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AbstractThe general approach to modeling binary data for the purpose of estimating thepropagation of an internal solitary wave (ISW) is based on the maximum likelihoodestimate (MLE) method. In cases where the number of observations in the data is small,any inferences made based on the asymptotic distribution of changes in the deviance may be unreliable for binary data (themodel's lack of fit is described in terms of a quantity known as the deviance). Thedeviance for the binary data is given by D. Collett (2003).

May be unreliable for binary data.Logistic regression shows that the -values for the likelihood ratio test and the scoretest are both 0.05. However, the null hypothesis is not rejected in the Wald test.

Theseeming discrepancies in -values obtained between the Wald test and the other two testsare a sign that the large-sample approximation is not stable. We find that the parametersand the odds ratio estimates obtained via conditional exact logistic regression aredifferent from those obtained via unconditional asymptotic logistic regression. Usingexact results is a good idea when the sample size is small and the approximate -valuesare 0.10.

Thus in this study exact analysis is more appropriate. IntroductionInternal waves refer to the motion at the interfacebetween layers of water of different densities in a stratifiedwater body, such as the ocean.

The simplest oceanic density structure, wheredifferences in water density are mostly caused by differences in watertemperature or salinity, can be approximated by a two-layer model. Oceanic internalwaves typically have wavelengths ranging from hundreds of meters to tens ofkilometers, with periods from tens of minutes to tens of hours. In theAndaman and Sulu Sea they canhave amplitudes (peak to trough distance) exceeding 50m and in the SouthChina Sea the amplitude can exceed 110 m–.The mixing and dissipation generated by internal waves have important effectson the cross slope exchange processes, enhancement of bottom stress, andgeneration of the nepheloid layers. It has recently been proposed that internal waves may make a significant contributionto internal oceanic mixing and hence have an important influence on climaticchange. This is why it is necessary toscrutinize the interaction of nonlinear internal solitary waves (ISWs)with the seabed topography –.Several studies, including both simulations andlaboratory experiments, aimingat exploring the mechanisms for the generation, propagation, andevolution of ISWs, have already been carried out. However, since energydissipation plays such an important and varied role on water and sedimentary movementin coastal seas , we need a better fitting and more appropriate model for predictingISWpropagation. Apreliminary approach has recently been made in which the effects of weightedparameters on the amplitude and reflection of energy-based ISWs from uniformslopes in a two layered fluid system were investigated.

The results are quite consistent with other experimentalresults, and are applicable to the naturally occurring reflection of ISWs fromsloping bottoms. More recently, Chen et al.

concluded the goodness-of-fitand predictive ability of the cumulative logistic regression models to bebetter than that of the binary logistic regressionmodels. However, in cases where the data are so small that there are someobservations with proportions close to zero or one, inferences based on the asymptoticdistribution of the change in deviance may be unreliable. In point of fact, reports on statisticalmanipulations related to this theme are rather rare.The restof the paper is organized as follows. In Section we describe the experimentalset-up and theoretical background needed to understand the hydrodynamicinteraction. We also discuss the analysis of the logistic regressionmodel, and introduce the exact conditional logistic model and the hypothesis on which the parameters are based.Sectionis devoted to a comparison of the conditional exact logistic regression model and theunconditional asymptotic logistic regression model. Finally, some conclusionsare made. It is noted that small sample sizemeans that there are some observations with proportions close to zero or oneand P-values of less than 0.10, whichis an indication that an exact analysis would be more appropriate.

Research FrameworkExperiments were carried out in the laboratory using atwo-layer fluid system of fresh and briny water in a 12 m long wave flume (rectangular incross-section). Theupper layer of water in the waveflume consisted of fresh water with a density and a depth, while the lower layer was comprised ofbrine with a density anda depth. The leading ISW was generated by the lifting of apneumatic sluice gate at one end of the flume. The wave propagated into themain section of the flume to the left-hand side (LHS) of the gate. The amplitude a and characteristic length of the ISW were predetermined by arranging the steplength L and step depth (see Figure ).

Six ultrasonic probes connected to an amplifierunit and A/D converter, then to a personal computer, gathered and processeddigital signals as the ISW propagated along the flume. Asthe ISW propagated from the RHS (right-hand side) to the LHS of theflume, the first ultrasonic probe (P1) recorded the properties of the incidentISW, the wave amplitude and characteristic length, while the second probe (P2)collected reflected signals showing the wave-obstacle interaction. The methodology formeasuring the physical properties related to the propagation and dissipation ofthe ISW has been reported in detail by Chenet al. Theamplitude-based transmission rate during the wave-ridge interaction wasdependent on two factors, ridge height and potential energy. A schematicview showing the set up for ISW propagation in a two-layer fluid system over a single obstacle. ExactConditional Logistic Regression ModelThe theoretical basis for the exact conditional logisticregression model was originally laid down by Cox , but recent algorithmicadvances in computing the exact distributions have made the methodology morepractical.

Since then Hirji et al. have developed an efficient algorithm for generating the requiredconditional distributions.

Cox and Snell noted that it has been knownsince the 1970’show to extend the theory of Fisher’s exact test to logistic regression models.The interested reader may refer to Mehta and Patel for a useful summary ofexact logistic regression. A complete discussion of the exact logisticregression methodology and more detailed applications can be found in a varietyof sources –.Here, letrepresent the probability of “success” for a binary response for theexplanatory variables.The notation can be simplified by usingto represent the conditional mean of given when a logistic distribution isutilized: such that The transformation of,which is central to this study of logistic regression, is the logittransformation. Deviance and Pearson goodness-of-fit statistics. Regression DiagnosticsThereare a number of different ways to plot the regression diagnostics, eachdirected at a particular aspect of the fit. For examples seeHosmer and Lemeshow ,and Landwehr et al. who discussedgraphical techniques for logistic regression diagnostics.

Generally suchtechniques offer a visual rather than numerical representation that may be moreintuitively appealing to some researchers. Index plots are useful for the identificationof extreme values.

An examination of the index plots of the Pearsonresiduals (Figure ) and the deviance residuals (Figure ) for our data indicatesthat case 11 and case 27 are poorly accounted for by the model. It can be seenin the index plot of the diagonal elements of the hat matrix (Figure ) that case 49 is at theextreme point in the design space. Plot of hat diagonal (Resdev) versus casenumber index. Outliers andInfluential ObservationsThe values of outliers can be quite substantial and influential. A look atTable shows the advantage of removing such observations from the data (here, case11, case 27, and case 49), then refitting the newly revised model to theremaining observations.The goodness-of-fit statistics are presented in Table.The estimates of deviance are shown in the column marked value/DF. Thedispersion parameter (value/DF) is 0.3376 and the Pearson Chi-square dispersion parameter is 0.4752.

The values of the devianceand Pearson Chi-square are less than thedegrees of freedom, while the P-valuesof the deviance and Pearson Chi-square are all 0.05 (i.e., 1.0000, 0.9993,resp.). These indicate that this model seems to have an acceptable fit with thedata. Deviance and Pearson goodness-of-fit statistics. Testing the GlobalNull Hypothesis:When testing the null hypothesis for large samples, theexplanatory variables have coefficients of zero. According to the Chi-squaresanalysis, the associated P-values areall approximately zero, suggesting that the explanatory coefficients are all zero.The results obtained after rerunningthe unconditional asymptotic logisticregression after the removalof some of the observations from the data (i.e., case 11, case 27, and case 49) (seeTable ) still contain some unconditional asymptoticresults. These results are obtained by deriving the Chi-square statistics whiletesting for the global null hypothesis(likelihood ratio, score, and Wald tests).

For the likelihood ratio and scoretests, the null hypothesis that is zero is rejected, but notfor the Wald test. The seeming discrepancies in P-values obtained between the Wald test and the other two tests area sign that the large-sample approximation is not stable. Testing of the global null hypothesis: β = 0. Exact Logistic Regression ModelExactlogistic regression for binary outcomes can be utilized to provide an exactscore test and an exact probability test for hypotheses where the parametersare equal to zero; these tests produce an exact P-value and a mid P-value.Totest whether individual parameter estimates are zero, we also require pointestimates of the parameters, an odds ratio that contains two-sided confidencelimits, and the P-value. Conditional Exact Tests:The results of exact conditionalanalysis obtained using the exact logistic regression model are shown inTable. The results for the exact score conditional test and the probability testare also reported in this table. For the joint test it is required that all theparameters for the exact statement be simultaneously equal to zero, that is,the null hypothesis is.

Analysis of MLEs.In the joint test results an exact P-value of. Odds ratio estimates.Parameter Estimation for Conditional Exact LogisticRegressionThe analyticalresults of the exact parameter estimates and exact odds ratio estimates arepresented in Tables and, respectively.

The ridge height, lower layer water depth, and potential energy are all significantfactors affecting the amplitude-based incident rate ( P. Exactodds ratios. ConclusionsA laboratory experiment is designed to investigate the propagationof an internal solitary wave over a submerged ridge.

Analytical methods and a logisticregression model are employed to examine the amplitude-based incident rate.Large sample theory may not be suitable for data with small cell counts. Thistends to make tests based on the asymptotic normality of the MLEs unreliable.The ridge height, lower layer water depth, and potentialenergy are considered in the regression model. Once a model has been fitted tothe observed values of a binary response variable, it is essential to check thevalidity of the fit. We discuss some methods for exploring the adequacy of the modeland some diagnostic methods. The techniques used to examine the adequacy of afitted unconditional asymptotic logistic regression model and conditional exactlogistic regressions are known as diagnostics methods for testing the global nullhypothesis. Based on the analytical results we can draw the followingconclusions.(1)The unconditional asymptotic logistic model resultslead us to the conclusion that the three explanatory variables (ridge height,lower layer water depth, and potential energy) are significant factorsaffecting the amplitude-based incident rate. Both deviance and Pearson Chi-squaretests are used to examine the goodness-of-fit of the model.

The dispersionparameter for the estimate of deviance (value/DF) is 0.7268, and the Pearson Chi-squaredispersion parameter is 1.2157. Preferably, this value should be very close to1.00. The Pearson parameter is slightly larger than the degrees of freedom. Wenote that there is still a little overdispersion with this model which meansthat it needs to be modified.(2)A look at the index plotsfor the Pearson residuals (Figure ) and the deviance residuals (Figure )shows that case 11 and case 27 are poorly accounted for by the model. In theindex plot of the diagonal elements of the hat matrix (Figure ), case 49 is anextreme point in the design space. After these observations (case 11, case 27,and case 49) are removed from the data, the new revised model is refitted basedon the remaining observations. The values ofthe deviance and Pearson Chi-squares are now less than the degrees of freedom,and the P-values for deviance andPearson Chi-square are all 0.05 (1.0000, 0.9993, resp.).

In other words,this revised model seems to fit the data acceptably well.(3)When testing the global null hypothesis, only threeChi-square statistics (likelihood ratio, score, and Wald tests) are generated. The P-values obtained by logisticregression for the likelihood ratio test and score test are both. References.

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Modelling Binary Data Collett Pdf Printer Manual

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This book shows how binary data, that is data that can take one of two possible forms, such as alive or dead, and success or failure can be analyzed using statistical modeling. The role of the linear logistic model is particularly stressed, but models based on the complementary log-log transformations are also introduced. Throughout this book, the practical aspects of the modeling approach are emphasized. Indeed the book begins by describing a number of studies in which binary data were recorded. 17.Ergodebooks Ships From UsaviaUnited StatesSoftcover, ISBN 002Publisher: Chapman and Hall/CRC, 1991384 pages. Usually ships within 6 - 10 business days, Edition: 1; Buy with confidence. Excellent Customer Service & Return policy.Ships from USA.

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