Multinomial Setting Research Articles

Deletion Diagnostics in Logistic Regression

Today, there are not many good measures for detecting influential observations in case of fitting a logistic regression model. So, the purpose of this article is to extrapolate from the pre-existing deletion diagnostics defined for detecting influential points for multiple linear regression, i.e. the DFFITS, DFBETAS and Cook's Distance to the scenario of a binary logistic regression model and then view the multinomial model as a special case of the same. The threshold for determining whether an observation is an influential observation or not is judged using the asymptotic distribution of the Cook's Distance in the multinomial setting, both for the single and the group deleted case. The results are examined under various simulation scenarios as well as over the modified Kyphosis data-set.

Poisson Limits for Sequential Multivariate Multinomial Data

A well known result in probability is the Poisson limit for rare independent Bernoulli trials. The asymptotic result also holds for a multinomial setting where events are dependent. An interesting data type is a survival process in which one observes many individuals over time periods. The risk set is all those available at the beginning of each time period, thus the same individual can appear in successive risk sets. Individuals exit rarely. In our motivating example these are corporations who exit a public trading system by default or merger, so there are several exit types, hence the multinomial setting. There are also covariates available at the beginning of each period. We study the numbers of exits over time. Under rare multinomial conditions we show that the exits types converge to independent Poisson laws with respect to the exit types and also with respect to time. An immediate application is the construction of one step ahead predictions which may then be tabulated or plotted, giving a convenient tool to study themodel behaviour with respect to time. Thus one can obtain one step ahead prediction intervals for the number of exits of each type, in our case bankruptcy or merger. This is a tool that is useful for large institutional investors such as pension plans.

Asymmetric, closed-form, finite-parameter models of multinomial choice

Class imbalance, where there are great differences between the number of observations associated with particular discrete outcomes, is common within transportation and other fields. In the statistics literature, one explanation for class imbalance that has been hypothesized is an asymmetric (rather than the typically symmetric) choice probability function. Unfortunately, few relatively simple models exist for testing this hypothesis in transportation settings—settings that are inherently multinomial. Our paper fills this gap.In particular, we address the following questions: “how can one construct asymmetric, closed-form, finite-parameter models of multinomial choice” and “how do such models compare against commonly used symmetric models?” Methodologically, we introduce (1) a new class of closed-form, finite-parameter, multinomial choice models, (2) a procedure for using these models to extend existing binary choice models to the multinomial setting, and (3) a procedure for creating new binary choice models (both symmetric and asymmetric). Together, our contributions allow us to create new asymmetric, closed-form, finite-parameter multinomial choice models. We demonstrate our methods by developing four new asymmetric, multinomial choice models. Empirically, most of our models strongly dominate the multinomial logit (MNL) model in terms of in-sample and out-of-sample log-likelihoods. Moreover, analyzing two policy applications, we find practical differences between the MNL and our new asymmetric models. Our results suggest that while asymmetric models may not always outperform symmetric ones, asymmetric choice models are worth testing because they might have better statistical performance and entail substantively different policy and financial implications when compared with traditional symmetric models, such as the MNL.

What determines outstanding corporate debt mix? Evidence from fractional multinomial logit estimation

ABSTRACTThis study employs the fractional multinomial logit setting proposed by Papke and Wooldridge (1996) to examine factors driving the choice among nonbank private (144A) debt, bank loans and public debt made by 988 nonfinancial firms during 1993–2007. We document that the majority of firm-level factors have persistent effects on corporate outstanding debt mix across economic conditions. We also highlight the importance of macroeconomic variables on firms’ borrowing decisions as predicted by Diamond (1991). Finally, we document a substitution effect among debt financing sources due to credit rating downgrades, which is inconsistent with Rauh and Sufi (2010).

Salaried employment and earnings in Indonesia: new evidence on the selection bias

This article uses household survey data to estimate the determinants of earnings in Indonesia, a country where nonsalaried work is widespread and earnings data are available for salaried employees only. We deal with the selection bias by estimating a Full-Information Maximum Likelihood (FIML) system of equations, where selection into the labour market is modelled in a multinomial setting. We find that some estimated parameters of the earnings equation differ from a binomial selection procedure by Heckman (1979), in particular for those variables with the strongest impact on the selection into the different labour-market statuses. However, the estimated returns to education are unaffected, even when we deal with the endogeneity of educational attainment following Duflo (2001). Overall, our findings show that the choice of the selection rule affects the estimates of the earnings determinants in the Indonesian labour market.

Building and using semiparametric tolerance regions for parametric multinomial models

We introduce a semiparametric “tubular neighborhood” of a parametric model in the multinomial setting. It consists of all multinomial distributions lying in a distance-based neighborhood of the parametric model of interest. Fitting such a tubular model allows one to use a parametric model while treating it as an approximation to the true distribution. In this paper, the Kullback–Leibler distance is used to build the tubular region. Based on this idea one can define the distance between the true multinomial distribution and the parametric model to be the index of fit. The paper develops a likelihood ratio test procedure for testing the magnitude of the index. A semiparametric bootstrap method is implemented to better approximate the distribution of the LRT statistic. The approximation permits more accurate construction of a lower confidence limit for the model fitting index.

A method for the meta‐analysis of mutually exclusive binary outcomes

Meta-analyses of multiple outcomes need to take into account the within-study correlation across the different outcomes. Here we focus on the meta-analysis of dichotomous outcomes that are mutually exclusive and exhaustive. Correlations between effect sizes for mutually exclusive outcomes are negative and can be obtained from data already available. We present both fixed-effects and random-effects methods that account for the negative correlations and yield correct simultaneous confidence intervals for both the marginal outcome-specific effect sizes and the relative effect sizes between outcomes. Formulae for the odds ratio, risk ratio, risk difference, and the differences in the arcsin-transformed risks are provided. An example of a meta-analysis of randomized trials of radiotherapy and mastectomy with axillary lymph node clearance versus only mastectomy with axillary clearance for early breast cancer is presented. The mutually exclusive outcomes of breast cancer deaths and deaths secondary to other causes are examined in separate meta-analyses, and also by taking the between-outcome correlation into account. We argue that mutually exclusive outcomes in the meta-analyses of binary data are optimally analyzed in a multinomial setting. This may also be applicable when a meta-analysis examines only one out of several mutually exclusive outcomes. For large sample sizes and/or low event counts, the covariances between outcome-specific effect sizes are small, and either ignoring them or accounting for them would result in similar estimates for any practical purpose. However, meta-analysts should explore the robustness of the findings from individual meta-analyses when mutually exclusive outcomes are assessed.

The Log Multinomial Regression Model for Nominal Outcomes with More than Two Attributes

An estimate of the risk or prevalence ratio, adjusted for confounders, can be obtained from a log binomial model (binomial errors, log link) fitted to binary outcome data. We propose a modification of the log binomial model to obtain relative risk estimates for nominal outcomes with more than two attributes (the "log multinomial model"). Extensive data simulations were undertaken to compare the performance of the log multinomial model with that of an expanded data multinomial logistic regression method based on the approach proposed by Schouten et al. (1993) for binary data, and with that of separate fits of a Poisson regression model based on the approach proposed by Zou (2004) and Carter, Lipsitz and Tilley (2005) for binary data. Log multinomial regression resulted in "inadmissable" solutions (out-of-bounds probabilities) exceeding 50% in some data settings. Coefficient estimates by the alternative methods produced out-of-bounds probabilities for the log multinomial model in up to 27% of samples to which a log multinomial model had been successfully fitted. The log multinomial coefficient estimates generally had lesser relative bias and mean squared error than the alternative methods. The practical utility of the log multinomial regression model was demonstrated with a real data example. The log multinomial model offers a practical solution to the problem of obtaining adjusted estimates of the risk ratio in the multinomial setting, but must be used with some care and attention to detail.

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

A problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any r ( r ≥ 1 ) probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering.

Non-parametric Back-projection of HIV Positive Tests using Multinomial and Poisson Settings

Back-projection is a commonly used method in reconstructing HIV incidence. Instead of using AIDS incidence data in back-projection, this paper uses HIV positive tests data. Both multinomial and Poisson settings are used. The two settings give similar results when a parametric form or step function is assumed for the infection curve. However, this may not be true when the HIV infection in each year is characterized by a different parameter. This paper attempts to use simulation studies to compare these two settings by constructing various scenarios for the infection curve. Results show that both methods give approximately the same estimates of the number of HIV infections in the past, whilst the estimates for HIV infections in the recent past differ a lot. The multinomial setting always gives a levelling-off pattern for the recent past, while the Poisson setting is more sensitive to the change in the shape of the HIV infection curve. Nonetheless, the multinomial setting gives a relatively narrower point-wise probability interval. When the size of the epidemic is large, the narrow probability interval may be under-estimating the true underlying variation.

Power and sample size for ordered categorical data.

We propose a new method for computing power and sample size for linear rank tests of differences between two ordered multinomial populations. The method is flexible in that it is applicable to any general alternative hypothesis and for any choice of rank scores. We show that the method, though asymptotic, closely approximates existing exact methods. At the same time it overcomes the computational limitations of the exact methods. This advantage makes our asymptotic approach more practical for sample size computations at the planning stages of a large study. We illustrate the method with data arising from both proportional and non-proportional odds models in the two ordered multinomial setting.

Multinomial estimation procedures for two stochastically ordered distributions

Standard estimation procedures in a multinomial setting are the methods of maximum likelihood, minimum chi-square, Neyman modified minimum chi-square, minimum discrimination information, and the Freeman-Tukey criteria. In this paper it is shown that if the parameter region of interest is restricted to be that of stochastic ordering between two multinomial parameters, then all these procedures can also be related as in Dykstra and Lee (1991) and the corresponding estimates can be expressed in terms of a simple weighted least square projection.

Testing for and against a set of linear inequality constraints in a multinomial setting

AbstractThere are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one‐sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely pij = Pji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood‐ratio results to cover more general situations. To be specific, we consider testing Ht,0 against H1 ‐ H0 and H1 against H2 ‐ H 1 when H0:k × i=1 pixji = 0, j = 1,…, s, H1:k × i=1 pixji × 0, j = 1,…, s, and does not impose any restrictions on p. The xji's are known constants, and s × k ‐ 1. We show that the asymptotic distributions of the likelihood‐ratio tests are of chi‐bar‐square type, and provide expressions for the weighting values.

Selecting among the multinomial losers

In a multinomial setting with a fixed number k of cells. the problem of screening out cells to find the cell, i.e., the one with the smallest cell probability, or looking for a (small) subset of cells containing the best cell is revisited. An inverse sampling procedure is used, unlike past work on this problem ([l], [2], [3], and [4]). Finding the cell with the smallest cell probability is clearly more difficult than finding the one with the argest cell probability. The proposed procedure takes one observation at a time (as usual) and igns a zero to all those (and only those) k - 1 cells into which the observation does not fall Sampling continues sequentially and stops as soon as any one cell has accumulated r zeros. For any given integer c (with 0 ≤ c < r), we put into the selected subset (SS) all those cells with at least r - c zeros and assert that this selected subset contains the best cell. It is important to note that for the slippage configuration (SC) we can attain any specified lower bound...

Using Discrete Distributions to Compute Powers for Goodness-of-Fit Test Statistics in a One-Way Multinomial Setting

Both the Bionomial and Poisson distributions are employed in this study to compute approximate powers for goodness-of-fit test statistics. The procedure adopted involves simulating 1000 samples from each of these distributions. These samples are then employed to compute both the randomized nominal critical values and estimated powers. The type I error rates returned from the use of the randomized critical levels Cα fall within the acceptable regions. We illustrate the use of the procedure with the Pearson's X2 test statistic and show that this can readily be extended to any of the other well known goodness-of-fit test statistics.

Multinomial estimation procedures for isotonic cones

Standard estimation procedures in a multinomial setting are the methods of maximum likelihood, minimum chi-square, Neyman modified minimum chi-square, minimum discriminate information, and the Freeman–Tukey criteria: In this paper it is shown that if the parameter region of interest is restricted by an isotonic cone, then all these procedures can be related and the corresponding estimates expressed in terms of equal weights, least squares projections.