Final generalized linear modeling by idrees waris iugc

Editor's Notes

• #3: Correlated or clustered dataThe standard GLM assumes that the observations are uncorrelated. Extensions have been developed to allow for correlation between observations, as occurs for example in longitudinal studies and clustered designs:Generalized estimating equations (GEEs) allow for the correlation between observations without the use of an explicit probability model for the origin of the correlations, so there is no explicit likelihood. They are suitable when the random effects and their variances are not of inherent interest, as they allow for the correlation without explaining its origin. The focus is on estimating the average response over the population ("population-averaged" effects) rather than the regression parameters that would enable prediction of the effect of changing one or more components of X on a given individual. GEEs are usually used in conjunction with Huber-White standard errors. [4][5]Generalized linear mixed models (GLMMs) are an extension to GLMs that includes random effects in the linear predictor, giving an explicit probability model that explains the origin of the correlations. The resulting "subject-specific" parameter estimates are suitable when the focus is on estimating the effect of changing one or more components of X on a given individual. GLMMs are a particular type of multilevel model (mixed model). In general, fitting GLMMs is more computationally complex and intensive than fitting GEEs.Hierarchical generalized linear models (HGLMs) are similar to GLMMs apart from two distinctions:The random effects can have any distribution in the exponential family, whereas current GLMMs nearly always have normal random effects;They are not as computationally intensive, as instead of integrating out the random effects they are based on a modified form of likelihood known as the hierarchical likelihood or h-likelihood.The theoretical basis and accuracy of the methods used in HGLMs have been the subject of some debate in the statistical literature. As of 2008, the method is only available in one statistical software package, namely
• #5: Canonical means: conforming to a general rule reduced to the simplest or clearest scheme possible the simplest form of a matrix (specifically the form of a square matrix that has zero off-diagonals).
• #6: AssumptionsNot assumed. GZLM/GEE, compared to GLM, do not assume a normally distributed dependent variable (or normally distributed independents), nor linearity between the predictors and the dependent, nor homogeneity of variance for the range of the dependent variable. Linearity of the link function. The researcher still must select a link function such that there is a linear relation between the linear predictor (the right-hand side of the model equation) and the link function of the dependent. For example, in logistic regression models, one must check to see if there is "linearity in the logit," meaning the linear predictor is in fact linearly related to the logit of the dependent. Absence of high multicollinearity. As in other linear models, presence of high multicollinearity among the independents will inflate standard errors and confound interpretation of the relative contributions of the independents. Centered data. As in regression, centering may be necessary either to reduce multicollinearity or to make interpretation of coefficients meaningful. Centering is almost always recommended for independent variables which are components of interaction terms in a logistic model. Data distribution. The independents may be of any distribution so long as linearity of the link function is maintained. The dependent may assume any of a wide variety of distributions, including normal, inverse normal (inverse Gaussian), binomial, multinomial, and Poisson. Independent vs. correlated data . In GZLM, observations are assumed to be independent. In GEE, observations are assumed to be independent between subjects within any given time period, cluster, or repeated measures factor, but are assumed to be dependent within the same subject across repeated measures factors. That is, GEE assumes between-subjects independence and within-subjects dependence. In practical terms, if there are repeated measures, GEE, not GZLM, should be chosen and the repeated measures properly specified. Data levels. In both GZLM and GEE, the dependent (response) variable may be binary, counts, scale, or events-in-trials. "Scale" means interval or ordinal if it can be assumed that ordinal values represent ordered categories with a meaningful metric, so that distance comparisons between values are appropriate. For ordinal response variables, the model type should be multinomial (ordinal) logistic or ordinal probit. Factors are categorical. Covariates are scale variables. Weight and offset variables, if any, are also assumed to be scale. In GEE, the variables specified as repeated measures within-subjects effects, or the variables used to define subjects, cannot be used as dependent variables. Missing data. As in other forms of statistical analysis, missing data can lead to biased coefficients unless data are missing completely at random.  
• #7: Interpretation: There seems to be a Shopping style effect; on average, "biweekly" customers spend $378.52, while "weekly" customers spend $404.55, and "often" customers spend $406.76. There also appears to be a Gender effect; on average, males in the sample spend $430.30 compared to $365.66 for females. Lastly, there may be an interaction effect between Gender and Shopping style, because the mean differences in amount spent by shopping style vary between genders. For example, "biweekly" male customers tend to spend more than "often" male customers, but this trend is reversed for "biweekly" and "often" female customers. The N column in the table shows there are unequal cell sizes. Most customers prefer to shop on a weekly basis. The standard deviations appear relatively homogenous, although you should check Levene's test and the spread-versus-level plots to be sure. Dependent variable Amount spent: This table tests the null hypothesis that the variance of the error term is constant across the cells defined by the combination of factor levelsSince the significance value of the test, 0.330, is greater than 0.10, there is no reason to believe that the equal variances assumption is violated. Thus, the small differences in group standard deviations observed in the descriptive statistics table are due to random variationThe spread-versus-level plot is a scatterplot of the cell means and standard deviations from the descriptive statistics tableIt provides a visual test of the equal variances assumption, with the added benefit of helping you to assess whether violations of the assumption are due to a relationship between the cell means and standard deviationsThere is no apparent pattern in this plot, so there is no indication of such a relationship here. The tests of between-subjects effects help you to determine the significance of a factor. However, they do not indicate how the levels of a factor differ. The post hoc tests show the differences in model-predicted means for each pair of factor levels. The first column displays the different post hoc testsThe next two columns display the pair of factor levels being tested. When the significance value for the difference in Amount spent for a pair of factor levels is less than 0.05, an asterisk (*) is printed by the differenceIn this case, there do not appear to be significant differences in the spending habits of "biweekly", "weekly", or "often" customers. Tamhane's T2 is generally more appropriate than Tukey's HSD when there are unequal cell sizes, but the results in this case are largely the sameThe confidence intervals for Tamhane's T2 are only slightly wider than those for Tukey's HSD. Since the results of these two tests are not very different, it is safe for you to look at the results of the homogenous subsets, which are available for Tukey's HSD but not for Tamhane's T2. The homogenous subsets table takes the results of the post hoc tests and shows them in a more easily interpretable formIn the subset columns the factor levels that do not have significantly different effects are displayed in the same columnIn this example, the first subset contains the "biweekly", "weekly", and "often" customers.  These are all the customers, so there are no other subsets.  The post hoc tests suggest that efforts at enticing customers to shop more often than usual is wasted because they will not spend significantly more. However, the post hoc test results do not account for the levels of other factors, thus ignoring the possibility of an interaction effect with Gender seen in the descriptive statistics table. See the estimated marginal means to see how this might change your conclusionsThis table displays the model-estimated marginal means and standard errors of Amount spent at the factor combinations of Gender and Shopping style. This table is useful for exploring the possible interaction effect between these two factorsIn this example, a male customer who makes purchases weekly is expected to spend about $440.96, while one who makes purchases more often is expected to spend $407.77. A female customer who makes purchases weekly is expected to spend $361.72, while one who makes purchases more often is expected to spend $405.72. Thus, there is a difference between "weekly" and "often" customers, depending upon the gender of the customer. This fact suggests an interaction effect between Gender and Shopping style. If there were no interaction, you would expect the difference between shopping styles to remain constant for male and female customers. The interaction can be seen more easily in the profile plotsThe profile plot is a visual representation of the marginal means table.The factor levels of Shopping style are shown along the horizontal axisSeparate lines are produced for each level of Gender.  Alternately, the factor levels of Gender could be shown along the horizontal axis, with separate lines produced for each level of Shopping style.  If there were no interaction effect, the lines in the table would be parallel. Instead, you can see that the difference in spending between "weekly" and "often" customers is greater for female customers, as the line for male customers slopes downward and that for female customers slopes upward.  This is a strong interaction effect and is unlikely to be due to chance, but you should check the tests of between-subjects effects for confirmation of its significanceThis is an analysis of variance table. Each term in the model, plus the model as a whole, is tested for its ability to account for variation in the dependent variable. Note that variable labels are not displayed in this table.  The significance value for each term, except STYLE, is less than 0.05. Therefore each term, except STYLE, is statistically significant.  The partial eta squared statistic reports the "practical" significance of each term, based upon the ratio of the variation (sum of squares) accounted for by the term, to the sum of the variation accounted for by the term and the variation left to error.Larger values of partial eta squared indicate a greater amount of variation accounted for by the model term, to a maximum of 1. Here the individual terms, while statistically significant, do not have great effect on the value of Amount spent.  The GLM Univariate procedure is useful for modeling the linear relationship between a dependent scale variable and one or more categorical and scale predictors. If you have only one factor, you can alternatively use the One-Way ANOVA procedure. If you only have covariates, use the Linear Regression procedure for more model-building, residual-checking, and output options
• #8: 15 custom Link functionsIdentity. f(x)=x. The dependent variable is not transformed. This link can be used with any distribution.• Complementary log-log. f(x)=log(−log(1−x)). This is appropriate only with the binomial distribution.• Cumulative Cauchit. f(x) = tan(π (x – 0.5)), applied to the cumulative probability of each category of the response. This is appropriate only with the multinomial distribution.• Cumulative complementary log-log. f(x)=ln(−ln(1−x)), applied to the cumulative probability of each category of the response. This is appropriate only with the multinomial distribution.• Cumulative logit. f(x)=ln(x / (1−x)), applied to the cumulative probability of each category of the response. This is appropriate only with the multinomial distribution.• Cumulative negative log-log. f(x)=−ln(−ln(x)), applied to the cumulative probability of each category of the response. This is appropriate only with the multinomial distribution.• Cumulative probit. f(x)=Φ−1(x), applied to the cumulative probability of each category of the response, where Φ−1 is the inverse standard normal cumulative distribution function. This is appropriate only with the multinomial distribution.• Log. f(x)=log(x). This link can be used with any distribution.• Log complement. f(x)=log(1−x). This is appropriate only with the binomial distribution.• Logit. f(x)=log(x / (1−x)). This is appropriate only with the binomial distribution.• Negative binomial. f(x)=log(x / (x+k−1)), where k is the ancillary parameter of the negative binomial distribution. This is appropriate only with the negative binomial distribution.• Negative log-log. f(x)=−log(−log(x)). This is appropriate only with the binomial distribution.• Odds power. f(x)=[(x/(1−x))α−1]/α, if α ≠ 0. f(x)=log(x), if α=0. α is the required number specification and must be a real number. This is appropriate only with the binomial distribution.• Probit. f(x)=Φ−1(x), where Φ−1 is the inverse standard normal cumulative distribution function. This is appropriate only with the binomial distribution.• Power. f(x)=xα, if α ≠ 0. f(x)=log(x), if α=0. α is the required number specification and must be a real number. This link can be used with any distribution.