arXiv subjects

Paul T. von Hippel

Publications and source records attributed to Paul T. von Hippel.

12 recordsLinked to original sources

arXiv2026

Faster Estimates from Binned Test Score Data---and How Accurate They Are

Education agencies often summarize test score distributions by counting how many students scored in 3 to 5 different \textit{bins}. The HETOP model transforms bin counts into estimated means and standard deviations, assuming that scores follow a normal distribution within each school or district. Past HETOP implementations ran slowly, taking 3--60 minutes, if they finished, when given bin counts for all 1,151 districts in Texas. Our new function, \code{fast\_hetop()} in the R package \pkg{binest}, runs all Texas districts in less than a second. Users can choose between maximum likelihood, empirical Bayes, sample-based or population-based estimates. Estimates are strongly correlated with true values, but have bias when the score distribution is skewed and scores are concentrated in the lowest or highest bin.

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arXiv2025

Imputing With Predictive Mean Matching Can Be Severely Biased When Values Are Missing At Random

Predictive mean matching (PMM) is a popular imputation strategy that imputes missing values by borrowing observed values from other cases with similar expectations. We show that, unlike other imputation strategies, PMM is not guaranteed to be consistent -- and in fact can be severely biased -- when values are missing at random (when the probability a value is missing depends only on values that are observed). We demonstrate the bias in a simple situation where a complete variable $X$ is both strongly correlated with $Y$ and strongly predictive of whether $Y$ is missing. The bias in the estimated regression slope can be as large as 80 percent, and persists even when we reduce the correlation between $X$ and $Y$. To make the bias vanish, the sample must be large ($n$=1,000) \emph{and} $Y$ values must be missing independently of $X$ (i.e., missing completely at random). Compared to other imputation methods, it seems that PMM requires larger samples and is more sensitive to the pattern of missing values. We cannot recommend PMM as a default approach to imputation.

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arXiv2019

Maximum likelihood multiple imputation: Faster imputations and consistent standard errors without posterior draws

Multiple imputation (MI) is a method for repairing and analyzing data with missing values. MI replaces missing values with a sample of random values drawn from an imputation model. The most popular form of MI, which we call posterior draw multiple imputation (PDMI), draws the parameters of the imputation model from a Bayesian posterior distribution. An alternative, which we call maximum likelihood multiple imputation (MLMI), estimates the parameters of the imputation model using maximum likelihood (or equivalent). Compared to PDMI, MLMI is less computationally intensive, faster, and yields slightly more efficient point estimates. A past barrier to using MLMI was the difficulty of estimating the standard errors of MLMI point estimates. We derive, implement, and evaluate three consistent standard error formulas: (1) one combines variances within and between the imputed datasets, (2) one uses the score function, and (3) one uses the bootstrap to estimate variance components due to sampling and imputation. Formula (1) modifies for MLMI a formula that has long been used under PDMI, while formulas (2) and (3) can be used without modification under either PDMI or MLMI. We have implemented MLMI and the standard error estimators in the mlmi and bootImpute packages for R.

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arXiv2018

How many imputations do you need? A two-stage calculation using a quadratic rule

When using multiple imputation, users often want to know how many imputations they need. An old answer is that 2 to 10 imputations usually suffice, but this recommendation only addresses the efficiency of point estimates. You may need more imputations if, in addition to efficient point estimates, you also want standard error (SE) estimates that would not change (much) if you imputed the data again. For replicable SE estimates, the required number of imputations increases quadratically with the fraction of missing information (not linearly, as previous studies have suggested). I recommend a two-stage procedure in which you conduct a pilot analysis using a small-to-moderate number of imputations, then use the results to calculate the number of imputations that are needed for a final analysis whose SE estimates will have the desired level of replicability. I implement the two-stage procedure using a new Stata command called how_many_imputations (available from SSC) and a new SAS macro called %mi_combine (available from the website missingdata.org).

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arXiv2017

Better estimates from binned income data: Interpolated CDFs and mean-matching

Researchers often estimate income statistics from summaries that report the number of incomes in bins such as \$0-10,000, \$10,001-20,000,...,\$200,000+. Some analysts assign incomes to bin midpoints, but this treats income as discrete. Other analysts fit a continuous parametric distribution, but the distribution may not fit well. We fit nonparametric continuous distributions that reproduce the bin counts perfectly by interpolating the cumulative distribution function (CDF). We also show how both midpoints and interpolated CDFs can be constrained to reproduce the mean of income when it is known. We compare the methods' accuracy in estimating the Gini coefficients of all 3,221 US counties. Fitting parametric distributions is very slow. Fitting interpolated CDFs is much faster and slightly more accurate. Both interpolated CDFs and midpoints give dramatically better estimates if constrained to match a known mean. We have implemented interpolated CDFs in the binsmooth package for R. We have implemented the midpoint method in the rpme command for Stata. Both implementations can be constrained to match a known mean.

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arXiv2017

Should a Normal Imputation Model Be Modified to Impute Skewed Variables?

Researchers often impute continuous variables under an assumption of normality, yet many incomplete variables are skewed. We find that imputing skewed continuous variables under a normal model can lead to bias; the bias is usually mild for popular estimands such as means, standard deviations, and linear regression coefficients, but the bias can be severe for more shape-dependent estimands such as percentiles or the coefficient of skewness. We test several methods for adapting a normal imputation model to accommodate skewness, including methods that transform, truncate, or censor (round) normally imputed values, as well as methods that impute values from a quadratic or truncated regression. None of these modifications reliably reduces the biases of the normal model, and some modifications can make the biases much worse. We conclude that, if one has to impute a skewed variable under a normal model, it is usually safest to do so without modifications -- unless you are more interested in estimating percentiles and shape that in estimated means, variance, and regressions. In the conclusion, we briefly discuss promising developments in the area of continuous imputation models that do not assume normality.

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arXiv2016

Robust estimation of inequality from binned incomes

Researchers must often estimate income inequality using data that give only the number of cases (e.g., families or households) whose incomes fall in "bins" such as $0-9,999, $10,000-14,999,..., $200,000+. We find that popular methods for estimating inequality from binned incomes are not robust in small samples, where popular methods can produce infinite, undefined, or arbitrarily large estimates. To solve these and other problems, we develop two improved estimators: the robust Pareto midpoint estimator (RPME) and the multimodel generalized beta estimator (MGBE). In a broad evaluation using US national, state, and county data from 1970 to 2009, we find that both estimators produce very good estimates of the mean and Gini, but less accurate estimates of the Theil and mean log deviation. Neither estimator is uniformly more accurate, but the RPME is much faster, which may be a consideration when many estimates must be obtained from many datasets. We have made the methods available as the rpme and mgbe commands for Stata and the binequality package for R.

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arXiv2016

Regression with missing Ys: An improved strategy for analyzing multiply imputed data

When fitting a generalized linear model -- such as a linear regression, a logistic regression, or a hierarchical linear model -- analysts often wonder how to handle missing values of the dependent variable Y. If missing values have been filled in using multiple imputation, the usual advice is to use the imputed Y values in analysis. We show, however, that using imputed Ys can add needless noise to the estimates. Better estimates can usually be obtained using a modified strategy that we call multiple imputation, then deletion (MID). Under MID, all cases are used for imputation, but following imputation cases with imputed Y values are excluded from the analysis. When there is something wrong with the imputed Y values, MID protects the estimates from the problematic imputations. And when the imputed Y values are acceptable, MID usually offers somewhat more efficient estimates than an ordinary MI strategy.

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arXiv2015

New Confidence Intervals and Bias Comparisons Show that Maximum Likelihood Can Beat Multiple Imputation in Small Samples

When analyzing incomplete data, is it better to use multiple imputation (MI) or full information maximum likelihood (ML)? In large samples ML is clearly better, but in small samples ML's usefulness has been limited because ML commonly uses normal test statistics and confidence intervals that require large samples. We propose small-sample t-based ML confidence intervals that have good coverage and are shorter than t-based confidence intervals under MI. We also show that ML point estimates are less biased and more efficient than MI point estimates in small samples of bivariate normal data. With our new confidence intervals, ML should be preferred over MI, even in small samples, whenever both options are available.

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arXiv2014

Estimates of heterogeneity (I2) can be biased in small meta-analyses

In meta-analysis, the fraction of variance that is due to heterogeneity is known as I2. We show that the usual estimator of I2 is biased. The bias is largest when a meta-analysis has few studies and little heterogeneity. For example, with 7 studies and the true value of I2 at 0, the average estimate of I2 is .124. Estimates of I2 should be interpreted cautiously when the meta-analysis is small and the null hypothesis of homogeneity (I2=0) has not been rejected. In small meta-analyses, confidence intervals may be preferable to point estimates for I2.

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arXiv2013

The Bias and Efficiency of Incomplete-Data Estimators in Small Univariate Normal Samples

Widely used methods for analyzing missing data can be biased in small samples. To understand these biases, we evaluate in detail the situation where a small univariate normal sample, with values missing at random, is analyzed using either observed-data maximum likelihood (ML) or multiple imputation (MI). We evaluate two types of MI: the usual Bayesian approach, which we call posterior draw (PD) imputation, and a little-used alternative, which we call ML imputation, in which values are imputed conditionally on an ML estimate. We find that observed-data ML is more efficient and has lower mean squared error than either type of MI. Between the two types of MI, ML imputation is more efficient than PD imputation, and ML imputation also has less potential for bias in small samples. The bias and efficiency of PD imputation can be improved by a change of prior.

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arXiv2012

Estimation with Binned Data

Variables such as household income are sometimes binned, so that we only know how many households fall in each of several bins such as $0-10,000, $10,000-15,000, or $200,000+. We provide a SAS macro that estimates the mean and variance of binned data by fitting the extended generalized gamma (EGG) distribution, the power normal (PN) distribution, and a new distribution that we call the power logistic (PL). The macro also implements a "best-of-breed" estimator that chooses from among the EGG, PN, and PL estimates on the basis of likelihood and finite variance. We test the macro by estimating the mean family and household incomes of approximately 13,000 US school districts between 1970 and 2009. The estimates have negligible bias (0-2%) and a root mean squared error of just 3-6%. The estimates compare favorably with estimates obtained by fitting the Dagum, generalized beta (GB2), or logspline distributions.

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