Second, it can be used to forecast effects or impact of changes. That is, the regression analysis helps us to understand how much the dependent variable changes with a change in one or more independent variables. Third, regression analysis predicts trends and future values. The regression analysis can be used to get point estimates.

When selecting the model for the analysis, an important consideration is model fitting. However, overfitting can occur by adding too many variables to the model, which reduces model generalizability. Statistically, if a model includes a large number of variables, some of the variables will be statistically significant due to chance alone.

Statistics Solutions can assist with your quantitative analysis by editing your methodology and results chapters. For more information on how we can assist, please click here. Video 8 videos. Introduction 58s. Correlation 9m. Residuals 1m.

## Regression analysis

Least Squares Line 11m. Prediction and Extrapolation 3m. Conditions for Linear Regression 10m. R Squared 4m. Regression with Categorical Explanatory Variables 5m. Reading 3 readings.

Lesson Learning Objectives 10m. Week 1 Suggested Readings and Practice 10m. Quiz 2 practice exercises. Week 1 Practice Quiz 8m. Week 1 Quiz 18m. Video 3 videos. Outliers in Regression 6m. Inference for Linear Regression 11m. Variability Partitioning 5m. Reading 5 readings.

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Week 2 Suggested Readings and Exercises 10m. About Lab Choices 10m. Quiz 3 practice exercises. Week 2 Practice Quiz 6m. Week 2 Quiz 16m. Video 7 videos. Introduction 2m. Multiple Predictors 11m. Adjusted R Squared 10m. Collinearity and Parsimony 3m. Inference for MLR 11m. Model Selection 11m. Diagnostics for MLR 7m. Week 3 Suggested Readings and Exercises 10m.

Week 3 Lab Instructions RStudio 10m. Discussion of the ways in which the linear regression model is extended by the general linear model can be found in the General Linear Models topic. A good theory is the end result of a winnowing process. We start with a comprehensive model that includes all conceivable, testable influences on the phenomena under investigation. Then we test the components of the initial comprehensive model, to identify the less comprehensive submodels that adequately account for the phenomena under investigation. Finally from these candidate submodels, we single out the simplest submodel, which by the principle of parsimony we take to be the "best" explanation for the phenomena under investigation.

We prefer simple models not just for philosophical but also for practical reasons. Simple models are easier to put to test again in replication and cross-validation studies. Simple models are less costly to put into practice in predicting and controlling the outcome in the future. The philosophical reasons for preferring simple models should not be downplayed, however. Simpler models are easier to understand and appreciate, and therefore have a "beauty" that their more complicated counterparts often lack. The entire winnowing process described above is encapsulated in the model-building techniques of stepwise and best-subset regression.

The use of these model-building techniques begins with the specification of the design for a comprehensive "whole model. Finally, the simplest of the adequate is adopted as the "best. Unlike the multiple regression model, which is used to analyze designs with continuous predictor variables, the general linear model can be used to analyze any ANOVA design with categorical predictor variables, any ANCOVA design with both categorical and continuous predictor variables, as well as any regression design with continuous predictor variables.

Effects for categorical predictor variables can be coded in the design matrix X using either the overparameterized model or the sigma-restricted model. Only the sigma-restricted parameterization can be used for model-building. True to its description as general, the general linear model can be used to analyze designs with effects for categorical predictor variables which are coded using either parameterization method. In many uses of the general linear model, it is arbitrary whether categorical predictors are coded using the sigma-restricted or the overparameterized coding.

When one desires to build models, however, the use of the overparameterized model is unsatisfactory; lower-order effects for categorical predictor variables are redundant with higher-order containing interactions, and therefore cannot be fairly evaluated for inclusion in the model when higher-order containing interactions are already in the model. This problem does not occur when categorical predictors are coded using the sigma-restricted parameterization, so only the sigma-restricted parameterization is necessary in general stepwise regression. Designs which cannot be represented using the sigma-restricted parameterization.

The sigma-restricted parameterization can be used to represent most, but not all types of designs. Specifically, the designs which cannot be represented using the sigma-restricted parameterization are designs with nested effects, such as nested ANOVA and separate slope , and random effects. Model building for designs with multiple dependent variables. Stepwise and best-subset model-building techniques are well-developed for regression designs with a single dependent variable e.

Using the sigma-restricted parameterization and general linear model methods, these model-building techniques can be readily applied to any ANOVA design with categorical predictor variables, any ANCOVA design with both categorical and continuous predictor variables, as well as any regression design with continuous predictor variables. Building models for designs with multiple dependent variables, however, involves considerations that are not typically addressed by the general linear model. Model-building techniques for designs with multiple dependent variables are available with Structural Equation Modeling.

To index. A wide variety of types of designs can be represented using the sigma-restricted coding of the design matrix X , and any such design can be analyzed using the general linear model.

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The following topics describe these different types of designs and how they differ. Some general ways in which designs might differ can be suggested, but keep in mind that any particular design can be a "hybrid" in the sense that it could have combinations of features of a number of different types of designs.

The levels or values of the predictor variables in an analysis describe the differences between the n subjects or the n valid cases that are analyzed. Thus, when we speak of the between subject design or simply the between design for an analysis, we are referring to the nature, number, and arrangement of the predictor variables. Concerning the nature or type of predictor variables, between designs which contain only categorical predictor variables can be called ANOVA analysis of variance designs, between designs which contain only continuous predictor variables can be called regression designs, and between designs which contain both categorical and continuous predictor variables can be called ANCOVA analysis of covariance designs.

Between designs may involve only a single predictor variable and therefore be described as simple e. Concerning the arrangement of predictor variables, some between designs employ only "main effect" or first-order terms for predictors, that is, the values for different predictor variables are independent and raised only to the first power.

Other between designs may employ higher-order terms for predictors by raising the values for the original predictor variables to a power greater than 1 e.

A common arrangement for ANOVA designs is the full-factorial design, in which every combination of levels for each of the categorical predictor variables is represented in the design. Designs with some but not all combinations of levels for each of the categorical predictor variables are aptly called fractional factorial designs. These basic distinctions about the nature, number, and arrangement of predictor variables can be used in describing a variety of different types of between designs.

Some of the more common between designs can now be described. Simple Regression. Simple regression designs involve a single continuous predictor variable. If there were 3 cases with values on a predictor variable P of, say, 7, 4, and 9, and the design is for the first-order effect of P , the X matrix would be.

### Introduction

If the simple regression design is for a higher-order effect of P, say the quadratic effect, the values in the X 1 column of the design matrix would be raised to the 2nd power, that is, squared. In regression designs, values on the continuous predictor variables are raised to the desired power and used as the values for the X variables. No recoding is performed. It is therefore sufficient, in describing regression designs, to simply describe the regression equation without explicitly describing the design matrix X.

Multiple Regression. Multiple regression designs are to continuous predictor variables as main effect ANOVA designs are to categorical predictor variables, that is, multiple regression designs contain the separate simple regression designs for 2 or more continuous predictor variables. The regression equation for a multiple regression design for the first-order effects of 3 continuous predictor variables P , Q , and R would be. A discussion of multiple regression methods is also provided in the Multiple Regression topic.

Factorial Regression. Factorial regression designs are similar to factorial ANOVA designs, in which combinations of the levels of the factors are represented in the design. In factorial regression designs, however, there may be many more such possible combinations of distinct levels for the continuous predictor variables than there are cases in the data set. To simplify matters, full-factorial regression designs are defined as designs in which all possible products of the continuous predictor variables are represented in the design.

For example, the full-factorial regression design for two continuous predictor variables P and Q would include the main effects i. The regression equation would be. Factorial regression designs can also be fractional, that is, higher-order effects can be omitted from the design.

A fractional factorial design to degree 2 for 3 continuous predictor variables P , Q , and R would include the main effects and all 2-way interactions between the predictor variables. Polynomial Regression. Polynomial regression designs are designs which contain main effects and higher-order effects for the continuous predictor variables but do not include interaction effects between predictor variables.

For example, the polynomial regression design to degree 2 for three continuous predictor variables P, Q, and R would include the main effects i.

## Multiple Regression and Machine Learning

Polynomial regression designs do not have to contain all effects up to the same degree for every predictor variable. For example, main, quadratic, and cubic effects could be included in the design for some predictor variables, and effects up the fourth degree could be included in the design for other predictor variables. Response Surface Regression. Quadratic response surface regression designs are a hybrid type of design with characteristics of both polynomial regression designs and fractional factorial regression designs.

Quadratic response surface regression designs contain all the same effects of polynomial regression designs to degree 2 and additionally the 2-way interaction effects of the predictor variables. The regression equation for a quadratic response surface regression design for 3 continuous predictor variables P, Q, and R would be.