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PQL Estimation Biases in Generalized Linear Mixed Models

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PQL Estimation Biases in Generalized Linear Mixed ModelsWoncheol Jang∗Johan Lim†March18,2006AbstractThe penalized quasi-likelihood(PQL)approach is the most com-mon estimation procedure for the generalized linear mixed model(GLMM).However,it has been noticed that the PQL tends to un-derestimate variance components as well as regression coefficients inthe previous literature.In this paper,we numerically show that thebiases of variance component estimates by PQL are systematicallyrelated to the biases of regression coefficient estimates by PQL,andalso show that the biases of variance component estimates by PQLincrease as random effects become more heterogeneous.Keywords and Phrases:Generalized linear mixed models;heterogeneity; penalized quasi-likelihood estimator;variance components.1IntroductionRepeatedly measured outcomes of each subject are the data scheme which we commonly encounter in medical studies.In the situation that there is only one measurement for each subject,the generalized linear model(GLM)is a common assumption for a broad variety of continuous and discrete data.A major issue in extending the GLM for single observation per subject to mod-els of repeatedly measured data is intra-subject correlation.In the previousliterature,two most common approaches to take into account the correlation within a subject are the marginal model approach(for example,general-ized estimation equation)by Zeger and Liang(1986)and the model with random effects to explain individual heterogeneity such as the generalized linear mixed effects model(GLMM),where the variance components are the parameters related to the random effects.In many of repeatedly measured data examples,the primary research in-terest is given to the change of response outcomes over time or covariates (equivalently estimating the regression coefficients)rather than the variance components.However,knowing the variance components is often as much of interest as knowing the regression coefficients.For example,in animal breed-ing where the GLMM has been widely used,the random effects explain the subject specific variation and the heritability of the genetic correlation can be represented as a function of variance components.In addition,although the role of variance components are not clearly specified in the marginal model approach,the optimal weight for the well known generalized estima-tion equation(GEE)method is a function of the variance components and their estimation is important in obtaining efficient regression coefficient esti-mates.When estimating the parameters in the GLMM,the exact likelihood func-tion involves an intractable high–dimensional integration and is hard to com-pute.Accordingly,several approximations to the likelihood function and ap-proximate maximum likelihood estimators(MLE)have been proposed in the previous literature(Schall,1991;Breslow and Clayton,1993;Wolfinger, 1993).Among them,the penalized quasi-likelihood(PQL)by Breslow and Clayton(1993)is the most popular for the GLMM.It approximates the high-dimensional integration using the well–known Laplace approximation and the approximated likelihood function has that of a Gaussian distribution. Subsequently,it suggests to apply linear mixed model restricted maximum likelihood(REML)estimation to the normal theory problem introduced in Harville(1977).Even though the penalized likelihood approach is widely used in many dif-ferent applications,it has known that estimating the variance components is quite challenging due to their non-observability.Accordingly,several different2types of likelihood functions of the variance components have been suggested including maximum adjusted profile h-likelihood estimator(MAPHLE)by Lee and Nelder(1996).However,most such variance component estimators have not received as much attention as regression coefficient estimators have. Furthermore they are not understood well when the observations are from non-Gaussian distributions.A main theme of this paper is to study the performance of the PQL variance component estimators,in particular,when the random effects are heterogeneous in the sense that the distributions of the random effects vary over subjects.The heterogeneity in random effects may arise in many differ-ent situations.The two common cases are:(1)the variability of individual effects can be affected by covariates such as sex,race,origin of each subject as in the(mean variance)joint-model of the generalized linear model(McCul-lagh and Nelder,1989);(2)the random effects also become heterogeneous when there exist several sources of variations(Li and Zhong,2002).In this paper,we address two specific questions for the performance of the PQL estimators when the random effects are heterogeneous.First,a simulation study in Section3.1shows that the biases of regression coefficient estimates by PQL are closely related to those of the variance com-ponent estimates by PQL,and,also shows that the PQL regression coefficient estimators are biased even we accurately estimate the variance components.Second,Section3.2shows that the PQL underestimates the variance components with the heterogeneity in random effects and the biases increase as the random effects become more heterogeneous;accordingly,the biases of PQL random coefficient estimators also increase.This paper is organized as follows.Section2summarizes the GLMM, REML and the PQL.Section3implements two simulation studies on the performance of the PQL estimators.Section4concludes the paper with discussions of other issues on the heterogeneous random effects not covered in this paper.32GLMM,REML and PQLLet us consider the odds ratio inference problem for a series of2×2tables from eight clinical centers reported by Beitler and Landis(1985).Each table has counts of“success”and“failures”among293patients distributed in treatment and control groups at each clinical center.The experimenters were interested whether if there was a clinic specific treatment effect.Let y ij be a binary indicator which is1for success and0for failure for j th subjectin the i th clinic.Let the covariate x ij be−12for treatment.A simple model for analysis is a simple random intercept model:logitE(y ij|b i)=α0+α1x ij+b i, where logit(x)=log xa ijV(µb ij),where g=h−1is the link function;φis a dispersion parameter;a ij is a prior weight and V(·)is a variance function,subjectively specified.4Here,αis a q×1vector offixed effects and the random effects b follows a multivariate normal distribution with mean0and covariance matrix D= D(θ)andθis a c×1unknown vector of variance components.It is usually assumed that the random effects are independent of each other which means D=diag(θs⊕I q s)for s=1,...,c and c s=1q s.The link function can be expressed as using matrix notation as:g(µb i)=X T iα+Z T i b.whereµb i= µi1,...,µb in i T and the design matrix X i and Z i have rows x T ij and z T ij.To estimate the parameters,one can consider the integrated quasi-likelihood L(α,θ)which is given byL=1(2π)q|D(θ)| exp−12b T D−1(θ)b db,where d ij(y,µ)=−2a ij µy y−u2φNi=1n i j=1d ij(y ij,µb ij)−1where i∼N(0,W−1i)and W i is the diagonal matrix ofw ij= V( µb ij) g ( µb ij) 2 −1,for j=1,...,n i.We briefly summarize the procedures as follows:Step1Givenθand b,we can estimatefixed effectαby solving the normal equationNi=1X T i V−1i X iα=N i=1X T i V−1i Y∗i,where V i=W−1i+Z i DZ T i.Step2The random effect b can be estimated asb=N i=1DZ T i V−1i Y∗i−X i α .Step3Subsequently,the REML estimator forθisθs= n∈Q s b2nHere H has components1h st=1−x orΦ−1(x)withΦis the normal cumulative distribu-tion function.Here,x ij1and x ij2were independently from N(0,1)for i= 1,...,50and j=1,...,4.Thefixed effects were set to beαT=(0.5,2.0,0.0) and the random effects b were generated from a multivariate normal distri-bution with mean0and covariance D=diag(θ1,θ2)⊕I25.In other words, thefirst25subjects had random intercepts of a varianceθ1while those in the second half of the subjects had a varianceθ2.For the Poisson regression,random variables were generated with condi-tional meanµb ij given bylog(µb ij)=b i+α0+α1x ij1+α2x ij2.7α0α2Binary0.4456(0.2166)-0.0038(0.1766) (Logit)0.4438(0.2286)-0.0057(0.1821)0.4602(0.2238)-0.0030(0.1896)0.4657(0.2304)0.0076(0.2123)0.4959(0.2397)0.0001(0.2031) Binary0.3790(0.1591)0.0090(0.1191) (Probit)0.4267(0.1778)0.0032(0.1296)0.4533(0.1832)0.0098(0.1339)0.4657(0.2304)0.0076(0.2123)0.4942(0.1987)-0.0027(0.1569) Poisson0.7441(0.1538)-0.0004(0.0483)0.5604(0.1427)-0.0001(0.0334)0.5498(0.1643)0.0015(0.0357)0.5224(0.1791)0.0011(0.1421)0.4999(0.1554)0.0018(0.0352)True value 2.0First,while the PQL estimates workfine for the Poisson regression when the true variance component values are given,it can be found that the PQL estimates of the regression coefficient for binary regressions are still underes-timated even we set the variance components as their true values.Second,it can be found that,for binary regressions,as thefixed values of the variance component decrease,the regression coefficient estimate ofα1 also decrease toward0.As pointed out in Section1,theses observations are compatible with the well known results that the regression coefficients are downward biased to0when the random effects are mistakenly disregarded (Neuhaus,1998;Henderson and Oman,1999).3.2Biases of the PQL variance component estimators To see the effect of heterogeneous random effects,five hundred data sets of 200observations were generated with the same data structure(50subjects with4replications).A simulation study implemented under the following4 scenarios for logistic and Poisson regressions,respectively.As we observed from the simulation in Section3.1,the logistic and the probit regressions were not different much in their results,hence,we only considered the logistic regression for binary outcomes.The four scenarios for logistic regression were:Scenario1:θ=(0.5,0.5);Scenario2:θ=(0.5,1.0);Scenario3:θ=(0.5,1.5);Scenario4:θ=(0.5,2.0).For Poisson regression,Scenario1:θ=(0.5,0.5)Scenario2:θ=(0.5,0.75);Scenario3:θ=(0.5,1.0);9α0 α2 θ2 Binary0.4971-0.00370.4782(Logit)0.4824-0.00630.78810.4716-0.0049 1.09580.4653-0.0045 1.3863Poisson0.5327-0.00050.47610.5345-0.00160.70360.53730.00290.96660.5277-0.0009 1.21740.50.0Table2:Average offixed effects and variance component estimates over500 data sets.Scenario4:θ=(0.5,1.25).Table2contains the average of the estimates offixed effects and variance components.Thefirst four rows in Table2correspond to these averages for the four different scenarios,while the last row contains the true parameter values forα0,α1,α2andθ1.Since the values ofθ2are different for each scenarios,we report them inside the parentheses after the estimatedθing the estimatedα1as a typical example in the estimatedfixed effects,one may find that the percentage of bias in α1is2.6%in scenario1,5.1%in scenario 2,7%in scenario3,and8.4%in scenario4.In other words,the largerθ2is ,the larger the biases offixed effect estimates are.In variance component estimates,the percentage of biases in θ2dramatically increases from0.4%to 30.7%asθ2increases.The effect of increasingθ2on the direction of biases in θ1is almost the same as on θ2,but the percentage of biases in θ1are much smaller than those inθ2asθ2increases.Table3shows the ratios of the estimated and the Monte Carlo simu-lated standard errors.The ratios are defined as Est./Monte where“Est”and “Monte”correspond to the estimated and Monte Carlo standard errors.The Monte Carlo standard errors are sample standard deviations of the PQL es-timators and the estimated standard errors are calculated from the equation (1).10α0 α2 θ2 Binary0.95940.9969 1.26770.9626 1.0211 1.12390.9582 1.0275 1.07200.9597 1.0312 1.0423(0.5,0.5)0.97790.9871(0.5,0.75)0.9921 1.0692(0.5,1.0)0.94560.9876(0.5,1.25)0.9865 1.0271Table3:Ratios of Estimated and Monte Carlo simulated standard errors.Infixed effects of the logit regression,the ratios are between0.95and1.04. However,for variance components of the binary regression,the Monte Carlo simulated standard errors are much smaller than the estimated standard errors.One may note that the ratios of Monte Carlo and estimated standard errors inθ2decreases while ratios inθ1increase asθ2increases.For variance components of the Poisson regression,one canfind similar results;the ratios of Monte Carlo and estimated standard errors inθ2decrease asθ2increases,but the ratios inθ1remains around1. Figure1shows the distributions of θfor logistic regression.Density es-timates are computed by kernel smoothing and the true parameter values are as indicated in each plot.Long right tails are observed in all θwhich is consistent with Lin(1997).Lin also pointed out that θis not exactly normally distributed,unless that the number of clusters is really large and theθare bounded away from the boundary,0.4DiscussionThis paper numerically studies the performance of the PQL estimates in the GLMM when the random effects are heterogeneous.The two main results for binary outcomes are:(1)the PQL regression coefficient estimates are biased even though we precisely estimate the variance components;(2)using PQL,the variance components are underestimated while the standard er-11theta=(0.5,0.5)0123450.00.20.40.60.81.0theta=(0.5,1.0)0123450.00.20.40.60.81.0theta=(0.5,1.5)01234560.00.20.40.60.81.0theta=(0.5,2.0)01234560.00.20.40.60.81.0Figure 1:Distribution of θrors of variance components are overestimated when the random effects areheterogeneous;such phenomena is clear in all binary regression.However,as pointed out in Lin (1997)and Breslow (2005),the biases of the PQL estimators decrease as the cluster size n i increases.In the remainder,we finish the paper with a few discussions of further issues on heterogeneous random effects which are not covered in this paper.Firstly,bias correction of the PQL estimates has been suggested by several authors in the previous literature.Some of them are based on higher oder Laplace approximation (Breslow and Lin ,1995;Lin and Breslow ,1996;Lin ,1997)However,their adjustments do not seem to take care of theheterogeneity of the random effects in the sense that θADJ θP QL does not change over different level of heterogeneity (see Equation (20)in Lin and12α0 α2 θ2 Logit0.45380.00230.5912(0.2248)(0.1784)(0.3384)0.4824-0.00630.7881(0.2248)(0.1897)(0.6681)Home. 1.99530.6020(0.0526)(0.1563)Hetero. 1.99390.4866(0.0511)(0.1590)True0.50.0the sense that,when we mistakenly assume a shorter tail distribution than the true random effect distribution,the regression coefficient estimates are downward biased to0.However,theoretical justification on the conjecture is demanded.ReferencesBeitler,P.J.and Landis,J.R.(1985).A mixed-effects model for categorical data.Biometrics.41,991-1000.Breslow,N.E.(2005).Whither PQL?In Proceedings of the Second Seattle Symposium in Biostatistics,Springer.Breslow,N.E.and Clayton,D.G.(1993).Approximate inference in general-ized linear mixed models.Journal of American Statistical Association,88, 9-25.Breslow,N.E.and Lin,X.(1995)Bias correction in generalized linear mixed models with a single component of dispersion.Biometrika,82,81-91.Harville,D.A.(1977).Maximum likelihood approaches to variance compo-nent estimation and to related problems.Journal of American Statistical Association,72,320-340.Henderson,R.and Oman,P.(1999).Effect of frailty on marginal regres-sion estimates in survival analysis.Journal of the Royal Statistical Society: Series B,61,367-379.Lee,Y.and Nelder,J.A.(1996),Hierarchical generalized linear models.Jour-nal of the Royal Statistical Society:Series B,58,619-678.Li,H.and Zhong,X.(2002),Multivariate survival models induced by genetic frailties,with applications to linkage analysis.Biostatistics,3,57-75. Lin,X.(1997)Variance component testing in generalised linear models with random effects.Biometrika,84,309-326.14Lin,X and Breslow,N.E.(1996)Bias Correction in Generalized Linear Mixed Models with Multiple Components of Dispersion.Journal of American Statistical Association,91,1007-1016.McCullagh,P.and Nelder,J.A.(1989).Generalized Linear Models.Chapman and Hall,London.Neuhaus,J.M.(1998).Estimation efficiency with omitted covariates in gen-eralized linear models.Journal of American Statistical Association,93, 1124-1129.Schall,R.(1991).Estimation in generalized linear models with random ef-fects.Biometrika,78,719-727.Wolfinger,R.(1993).Laplace’s approximation for nonlinear mixed models. Biometrika,80,791-795.Zeger,S.L.and Liang,K.Y.(1986)Longitudinal data analysis for discrete continuous outcomes.Biometrics,42,121-130.15。

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STATA软件中xtreg指令的后处理工具-xtreg后处理指令详述说明书

STATA软件中xtreg指令的后处理工具-xtreg后处理指令详述说明书

TitleDescriptionThe following postestimation commands are of special interest after xtreg: command descriptionxttest0Breusch and Pagan LM test for random effectsFor information about this command,see below.In addition,the following standard postestimation commands are available:command descriptionadjust1adjusted predictions of xβ∗estat AIC,BIC,VCE,and estimation sample summaryestimates cataloging estimation resultshausman Hausman’s specification testlincom point estimates,standard errors,testing,and inference for linear combinationsof coefficientslrtest likelihood-ratio testmfx marginal effects or elasticitiesnlcom point estimates,standard errors,testing,and inference for nonlinear combinations of coefficientspredict predictions,residuals,influence statistics,and other diagnostic measurespredictnl point estimates,standard errors,testing,and inference for generalized predictions test Wald tests for simple and composite linear hypothesestestnl Wald tests of nonlinear hypotheses1adjust does not work with time-series operators.∗estat ic may not be used after xtreg with the be,pa,or re options.See the corresponding entries in the Stata Base Reference Manual for details.Special-interest postestimation commandsxttest0,for use after xtreg,re,presents the Breusch and Pagan(1980)Lagrange-multiplier test for random effects,a test that Var(νi)=0.304xtreg postestimation—Postestimation tools for xtreg305 Syntax for predictFor all but the population-averaged modelpredict type newvar if in ,statistic nooffsetPopulation-averaged modelpredict type newvar if in ,PA statistic nooffsetstatistic descriptionxb x j b,fitted values;the defaultstdp standard error of thefitted valuesue u i+e it,the combined residual∗xbu x j b+u i,prediction including effect∗u u i,thefixed-or random-error component∗e e it,the overall error componentUnstarred statistics are available both in and out of sample;type predict...if e(sample)...if wantedonly for the estimation sample.Starred statistics are calculated only for the estimation sample,even whenif e(sample)is not specified.PA statistic descriptionxb linear predictionstdp standard error of the linear predictionscorefirst derivative of the log likelihood with respect to x jβThese statistics are available both in and out of sample;type predict...if e(sample)...if wanted onlyfor the estimation sample.Options for predictxb calculates the linear prediction,that is,a+bx it.This is the default for all except the population-averaged model.stdp calculates the standard error of the linear prediction.Note that,in the case of thefixed-effects model,this excludes the variance due to uncertainty about the estimate of u i.ue calculates the prediction of u i+e it.xbu calculates the prediction of a+bx it+u i,the prediction including thefixed-or random-component.u calculates the prediction of u i,the estimatedfixed-or random-effect.e calculates the prediction of e it.score calculates the equation-level score,u j=∂ln L j(x jβ)/∂(x jβ).nooffset is relevant only if you specified offset(varname)for xtreg,pa.It modifies the calculations made by predict so that they ignore the offset variable;the linear prediction is treated as x it b rather than x it b+offset it.306xtreg postestimation—Postestimation tools for xtregSyntax for xttest0xttest0RemarksExample1Continuing with our xtreg,re estimation example(example4)in xtreg,we can see that xttest0 will report a test ofνi=0.In case we have any doubts,we could type.xttest0Breusch and Pagan Lagrangian multiplier test for random effects:ln_wage[idcode,t]=Xb+u[idcode]+e[idcode,t]Estimated results:Var sd=sqrt(Var)ln_wage.2283326.4778416e.0845038.2906954u.066514.2579031Test:Var(u)=0chi2(1)=14779.98Prob>chi2=0.0000Example2More importantly,after xtreg,re estimation,hausman will perform the Hausman specification test.If our model is correctly specified,and ifνi is uncorrelated with x it,the(subset of)coefficients that are estimated by thefixed-effects estimator and the same coefficients that are estimated here should not statistically differ:.xtreg ln_w grade age*ttl_exp*tenure*black not_smsa south,re(output omitted).estimates store random_effects.xtreg ln_w grade age*ttl_exp*tenure*black not_smsa south,fe(output omitted)xtreg postestimation—Postestimation tools for xtreg307.hausman.random_effectsCoefficients(b)(B)(b-B)sqrt(diag(V_b-V_B)).random_eff~s Difference S.E.age.0359987.036806-.0008073.0013177age2-.000723-.0007133-9.68e-06.0000184ttl_exp.0334668.0290207.0044461.001711ttl_exp2.0002163.0003049-.0000886.000053tenure.0357539.039252-.0034981.0005797tenure2-.0019701-.0020035.0000334.0000373not_smsa-.0890108-.1308263.0418155.0062745south-.0606309-.0868927.0262618.0081346b=consistent under Ho and Ha;obtained from xtregB=inconsistent under Ha,efficient under Ho;obtained from xtreg Test:Ho:difference in coefficients not systematicchi2(8)=(b-B)’[(V_b-V_B)^(-1)](b-B)=149.44Prob>chi2=0.0000We can reject the hypothesis that the coefficients are the same.Before turning to what this means, note that hausman listed the coefficients estimated by the two models.It did not,however,list grade and race.hausman did not make a mistake;in the Hausman test,we compare only the coefficients estimated by both techniques.What does this mean?We have an unpleasant choice:we can admit that our model is misspecified—that we have not parameterized it correctly—or we can hold that our specifica-tion is correct,in which case the observed differences must be due to the zero-correlation ofνi and the x it assumption.u Technical NoteWe can also mechanically explore the underpinnings of the test’s dissatisfaction.In the comparison table from hausman,note that it is the coefficients on not smsa and south that exhibit the largest differences.In equation(1 )of[XT]xtreg,we showed how to decompose a model into within and between effects.Let us do that with these two variables,assuming that changes in the average have one effect while transitional changes have another:.egen avgnsmsa=mean(not_smsa),by(idcode).generate devnsma=not_smsa-avgnsmsa(8missing values generated).egen avgsouth=mean(south),by(idcode).generate devsouth=south-avgsouth(8missing values generated)308xtreg postestimation—Postestimation tools for xtreg.xtreg ln_w grade age*ttl_exp*tenure*black avgnsm devnsm avgsou devsouRandom-effects GLS regression Number of obs=28091Group variable(i):idcode Number of groups=4697R-sq:within=0.1723Obs per group:min=1 between=0.4809avg= 6.0overall=0.3737max=15 Random effects u_i~Gaussian Wald chi2(12)=9319.69corr(u_i,X)=0(assumed)Prob>chi2=0.0000 ln_wage Coef.Std.Err.z P>|z|[95%Conf.Interval]grade.0631716.001790335.290.000.0596627.0666805age.0375196.003118612.030.000.0314072.043632age2-.0007248.00005-14.500.000-.0008228-.0006269 ttl_exp.0286542.002420711.840.000.0239097.0333987ttl_exp2.0003222.0001162 2.770.006.0000945.0005499tenure.0394424.00175422.490.000.0360045.0428803tenure2-.0020081.0001192-16.850.000-.0022417-.0017746black-.0545938.0102099-5.350.000-.0746048-.0345827 avgnsmsa-.1833238.0109337-16.770.000-.2047533-.1618942devnsma-.0887596.0095071-9.340.000-.1073932-.070126avgsouth-.1011235.0098787-10.240.000-.1204855-.0817616devsouth-.0598538.0109054-5.490.000-.081228-.0384796 _cons.268298.0495776 5.410.000.1711277.3654683sigma_u.25791607sigma_e.29069544rho.44046285(fraction of variance due to u_i)We will leave the reinterpretation of this model to you,except to note that if we were really going to sell this model,we would have to explain why the between and within effects are different.Focusing on residence in a non SMSA,we might tell a story about rural people being paid less and continuing to get paid less when they move to the SMSA.Given our panel data,we could create variables to measure this(an indicator for moved from non SMSA to SMSA)and to measure the effects.In our assessment of this model,we should think about women in the cities moving to the country and their relative productivity in a bucolic setting.xtreg postestimation—Postestimation tools for xtreg309 In any case,the Hausman test now is.estimates store new_random_effects.xtreg ln_w grade age*ttl_exp*tenure*black avgnsm devnsm avgsou devsou,fe(output omitted).hausman.new_random_effectsCoefficients(b)(B)(b-B)sqrt(diag(V_b-V_B)).new_random~s Difference S.E.age.0359987.0375196-.001521.0013198age2-.000723-.0007248 1.84e-06.0000184ttl_exp.0334668.0286542.0048126.0017127ttl_exp2.0002163.0003222-.0001059.0000531tenure.0357539.0394424-.0036885.0005839tenure2-.0019701-.0020081.000038.0000377devnsma-.0890108-.0887596-.0002512.0006826devsouth-.0606309-.0598538-.0007771.0007612b=consistent under Ho and Ha;obtained from xtregB=inconsistent under Ha,efficient under Ho;obtained from xtreg Test:Ho:difference in coefficients not systematicchi2(8)=(b-B)’[(V_b-V_B)^(-1)](b-B)=92.52Prob>chi2=0.0000We have mechanically succeeded in greatly reducing theχ2,but not by enough.The major differences now are in the age,experience,and tenure effects.We already knew this problem existed because of the ever-increasing effect of experience.More careful parameterization work rather than simply including squares needs to be done.uMethods and FormulasAll postestimation commands listed above are implemented as ado-files.xttest0xttest0reports the Lagrange-multiplier test for random effects developed by Breusch and Pagan (1980)and as modified by Baltagi and Li(1990).The modely it=α+x itβ+νitis estimated via OLS,and then the quantityλLM=(nT)22 A21( i T2i)−nTis calculated,whereA1=1− n i=1( T i t=1v it)2i t v2it310xtreg postestimation—Postestimation tools for xtregThe Baltagi and Li modification allows for unbalanced data and reduces to the standard formulaλLM=nT2(T−1)i( t v it)2i t v2it−1 2when T i=T(balanced data).Under the null hypothesis,λLM is distributedχ2(1).ReferenceBaltagi,B.H.and Q.Li.1990.A Lagrange-multiplier test for the error components model with incomplete panels.Econometric Reviews9(1):103–107.Breusch,T.and A.Pagan.1980.The Lagrange-multiplier test and its applications to model specification in econometrics.Review of Economic Studies47:239–253.Hausman,J.A.1978.Specification tests in econometrics.Econometrica46:1251–1271.Also SeeComplementary:[XT]xtreg,[R]adjust,[R]estimates,[R]hausman,[R]lincom,[R]lrtest,[R]mfx,[R]nlcom,[R]predictnl,[R]test,[R]testnlBackground:[U]13.5Accessing coefficients and standard errors,[U]20Estimation and postestimation commands,[R]estat,[R]predict。

统计学术语中英文对照

统计学术语中英文对照

population 母体sample 样本census 普查sampling 抽样quantitative 量的qualitative/categorical质的discrete 离散的continuous 连续的population parameters 母体参数sample statistics 样本统计量descriptive statistics 叙述统计学inferential/inductive statistics 推论 ...抽样调查（sampliing survey单纯随机抽样（simple random sampling 系统抽样（systematic sampling分层抽样（stratified sampling整群抽样（cluster sampling多级抽样（multistage sampling常态分配(Parametric Statistics)无母数统计学(Nonparametric Statistics) 实验设计(Design of Experiment)参数(Parameter)Data analysis 资料分析Statistical table 统计表Statistical chart 统计图Pie chart 圆饼图Stem-and-leaf display 茎叶图Box plot 盒须图Histogram 直方图Bar Chart 长条图Polygon 次数多边图Ogive 肩形图Descriptive statistics 叙述统计学Expectation 期望值Mode 众数Mean 平均数Variance 变异数Standard deviation 标准差Standard error 标准误Covariance matrix 共变异数矩阵Inferential statistics 推论统计学Point estimation 点估计Interval estimation 区间估计Confidence interval 信赖区间Confidence coefficient 信赖系数Testing statistical hypothesis 统计假设检定Regression analysis 回归分析Analysis of variance 变异数分析Correlation coefficient 相关系数Sampling survey 抽样调查Census 普查Sampling 抽样Reliability 信度Validity 效度Sampling error 抽样误差Non-sampling error 非抽样误差Random sampling 随机抽样Simple random sampling 简单随机抽样法Stratified sampling 分层抽样法Cluster sampling 群集抽样法Systematic sampling 系统抽样法Two-stage random sampling 两段随机抽样法Convenience sampling 便利抽样Quota sampling 配额抽样Snowball sampling 雪球抽样Nonparametric statistics 无母数统计The sign test 等级检定Wilcoxon signed rank tests 魏克森讯号等级检定Wilcoxon rank sum tests 魏克森等级和检定Run test 连检定法Discrete uniform densities 离散的均匀密度Binomial densities 二项密度Hypergeometric densities 超几何密度Poisson densities 卜松密度Geometric densities 几何密度Negative binomial densities 负二项密度Continuous uniform densities 连续均匀密度Normal densities 常态密度Exponential densities 指数密度Gamma densities 伽玛密度Beta densities 贝他密度Multivariate analysis 多变量分析Principal components 主因子分析Discrimination analysis 区别分析Cluster analysis 群集分析Factor analysis 因素分析Survival analysis 存活分析Time series analysis 时间序列分析Linear models 线性模式Quality engineering 品质工程Probability theory 机率论Statistical computing 统计计算Statistical inference 统计推论Stochastic processes 随机过程Decision theory 决策理论Discrete analysis 离散分析Mathematical statistics 数理统计统计学: Statistics母体: Population样本: Sample资料分析: Data analysis统计表: Statistical table统计图: Statistical chart圆饼图: Pie chart茎叶图: Stem-and-leaf display盒须图: Box plot直方图: Histogram长条图: Bar Chart次数多边图: Polygon肩形图: Ogive叙述统计学: Descriptive statistics 期望值: Expectation众数: Mode平均数: Mean变异数: Variance标准差: Standard deviation标准误: Standard error共变异数矩阵: Covariance matrix推论统计学: Inferential statistics点估计: Point estimation区间估计: Interval estimation信赖区间: Confidence interval信赖系数: Confidence coefficient统计假设检定: Testing statisticalhypothesis回归分析: Regression analysis变异数分析: Analysis of variance相关系数: Correlation coefficient抽样调查: Sampling survey普查: Census抽样: Sampling信度: Reliability效度: Validity抽样误差: Sampling error非抽样误差: Non-sampling error随机抽样: Random sampling简单随机抽样法: Simple randomsampling分层抽样法: Stratified sampling群集抽样法: Cluster sampling系统抽样法: Systematic sampling两段随机抽样法: Two-stage randomsampling便利抽样: Convenience sampling配额抽样: Quota sampling雪球抽样: Snowball sampling无母数统计: Nonparametric statistics等级检定: The sign test魏克森讯号等级检定: Wilcoxon signedrank tests魏克森等级和检定: Wilcoxon rank sumtests连检定法: Run test离散的均匀密度: Discrete uniformdensities二项密度: Binomial densities超几何密度: Hypergeometric densities卜松密度: Poisson densities几何密度: Geometric densities负二项密度: Negative binomial densities连续均匀密度: Continuous uniformdensities常态密度: Normal densities指数密度: Exponential densities伽玛密度: Gamma densities贝他密度: Beta densities多变量分析: Multivariate analysis主因子分析: Principal components区别分析: Discrimination analysis群集分析: Cluster analysis因素分析: Factor analysis存活分析: Survival analysis时间序列分析: Time series analysis线性模式: Linear models品质工程: Quality engineering机率论: Probability theory统计计算: Statistical computing统计推论: Statistical inference随机过程: Stochastic processes决策理论: Decision theory离散分析: Discrete analysis数理统计: Mathematical statistics统计名词市调辞典众数(Mode) 普查(census)指数(Index) 问卷(Questionnaire)中位数(Median) 信度(Reliability)百分比(Percentage) 母群体(Population)信赖水准(Confidence level) 观察法(Observational Survey)假设检定(Hypothesis Testing) 综合法(Integrated Survey)卡方检定(Chi-square Test) 雪球抽样(Snowball Sampling)差距量表(Interval Scale) 序列偏差(Series Bias)类别量表(Nominal Scale) 次级资料(Secondary Data)顺序量表(Ordinal Scale) 抽样架构(Sampling frame)比率量表(Ratio Scale) 集群抽样(Cluster Sampling)连检定法(Run Test) 便利抽样(Convenience Sampling)符号检定(Sign Test) 抽样调查(SamplingSur)算术平均数(Arithmetic Mean) 非抽样误差(non-sampling error)展示会法(Display Survey)调查名词准确效度(Criterion-RelatedValidity)元素(Element) 邮寄问卷法(Mail Interview)样本(Sample) 信抽样误差(Sampling error)效度(Validity) 封闭式问题(Close Question)精确度(Precision) 电话访问法(TelephoneInterview)准确度(Validity) 随机抽样法(RandomSampling)实验法(Experiment Survey)抽样单位(Sampling unit) 资讯名词市场调查(Marketing Research) 决策树(Decision Trees)容忍误差(Tolerated erro) 资料采矿(DataMining)初级资料(Primary Data) 时间序列(Time-Series Forecasting)目标母体(Target Population) 回归分析(Regression)抽样偏差(Sampling Bias) 趋势分析(TrendAnalysis)抽样误差(sampling error) 罗吉斯回归(Logistic Regression)架构效度(Construct Validity) 类神经网络(Neural Network)配额抽样(Quota Sampling) 无母数统计检定方法(Non-Parametric Test)人员访问法(Interview) 判别分析法(Discriminant Analysis)集群分析法(cluster analysis) 规则归纳法(Rules Induction)内容效度(Content Validity) 判断抽样(Judgment Sampling)开放式问题(Open Question) OLAP(OnlineAnalytical Process)分层随机抽样(Stratified Randomsampling) 资料仓储(Data Warehouse)非随机抽样法(Nonrandom Sampling) 知识发现(Knowledge DiscoveryAbsolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additive Noise, 加性噪声Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alpha factoring,α因子法Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis Of Effects, 效应分析Analysis Of Variance, 方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型ANOVA table and eta, 分组计算方差分析Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area 区域图Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs),BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector,卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Collinearity, 共线性Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate,相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照, 质量控制图Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation, 相关性Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Crosstabs 列联表分析Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve Estimation, 曲线拟合Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Direct Oblimin, 斜交旋转Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Error Bar, 均值相关区间图Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explained variance （已说明方差）Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查Generalized least squares, 综合最小平方法GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 通用线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点High-Low, 低区域图Higher Order Interaction Effects，高阶交互作用HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Image factoring,, 多元回归法Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K-Means Cluster逐步聚类分析K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least Squared Criterion，最小二乘方准则Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Leveage Correction，杠杆率校正Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logi t转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形MSC（多元散射校正）Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal P-P, 正态概率分布图Normal Q-Q, 正态概率单位分布图Normal ranges, 正常范围Normal value, 正常值Normalization 归一化Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Pareto, 直条构成线图（又称佩尔托图）Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式PCA（主成分分析）Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 构成图,饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡PLS（偏最小二乘法）Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal axis factoring,主轴因子法Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified randomsampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q 图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和residual variance (剩余方差)Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequence, 普通序列图Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significant Level, 显著水平Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Test(检验)Test of linearity, 线性检验Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数。

计量经济学导论第四版部分课后答案中文翻译

计量经济学导论第四版部分课后答案中文翻译

2.10(iii) From (2.57), Var(1ˆβ) = σ2/21()n i i x x =⎛⎫- ⎪⎝⎭∑. 由提示：： 21n i i x=∑ ≥21()n i i x x =-∑, and so Var(1β ) ≤ Var(1ˆβ). A more direct way to see this is to write(一个更直接的方式看到这是编写) 21()n ii x x =-∑ = 221()n i i x n x =-∑, which is less than21n i i x=∑unless x = 0.(iv)给定的c 2i x 但随着x 的增加， 1ˆβ的方差与Var(1β )的相关性也增加.0β小时1β 的偏差也小.因此, 在均方误差的基础上不管我们选择0β还是1β 要取决于0β,x ,和n 的大小 (除了 21n i i x=∑的大小).3.7We can use Table 3.2. By definition, 2β > 0, and by assumption, Corr(x 1,x 2) < 0. Therefore, there is anegative bias in 1β : E(1β ) < 1β. This means that, on average across different random samples, the simple regression estimator underestimates the effect of the training program. It is even possible that E(1β ) is negative even though 1β > 0. 我们可以使用表3.2。

根据定义，> 0，由假设，科尔（X1，X2）<0。

因此，有一个负偏压为：E （）<。

Stata 统计软件：biprobit 双因变量回归分析的 postestimation 工具说明书

Stata 统计软件：biprobit 双因变量回归分析的 postestimation 工具说明书

Title biprobit postestimation—Postestimation tools for biprobitPostestimation commands predict margins Also seePostestimation commandsThe following postestimation commands are available after biprobit:Command Descriptioncontrast contrasts and ANOV A-style joint tests of estimatesestat ic Akaike’s,consistent Akaike’s,corrected Akaike’s,and Schwarz’s Bayesian in-formation criteria(AIC,CAIC,AIC c,and BIC)estat summarize summary statistics for the estimation sampleestat vce variance–covariance matrix of the estimators(VCE)estat(svy)postestimation statistics for survey dataestimates cataloging estimation resultsetable table of estimation results∗hausman Hausman’s specification testlincom point estimates,standard errors,testing,and inference for linear combinations ofcoefficients∗lrtest likelihood-ratio testmargins marginal means,predictive margins,marginal effects,and average marginal effects marginsplot graph the results from margins(profile plots,interaction plots,etc.)nlcom point estimates,standard errors,testing,and inference for nonlinear combinationsof coefficientspredict probabilities for joint,marginal,and conditional outcomespredictnl point estimates,standard errors,testing,and inference for generalized predictions pwcompare pairwise comparisons of estimatessuest seemingly unrelated estimationtest Wald tests of simple and composite linear hypothesestestnl Wald tests of nonlinear hypotheses∗hausman and lrtest are not appropriate with svy estimation results.12biprobit postestimation—Postestimation tools for biprobitpredictDescription for predictpredict creates a new variable containing predictions such as probabilities,linear predictions, and standard errors.Menu for predictStatistics>PostestimationSyntax for predictpredicttypenewvarifin,statistic nooffsetpredicttypestub*ifin,scoresstatistic DescriptionMainp11Φ2(x j b,z j g,ρ),predicted probability Pr(y1j=1,y2j=1);the defaultp10Φ2(x j b,−z j g,−ρ),predicted probability Pr(y1j=1,y2j=0)p01Φ2(−x j b,z j g,−ρ),predicted probability Pr(y1j=0,y2j=1)p00Φ2(−x j b,−z j g,ρ),predicted probability Pr(y1j=0,y2j=0)pmarg1Φ(x j b),marginal success probability for equation1pmarg2Φ(z j g),marginal success probability for equation2pcond1Φ2(x j b,z j g,ρ)/Φ(z j g),conditional probability of success for equation1pcond2Φ2(x j b,z j g,ρ)/Φ(x j b),conditional probability of success for equation2 xb1x j b,linear prediction for equation1xb2z j g,linear prediction for equation2stdp1standard error of the linear prediction for equation1stdp2standard error of the linear prediction for equation2whereΦ(·)is the standard normal-distribution function andΦ2(·)is the bivariate standardnormal-distribution function.These statistics are available both in and out of sample;type predict...if e(sample)...if wanted only for the estimation sample.Options for predict££Main p11,the default,calculates the bivariate predicted probability Pr(y1j=1,y2j=1).p10calculates the bivariate predicted probability Pr(y1j=1,y2j=0).p01calculates the bivariate predicted probability Pr(y1j=0,y2j=1).biprobit postestimation—Postestimation tools for biprobit3p00calculates the bivariate predicted probability Pr(y1j=0,y2j=0).pmarg1calculates the univariate(marginal)predicted probability of success Pr(y1j=1).pmarg2calculates the univariate(marginal)predicted probability of success Pr(y2j=1).pcond1calculates the conditional(on success in equation2)predicted probability of success Pr(y1j=1,y2j=1)/Pr(y2j=1).pcond2calculates the conditional(on success in equation1)predicted probability of success Pr(y1j=1,y2j=1)/Pr(y1j=1).xb1calculates the probit linear prediction x j b.xb2calculates the probit linear prediction z j g.stdp1calculates the standard error of the linear prediction for equation1.stdp2calculates the standard error of the linear prediction for equation2.nooffset is relevant only if you specified offset1(varname)or offset2(varname)for biprobit.It modifies the calculations made by predict so that they ignore the offset variables;the linear predictions are treated as x j b rather than as x j b+offset1j and z jγrather than as z jγ+offset2j. scores calculates equation-level score variables.Thefirst new variable will contain∂ln L/∂(x jβ).The second new variable will contain∂ln L/∂(z jγ).The third new variable will contain∂ln L/∂(atanhρ).4biprobit postestimation—Postestimation tools for biprobitmarginsDescription for marginsmargins estimates margins of response for probabilities and linear predictions.Menu for marginsStatistics>PostestimationSyntax for marginsmarginsmarginlist,optionsmarginsmarginlist,predict(statistic...)predict(statistic...)...optionsstatistic Descriptionp11Φ2(x j b,z j g,ρ),predicted probability Pr(y1j=1,y2j=1);the default p10Φ2(x j b,−z j g,−ρ),predicted probability Pr(y1j=1,y2j=0)p01Φ2(−x j b,z j g,−ρ),predicted probability Pr(y1j=0,y2j=1)p00Φ2(−x j b,−z j g,ρ),predicted probability Pr(y1j=0,y2j=0)pmarg1Φ(x j b),marginal success probability for equation1pmarg2Φ(z j g),marginal success probability for equation2pcond1Φ2(x j b,z j g,ρ)/Φ(z j g),conditional probability of success for equation1 pcond2Φ2(x j b,z j g,ρ)/Φ(x j b),conditional probability of success for equation2 xb1x j b,linear prediction for equation1xb2z j g,linear prediction for equation2stdp1not allowed with marginsstdp2not allowed with marginsStatistics not allowed with margins are functions of stochastic quantities other than e(b).For the full syntax,see[R]margins.Also see[R]biprobit—Bivariate probit regression[U]20Estimation and postestimation commandsStata,Stata Press,and Mata are registered trademarks of StataCorp LLC.Stata andStata Press are registered trademarks with the World Intellectual Property Organizationof the United Nations.Other brand and product names are registered trademarks ortrademarks of their respective companies.Copyright c 1985–2023StataCorp LLC,College Station,TX,USA.All rights reserved.®。