year 1979. The explosion in allocative uses of data has serious consequences for statistical activities. As allocation of funds forms a highly visible use of statistical data, there will be a tendency to emphasize the importance of quality in these series used for fund allocations to the detriment of other statistical series. Benefit-cost analysis is currently practicable for several kinds of statistical activities, including the production, analysis, and dissemination of forecasts and projections, data used for determining allocations, and physical monitoring data. More benefit-cost analysis needs to be carried out for the kinds of data programs just mentioned. Lack of benefit-cost analysis will likely resultmore » in overemphasis of statistical activities with the most obvious, but not the greatest, benefits. Support for many of the most expensive data programs will tend to erode if their benefits are not perceived. The greatest potential value of benefit-cost analysis for data may arise in connection with social monitoring data, such as price indexes, achievement test scores, and other social indicators. Benefit-cost may be more difficult in these cases but pilot studies are feasible and appropriate. 45 refs.« less

Educational and psychological tests traditionally have informed judgments concerning individuals, but recently they have been widely used for comparing groups. Such uses include program evaluations (e.g., to what extent does Title I improve achievement), analysis of test score trends (e.g., the SAT decline) (Harnischfeger & Wiley, 1975), and comparison of contemporary subpopulations (e.g., achievement levels of black and white youth) (Burton & Jones, 1982). Using test scores (or item responses) to make group comparisons involves summarizing individuals' characteristics into a measure for the group. In many if not most applications, the measure chosen has been the average (i.e., arithmetic mean). But averaging scores (ability levels) is invalid if the scores (or abilities) are ordinal rather than cardinal attributes. In particular, group comparisons can change completely under order-preserving transformations of the test scales. This is illustrated by real and artificial examples. The notion of stochastic ordering is applied to develop group comparisons that are invariant under order-preserving transformations of the test scale.

Commonly accepted hypotheses guide practical determination of needed data quality; for example, as the probability that a decision uses data increases, the needed data quality increases, and the more rudimentary the uses of the data, the less data quality is needed. These hypotheses are formally defined and analyzed in some decision-theoretic models. Conditions under which the hypotheses hold and fail are examined. Particular attention is given to determining needed data quality when the users of the data behave nonoptimally.