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Model Assessment
Published in Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson, Bayesian Thinking in Biostatistics, 2021
Gary L. Rosner, Purushottam W. Laud, Wesley O. Johnson
Residuals are observed deviations of yi from the fitted values , where is an estimate of β, typically , the ML (also LS) estimate. Studentized (also called standardized) residuals are residuals that are divided by their frequentist estimated standard deviations in repeated data sampling. The Studentized residual is
Testing for Average Bioequivalence
Published in Scott Patterson, Byron Jones, Bioequivalence and Statistics in Clinical Pharmacology, 2017
Figure 3.9 displays the Q-Q plots for the studentized residuals corresponding to logAUC and logCmax. Identified on the plots are the two most extreme residuals in each plot. The studentized residuals are the raw residuals (rijk) divided by their estimated standard error. The standardization is necessary because Var(rijk) is not a constant. If the plotted data are truly normally distributed, the plotted points should lie on or close to a straight line. We can see that this is mostly true in Figure 3.9, except for the logAUC values of two subjects (12 and 25). A more formal test of normality is one due to Shapiro and Wilk [1135]. For logAUC the p-value for this test is 0.497 and for logCmax it is 0.314. There is no evidence to suggest the studentized residuals are not normally distributed. The responses with the largest studentized residuals (in absolute value) may be outliers. These are values that are typically greater than 3. There is no evidence that our extreme residuals are outliers.
Multiple regression
Published in Pat Dugard, John Todman, Harry Staines, Approaching Multivariate Analysis, 2010
Pat Dugard, John Todman, Harry Staines
Since we requested casewise diagnostics in SPSS Dialog Box 4.2, we get a table listing cases where the residual is more than two standard deviations (shown in SPSS Output 4.3). There are three of these, with the largest (in absolute value) just under 3, which is no cause for concern with 110 cases. We would expect about 5% as large as 2, just by chance. SPSS can calculate a variety of transformations of the (raw or unstandardized) residuals. The standardized residual is the residual divided by its standard error. Hence observations with residuals with more than two standard deviations are those with a standardized residual greater than 2. The studentized residual is similar to the standardized residual but it is more sensitive to departures from the model. This is because its scaling factor takes into account how far the observation is from the mean. The deleted and studentized residuals for an observation are found as above but from the model that excludes that observation.
Obsessional slowness in obsessive-compulsive disorder: identifying characteristics and comorbidities in a clinical sample
Published in International Journal of Psychiatry in Clinical Practice, 2023
Erin Crowe, Maria C. Rosário, Ygor A. Ferrão, Lucy Albertella, Euripedes C. Miguel, Leonardo F. Fontenelle
Group differences on categorical variables were investigated using Pearson’s chi-square test of contingencies, whilst nonparametric Mann-Whitney U tests were used for continuous variables. Finally, binary logistic regression examined which variables were independently related to OS. Variables found to be univariately significant at alpha level and trend level (p < .10) were entered into the regression model, where the outcome variable was OS. Univariately significant variables with conceptual overlap such as YBOCS total scores and its associated obsession and compulsions scores, presence of symptoms on DYBOCS dimensions and DYBOCS total scores were removed. The remaining predictors were checked for unacceptable multicollinearity using variance inflation factor (VIF) statistics with a cut-off of ≥ 5. Logit linearity was also assessed by checking the significance of interactions between continuous predictors and their natural logarithms. Studentized residuals were used to assess for outliers, with a cut-off score of ±3.
Psychological Distress and Suicidal Ideation in Australian Online Help-Seekers: The Mediating Role of Perceived Burdensomeness
Published in Archives of Suicide Research, 2023
Christopher Rainbow, Peter Baldwin, Warwick Hosking, Peter Gill, Grant Blashki, Fiona Shand
Data analysis was conducted in SPSS Statistics version 25, with moderated mediation via the PROCESS macro (Hayes, 2018). Heteroscedasticity was found in examining studentized residuals against predicted values. This is partially due to the nature of thevariables being measured, in which the scores were not expected to be normally distributed: two of the variables contained an expected clustering of zero scores (abstainers on AUDIT, no suicidal ideation experienced on SIDAS). Six multivariate outliers were found, with standardized residuals of greater than +3 standard deviations. Conditional process analysis was run excluding these outliers, with no significant changes to the result; it was decided to proceed with these outliers included. Partial regression plots revealed linear relationships between suicidal ideation and all independent variables except alcohol consumption. Removing alcohol abstainer participants restored a linear relationship, but did not affect the significance of alcohol consumption correlations, direct or indirect effects; so it was decided to proceed with all alcohol consumption data intact. All other statistical assumptions were met.
Birnbaum–Saunders sample selection model
Published in Journal of Applied Statistics, 2021
Fernando de Souza Bastos, Wagner Barreto-Souza
We now conclude this section by discussing on residual analysis, which is frequently used to evaluate the validity of model assumptions. Residuals carry important information concerning the appropriateness of statistical models, and thereby play an important role in checking model adequacy. In the context of regression models, Pearson and studentized residuals are often used. Nevertheless, in a censored scenario, these residuals are not adequate even under normality; for instance, see [6]. In the Heckman-BS case, we consider the generalized Cox-Snell (GCS) residual given by 11], which is defined by F is the cumulative distribution function of the response variable. Under a correct model specification, the GCS and normalized quantile residuals are approximately standard exponential and standard normal distributed, respectively. We illustrate the usefulness of these residuals in the empirical application of this paper.