2. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 1 S052/§I.1(a): Applied Data Analysis Roadmap of the Course – What Is Today’s Topic? More details can be found in the “Course Objectives and Content” handout on the course webpage. Multiple Regression Analysis (MRA) Multiple Regression Analysis (MRA) Do your residuals meet the required assumptions? Test for residual normality Use influence statistics to detect atypical datapoints If your residuals are not independent, replace OLS by GLS regression analysis Use Individual growth modeling Specify a Multi-level Model If your sole predictor is continuous, MRA is identical to correlational analysis If your sole predictor is dichotomous, MRA is identical to a t-test If your several predictors are categorical, MRA is identical to ANOVA If time is a predictor, you need discrete- time survival analysis… If your outcome is categorical, you need to use… Binomial logistic regression analysis (dichotomous outcome) Multinomial logistic regression analysis (polytomous outcome) If you have more predictors than you can deal with, Create taxonomies of fitted models and compare them. Form composites of the indicators of any common construct. Conduct a Principal Components Analysis Use Cluster Analysis Use non-linear regression analysis. Transform the outcome or predictor If your outcome vs. predictor relationship is non-linear, How do you deal with missing data? Today’s Topic Area

3. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 2 S052/§I.1(a): Applied Data Analysis Where Does Today’s Topic Appear in the Printed Syllabus? In the future, I ask you to keep automatically tabs on the inter- connections among the Roadmap, the Daily Topic Area, the Printed Syllabus, and the content of the day’s class, when you first download and pre-read the required day’s class materials. Today’s topic, Deciding Which Regression Models to Fit, is from Syllabus Section I.1(a) and includes:Syllabus Slides 3-4: Introducing the ILLCAUSE data-example. Slides 5-6: Imagining a “Universe Of All Possible Models”. Slides 7-8: Two Strategies For Choosing Sensible Subsets of Regression Models To Fit. Slides 9-14: Exploratory Univariate & Bivariate Analyses in the ILLCAUSE Dataset. Slide 15: Establishing Priorities Among the Predictors. Slides 16-18: Fitting a Taxonomy of Regression Models in the ILLCAUSE Dataset. Slide 19: Decoding Standard Regression Output. Slide 20: APA-Style Table Displaying a Taxonomy Of Fitted Regression Models. Slides 21-22: Appendix 1. Slide 23: Appendix 2. Today’s topic, Deciding Which Regression Models to Fit, is from Syllabus Section I.1(a) and includes:Syllabus Slides 3-4: Introducing the ILLCAUSE data-example. Slides 5-6: Imagining a “Universe Of All Possible Models”. Slides 7-8: Two Strategies For Choosing Sensible Subsets of Regression Models To Fit. Slides 9-14: Exploratory Univariate & Bivariate Analyses in the ILLCAUSE Dataset. Slide 15: Establishing Priorities Among the Predictors. Slides 16-18: Fitting a Taxonomy of Regression Models in the ILLCAUSE Dataset. Slide 19: Decoding Standard Regression Output. Slide 20: APA-Style Table Displaying a Taxonomy Of Fitted Regression Models. Slides 21-22: Appendix 1. Slide 23: Appendix 2.

4. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 3 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit What Question, and Dataset, Will Drive Our Presentation Today? As an introductory example of this part of the syllabus -- “Deciding Which Multiple Regression Models To Fit” -- here’s a multiple regression analysis to address the following broad question: Do Chronically-ill Children Understand The Causes Of Illness Differently From Healthy Children? As an introductory example of this part of the syllabus -- “Deciding Which Multiple Regression Models To Fit” -- here’s a multiple regression analysis to address the following broad question: Do Chronically-ill Children Understand The Causes Of Illness Differently From Healthy Children? Dataset on the website:  ILLCAUSE.txt ILLCAUSE.txt Codebook on the website:  ILLCAUSE_info ILLCAUSE_info Dataset on the website:  ILLCAUSE.txt ILLCAUSE.txt Codebook on the website:  ILLCAUSE_info ILLCAUSE_info DatasetILLCAUSE.txt OverviewData for investigating differences in children’s understanding of the causes of illness, by their health status. SourcePerrin E.C., Sayer A.G., and Willett J.B.Perrin E.C., Sayer A.G., and Willett J.B. (1991). Sticks And Stones May Break My Bones: Reasoning About Illness Causality And Body Functioning In Children Who Have A Chronic Illness, Pediatrics, 88(3), 608-19. Sample size301 children, including a sub-sample of 205 who were described as asthmatic, diabetic, or healthy. After further reductions due to the list-wise deletion of cases with missing data on one or more variables, the analytic sub-sample used in class ends up containing 33 diabetic children, 68 asthmatic children and 93 healthy children. More infoChronically-ill children were recruited into the study through their pediatricians; healthy children were a matched random sample drawn from the same schools as the ill children. UpdatedSeptember 16, 2005

5. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 4 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit What Variables Will We Focus On In Our Analyses?

6. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 5 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Even With A Few Predictors, There Are So Many Models You Can Possibly Fit! To address a RQ about children’s understanding of illness, you can: outcome  Identify ILLCAUSE as the outcome. predictors  Identify “HEALTH,” AGE, and SES as predictors. And proceed with regression analysis … To address a RQ about children’s understanding of illness, you can: outcome  Identify ILLCAUSE as the outcome. predictors  Identify “HEALTH,” AGE, and SES as predictors. And proceed with regression analysis … Now. what happens if we include non-linear expressions of the continuous predictors, or break them up into categories, or add another predictor like gender or race, or … How Many Potential Models Would Exist Then? until you begin to enumerate how many possible models you can actually specify using just these few predictors … The task appears reasonable until you begin to enumerate how many possible models you can actually specify using just these few predictors … Three models with a 1 main effect Three models with 2 main effects and 1 two-way interaction One model with 3 main effects Three models with 2 main effects Three models with 3 main effects and 1 two-way interaction Three models with 3 main effects and 2 two-way interactions One model with 3 main effects and 3 two-way interactions and so on...

7. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 6 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Really, How Big Is the Problem and How Can We Deal With It? Number of feasible regression models that can possibly be specified increases exponentially as the number of potential predictors rises.* “Initial” model contains main effect of the question predictor HEALTH? * * Second model adds the main effect of control predictor AGE? * * Third model adds two- way interaction of HEALTH and AGE? * * Fourth model adds the main effect of control predictor, SES? * * next ? You are here! With one outcome and thirteen predictors … The “universe of all possible models” contains 73,566,892 specifications. It seems plausible to ask, then … What strategy can we use to identify a subset of “essential” models”? It seems plausible to ask, then … What strategy can we use to identify a subset of “essential” models”?

8. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 7 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Two Broad Strategies For Deciding Which Models To Specify And Fit Two Broad Classes of Model-Specification Strategy More Thoughtful Methods:  Use your knowledge of the research question, the supporting substantive theory and logic to specify and fit a systematic taxonomy of regression models.  Make sure your decisions are driven by your need to answer specific research questions, to test reasonable hypotheses, and to tell a good story. More Thoughtful Methods:  Use your knowledge of the research question, the supporting substantive theory and logic to specify and fit a systematic taxonomy of regression models.  Make sure your decisions are driven by your need to answer specific research questions, to test reasonable hypotheses, and to tell a good story. Example follows, using ILLCAUSE data: Subsequently, we’ll use the same example to refine our ability to conduct regression analyses by adding new tools:  General linear hypothesis (GLH) test,  Influence statistics,  Innovations in residual analysis,  Strategies for improving interpretation of fitted models. Example follows, using ILLCAUSE data: Subsequently, we’ll use the same example to refine our ability to conduct regression analyses by adding new tools:  General linear hypothesis (GLH) test,  Influence statistics,  Innovations in residual analysis,  Strategies for improving interpretation of fitted models. Well-known “Automated” Methods: Forward selection. Backward elimination. Stepwise regression. All-possible-subsets regression. I don’t recommend these methods at all: Choice of models is abdicated to a computer. Choice of each subsequent model can be strongly impacted by inter-relationships among the predictors currently in the model and any potential additional predictors. Please don’t ever use these methods, but do read about them in the post-class readings so that you can recognize what to avoid!

9. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 8 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Example of Specifying a Sensible Taxonomy of Regression Models to Data Do children suffering from different types of chronic illness understand the causes of illness differently from each other, and from healthy children? Here starts the data-example – my illustrative data-analyses are contained in... Data-Analytic Handout I.1(a).1  Available on class website.  Features exploratory univariate and bivariate analyses of the ILLCAUSE data. Data-Analytic Handout I.1(a).2  Available on class website.  Features the fitting of a sensible taxonomy of regression models to the ILLCAUSE data. A “Do it Yourself” PC-SAS Manual is available on the course website:  Full_PCSAS_Manual.pdf.  Trimmed_PCSAS_Manual.pdf. All S052 Data-Analytic Handouts contain “model” PC-SAS Code and Statistical Output  I asked you to print these handouts out, and include them in your package of course materials for today’s class.  They will serve as “models” for your own future data-analyses, including our regular Data-Analytic Memos (DAMs), and your future research.  You should consult them carefully as you work on the DAM assignments. A few programming comments follow … All S052 Data-Analytic Handouts contain “model” PC-SAS Code and Statistical Output  I asked you to print these handouts out, and include them in your package of course materials for today’s class.  They will serve as “models” for your own future data-analyses, including our regular Data-Analytic Memos (DAMs), and your future research.  You should consult them carefully as you work on the DAM assignments. A few programming comments follow …

10. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 9 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code From The Exploratory Analysis OPTIONS Nodate Pageno=1; TITLE1 'S-052: APPLIED DATA ANALYSIS'; TITLE2 'I: Conducting Sensible Multiple Regression Analyses'; TITLE3 'I.1(a): Fitting Taxonomies of Multiple Regression Models'; TITLE4 'Handout #1: Introducing the Data'; TITLE5 'Understanding of Illness Causality and Chronic Illness in Children'; *-------------------------------------------------------------------------* Input data, name and label variables in dataset *-------------------------------------------------------------------------*; DATA ILLCAUSE; INFILE 'C:\DATA\S052\ILLCAUSE.txt'; INPUT ID ILLCAUSE SES PPVT AGE GENREAS HEALTH; IF HEALTH=3 OR HEALTH=5 OR HEALTH=6; LABEL ID = 'Subject Identification Code' ILLCAUSE = 'Understanding of Illness Causality' SES = 'Hollingshead SES' PPVT = 'Score on Peabody Picture Vocabulary Test' AGE = 'Chronological Age (Months)' GENREAS = 'General Reasoning Ability' HEALTH = 'Health Status'; PROC FORMAT; VALUE HFMT 3 = 'Diabetic' 5 = 'Asthmatic' 6 = 'Healthy'; Data-Analytic Handout I.1(a).1 … As many TITLEs as you like – the title text is in single quotes, and ends in a semicolon Comments can run over several lines – they start with an asterisk and end with a semicolon In the DATA step, raw data is read from an external file into a temporary SAS dataset, here called ILLCAUSE In this PROCedure, labels are created for the values of the categorical HEALTH variable. In the INPUT command, variables are listed in the order they appear in the dataset. This IF statement selects out the diabetic, asthmatic and healthy children. In the INFILE command, you specify the location of the raw data.

11. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 10 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting Exploratory Analysis *-------------------------------------------------------------------------* List data for subsample of 30 cases *-------------------------------------------------------------------------*; PROC PRINT LABEL DATA=ILLCAUSE(obs=35); TITLE6 'Outcome and Predictor Values For the First 35 Children'; FORMAT HEALTH HFMT.; VAR ID ILLCAUSE AGE HEALTH SES; *-------------------------------------------------------------------------* Display selected univariate and bivariate summary information *-------------------------------------------------------------------------*; PROC TABULATE DATA=ILLCAUSE; TITLE6 'Selected Univariate Summaries of ILLCAUSE, AGE and SES'; CLASS HEALTH; FORMAT HEALTH HFMT.; VAR ILLCAUSE AGE SES; TABLE (ILLCAUSE AGE SES)*(HEALTH ALL), N MEAN STD MIN MAX; PROC PLOT DATA=ILLCAUSE; TITLE6 'Bivariate Scatterplots for ILLCAUSE by AGE and SES'; PLOT ILLCAUSE*(AGE SES); PROC CORR NOSIMPLE DATA=ILLCAUSE; TITLE6 'Estimated Bivariate Correlations...'; VAR ILLCAUSE AGE SES; RUN; Print the first 35 cases for inspection and review Add a sixth title to the output for this procedure Format the values of the HEALTH variable, just for this procedure List the variables in the order you want them printing Inspect the bivariate relationships between outcome, ILLCAUSE, and the two continuous controls, AGE & SES, by plotting the values of the outcome against each predictor, in scatter-plots Create a tabulation of summary statistics (mean, standard deviation, minimum and maximum) for the three continuous variables in the analysis -- ILLCAUSE (outcome), AGE and SES (control predictors) – separately by the values of categorical (“CLASSification”) variable HEALTH. Estimate bivariate correlation coefficients to summarize the bivariate relationships between the outcome, ILLCAUSE, and the continuous controls, AGE & SES Don’t forget to “RUN” Data-Analytic Handout I.1(a).1 …

12. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 11 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Listing a Few Cases from the ILLCAUSE Datasest Subject Understanding Identification of Illness Chronological Health Hollingshead Obs Code Causality Age (Months) Status SES 1 301. 128 Diabetic 2 2 302 2.857 79 Diabetic 2 3 303 3.429 151 Diabetic 3 4 304 4.286 178 Diabetic 3 5 305 4.286 113 Diabetic 4 6 306 3.286 81 Diabetic 2 7 307 2.857 194 Diabetic 4... Notice that a period (.) is used in PC-SAS as the default missing value code. Notice that the children's values of SES are heterogeneous and remember that higher values mean lower SES!!! Notice the heterogeneous ages of the sampled children (in months) “Obs” represents the values of a SAS “system” variable, which counts and identifies the observations in the order in which they appear in the temporary SAS dataset Selected output from Data-Analytic Handout I.1(a).1 … data for a few early cases The values of the outcome, ILLCAUSE, are listed here for each child Notice that the health status of the children has been reformatted by my PC-SAS program from a numerical to an alphabetic label

13. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 12 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Exploratory Bivariate Tabulations of the ILLCAUSE Data Selected output from Data- Analytic Handout I.1(a).1 … What four things do you notice that may usefully inform the subsequent regression analysis? Selected output from Data- Analytic Handout I.1(a).1 … What four things do you notice that may usefully inform the subsequent regression analysis?

14. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 13 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Exploratory Bivariate Scatterplot of the ILLCAUSE/AGE Relationship Scatterplot of the bivariate relationship between outcome ILLCAUSE and control predictor AGE: What conclusions do you draw from this plot that may usefully inform subsequent regression analysis? Scatterplot of the bivariate relationship between outcome ILLCAUSE and control predictor AGE: What conclusions do you draw from this plot that may usefully inform subsequent regression analysis?

15. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 14 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Exploratory Bivariate Scatterplot of the ILLCAUSE/SES Relationship Scatterplot of the bivariate relationship between outcome ILLCAUSE and control predictor SES: What conclusions do you draw from this plot that may usefully inform subsequent regression analysis? Scatterplot of the bivariate relationship between outcome ILLCAUSE and control predictor SES: What conclusions do you draw from this plot that may usefully inform subsequent regression analysis?

16. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 15 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit How To Create A Thoughtful Taxonomy Of Fitted Regression Models?  Identify the outcome variable (here, ILLCAUSE) – this is usually not difficult.  Determine the “best” path through the universe of all possible regression models by deciding on an order that predictors must be added to the regression model.  Do this by establishing priorities among classes of predictors based on substance (the research questions and the theoretical framework), the research design, … etc.  For example, here’s one way of parsing out the problem … others appear in the appendices.  Identify the outcome variable (here, ILLCAUSE) – this is usually not difficult.  Determine the “best” path through the universe of all possible regression models by deciding on an order that predictors must be added to the regression model.  Do this by establishing priorities among classes of predictors based on substance (the research questions and the theoretical framework), the research design, … etc.  For example, here’s one way of parsing out the problem … others appear in the appendices. PriorityPredictorComment HighHEALTH is the key question predictor. Without health status in the final model, we cannot address the research questions! Medium AGE is a key control predictor because it represents the multi-cohort nature of the research design: The sample contains multiple sub-samples of children of different ages. By controlling for AGE, we can include all children in the same analysis, regardless of their age, rather than doing an “age- slice by age-slice” analysis, as was suggested by one ill-informed reviewer!!! Low SES is a subsidiary control predictor. It is often worth including because some twit will always ask you if it had an effect. In these data, descriptive analyses suggest that ill children have lower SES. So, if understanding illness also depends on home resources, the effect of SES could masquerade as an effect of HEALTH.

17. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 16 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting the Multiple Regression Analyses Read the data into a temporary SAS dataset called ILLCAUSE Access the temporary SAS dataset, ILLCAUSE, a second time to manipulate the data prior to regression analysis Convert the categorical HEALTH status variable into a system of dummy variables Create a set of two-way HEALTH by AGE interactions Create a set of two-way HEALTH by SES interactions Create an AGE by SES interactions Now, fit a taxonomy of regression models that takes into account the priorities you’ve established …here’s the DATA step from Data-Analytic Handout I.1(a).2 in which we first recode/create the variables we need …

18. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 17 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting the Multiple Regression Analyses PROC REG is the PC-SAS procedure for conducting multiple regression analysis Begin by listing all variables that will be used in any regression model in order to ensure listwise deletion of cases with missing values Optionally label each model so that you can refer to it later You can specify many regression models in one PROC REG “paragraph.” And here’s the PROC REG step … Indicate that you are about to specify a regression model Identify the variable that is the hypothesized outcome Specify the predictors you want included in each hypothesized model The categorical question predictor, HEALTH, is represented by only two of the health status dummies: D (for diabetics) and A (for asthmatics). For a discussion of this, see Appendix 1.

19. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 18 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Annotated PC-SAS Code For Conducting the Multiple Regression Analyses What logic drove the specification of the sequence of models in the taxonomy?

20. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 19 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit What Does Regular Multiple Regression Output Look Like? Question …Can you navigate, and interpret, the contents of typical regression output? What hypothesis does this pair of statistics test? Interpret the estimated intercept? What hypothesis does each of these pairs of statistics test? Conceptually, what is standard error? Interpret the “Sum of Squares Error” or SSError statistic? Interpret the R 2 statistic? Interpret the estimated coefficient associated with predictor D? Interpret the estimated coefficient associated with predictor A? Interpret the “Root MSE” statistic? Interpret the “Sum of Squares Model” or SSModel statistic? Interpret the “Sum of Squares Total” or SSTotal statistic?

21. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 20 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Finally, Assemble The Taxonomy of Fitted Models Into An APA-Style Table What are the critical features of APA formatting for tables? Consult the style manuals Consult the style manuals and exemplars on the course website We’ll dissect this taxonomy, and interpret its substantive story, in subsequent classes...

22. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 21 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Appendix 1: Why Can Two Dummy Predictors Distinguish Among Three Groups? The first fitted regression model (M1) from Data-Analytic Handout I.1(a).1 (Slide #19) is: From it, you can estimate the predicted value of ILLCAUSE in each health status group by substituting numerical values of the health status predictors that represent prototypical individuals in the dataset: Notice that the predicted outcome values corresponding to one of the groups – the reference, omitted or comparison group (here, healthy children) – are obtained when the two dichotomous predictors that distinguish the chronically-ill children are both set to zero. This means that, if you have an intercept in the model, you need one less dummy predictor in the model than there are groups compared, as the fitted value for the “reference (or omitted) group” is provided by the estimated intercept. Another way of thinking about this is to understand that, although there are three distinct health status groups present, only two independent pieces of information are needed to indicate the health status of a child because if a child is neither diabetic nor asthmatic then s/he must be healthy, by default. Of course, you get to choose which of the health status groups serves as the reference, because you are the one who picks which dummy predictor is omitted from the regression model. Typically, you make this choice for substantive, not statistical, reasons.

23. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 22 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Appendix 1: Why Can Two Dummy Predictors Distinguish Among Three Groups? Inspection of the fitted values computed on the previous slide indicate that the fitted regression parameters that we obtained in the analysis – that is, the estimated intercept parameter and the two estimated slope parameters associated with the dummy predictors representing health status, can be interpreted as follows: The fitted slope parameter associated with dummy predictor A represents the difference in the predicted value of ILLCAUSE between the asthmatic and “reference” healthy children – it is our best estimate of the difference between asthmatic and healthy children, on average, in the population (- 0.94). The fitted slope parameter associated with dummy predictor D represents the difference in the predicted value of ILLCAUSE between diabetic and “reference” healthy children – it is our best estimate of the difference between diabetic and healthy children, on average, in the population (-0.84). The fitted intercept represents the predicted value of ILLCAUSE (4.60) for those in the reference (or omitted) category –it is our best estimate of the understanding of healthy children, on average, in the population.

24. © Willett, Harvard University Graduate School of EducationS052/I.1(a) – Slide 23 S052/§I.1(a): Deciding Which Multiple Regression Models To Fit Appendix 2: Other Strategies For Specifying Taxonomies Of Regression Models “Baseline Control Model” Approach: Form a baseline control model, by sequentially adding control predictors, highest priority first, and testing for appropriate interactions as you go along. Then, add the main effects of the question predictors to the new baseline control model. Then, add interactions between the question predictors and the control predictors in the baseline control model, sequentially. Finally, add interactions between the question predictors. Here, the objective is to obtain a parsimonious model that controls away all extraneous variation first, and then focus attention on the impact of the question predictors. While this approach refines your view of the impact of the question predictors, removing that part of their effect that may depend on the inter-relationships with the controls, it never reveals the “total” impact of the question predictors on the outcome for a person who has been randomly selected from the population without regard to any of their other characteristics. “Work Back From The End” Approach: Include all possible predictors in the model, both their main effects and interactions. The, remove statistically unimportant predictors sequentially to achieve a more parsimonious model, starting with those of lowest declared priority that do not appear to have statistically significant effects (i.e., remove question predictors last). Make sure that you remove any statistically unimportant ahead of any of the main effects from which they are constituted. Here, the objective is to obtain a final parsimonious model by sequentially removing predictors that appear unimportant. The idea is that you get to see the impact of “everything” to start with, and then you can “slim down” the fitted model to a final model. However, the impact of main effects is always masked when interactions are present in the model, and you still may remove an important predictor whose correlation with another predictor makes it look unimportant. Devise Your Own Strategy? It’s acceptable to devise your own strategy, in fact it’s probably the best approach as you know the field the best!. It’s perfectly acceptable to devise your own strategy providing that it is systematic, sensible and that you explain it explicitly to your reader, describing the logic that underpins its construction.