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Chapter 3 Association: Contingency, Correlation, and Regression

Learning Objectives. Identify variable type: Response or ExplanatoryDefine AssociationContingency tablesCalculate proportions and conditional proportions. Learning Objective 1: Response and Explanatory variables. Response variable (Dependent Variable) the outcome variable on which comparisons ar

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Chapter 3 Association: Contingency, Correlation, and Regression

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    1. Chapter 3 Association: Contingency, Correlation, and Regression Section 3.1 How Can We Explore the Association between Two Categorical Variables?

    2. Learning Objectives Identify variable type: Response or Explanatory Define Association Contingency tables Calculate proportions and conditional proportions

    3. Learning Objective 1: Response and Explanatory variables Response variable (Dependent Variable) the outcome variable on which comparisons are made Explanatory variable (Independent variable) defines the groups to be compared with respect to values on the response variable Example: Response/Explanatory Blood alcohol level/# of beers consumed Grade on test/Amount of study time Yield of corn per bushel/Amount of rainfall

    4. Learning Objective 2: Association The main purpose of data analysis with two variables is to investigate whether there is an association and to describe that association An association exists between two variables if a particular value for one variable is more likely to occur with certain values of the other variable

    5. Learning Objective 3: Contingency Table A contingency table: Displays two categorical variables The rows list the categories of one variable The columns list the categories of the other variable Entries in the table are frequencies

    6. Learning Objective 3: Contingency Table

    7. Learning Objective 4: Calculate proportions and conditional proportions

    8. Learning Objective 4: Calculate proportions and conditional proportions What proportion of organic foods contain pesticides? What proportion of conventionally grown foods contain pesticides? What proportion of all sampled items contain pesticide residuals?

    9. Learning Objective 4: Calculate proportions and conditional proportions

    10. Learning Objective 4: Calculate proportions and conditional proportions If there was no association between organic and conventional foods, then the proportions for the response variable categories would be the same for each food type

    11. Chapter 3 Association: Contingency, Correlation, and Regression Section 3.2 How Can We Explore the Association between Two Quantitative Variables?

    12. Learning Objectives: Constructing scatterplots Interpreting a scatterplot Correlation Calculating correlation

    13. Learning Objective 1: Scatterplot Graphical display of relationship between two quantitative variables: Horizontal Axis: Explanatory variable, x Vertical Axis: Response variable, y

    14. Learning Objective 1: Internet Usage and Gross National Product (GDP) Data Set

    15. Learning Objective 1: Internet Usage and Gross National Product (GDP)

    16. Learning Objective 1: Baseball Average and Team Scoring

    17. Learning Objective 1: Baseball Average and Team Scoring

    18. Learning Objective 2: Interpreting Scatterplots You can describe the overall pattern of a scatterplot by the trend, direction, and strength of the relationship between the two variables Trend: linear, curved, clusters, no pattern Direction: positive, negative, no direction Strength: how closely the points fit the trend Also look for outliers from the overall trend

    19. Learning Objective 2: Interpreting Scatterplots: Direction/Association Two quantitative variables x and y are Positively associated when High values of x tend to occur with high values of y Low values of x tend to occur with low values of y Negatively associated when high values of one variable tend to pair with low values of the other variable

    20. Learning Objective 2: Example: 100 cars on the lot of a used-car dealership Would you expect a positive association, a negative association or no association between the age of the car and the mileage on the odometer? Positive association Negative association No association

    21. Learning Objective 2: Example: Did the Butterfly Ballot Cost Al Gore the 2000 Presidential Election?

    22. Learning Objective 3: Linear Correlation, r Measures the strength and direction of the linear association between x and y A positive r value indicates a positive association A negative r value indicates a negative association An r value close to +1 or -1 indicates a strong linear association An r value close to 0 indicates a weak association

    23. Learning Objective 3: Correlation coefficient: Measuring Strength & Direction of a Linear Relationship

    24. Learning Objective 3: Properties of Correlation Always falls between -1 and +1 Sign of correlation denotes direction (-) indicates negative linear association (+) indicates positive linear association Correlation has a unitless measure - does not depend on the variables’ units Two variables have the same correlation no matter which is treated as the response variable Correlation is not resistant to outliers Correlation only measures strength of linear relationship

    25. Leaning Objective 4: Calculating the Correlation Coefficient

    26. Learning Objective 4: Calculating the Correlation Coefficient

    27. Learning Objective 4: Internet Usage and Gross National Product (GDP) STAT CALC menu Choose 8: LinReg(a+bx) 1st number = x variable 2nd number = y variable Enter

    28. Learning Objective 4: Baseball Average and Team Scoring Enter x data into L1 Enter y data into L2 STAT CALC memu Choose 8: LinReg(a+bx) 1st number = x variable 2nd number = y variable Enter

    29. Learning Objective 4: Cereal: Sodium and Sugar

    30. Chapter 3 Association: Contingency, Correlation, and Regression Section 3.3 How Can We Predict the Outcome of a Variable?

    31. Learning Objectives Definition of a regression line Use a regression equation for prediction Interpret the slope and y-intercept of a regression line Identify the least-squares regression line as the one that minimizes the sum of squared residuals Calculate the least-squares regression line

    32. Learning Objectives Compare roles of explanatory and response variables in correlation and regression Calculate r2 and interpret

    33. Learning Objective 1: Regression Analysis The first step of a regression analysis is to identify the response and explanatory variables We use y to denote the response variable We use x to denote the explanatory variable

    34. Learning Objective 1: Regression Line A regression line is a straight line that describes how the response variable (y) changes as the explanatory variable (x) changes A regression line predicts the value of the response variable (y) for a given level of the explanatory variable (x) The y-intercept of the regression line is denoted by a The slope of the regression line is denoted by b

    35. Learning Objective 2: Example: How Can Anthropologists Predict Height Using Human Remains? Regression Equation: is the predicted height and is the length of a femur (thighbone), measured in centimeters

    36. Learning Objective 3: Interpreting the y-Intercept y-Intercept: The predicted value for y when x = 0 Helps in plotting the line May not have any interpretative value if no observations had x values near 0

    37. Learning Objective 3: Interpreting the Slope Slope: measures the change in the predicted variable (y) for a 1 unit increase in the explanatory variable in (x) Example: A 1 cm increase in femur length results in a 2.4 cm increase in predicted height

    38. Learning Objective 3: Slope Values: Positive, Negative, Equal to 0

    39. Learning Objective 3: Regression Line At a given value of x, the equation: Predicts a single value of the response variable But… we should not expect all subjects at that value of x to have the same value of y Variability occurs in the y values!

    40. Learning Objective 3: The Regression Line The regression line connects the estimated means of y at the various x values In summary, Describes the relationship between x and the estimated means of y at the various values of x

    41. Learning Objective 4: Residuals Measures the size of the prediction errors, the vertical distance between the point and the regression line Each observation has a residual Calculation for each residual: A large residual indicates an unusual observation

    42. Learning Objective 4: “Least Squares Method” Yields the Regression Line Residual sum of squares: The least squares regression line is the line that minimizes the vertical distance between the points and their predictions, i.e., it minimizes the residual sum of squares Note: the sum of the residuals about the regression line will always be zero

    43. Learning Objective 5: Regression Formulas for y-Intercept and Slope Slope: Y-Intercept:

    44. Learning Objective 5: Calculating the slope and y intercept for the regression line

    45. Learning Objective 5: Internet Usage and Gross National Product (GDP)

    46. Learning Objective 5: Internet Usage and Gross National Product Enter x data into L1 Enter y data into L2 STAT CALC menu Choose 8: LinReg(a+bx) 1st number = x variable 2nd number = y variable Enter

    47. Learning Objective 5: Baseball Average and Team Scoring

    48. Learning Objective 5: Baseball average and Team Scoring Enter x data into L1 Enter y data into L2 STAT CALC Choose 8: LinReg(a+bx) 1st number = x variable 2nd number = y variable Enter

    49. Learning Objective 5: Cereal: Sodium and Sugar

    50. Learning Objective 6: The Slope and the Correlation Correlation: Describes the strength of the linear association between 2 variables Does not change when the units of measurement change Does not depend upon which variable is the response and which is the explanatory

    51. Learning Objective 6: The Slope and the Correlation Slope: Numerical value depends on the units used to measure the variables Does not tell us whether the association is strong or weak The two variables must be identified as response and explanatory variables The regression equation can be used to predict values of the response variable for given values of the explanatory variable

    52. Learning Objective 7: The Squared Correlation When a strong linear association exists, the regression equation predictions tend to be much better than the predictions using only We measure the proportional reduction in error and call it, r2

    53. Learning Objective 7: The Squared Correlation measures the proportion of the variation in the y-values that is accounted for by the linear relationship of y with x A correlation of .9 means that 81% of the variation in the y-values can be explained by the explanatory variable, x

    54. Chapter 3 Association: Contingency, Correlation, and Regression Section 3.4 What Are Some Cautions in Analyzing Association?

    55. Learning Objectives: Extrapolation Outliers and Influential Observations Correlations does not imply causation Lurking variables and confounding Simpson’s Paradox

    56. Learning Objective 1: Extrapolation Extrapolation: Using a regression line to predict y-values for x-values outside the observed range of the data Riskier the farther we move from the range of the given x-values There is no guarantee that the relationship given by the regression equation holds outside the range of sampled x-values

    57. Learning Objective 2: Outliers and Influential Points Construct a scatterplot Search for data points that are well outside of the trend that the remainder of the data points follow

    58. Learning Objective 2: Outliers and Influential Points A regression outlier is an observation that lies far away from the trend that the rest of the data follows An observation is influential if Its x value is relatively low or high compared to the remainder of the data The observation is a regression outlier Influential observations tend to pull the regression line toward that data point and away from the rest of the data

    59. Learning Objective 2: Outliers and Influential Points Impact of removing an Influential data point

    60. Learning Objective 3: Correlation does not Imply Causation A strong correlation between x and y means that there is a strong linear association that exists between the two variables A strong correlation between x and y, does not mean that x causes y

    61. Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire Would you expect the correlation to be negative, zero, or positive? If the correlation is positive, does this mean that having more firefighters at a fire causes the damages to be worse? Yes or No Identify a third variable that could be considered a common cause of x and y: Distance from the fire station Intensity of the fire Size of the fire

    62. Learning Objective 4: Lurking Variables & Confounding A lurking variable is a variable, usually unobserved, that influences the association between the variables of primary interest Ice cream sales and drowning – lurking variable =temperature Reading level and shoe size – lurking variable=age Childhood obesity rate and GDP-lurking variable=time When two explanatory variables are both associated with a response variable but are also associated with each other, there is said to be confounding Lurking variables are not measured in the study but have the potential for confounding

    63. Learning Objective 5: Simpson’s Paradox Simpson’s Paradox: When the direction of an association between two variables changes after we include a third variable and analyze the data at separate levels of that variable

    64. Learning Objective 5: Simpson’s Paradox Example

    65. Learning Objective 5: Simpson’s Paradox Example

    66. Learning Objective 5: Simpson’s Paradox Example

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