The Use of Maximum Likelihood Estimates in {\chi^2} Tests for Goodness of Fit
01 Sep 1954-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 25, Iss: 3, pp 579-586
TL;DR: In this article, it was shown that the test statistic does not have a limiting χ2-distribution, but that it is stochastically larger than would be expected under the χ 2 theory.
Abstract: The usual test that a sample comes from a distribution of given form is performed by counting the number of observations falling into specified cells and applying the χ2 test to these frequencies. In estimating the parameters for this test, one may use the maximum likelihood (or equivalent) estimate based (1) on the cell frequencies, or (2) on the original observations. This paper shows that in (2), unlike the well known result for (1), the test statistic does not have a limiting χ2-distribution, but that it is stochastically larger than would be expected under the χ2 theory. The limiting distribution is obtained and some examples are computed. These indicate that the error is not serious in the case of fitting a Poisson distribution, but may be so for the fitting of a normal.
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TL;DR: In this paper, the power of the Kolmogorov-smirnov test is investigated and a table for testing whether a set of observations is from a normal population when the mean and variance are not specified but must be estimated from the sample.
Abstract: The standard tables used for the Kolmogorov-Smirnov test are valid when testing whether a set of observations are from a completely-specified continuous distribution. If one or more parameters must be estimated from the sample then the tables are no longer valid. A table is given in this note for use with the Kolmogorov-Smirnov statistic for testing whether a set of observations is from a normal population when the mean and variance are not specified but must be estimated from the sample. The table is obtained from a Monte Carlo calculation. A brief Monte Carlo investigation is made of the power of the test.
3,923 citations
TL;DR: This paper developed and adapted statistical models of counts (nonnegative integers) in the context of panel data and used them to analyze the relationship between patents and R&D expenditures. But their model is not suitable for the analysis of large-scale data sets.
Abstract: This paper focuses on developing and adapting statistical models of counts (nonnegative integers) in the context of panel data and using them to analyze the relationship between patents and R & D expenditures. Since a variety of other economic data come in the form of repeated counts of some individual actions or events, the methodology should have wide applications. The statistical models we develop are applications and generalizations of the Poisson distribution. Two important issues are (i) Given the panel nature of our data, how can we allow for separate persistent individual (fixed or random) effects? (ii) How does one introduce the equivalent of disturbances-in-the-equation into the analysis of Poisson and other discrete probability functions? The first problem is solved by conditioning on the total sum of outcomes over the observed years, while the second problem is solved by introducing an additional source of randomness, allowing the Poisson parameter to be itself randomly distributed, and compounding the two distributions. Lastly, we develop a test statistic for the presence of serial correlation when fixed effects estimators are used in nonlinear conditional models.
2,947 citations
TL;DR: In this article, several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model, which are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions.
Abstract: Several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model. The test statistics are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions. The null distributions of these statistics are examined by applying the theory for chi-square tests of Moore Spruill (1975) and through computer simulations. All statistics are shown to have a chi-square distribution or a distribution which can be well approximated by a chi-square. The degrees of freedom are shown to depend on the particular statistic and the distributional assumptions. The power of each of the proposed statistics is examined for the normal, linear, and exponential alternative models using computer simulations.
1,463 citations
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TL;DR: In this article, the authors developed and adapted statistical models of counts (nonnegative integers) in the context of panel data and used them to analyze the relationship between patents and RD persistent individual (fixed or random) effects, and "noise" or randomness in the Poisson probability function.
Abstract: This paper focuses on developing and adapting statistical models of counts (non-negative integers) in the context of panel data and using them to analyze the relationship between patents and RD persistent individual (fixed or random) effects, and "noise" or randomness in the Poisson probability function. We apply our models to a data set previously analyzed by Pakes and Griliches using observations on 128 firms for seven years, 1968-74. Our statistical results indicate clearly that to rationalize the data, we need both a disturbance in the conditional within dimension and a different one, with a different variance, in the marginal (between) dimension. Adding firm specific variables, log book value and a scientific industry dummy, removes most of the positive correlation between the individual firm propensity to patent and its R&D intensity. The other new finding is that there is an interactive negative trend in the patents - R&D relationship, that is, firms are getting less patents from their more recent R&D investments, implying a decline in the "effectiveness" or productivity of R&D.
1,093 citations
TL;DR: Data support for the first time the hypothesis that Fc gamma R-mediated ADCC plays an important role in the clinical effect of trastuzumab.
Abstract: Purpose The anti–HER-2/neu monoclonal antibody trastuzumab has been shown to engage both activatory (fragment C receptor [FcγR]IIIa; FcγRIIa) and inhibitory (FcγRIIb) antibody receptors and FcγR polymorphisms have been identified that may affect the antibody-dependent cell-mediated cytotoxicity (ADCC) of natural-killer cells/monocytes. In this study, we tested whether FcγR polymorphisms are associated with clinical outcome of patients with breast cancer who received trastuzumab. Patients and Methods Fifty-four consecutive patients with HER-2/neu–amplified breast cancer receiving trastuzumab plus taxane for metastatic disease were evaluated for genotype for the FcγRIIIa-158 valine(V)/phenylalanine(F), FcγRIIa-131 histidine(H)/arginine(R), and FcγRIIb-232 isoleucine(I)/threonine(T) polymorphisms. Trastuzumab-mediated ADCC of patients' peripheral blood mononuclear cells (PBMCs) was measured by chromium-51 release using a HER-2/neu–expressing human breast cancer cell line as a target. Controls comprised thirt...
963 citations
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TL;DR: In this paper, the asymptotic distribution of the likelihood ratio λ is examined when the value of the parameter is a boundary point of both the set of points corresponding to the hypothesis and the set corresponding to an alternative.
Abstract: A classical result due to Wilks [1] on the distribution of the likelihood ratio $\lambda$ is the following. Under suitable regularity conditions, if the hypothesis that a parameter $\theta$ lies on an $r$-dimensional hyperplane of $k$-dimensional space is true, the distribution of $-2 \log \lambda$ is asymptotically that of $\chi^2$ with $k - r$ degrees of freedom. In many important problems it is desired to test hypotheses which are not quite of the above type. For example, one may wish to test whether $\theta$ is on one side of a hyperplane, or to test whether $\theta$ is in the positive quadrant of a two-dimensional space. The asymptotic distribution of $-2 \log \lambda$ is examined when the value of the parameter is a boundary point of both the set of $\theta$ corresponding to the hypothesis and the set of $\theta$ corresponding to the alternative. First the case of a single observation from a multivariate normal distribution, with mean $\theta$ and known covariance matrix, is treated. The general case is then shown to reduce to this special case where the covariance matrix is replaced by the inverse of the information matrix. In particular, if one tests whether $\theta$ is on one side or the other of a smooth $(k - 1)$-dimensional surface in $k$-dimensional space and $\theta$ lies on the surface, the asymptotic distribution of $\lambda$ is that of a chance variable which is zero half the time and which behaves like $\chi^2$ with one degree of freedom the other half of the time.
747 citations
TL;DR: In this article, the distribution functions of a positive quadratic form in normal variates and the ratio of two independent forms of this type were derived using the method of mixtures.
Abstract: The method of mixtures, explained in Section 2, is applied to derive the distribution functions of a positive quadratic form in normal variates and of the ratio of two independent forms of this type.
157 citations