Logical Models and Basic Numeracy in Social Sciences http://www.psych.ut.ee/stk/Beginners_Logical_Models.pdf Rein Taagepera © 2015 REIN TAAGEPERA, professor emeritus at University of California, Irvine, and University of Tartu (Estonia), is the recipient of the Johan Skytte Prize in Political Science, 2008. He has 3 research articles in physics and over 120 in social sciences. Table of Contents Preface 8 A. Simple Models and Graphing 1. A Game with Serious Consequences 13 A guessing game. Skytte Prize 2008. An ignorance-based logical model. Exponents. 1a. Professionals Correct Their Own Mistakes 18 Means and median. Do not trust – try to verify by simple means. Pro- fessionals correct their own mistakes. But can this be so? 2. Pictures of Connections: How to Draw Graphs on Regular Scales 22 Example 1: Number of parties and cabinet duration. Constructing the framework. Placing data and theoretical curves on graphs. Making sense of the graph. Example 2: Linear patterns. How to measure the number of parties. 3. Science Walks on Two Legs: Observation and Thinking 29 Quantitatively predictive logical models. The invisible gorilla. Gravi- tation undetected. The gorilla moment for the number of seat-winning parties. What is “Basic Numeracy”? 4. The Largest Component: Between Mean and Total 36 Between mean and total. How to express absolute and relative differen- ces. Directional and quantitative models. Connecting the connections. How long has the geometric mean been invoked in political science? 5. Forbidden and Allowed Regions: Logarithmic Scales 42 Regular scale and its snags. Logarithmic scale. When numbers multiply, their logarithms add. Graphing logarithms. Logarithms of numbers between 1 and 10 – and beyond. 6. Duration of Cabinets: The Number of Communication Channels 49 The number of communication channels among n actors. Average duration of governmental cabinets. Leap of faith: A major ingredient in model building. The basic rule of algebra: balancing. Laws and models. Table of Contents 7. How to Use Logarithmic Graph Paper 57 Even I can find log2! Placing simple integer values on logarithmic scale. Fully logarithmic or log-log graph paper. Slopes of straight lines on log-log paper. Semilog graphs. Why spend so much time on the cabinet duration data? Regular, semilog and log-log graphs – when to use which? B. Some Basic Formats 8. Think Inside the Box – The Right Box 66 Always graph the data – and more than the data! Graph the equality line, if possible. Graph the conceptually allowed area – this is the right box to think in. The simplest curve joining the anchor points. Support for Democrats in US states: Problem. Support for Democrats in US states: Solution 9. Capitalism and Democracy in a Box 78 Support for democracy and capitalism: How can we get more out of this graph? Expanding on the democracy-capitalism box. Fitting with fixed exponent function Y=Xk. Why Y=Xk is simpler than y=a+bx ? Fitting with fixed exponent function 1-Y=(1-X)k. What is more basic: support or opposition? Logical formats and logical models. How can we know that data fits Y=Xk? A box with three anchor points: Seats and votes. 10. Science Means Connections Among Connections: Interlocking Relationships 89 Interlocking equations. Connections between constant values in relation- ships of similar form. Why linear fit lack connecting power. Many variables are interdependent, not “independent” or “dependent”. 11. Volatility: A Partly Open Box 96 The logical part of a coarse model for volatility. Make your models as simple as possible – but no simpler. Introducing an empirical note into the coarse model. Testing the model with data. Testing the model for logical consistency. 12. How to Test Models: Logical testing and Testing with Data 104 Logical testing. Testing with data. Many models become linear when logarithmic are taken. What can we see in this graph? The tennis match between data and models. Why would the simplest forms prevail? What can we see in this graph? – A sample list. 13. Getting a Feel for Exponentials and Logarithms 113 Exponents. Fractional exponents of 10. Decimal logarithms. What are logarithms good for? Logarithms on other bases than 10. 4 Table of Contents 14. When to Fit with What 118 Unbounded field – try linear fit. One quadrant allowed – try fixed exponent fit. Two quadrants allowed – try exponential fit. How to turn curves into straight lines. Calculating the parameters of fixed expo- nent equation in a single quadrant. Calculating the parameters of exponential equation in two quadrants. Constraints within quadrants: Two kinds of “drawn-out S” curves. C. Interaction of Logical Models and Statistical Approaches 15. The Basics of Linear Regression and Correlation 2 Coefficient R 129 Regression of y on x. Reverse regression of x on y. Directionality of the two OLS lines: A tall twin’s twin tends to be shorter than her twin. Non- transitivity of OLS regression. Correlation coefficient R2. The R2 measures lack of scatter: But scatter along which line? 16. Symmetric Regression and its Relationship to R2 142 From minimizing the sum of squares to minimizing the sum of rectangles. How R-squared connects with the slopes of regression lines. EXTRA: The mathematics of R2 and the slopes of regression lines. 17. When is Linear Fit Justified? 149 Many data clouds do not resemble ellipses. Grossly different patterns can lead to the same regression lines and R2. Sensitivity to outliers. Empirical configuration and logical constraints. 18. Federalism in a Box 157 Constitutional rigidity and judicial review. Degree of federalism and central bank independence. Bicameralism and degree of federalism. Conversion to scale 0 to 1. 19. The Importance of Slopes in Model Building 167 Notation for slopes.Equation for the slope of a parabola – and for y=xk. Cube root law of assembly sizes: Minimizing communication load. Expo- nential growth as an ignorance-based model: Slope proportional to size. Simple logistic model: Stunted exponential. How slopes combine – evidence for the slope of y=xk. Equations and constraints for some basic models. D. Further Examples and Tools 20. Interest Pluralism and the Number of Parties: Exponential Fit 181 Interest group pluralism. Fitting an exponential curve to interest group pluralism data. The slope of the exponential equation. Why not fit with fixed exponent format? EXTRA 1: Why not a fit with fixed expo- nent? EXTRA 2: Electoral disproportionality. 5 Table of Contents 21. Moderate Districts, Extreme Representatives: Competing Models 192 Graph more than the data. A model based on smooth fit of data to anchor points. A kinky model based on political polarization. Com- paring the smooth and kinky models. The envelopes of the data cloud. Why are representatives more extreme than their districts? 22. Centrist Voters, Leftist Elites: Bias Within the Box 202 23. Medians and Geometric Means 207 A two-humped camel’s median hump is a valley. Arithmetic mean and normal distribution. Geometric mean and lognormal distribution. The median is harder to handle than the means. The sticky case of almost lognormal distributions. How conceptual ranges, means, and forms of relationships are connected. 24. Fermi’s Piano Tuners: “Exact” Science and Approximations 215 As exact as possible – and as needed. How many piano tuners? The range of possible error. Dimensional consistency. 25. Examples of Models Across Social Sciences 220 Sociology: How many journals for speakers of a language? Political history: Growth of empires. Demography: World population growth over one million years. Economics: Trade/GDP ratio. 26. Comparing Models 229 An attempt at classifying quantitatively predictive logical models and formats. Some distinctions that may create more quandaries than they help to solve. Ignorance, except for constraints. Normal distribution – full ignorance. Lognormal distribution – a single constraint. The mean of the limits – two constraints. The dual source of exponential format: Constant rate of change, and constraint on range. Simple logistic change: Zone in two quadrants. Limits on ranges of two interrelated factors: Fixed exponent format. Zone in one quadrant: Anchor point plus floor or ceiling lead to exponential change. Box in one quadrant, plus two anchor points. Box in one quadrant, plus three anchor points. Communication channels and their consequences. Population and trade models. Substantive models: Look for connections among connections. Appendix A: What to Look for and Report in Multivariable Linear Regression 241 Making use of published multi-variable regression tables: A simple example. Guarding against colinearity. Running a multi-variable linear regression. Processing data prior to exploratory regression. Running exploratory regression. Lumping less significant variables: The need to report all medians and means. Re-running exploratory regression with fewer variables. Report the domains of all variables! Graphing the predictions of the regression equation against the actual outputs. Model- testing regression. Substantive vs. statistical significance. 6 Table of Contents Appendix B: Some Exam Questions 255 Appendix C: An Alternative Introduction to Logical Models: Basic “Graphacy” in Social Science Model Building 267 References 296 7 Preface This is a hands-one book that aims to add meaning to statistical ap- proaches, which often are the only quantitative methodology social science students receive. The scientific method includes stages where observation and statistical data fitting do not suffice – broader logical