Researchers conducted a poll to study the effect of context questions prior to a
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Researchers conducted a poll to study the effect of context questions prior to a target question. The target question concerns continuing U.S. aid to the Nicaraguan Contra rebels. 8 different versions of the interview were given separately to 1, 054 different subjects randomly assigned to 8 groups, representing all possible combinations of three factors each with two levels: Context (0: question about Cuba: 1 question about Vietnam), Mode (0: target question immediately followed the context question 1: two questions were scattered), and Level (0: low level wording of the context question: 1: high level wording of the context question) The response is e binomial proportions of subjects who have a favorable answer t the target question in the 8 groups. > fiti summary (fit 1) Deviance Residuals: 0.603 1-.252 1.358 0.539 0.999 0.767 -1.779 -0.096 Coefficients: Estimate Std. Error z value Pr(> |z|) Intercept -0.1621 0.1304 -1.243 0.21383 Context -0.6645. 0.1337 -4.969 6.72e-7*** Mode -0.3850 0.1333 -2.889 0.00387** Level -0.0179 0.1330 -0.135 0.89298 (Dispersion parameter for binomial family taken to be 1) Null deviance: 41.9151 on 7 degrees of freedom Residual deviance 8.8251 on 4 degrees of freedom Suppose I want to fit a quasi-binomial model. (a) Give an estimate, psi^cap, for the dispersion parameter, psi, in the model manually. [To answer the rest of this problem, you may need psi. If you were not able to obtain it in i, assume psi^cap=2.] (b) Let beta be the coefficient of Mode. What is the estimate for beta in the quasi-binomial model? (c) Perform a hypothesis test to check if beta is zero in the quasi-binomial model. Make sure to provide the test statistic and the sampling distribution you (The 90%, 95%, and 99% percentiles of the compare the test statistic with standard normal distribution are 1.28, 1.64, and 2 respectively. (d) Interpret the results of (b) and (c) in the context of the research study.Explanation / Answer
Test statistics for evaluating the significance of added variables in a regression equation are developed for mixed Poisson models, where the structural parameter that determines the mean/variance relationship var(; ) = + · 2 is estimated by the method of moments and the regression coefficients are estimated by quasi-likelihood. The formulas presented for test statistics and related estimating equations are applicable generally to quasi-likelihood models specified by an arbitrary mean value function (x; ), together with a variance function V(; ) that contains one or more unknown parameters. Two versions of the Wald and score tests are investigated—one calculated from the usual model-based covariance matrix whose validity depends on correct specification of the variance function, and another using an “empirical” covariance matrix that has a more general asymptotic justification. Monte Carlo simulations demonstrate that the quasi-likelihood/method of moments (QL/M) procedures yield approximately unbiased estimates of regression coefficients and their standard errors and that model-based Wald, score, and deviance tests approximate the nominal size at the 5% level for moderate sample sizes. The simpler Poisson analysis also produces approximately unbiased regression coefficients, even though the overdispersion is not accounted for. Although tests and standard errors based on Poisson theory are seriously in error in the presence of overdispersion, the empirical standard errors and especially the empirical score test obtained in conjunction with the Poisson analysis perform reasonably well with overdispersed data provided the sample size is sufficiently large. They perform less well in small samples than do the model-based QL/M procedures, probably because of the lack of precision in the empirical variances. These methods have important applications in epidemiology, toxicology, and related areas.
To define a likelihood we have to specify the form of distribution of the observations, but to define a quasi-likelihood function we need only specify a relation between the mean and variance of the observations and the quasi-likelihood can then be used for estimation. For a one-parameter exponential family the log likelihood is the same as the quasi-likelihood and it follows that assuming a one-parameter exponential family is the weakest sort of distributional assumption that can be made. The Gauss-Newton method for calculating nonlinear least squares estimates generalizes easily to deal with maximum quasi-likelihood estimates, and a rearrangement of this produces a generalization of the method described by Nelder & Wedderburn (1972).
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