Sociology 6Z03 Review I

Transcription

1 Sociology 6Z03 Review I John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Outline: Review I Introduction Displaying Distributions Describing Distributions with Numbers The Normal Distributions Scatterplots and Correlation Least-Squares Regression Multiple Regression Contingency Tables Statistical Issues in Research Design John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

2 Introduction Organization of data the data table: Observations ( individuals, rows) by variables (characteristics of the individuals, columns) Kinds of variables: Quantitative variables (have a unit of measurement) Categorical variables (ordered or unordered categories; no unit of measurement) John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Displaying Distributions Categorical variables Bar graphs Pie charts Quantitative variables Histograms Stemplots (stem and leaf displays) Interpretation Time plots centre spread shape: symmetric, skewed, irregular outliers John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

3 Describing Distributions with Numbers Measuring centre Mean: x = 1 n x i Median: position of M = (n + 1)/2 Resistance to outliers Measuring spread Quartiles: position of Q 1 and Q 3 is (n + 1)/2 where n n/2 Variance: s 2 = 1 n 1 (x i x) 2 Standard deviation: s = s 2 Degrees of freedom: n 1 Five-number summaries (minimum, Q 1, median, Q 3, maximum) and boxplots John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 The Normal Distributions Density curves Mean µ, median, and standard deviation σ Family of normal distributions, x N(µ, σ) The standard normal distribution, z N(0, 1) Standardization: Normal distribution calculations Finding areas given z and x-values Finding z and x-values given areas z = x µ σ x = µ + zσ John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

4 Scatterplots and Correlation Scatterplots Explanatory variable (x) vs. response variable (y) Interpretation Clusters Outliers Relationship Direction (positive, negative, neither) Form (linear, nonlinear) Strength Coding values of a categorical variable on a scatterplot with colours or plotting symbols John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Scatterplots and Correlation Correlation The correlation coefficient r = 1 ( ) ( ) n 1 xi x yi y s x s y Measures strength and direction of linear relationship 1 r 1 John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

5 Least-Squares Regression Equation of a straight line, y = a + bx Intercept: a Slope: b The least-squares line Fitted (predicted) values: ŷ i = a + bx i Residuals: residual i = y i ŷ i Find a, b to minimize residual 2 i Slope and intercept b = r s y s x a = y bx a is predicted y when x = 0 b is average change in y associated with a one-unit increase in x John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Least-Squares Regression Correlation and Regression When there is no linear relationship, r and b are both 0 r = b when x and y are standardized variables r is symmetric in x and y, but b is not The two regression lines: of y on x and of x on y Squared correlation: r 2 is proportion of variation in y accounted for by linear regression of y on x John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

6 Least-Squares Regression Problems in Regression Outliers and influential data Nonlinearity Non-constant residual spread These problems can be detected in the scatterplot of y vs. x, or in a scatterplot of residuals vs. x. John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Least-Squares Regression Interpreting Correlation and Regression Dangers of extrapolation Lurking variables: Association is not causation John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

7 Multiple Regression Motivation: Decrease size of residuals Hold lurking variables constant The least-squares plane for two explanatory variables ŷ = a + b 1 x 1 + b 2 x 2 a, b 1, b 2 selected to minimize residual 2 a is predicted y when both x s are 0 b 1 is the slope of the plane in the direction of x 1 : average change in y when x 1 increases by 1, holding x 2 constant Multiple correlation, R R 2 is proportion of variation in y accounted for by linear regression of y on x 1 and x 2 Extension to several explanatory variables: ŷ = a + b 1 x 1 + b 2 x b k x k Residual plots John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Contingency Tables Two-Way Tables Two-way frequency tables of counts Percentage tables Calculate percentages within categories of the explanatory variable Make comparisons across categories of the explanatory variable John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

8 Contingency Tables Three-Way Tables Partial tables and partial associations Partial vs. marginal associations Partial relationships expected to disappear when: Control variable (z) intervenes causally between the explanatory variable (x) and the response (y): x z y Control variable is a common prior cause of the explanatory variable and the response: z x y Simpson s paradox: marginal and partial relationships can have different directions John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Statistical Issues in Research Design Experimental vs. Observational Data Fundamental distinction between experimental and observational data In an observational study, the researcher collects naturally occurring data, without trying to influence the value of the explanatory variable or variables. Example: Social surveys. Causal inference in observational studies is intrinsically ambiguous, because a relationship could be due to lurking variables. In an experimental study, the explanatory variable or variables are under the direct control of the researcher. In a properly designed experiment, causal inference is much less ambiguous than in observational research. John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

9 Statistical Issues in Research Design Sample Surveys Populations and samples Bias in study design Voluntary response samples (self-selection) Convenience samples Simple random sampling (SRS): Each possible sample of size n has equal chance of selection Other probability sampling designs Stratified random sampling Cluster sampling Multistage sampling John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19 Statistical Issues in Research Design Sample Surveys Telephone surveys: Random digit dialing Problems in survey design Undercoverage Nonresponse (global and item) Response biases Question-wording effects John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19

10 Statistical Issues in Research Design Experimental Design Principles of sound experimental design Control through comparison Random assignment of subjects Use enough subjects Problems in experimentation Hidden biases (and double-blind experimentation) Lack of realism Ethical issues John Fox (McMaster University) Sociology 6Z03 Review I Fall / 19