Validity and Reliability: 2016 Edition (Statistical Associates Blue Book Series 12)
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The marginsplot command How do I code dummy variables in regression? What is "attenuation" in the context of regression? Is multicollinearity only relevant if there are significant findings? What can be done to handle multicollinearity?
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What can be done to handle autocorrelation? How does stepwise multiple regression relate to multicollinearity? What are forward inclusion and backward elimination in stepwise regression? Should I keep dropping non-significant independent variables one at a time until only significant ones remain in my model? What are different types of sums of squares used in F tests? Can regression be used in place of Anova for analysis of categorical independents affecting an interval dependent?
How can you test the significance of the difference between two R-squareds? How do I compare b coefficients after I compute a model with the same variables for two subgroups of my sample? How do I compare regression results obtained for one group of subjects to results obtained in another group, assuming the same variables were used in each regression model? What do I do if I have censored, truncated, or sample-selected data? What do I do if I am measuring the same independent variable at both the individual and group level?
What is a relative effects regression model? How do I test to see what effect a quadratic or other nonlinear term makes in my regression model? What is "smoothing" in regression and how does it relate to dealing with nonlinearities in OLS regression? What is nonparametric regression for nonlinear relationships? What is Poisson regression? SPSS questions What is the command syntax for linear regression in SPSS?
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All I want is a simple scatterplot with a regression line. Why won't SPSS give it to me? What is categorical regression in SPSS? Often called OLS regression because of its reliance on ordinary least squares estimation, multiple regression can establish whether a set of independent variables explains a proportion of the variance in a dependent variable through R 2, which is percent of variance explained at a significant level through a significance test of R 2 , and can establish the relative predictive importance of the independent variables by comparing beta weights, which are standardized regression coefficients.
In multiple regression Power terms can be added as independent variables to explore curvilinear effects. Cross-product terms can be added as independent variables to explore interaction effects.
The researcher can test the significance of the difference between two models, with and without a given predictor variable, to determine if adding an independent variable to the model would improve the model significantly. Using hierarchical regression a. Finally, the parameter estimates the b coefficients and the constant can be used to construct a prediction equation and generate predicted scores of the dependent variable for further analysis. The b parameter estimates are the regression coefficients, representing the amount the dependent variable y changes when the corresponding independent variable by G.
The c is the constant, indicating where the regression line intercepts the y axis, representing the average magnitude of the dependent variable when all the independent variables are held to 0.
The the data have been centered, this will be controlling all other variables at their means. The standardized version of the b coefficients are the beta weights, and the ratio of the beta weights is often interpreted as the ratio of the relative predictive power of the independent variables. Associated with multiple regression is R 2 a. Multiple regression shares all the assumptions of correlation: linearity of relationships, the same level of relationship throughout the range of the dependent variable homoscedasticity , interval or near-interval measurement level, absence of outliers, and data whose range is not truncated.
In addition, it is important that the model being tested is correctly specified. The exclusion of important causal variables or the inclusion of extraneous variables can change markedly the b coefficients and beta weights and hence the interpretation of the importance of the independent variables.
There are many of alternatives to ordinary least squares OLS regression. These are treated in separate volumes of the Statistical Associates "Blue Book" series. Some of these topics are listed alphabetically below. Cox regression may be used to analyze time-to-event as well as proximity, and preference data.
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Curve estimation lets the researcher explore how linear regression compares to nonlinear models, useful for exploring which statistical procedures and models may be appropriate for relationships in one's data. Discriminant function analysis. The general linear model multivariate GLM. Multivariate GLM can implement regression models with multiple dependent variables. Generalized linear models GZLM is the generalization of linear modeling to a form covering almost any dependent distribution with almost any link function, thus supporting linear regression, Poisson regression, gamma regression, and many others.
See above. Linear mixed models LMM , also called hierarchical linear models, implement regression in the context of multilevel data and linear effects of higher levels on lower levels. See Garson Generalized linear mixed models GLMM implement regression for multilevel data, supporting a variety of nonlinear link functions. Logistic regression is used for dichotomous and multinomial dependent variables, implemented with stand-alone logistic procedures or through generalized linear models. Logit regression is an equivalent to logistic regression, using log-linear techniques to predict one or more categorical dependent variables.
Multinomial regression handles research where the dependent variable is categorical. Nonlinear regression is used when the model is inherently nonlinear nonlinearities cannot be dealt with using link functions in generalized linear models or by power and other transformations in general linear models, including regression. Partial least squares regression, which merges regression and factor analysis techniques, may be used even with small datasets to predict a set of response variables from a set of independent variables.
Partial proportional odds PPO regression is an alternative to ordinal regression when the parallel lines test fails. Poisson regression is used for count data in survival event history analysis and other techniques, implemented with stand-alone general loglinear procedures and through generalized linear models. Two-stage least squares regression 2SLS may be used when one or more independent variables are correlated with the error term, when omitted variables may be treated using instrumental variables, and when the model involves interdependence among predictor variables and thus is recursive.
Weighted least squares WLS regression may be used when the OLS regression assumption of homoscedasticity has been violated.
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Data examples in this volume The example datasets used in this volume are listed below in order of use, with versions for SPSS. Except where otherwise noted, examples in this volume are from a modified version of the GSS93subset. Click here to download GSS93subset.
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Click here to download cars. A section below on quantile regression uses the auto dataset in a version based on that provided by Stata. Click here to download auto.
A section below on difference-in-difference regression uses the kielmcclainsubset dataset in a version based on that provided by Stata. Click here to download kielmcclainsubset. The name is derived from the criterion used to draw the best fit regression line: a line such that the sum of the squared deviations of the distances of all observed points to the line is minimized. In the illustration below, Y is regressed on X for three data points.
Equations such this, with no interaction effects see below , are called main effects models. OLS parameter estimates: Shown in blue in the figure above, the b coefficient for X is 1. Residual sum of squares: This is the sum of the squared residuals, where residuals are the differences between the observed Y and the predicted Y by G.
Model sum of squares: The total sum of squares minus the residual sum of squares is the model sum of squares. R 2 : R-square is the model sum of squares as a percentage of the total sum of squares and is interpreted as the percent of variance in Y explained by the regression model. Here, Second example. In the figure below, using SPSS output, the highest year of school completed educ is predicted from total family income income91 and age when first married agewed.
Dependent variable The dependent variable is the predicted variable in the regression equation educ in the example above. Also called response, outcome, or criterion variables, dependent variables are assumed to be continuous, interval variables, though it is common to see binary or ordinal independent variables in linear regression. Use of binary variables as dependent variables is no longer acceptable since such variables cannot meet regression's normal distribution assumption - logistic regression is commonly used instead.
Likewise, use of ordinal variables as dependent variables is now derogated in favor of ordinal regression and proportional odds models treated in the Ordinal Regression title of the Statistical Associates Blue Book series by G. Predictor variables usually are continuous variables. It is, however, common to see the use of ordinal predictor variables in linear regression even though this violates the assumptions of OLS regression.
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Use of ordinal predictor variables with fewer than five levels is particularly derogated. It is acceptable to use binary predictor variables and to transform nominal and ordinal categorical variables into sets of dummy variables coded 0, 1, and leaving one level out as the reference category to avoid perfect multicollinearity.
See the discussion in the "Assumptions" section below. Dummy variables Dummy variables are a way of adding the values of a nominal or ordinal variable to a regression equation. The standard approach to modeling categorical variables is to include the categorical variables in the regression equation by converting each level of each categorical variable into a variable of its own, usually coded 0 or 1. For instance, the categorical variable "region" may be converted into dummy variables such as "East," "West," "North," or "South.
Once a set of dummy variables is created, if we know an observation s value on all the levels of a categorical variable except one, that last one is determined. We have to leave one of the levels out of the regression model to avoid perfect multicollinearity a. For example, we may leave out "West" to avoid singularity. The reference level should be the level of greatest interest or at least a level with known characteristics. Letting a residual category e.
The omitted category is the reference category because b coefficients must be interpreted with reference to it.