class: center, middle, inverse, title-slide .title[ # MT611 - Quantitative Research Methods ] .subtitle[ ## Lecture 2: Understanding Models and Equations ] .author[ ### Damien Dupré ] .date[ ### Dublin City University ] --- # Essential Concepts to Master In Academic Reports, all sections are linked: .center[**Introduction ➡️ Literature Review ➡️ Method ➡️ Results ➡️ Discussion & Conclusion**] -- To understand the statistics in the results section, it is essential to identify the concepts presented in each section: <div class="nomnoml html-widget html-fill-item" id="htmlwidget-9eceb20dc2a931752ae4" style="width:864px;height:216px;"></div> <script type="application/json" data-for="htmlwidget-9eceb20dc2a931752ae4">{"x":{"code":"\n#fill: #FEFEFF\n#lineWidth: 1\n#zoom: 4\n#direction: right\n\n#stroke: black\n#direction: right\n#align: center\n[Introduction | Variables]->[Literature Review | Hypotheses]\n[Literature Review | Hypotheses]->[Method | Model & Equation]\n[Method | Model & Equation]->[Results | Statistical Test]\n[Results | Statistical Test]->[Discussion & Conclusion | Interpretation]","svg":false,"png":null},"evals":[],"jsHooks":[]}</script> --- class: inverse, mline, center, middle # 1. Model Representation in the Method Section of Academic Reseach Paper --- # Method Section in Academic Papers The method section is always structured in the same way: #### 1. Observations > Short section presenting where the data are coming from. If they are coming from human participants, then their average age and gender is indicated. #### 2. Variables > Short section presenting each variable as well as their type and role. #### 3. Procedure > Short section presenting how data were collected. #### 4. Data Analytics > Short section to display how the hypotheses are tested by displaying a graphical representation of the Model and its corresponding Equation(s). **While 1, 2, and 3 are straight forward, we will focus on the last point "Data Analytics" as it is uses rules that can be tricky.** --- # Model Representation Models are an overview of the predicted relationship between variables stated in the hypotheses You must follow these rules: - Rule 1: All the arrows correspond to an hypothesis to be tested - Rule 2: All the tested hypotheses have to be represented with an arrow - Rule 3: Hypotheses using the same Outcome variable should be included in the same model - Rule 4: Only one Outcome variable is included in each model (except for SEM model) --- # Model Representation .pull-left[ .center[**A simple arrow is a main effect**] <div id="htmlwidget-f30d7598595affc4f25a" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-f30d7598595affc4f25a">{"x":{"diagram":"\ndigraph rmarkdown {\n graph [rankdir = LR]\n \n node [shape = box]\n Predictor; Outcome\n \n Predictor -> Outcome [label= b1]\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] .pull-right[ .center[**A crossing arrow is an interaction effect**] <div id="htmlwidget-b85bf888dc62471906a9" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-b85bf888dc62471906a9">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n \"Predictor 1\"; Outcome; \"Predictor 2\"\n node [shape = point, width = 0, height = 0]\n \"\"\n \n \"Predictor 2\" -> \"\" [label= b2]\n \"Predictor 1\" -> \"\" [arrowhead = none] [label= b1]\n \"\"-> Outcome [label= b3]\n \n subgraph {\n rank = same; \"Predictor 2\"; \"\";\n }\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> .center[Note: By default, an interaction effect involves the test of the main effect hypotheses of all Predictors involved] ] --- # Structure of Models Distinguish square and circles - **squares** are actual **measures/items** - **circles** are **latent variables** related to measures/items Example: - `\(Salary\)` is directly measured (in $, €, or £) so it's a square. - `\(Job\,Satisfaction\)` is a latent variable with several questions so it's a circle. Items used for latent variables can be omitted in a model, variables are the most important. We can distinguish 2 types of relationship in a model: - Main effect relationship - Interaction effect relationship --- # Main Effect Relationship .pull-left[ .center[Relationship between one Predictor and one Outcome variable] <div id="htmlwidget-88fdd2a34d502a66744b" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-88fdd2a34d502a66744b">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n node [shape = box]\n \n Predictor -> Outcome\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> This model tests one hypothesis: - 1 main effect ] .pull-right[ .center[Relationship between two Predictors and one Outcome variable] <div id="htmlwidget-8e1fe96c0688a849169a" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-8e1fe96c0688a849169a">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n node [shape = box]\n \n \"Predictor 1\" -> Outcome\n \"Predictor 2\" -> Outcome\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> This model tests two hypotheses: - 2 main effects ] --- # Interaction Effect Relationship An interaction means that **the effect of a Predictor 1 on the Outcome variable will be different according the possibilities of a Predictor 2** (also called Moderation). .pull-left[ classic representation: <div id="htmlwidget-bb3486349adfa5d821dc" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-bb3486349adfa5d821dc">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n \"Predictor 1\"; \"Predictor 2\"; Outcome\n node [shape = point, width = 0, height = 0]\n \"\"\n \n \"Predictor 2\" -> \"\"\n \"Predictor 1\" -> \"\" [arrowhead = none]\n \"\"-> Outcome\n \n subgraph {\n rank = same; \"Predictor 2\"; \"\";\n }\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] .pull-right[ is the same as: <div id="htmlwidget-44474b3e8cd160f85a98" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-44474b3e8cd160f85a98">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n node [shape = box]\n\n \n \"Predictor 1\" -> Outcome\n \"Predictor 2\" -> Outcome\n \"Predictor 1 X Predictor 2\" -> Outcome\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] This model tests three hypotheses: - 2 main effects - 1 interaction effect --- # Types of Model ## Simple Model - One or more predictors - Only one outcome - Made of main or/and interaction effects ## Mediation Model (simple or moderated) - At least 2 predictors (one call Mediator) - Only one outcome - Made of main effects only for simple mediation / main and interaction effects for moderated mediation ## Structural Equation Model (SEM) - At least 2 predictors (usually latent variables) - One or more outcome - Made of main or/and interaction effects --- # Simple Model Simple Models are the most statistically powerful, easy to test and reliable models. Always prefer a simple model compared to a more complicated solution. Warning, including interaction effect requires a significantly higher sample size. Example: <div id="htmlwidget-2514ce28ae593bde0f73" style="width:800px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-2514ce28ae593bde0f73">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n \"Salary\"; \"Gender\"; \"Age\"\n node [shape = circle]\n \"Job Satisfaction\"\n node [shape = point, width = 0, height = 0]\n \"\"\n \n \"Age\" -> \"Job Satisfaction\"\n \"Gender\" -> \"\"\n \"Salary\" -> \"\" [arrowhead = none]\n \"\" -> \"Job Satisfaction\"\n \n subgraph {\n rank = same; \"Gender\"; \"\";\n }\n subgraph {\n rank = same; \"Age\"; \"Salary\";\n }\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> This model tests four hypotheses: - 3 main effects - 1 interaction effect --- # Mediation Models A Mediation model is a complex path analysis between 3 variables, where one of them explains the relationship between the other two. It is usually used to identify cognitive process in psychology. Example: <div id="htmlwidget-5d05eac75555269eefd0" style="width:800px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-5d05eac75555269eefd0">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n \"Exam Results\"\n node [shape = oval]\n \"Self-Esteem\"; Happiness\n \n \"Exam Results\" -> {Happiness \"Self-Esteem\"}\n \"Self-Esteem\" -> Happiness\n\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> This model tests one hypothesis: - 1 mediation effect - but it requires 2 main effects --- # Structural Equation Model A Structural Equation Model (SEM) is a complex path analysis between multiple variables including multiple Outcomes and using factor analysis for latent variable estimation. <div id="htmlwidget-7531f021030c44f57ef2" style="width:800px;height:300px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-7531f021030c44f57ef2">{"x":{"diagram":"\ndigraph rmarkdown {\n graph [rankdir = LR]\n \n node [shape = oval]\n \"Perceived Ease-of-use\"; \"Perceived Usefulness\"; \"Intention to Use\"\n \n node [shape = box]\n PU1; PU2; PU3; PU4; PU5; PEOU1; PEOU2; PEOU3; PEOU4; BI1; BI2; \"Actual Use\"\n \n {PU1 PU2 PU3 PU4 PU5} -> \"Perceived Usefulness\" [arrowhead = none]\n {PEOU1 PEOU2 PEOU3 PEOU4} -> \"Perceived Ease-of-use\" [arrowhead = none]\n {BI1 BI2} -> \"Intention to Use\" [arrowhead = none]\n \n \"Perceived Usefulness\" -> \"Intention to Use\"\n \"Perceived Ease-of-use\" -> {\"Perceived Usefulness\" \"Intention to Use\"}\n \"Intention to Use\" -> \"Actual Use\"\n \n subgraph {\n rank = same; \"Perceived Usefulness\"; \"Perceived Ease-of-use\";\n }\n \n subgraph {\n rank = same; PU1; PU2; PU3; PU4; PU5; PEOU1; PEOU2; PEOU3; PEOU4; BI1; BI2;\n }\n\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> This model tests four hypotheses: - 4 main effects - because the relationship between items of a scale and their corresponding latent variable is considered as significant by default if the scale is valid --- # A Good Model - Comprehensiveness: Explains a wide range of phenomena - Internal Consistency: Propositions and assumptions are consistent and fit together in a coherent manner - Parsimony: Contains only those concepts and assumptions essential for the explanation of a phenomenon - Testability: Concepts and relational statements are precise. - Empirical Validity: Holds up when tested in the real world. Example: <div id="htmlwidget-983315589fed18d622a7" style="width:800px;height:300px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-983315589fed18d622a7">{"x":{"diagram":"\ndigraph rmarkdown {\n graph [rankdir = LR]\n \n node [shape = oval]\n \"Perceived Ease-of-use\"; \"Perceived Usefulness\"; \"Intention to Use\"\n \n node [shape = box]\n \"Actual Use\"\n \n \"Perceived Usefulness\" -> \"Intention to Use\"\n \"Perceived Ease-of-use\" -> {\"Perceived Usefulness\" \"Intention to Use\"}\n \"Intention to Use\" -> \"Actual Use\"\n \n subgraph {\n rank = same; \"Perceived Usefulness\"; \"Perceived Ease-of-use\";\n }\n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> --- # A Bad Theory/Model - Too complicated - Does not explain many things - Cannot be tested Is it bad?  --- # Representing a Model .pull-left[ The representation of a model can easily be done directly in a manuscript written with Microsoft Words For more details, is it also possible to draw the model in Microsoft PowerPoint and to copy-paste it in Microsoft Words <span><i class="fas fa-exclamation-triangle faa-flash animated faa-slow " style=" color:red;"></i></span> **Warning**: - Do NOT fill boxes with any color, use only black and white colors. - Use line arrow, no thick arrows allowed. ] .pull-right[ <img src="https://www.myofficetricks.com/wp-content/uploads/2020/09/myofficetricks.com_2020-09-23_01-58-57.gif" style="display: block; margin: auto;" /> ] --- # Representing a Model Beside MS Words and PowerPoint, there are some ways to draw models in a nicer way. <span><i class="fas fa-arrow-circle-right faa-horizontal animated faa-slow " style=" color:blue;"></i></span> **Flowchart Software/Websites** There are many of them and google would find them very quickly but to my knowledge, https://www.diagrams.net/ is free, easy to use and very efficient. <span><i class="fas fa-arrow-circle-right faa-horizontal animated faa-slow " style=" color:blue;"></i></span> **Flowchart Coding Languages** Going further into details of how to design academic models, flowchart coding languages are the ultimate steps. Instead of using a GUI, it is possible to draw models from a couple of lines of code which is faster after practising a lot. To my knowledge, the main flowchart coding language tool implementing the [DOT style](https://en.wikipedia.org/wiki/DOT_(graph_description_language)) are: - [mermaid-js](http://mermaid-js.github.io/mermaid/#/) - [viz-js](http://viz-js.com/) - [nomnoml](https://www.nomnoml.com/) [But many more alternative can be used](https://modeling-languages.com/text-uml-tools-complete-list/) --- # DOT Language It is easy to start by following these rules: - Variable names must not include space between words - Arrows are represented by the characters - and > .pull-left[ ```r Predictor1 -> Outcome Predictor2 -> Outcome ``` ] .pull-right[ <div id="htmlwidget-6f1d8b9e786d9d3fe28d" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-6f1d8b9e786d9d3fe28d">{"x":{"diagram":"\n digraph {\n Predictor1 -> Outcome\n Predictor2 -> Outcome\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] -- To modify the orientation and the shape of the box, add the following options: .pull-left[ ```r graph [rankdir = LR] node [shape = box] Predictor1 -> Outcome Predictor2 -> Outcome ``` ] .pull-right[ <div id="htmlwidget-dcdfc64296f5794e54a4" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-dcdfc64296f5794e54a4">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n node [shape = box]\n \n Predictor1 -> Outcome\n Predictor2 -> Outcome\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] --- # DOT Language Unfortunately the way to plot an internaction is much trickier. You have to include an invisible box to design the arrow crossing the main arrow: .pull-left[ ```r graph [rankdir=LR] node [shape=box] Predictor1; Predictor2; Outcome node [shape=point, width=0, height=0] '' Predictor1 -> '' [arrowhead=none] Predictor2 -> '' ''-> Outcome ``` ] .pull-right[ <div id="htmlwidget-873249b169eb6149d57e" style="width:400px;height:200px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-873249b169eb6149d57e">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n Predictor1; Predictor2; Outcome\n node [shape = point, width = 0, height = 0]\n \"\"\n \n Predictor2 -> \"\"\n Predictor1 -> \"\" [arrowhead = none]\n \"\"-> Outcome\n \n subgraph {\n rank = same; Predictor2; \"\";\n }\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] There are many more rules to make more complicated models, see for more details https://graphviz.org/doc/info/lang.html --- # DotUML Extension for Google Docs This DOT language can easily be implemented with the DotUML extension in Google Docs. DotUML is developed by a company called BML Solutions but as far as I know it is a free plug-in (at least free of money). See https://dotuml.com for more details. After installed the extension, use GraphViz and remove the code corresponding to the default example. <img src="https://lh3.googleusercontent.com/-Z9ItvyBfmkU/Xrd4cCsLmlI/AAAAAAAABfI/8voEH5bxHYktwF6pM3onNwG2X9JZd9jWwCNcBGAsYHQ/s640-w640-h400/Capture%2Bd%25E2%2580%2599%25C3%25A9cran%2B-%2B2020-05-09%2B%25C3%25A0%2B21.38.49.png" style="display: block; margin: auto;" /> --- class: title-slide, middle ## Homework Exercise In the research paper you have selected, **draw the model(s) tested.** Remember, there is only 1 Outcome variable per model and it is not possible to draw two models with the same Outcome variable. Send me your figure by email at damien.dupre@dcu.ie **before the next lecture.** --- class: inverse, mline, center, middle # 2. Understanding the Equation used to Test Hypotheses --- # A Basic Equation Let's imagine the perfect scenario: **your predictor Predictor variable explains perfectly the outcome variable**. The corresponding equation is: `\(Outcome = Predictor\)` .pull-left[ <table class="table table-striped" style="font-size: 14px; color: black; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:center;"> Observation </th> <th style="text-align:center;"> Outcome </th> <th style="text-align:center;"> Predictor </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> a </td> <td style="text-align:center;"> 10 </td> <td style="text-align:center;"> 10 </td> </tr> <tr> <td style="text-align:center;"> b </td> <td style="text-align:center;"> 9 </td> <td style="text-align:center;"> 9 </td> </tr> <tr> <td style="text-align:center;"> c </td> <td style="text-align:center;"> 8 </td> <td style="text-align:center;"> 8 </td> </tr> <tr> <td style="text-align:center;"> d </td> <td style="text-align:center;"> 7 </td> <td style="text-align:center;"> 7 </td> </tr> <tr> <td style="text-align:center;"> e </td> <td style="text-align:center;"> 6 </td> <td style="text-align:center;"> 6 </td> </tr> <tr> <td style="text-align:center;"> f </td> <td style="text-align:center;"> 5 </td> <td style="text-align:center;"> 5 </td> </tr> <tr> <td style="text-align:center;"> g </td> <td style="text-align:center;"> 4 </td> <td style="text-align:center;"> 4 </td> </tr> <tr> <td style="text-align:center;"> h </td> <td style="text-align:center;"> 3 </td> <td style="text-align:center;"> 3 </td> </tr> <tr> <td style="text-align:center;"> i </td> <td style="text-align:center;"> 2 </td> <td style="text-align:center;"> 2 </td> </tr> <tr> <td style="text-align:center;"> j </td> <td style="text-align:center;"> 1 </td> <td style="text-align:center;"> 1 </td> </tr> <tr> <td style="text-align:center;"> k </td> <td style="text-align:center;"> 0 </td> <td style="text-align:center;"> 0 </td> </tr> </tbody> </table> ] .pull-right[ <img src="lecture_2_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" /> ] --- # A Basic Equation In the equation `\(Outcome = Predictor\)`, **three coefficients are hidden** because they are unused: - the **intercept coefficient** `\(b_{0}\)` (i.e., the value of the Outcome when the Predictor = 0) which is 0 in our case - the **estimate coefficient** `\(b_{1}\)` (i.e., how much the Outcome increases when the Predictor increases by 1) which is 1 in our case - the **error coefficient** `\(e\)` (i.e., how far from the prediction line the values of the Outcome are) which is 0 in our case So in general, the relation between a predictor and an outcome can be written as: `$$Outcome = b_{0} + b_{1}\,Predictor + e$$` which is in our case: `$$Outcome = 0 + 1 * Predictor + 0$$` --- # A Basic Equation The equation `\(Outcome = b_{0} + b_{1}\,Predictor + e\)` is the same as the good old `\(y = ax + b\)` (here ordered as `\(y = b + ax\)`) where `\(b_{0}\)` is `\(b\)` and `\(b_{1}\)` is `\(a\)`. It is very important to know that under **EVERY** statistical test, a similar equation is used (t-test, ANOVA, Chi-square are all linear regressions). <div class="plotly html-widget html-fill-item" id="htmlwidget-cbd3ad10bd2d35a2e76d" style="width:396px;height:396px;"></div> <script type="application/json" data-for="htmlwidget-cbd3ad10bd2d35a2e76d">{"x":{"data":[{"x":[0,1,2,3,4,5,6,7,8,9,10],"y":[0,1,2,3,4,5,6,7,8,9,10],"text":["Predictor: 0<br />Outcome: 0","Predictor: 1<br />Outcome: 1","Predictor: 2<br />Outcome: 2","Predictor: 3<br />Outcome: 3","Predictor: 4<br />Outcome: 4","Predictor: 5<br />Outcome: 5","Predictor: 6<br />Outcome: 6","Predictor: 7<br />Outcome: 7","Predictor: 8<br />Outcome: 8","Predictor: 9<br />Outcome: 9","Predictor: 10<br />Outcome: 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2.025316455696203554737","Predictor: 2.1518987<br />Outcome: 2.151898734177215999352","Predictor: 2.2784810<br />Outcome: 2.278481012658228888057","Predictor: 2.4050633<br />Outcome: 2.405063291139241776762","Predictor: 2.5316456<br />Outcome: 2.531645569620254221377","Predictor: 2.6582278<br />Outcome: 2.658227848101266665992","Predictor: 2.7848101<br />Outcome: 2.784810126582279554697","Predictor: 2.9113924<br />Outcome: 2.911392405063292443401","Predictor: 3.0379747<br />Outcome: 3.037974683544304443927","Predictor: 3.1645570<br />Outcome: 3.164556962025316888543","Predictor: 3.2911392<br />Outcome: 3.291139240506329777247","Predictor: 3.4177215<br />Outcome: 3.417721518987342665952","Predictor: 3.5443038<br />Outcome: 3.544303797468355110567","Predictor: 3.6708861<br />Outcome: 3.670886075949367555182","Predictor: 3.7974684<br />Outcome: 3.797468354430380443887","Predictor: 3.9240506<br />Outcome: 3.924050632911393332591","Predictor: 4.0506329<br />Outcome: 4.050632911392405333117","Predictor: 4.1772152<br />Outcome: 4.177215189873417777733","Predictor: 4.3037975<br />Outcome: 4.303797468354430222348","Predictor: 4.4303797<br />Outcome: 4.430379746835443555142","Predictor: 4.5569620<br />Outcome: 4.556962025316455999757","Predictor: 4.6835443<br />Outcome: 4.683544303797468444372","Predictor: 4.8101266<br />Outcome: 4.810126582278481777166","Predictor: 4.9367089<br />Outcome: 4.936708860759494221782","Predictor: 5.0632911<br />Outcome: 5.063291139240506666397","Predictor: 5.1898734<br />Outcome: 5.189873417721519111012","Predictor: 5.3164557<br />Outcome: 5.316455696202531555628","Predictor: 5.4430380<br />Outcome: 5.443037974683544888421","Predictor: 5.5696203<br />Outcome: 5.569620253164557333037","Predictor: 5.6962025<br />Outcome: 5.696202531645569777652","Predictor: 5.8227848<br />Outcome: 5.822784810126583110446","Predictor: 5.9493671<br />Outcome: 5.949367088607595555061","Predictor: 6.0759494<br />Outcome: 6.075949367088607999676","Predictor: 6.2025316<br />Outcome: 6.202531645569620444292","Predictor: 6.3291139<br />Outcome: 6.329113924050632888907","Predictor: 6.4556962<br />Outcome: 6.455696202531645333522","Predictor: 6.5822785<br />Outcome: 6.582278481012657778138","Predictor: 6.7088608<br />Outcome: 6.708860759493670222753","Predictor: 6.8354430<br />Outcome: 6.835443037974683555547","Predictor: 6.9620253<br />Outcome: 6.962025316455696000162","Predictor: 7.0886076<br />Outcome: 7.088607594936708444777","Predictor: 7.2151899<br />Outcome: 7.215189873417720889393","Predictor: 7.3417722<br />Outcome: 7.341772151898733334008","Predictor: 7.4683544<br />Outcome: 7.468354430379746666802","Predictor: 7.5949367<br />Outcome: 7.594936708860759111417","Predictor: 7.7215190<br />Outcome: 7.721518987341771556032","Predictor: 7.8481013<br />Outcome: 7.848101265822784888826","Predictor: 7.9746835<br />Outcome: 7.974683544303797333441","Predictor: 8.1012658<br />Outcome: 8.101265822784808889878","Predictor: 8.2278481<br />Outcome: 8.227848101265822222672","Predictor: 8.3544304<br />Outcome: 8.354430379746833779109","Predictor: 8.4810127<br />Outcome: 8.481012658227847111903","Predictor: 8.6075949<br />Outcome: 8.607594936708858668339","Predictor: 8.7341772<br />Outcome: 8.734177215189872001133","Predictor: 8.8607595<br />Outcome: 8.860759493670885333927","Predictor: 8.9873418<br />Outcome: 8.987341772151896890364","Predictor: 9.1139241<br />Outcome: 9.113924050632910223158","Predictor: 9.2405063<br />Outcome: 9.240506329113923555951","Predictor: 9.3670886<br />Outcome: 9.367088607594935112388","Predictor: 9.4936709<br />Outcome: 9.493670886075948445182","Predictor: 9.6202532<br />Outcome: 9.620253164556961777976","Predictor: 9.7468354<br />Outcome: 9.746835443037973334413","Predictor: 9.8734177<br />Outcome: 9.873417721518986667206","Predictor: 10.0000000<br />Outcome: 9.999999999999998223643","Predictor: 10.0000000<br />Outcome: 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8.101265822784808889878","Predictor: 7.9746835<br />Outcome: 7.974683544303797333441","Predictor: 7.8481013<br />Outcome: 7.848101265822784888826","Predictor: 7.7215190<br />Outcome: 7.721518987341771556032","Predictor: 7.5949367<br />Outcome: 7.594936708860759111417","Predictor: 7.4683544<br />Outcome: 7.468354430379746666802","Predictor: 7.3417722<br />Outcome: 7.341772151898733334008","Predictor: 7.2151899<br />Outcome: 7.215189873417720889393","Predictor: 7.0886076<br />Outcome: 7.088607594936708444777","Predictor: 6.9620253<br />Outcome: 6.962025316455696000162","Predictor: 6.8354430<br />Outcome: 6.835443037974683555547","Predictor: 6.7088608<br />Outcome: 6.708860759493670222753","Predictor: 6.5822785<br />Outcome: 6.582278481012657778138","Predictor: 6.4556962<br />Outcome: 6.455696202531645333522","Predictor: 6.3291139<br />Outcome: 6.329113924050632888907","Predictor: 6.2025316<br />Outcome: 6.202531645569620444292","Predictor: 6.0759494<br />Outcome: 6.075949367088607999676","Predictor: 5.9493671<br />Outcome: 5.949367088607595555061","Predictor: 5.8227848<br />Outcome: 5.822784810126583110446","Predictor: 5.6962025<br />Outcome: 5.696202531645569777652","Predictor: 5.5696203<br />Outcome: 5.569620253164557333037","Predictor: 5.4430380<br />Outcome: 5.443037974683544888421","Predictor: 5.3164557<br />Outcome: 5.316455696202531555628","Predictor: 5.1898734<br />Outcome: 5.189873417721519111012","Predictor: 5.0632911<br />Outcome: 5.063291139240506666397","Predictor: 4.9367089<br />Outcome: 4.936708860759494221782","Predictor: 4.8101266<br />Outcome: 4.810126582278481777166","Predictor: 4.6835443<br />Outcome: 4.683544303797468444372","Predictor: 4.5569620<br />Outcome: 4.556962025316455999757","Predictor: 4.4303797<br />Outcome: 4.430379746835443555142","Predictor: 4.3037975<br />Outcome: 4.303797468354430222348","Predictor: 4.1772152<br />Outcome: 4.177215189873417777733","Predictor: 4.0506329<br />Outcome: 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2.025316455696203554737","Predictor: 1.8987342<br />Outcome: 1.898734177215191110122","Predictor: 1.7721519<br />Outcome: 1.772151898734178443462","Predictor: 1.6455696<br />Outcome: 1.645569620253165776802","Predictor: 1.5189873<br />Outcome: 1.518987341772153110142","Predictor: 1.3924051<br />Outcome: 1.392405063291140443482","Predictor: 1.2658228<br />Outcome: 1.265822784810127998867","Predictor: 1.1392405<br />Outcome: 1.139240506329115332207","Predictor: 1.0126582<br />Outcome: 1.012658227848102665547","Predictor: 0.8860759<br />Outcome: 0.886075949367089998887","Predictor: 0.7594937<br />Outcome: 0.759493670886077443249","Predictor: 0.6329114<br />Outcome: 0.632911392405064776590","Predictor: 0.5063291<br />Outcome: 0.506329113924052109930","Predictor: 0.3797468<br />Outcome: 0.379746835443039554292","Predictor: 0.2531646<br />Outcome: 0.253164556962026887632","Predictor: 0.1265823<br />Outcome: 0.126582278481014276483","Predictor: 0.0000000<br />Outcome: 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(y) ","x"],"9cea315e689d":["function (y) ","x"],"9cea2d2634bd":["function (y) ","x"],"9cea27507b73":["function (y) ","x"],"9cea14d8cf2f":["function (y) ","x"],"9cea260a215a":["function (y) ","x"],"9cea6313af38":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- # Relationship between Variables Relationship between a `\(Predictor\)` and an `\(Outcome\)` variable (stated in a main effect hypothesis or in an interaction effect hypothesis) is analysed in terms of: .center[**"How many units of the Outcome variable increases/decreases/changes when the Predictor increases by 1 unit?"**] For example: > How much Job Satisfaction increases when the Salary increases by €1? The value of how much of the Outcome variable changes: - Is called the **Estimate** (also called Unstandardised Estimate) - Uses the letter `\(b\)` in equations (e.g., `\(b_1\)`, `\(b_2\)`, `\(b_3\)`, ...) For example: > If Job Satisfaction increases by 0.1 on a scale from 0 to 5 when the Salary increases by €1, then *b* associated to Salary is 0.1 --- # Significance of Relationships To evaluate if the strength of the relationship `\(b\)` between a Predictor and an Outcome variable is significant, an equation is statistically tested using all the predictors related to the same Outcome. The basic equation of a statistical model is: `$$Outcome = b_0 + b_n \,Predictors + Error$$` where the `\(Predictors\)` includes all the `\(n\)` variables used as predictor in formulated hypotheses using this specific `\(Outcome\)` variable and being associated to a specific `\(b\)` estimate. This expresses the idea that: - The Outcome can be described by one or multiple predictors. - The remaining part of the Outcome's variability that is not explained by the predictors is call the Error. --- # Equations, Variables and Effect Types Except in special cases: - An Outcome (or Dependent Variable) has to be Continuous - A Predictor can be Continuous or Categorical Example: `\(Job\,Satisfaction = b_{0} + b_{1}\,Salary + b_{2}\,Origin + e\)` In this equation: - `\(Salary\)` is continuous with a main effect on `\(Job\,Satisfaction\)` ( `\(b_{1}\)`) - `\(Origin\)` is categorical with a main effect on `\(Job\,Satisfaction\)` ( `\(b_{2}\)`) --- # Equations, Variables and Effect Types An interaction effect is represented by multiplying the 2 predictors involved: `$$Job\,Satisfaction = b_{0} + b_{1}\,Salary + b_{2}\,Origin + b_{3}\,Salary*Origin + e$$` In this equation: - `\(Salary\)` is continuous with a main effect on `\(Job\,Satisfaction\)` ( `\(b_{1}\)`) - `\(Origin\)` is categorical with a main effect on `\(Job\,Satisfaction\)` ( `\(b_{2}\)`) - `\(Salary\)` and `\(Origin\)` have an interaction effect on `\(Job\,Satisfaction\)` ( `\(b_{3}\)`) --- # Relevance of the Intercept To test hypotheses, only the `\(b\)` values associated to Predictors / Independent Variables are important. The intercept is always included in an equation but its result is useless for hypothesis testing. Let's see why the intercept is always included but discarded most of the time. Imagine we want to test the relationship between GDP per Capita and Life Expectancy of countries in the world. Let's compare a model without and a model with intercept: - Without intercept: `\(Life\,Expectancy = b_{1}\,GDP\,per\,Capita + e\)` - With intercept: `\(Life\,Expectancy = b_{0} + b_{1}\,GDP\,per\,Capita + e\)` --- # Relevance of the Intercept <img src="lecture_2_files/figure-html/unnamed-chunk-24-1.png" width="504" style="display: block; margin: auto;" /> If the intercept is not included, the intercept is zero and can lead to estimation errors --- # Notes on the Equations #### 1. Greek or Latin alphabet? `$$Y = \beta_{0} + \beta_{1}\,X_{1} + \epsilon \; vs. \; Y = b_{0} + b_{1}\,X_{1} + e$$` #### 2. Subscript `\(i\)` or not? `$$Y = b_{0} + b_{1}\,X_{1} + e \; vs. \; Y_{i} = b_{0} + b_{1}\,X_{1_{i}} + e_{i}$$` #### 3. Which sign between estimates and predictors? `$$Y = b_{0} + b_{1}.X_{1} + b_{2}*X_{2} + b_{3}\,X_{3} + e$$` #### 4. Hat on `\(Y\)` or not? Capital letter or not? .center[ `$$\hat{Y}\; or\; \hat{y}\; vs.\; Y\; or\; y$$` ] --- # Representing an Equation Exactly like with models, there are different ways to communicate an equation in Academic research outputs. The least sophisticated approach would be to type the equation in Microsoft Words and to apply some italics and subscript style a posteriori. While there is nothing wrong with this approach, note that Microsoft Words has a tool to insert equations (Insert -> Equations), then a GUI will help you to design special characters in equations. <img src="https://i.ytimg.com/vi/SRGaW3maK38/maxresdefault.jpg" style="display: block; margin: auto;" /> --- # Representing an Equation Now there is a better way, which is also more complicated called `\(\LaTeX\)` LaTex is used for entire manuscripts with all the specific design requirements imposed by ths style of academic journals. LaTex is the hell and we will see a specific approach to avoid it but the LaTex style for equations is the best. Here are the most basic rules: - A latex equation starts with `\begin{equation}` and ends with `\end{equation}` - There is no white space between words, a space can be added with `\,` - To subscript a number use the underscore sign `_` For example: ```r \begin{equation} Outcome = b_0 + b_1\,Predictor1 + b_2\,Predictor2 + e \end{equation} ``` Is translated as: `$$Outcome = b_0 + b_1\,Predictor1 + b_2\,Predictor2 +e$$` --- # Auto-LaTeX Equations for Google Docs Another fantastic extension is available for Google Docs called Auto-LaTeX Equations Instead of using `\begin{equation}` and `\end{equation}`, use `$$` to open and close the equation Then click on "Render Equation", for the equation to be transformed in LaTex style Note that some website can help to create the latex code, see for example https://latex.codecogs.com/eqneditor/editor.php <img src="https://lh3.googleusercontent.com/1UoEk_1Ej273-jmXBFNJV2GPyyWenrFtwm2jgoq0FxIs5WZeaRnij_3tfW8z6s73wh-q7mjdgA=w1280-h800" style="display: block; margin: auto;" /> --- class: title-slide, middle ## Exercise: Predicates of Statistical Analyses **Using the results obtained, identify the role of variables, formulate the tested hypotheses, draw the corresponding model, and translate it in an equation** --- # Example 1 **Using the results obtained, identify the role of variables, formulate the tested hypotheses, draw the corresponding model, and translate it in an equation** .pull-left[ .center[Data] |Participant | Sleep Time| Exam Results| |:-----------|----------:|------------:| |ppt1 | 9.0| 89| |ppt2 | 5.0| 64| |ppt3 | 8.5| 71| |ppt4 | 7.0| 77| |ppt5 | 6.5| 78| |ppt6 | 5.5| 69| ] .pull-right[ .center[Visualisation] <img src="lecture_2_files/figure-html/unnamed-chunk-29-1.png" width="504" style="display: block; margin: auto;" /> ] <div class="countdown" id="timer_21af9b68" data-warn-when="60" data-update-every="1" tabindex="0" style="right:1;bottom:0;"> <div class="countdown-controls"><button class="countdown-bump-down">−</button><button class="countdown-bump-up">+</button></div> <code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code> </div> --- # Example 1 Variables: - Outcome = Exam Results (from 0 to 100) - Predictor = Sleep Time (from 0 to Inf.) Alternative Hypothesis: - `\(H_a\)`: **Exam Results** *increases* when **Sleep Time** increases - ( `\(H_0\)`: **Exam Results** *stay the same* when **Sleep Time** increases) Model: <div id="htmlwidget-f25d4e1770d66a86dcce" style="width:500px;height:100px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-f25d4e1770d66a86dcce">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n node [shape = box]\n \n \"Sleep Time\" -> \"Exam Results\" [label= b1]\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> Equation: - `\(Exam\,Results = b_{0} + b_{1}\,Sleep\,Time + e\)` --- # Example 2 **Using the results obtained, identify the role of variables, formulate the tested hypotheses, draw the corresponding model, and translate it in an equation** .pull-left[ .center[Data] |Participant | Sleep Time| Exam Results|Age_c | |:-----------|----------:|------------:|:-----------| |ppt1 | 9.0| 89|experienced | |ppt2 | 5.0| 64|experienced | |ppt3 | 8.5| 71|experienced | |ppt4 | 7.0| 77|beginner | |ppt5 | 6.5| 52|beginner | |ppt6 | 5.5| 69|beginner | ] .pull-right[ .center[Visualisation] <img src="lecture_2_files/figure-html/unnamed-chunk-33-1.png" width="504" style="display: block; margin: auto;" /> ] <div class="countdown" id="timer_322e2d59" data-warn-when="60" data-update-every="1" tabindex="0" style="right:1;bottom:0;"> <div class="countdown-controls"><button class="countdown-bump-down">−</button><button class="countdown-bump-up">+</button></div> <code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code> </div> --- # Example 2 Variables: - Outcome = Exam Results (from 0 to 100) - Predictor 1 = Sleep Time (from 0 to Inf.) - Predictor 2 = Age (experienced vs beginner) Alternative Hypotheses: - `\(H_{a_{1}}\)`: **Exam Results** *increases* when **Sleep Time** increases - `\(H_{a_{2}}\)`: **Exam Results** of **experienced students** are *higher* than for **beginner students** - `\(H_{a_{3}}\)`: The effect of **Sleep Time** on **Exam Results** is *higher* for **experienced** than for **beginner students** --- # Example 2 Model: .pull-left[ .center[Classic Representation] <div id="htmlwidget-969965edfe9f869aa904" style="width:400px;height:300px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-969965edfe9f869aa904">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n \"Sleep Time\"; \"Exam Results\"; Age\n node [shape = point, width = 0, height = 0]\n \"\"\n \n Age -> \"\"\n \"Sleep Time\" -> \"\" [arrowhead = none]\n \"\"-> \"Exam Results\"\n \n subgraph {\n rank = same; Age; \"\";\n }\n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] .pull-right[ .center[Effects Correspondence] <div id="htmlwidget-450f171f14364aa3ebed" style="width:300px;height:300px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-450f171f14364aa3ebed">{"x":{"diagram":"\n digraph {\n graph [rankdir = LR]\n \n node [shape = box]\n \"Sleep Time\"; \"Exam Results\"; Age; \"Sleep Time * Age\"\n\n Age -> \"Exam Results\" [label= b2]\n \"Sleep Time\" -> \"Exam Results\" [label= b1]\n \"Sleep Time * Age\" -> \"Exam Results\" [label= b3]\n \n }","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> ] Equations: - `\(Exam\,Results = b_{0} + b_{1}\,Sleep\,Time + b_{2}\,Age + b_{3}\,Interaction + e\)` which corresponds to: `\(Exam\,Results = b_{0} + b_{1}\,Sleep\,Time + b_{2}\,Age + b_{3}\,Sleep\,Time*Age + e\)` --- class: title-slide, middle ## Homework Exercise In the research paper you have selected, **write the equation(s) corresponding to the model(s)** Send me your equation(s) by email at damien.dupre@dcu.ie **before the next lecture** --- class: inverse, mline, center, middle # Conclusion --- # Follow the Rules - Variables - Identify the role and type of each of your variables - Hypotheses - Formulate your alternative hypotheses by using the proposed templates - Any other formulation, even if it make sense, is not good practice (e.g. using the terms "has an impact", "is related to", "influences", ...) - An interaction hypothesis requires the formulation of the main effect hypothesis of each predictor involved - Model - A model should represent all your hypotheses and only your hypotheses - Draw only model per Outcome variable (except is doing SEM analyses) - Use the same names or acronyms as your hypotheses - Equation - Formulate your equation(s) in every paper that you want to submit (if reviewers want to remove it, they will tell you) --- class: inverse, mline, left, middle <img class="circle" src="https://github.com/damien-dupre.png" width="250px"/> # Thanks for your attention and don't hesitate to ask if you have any questions! 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