Chapter 14. Statistical versus Deterministic Relationships. Distance versus Speed. Describing Relationships: Scatterplots and Correlation

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1 Chapter 14 Describing Relationships: Scatterplots and Correlation Chapter 14 1 Statistical versus Deterministic Relationships Distance versus Speed (when travel time is constant). Income (in millions of dollars) versus total assets of banks (in billions of dollars). Distance versus Speed Distance = Speed Time Suppose time = 1.5 hours Each subject drives a fixed speed for the 1.5 hrs speed chosen for each subject varies from 10 mph to 50 mph Distance does not vary for those who drive the same fixed speed Deterministic relationship 1

2 Income versus Assets Income = a + b Assets Assets vary from 3.4 billion to 49 billion Income varies from bank to bank, even among those with similar assets Statistical relationship A scatter plot shows a linear relationship if the points follow, more or less, along a straight line Example - heights and weights of 165 students in a college statistics course: Positive association: High values of one variable tend to occur together with high values of the other variable. Negative association: High values of one variable tend to occur together with low values of the other variable. 2

3 No relationship: x and y vary independently. Knowing x tells you nothing about y. One way to remember this: The equation for this line is y = 5. x is not involved. The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form. With a strong relationship, you can get a pretty good estimate of y if you know x. With a weak relationship, for any x you might get a wide range of y values. Correlation measures the strength and direction of a linear relationship between two quantitative variables 3

4 Negative correlation X Y X Y X,Y behave oppositely Positive correlation X Y X Y X,Y behave similarly r Pearson correlation coefficient (r) describes the direction and strength of a linear relationship between two variables. -1 r -0.8 strong negative correlation -0.8 < r < -0.2 weak to moderate negative cor r 0.2 negligible correlation 0.2 < r < 0.8 weak to moderate positive cor. 0.8 r 1 strong positive correlation 4

5 Problems with Correlations Outliers can inflate or deflate correlations Groups combined inappropriately may mask relationships (a third variable) groups may have different relationships when separated Not an outlier: Outliers The upper right-hand point here is not an outlier of the relationship it is what you would expect for this many beers given the linear relationship between beers/weight and blood alcohol. This point is not in line with the others, so it is an outlier of the relationship. What does statistical significance mean? 5.Statistics. Of or relating to observations or occurrences that are too closely correlated to be attributed to chance and therefore indicate a systematic relationship 5

6 Strength and Statistical Significance A strong relationship seen in the sample may indicate a strong relationship in the population. The sample may exhibit a strong relationship simply by chance and the relationship in the population is not strong or is zero. The observed relationship is considered to be statistically significant if it is stronger than a large proportion of the relationships we could expect to see just by chance. Warnings about Statistical Significance Statistical significance does not imply the relationship is strong enough to be considered practically important. Even weak relationships may be labeled statistically significant if the sample size is very large. Even very strong relationships may not be labeled statistically significant if the sample size is very small. Chapter 15 Describing Relationships: Regression, Prediction, and Causation Chapter

7 y = a + bx a = y intercept b = slope Straight lines (lines: a quick review) Slope = y / x = rise/run e.g. slope is - 2, y decreases 2 units for every one unit increase in x y = 3-2x A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x. 7

8 The least-squares regression line is the unique line such that the sum of the squared vertical (y) distances between the data points and the line is the smallest possible. Distances between the points and line are squared so all are positive values. This is done so that distances can be properly added. The least-squares regression line can be shown to have this equation: y ˆ = a + bx y ˆ is the predicted y value (y hat) b is the slope a is the y-intercept Making predictions The equation of the least-squares regression allows you to predict y for any x within the range studied. This is called interpolating. y ˆ = x Nobody in the study drank 6.5 beers, but by finding the value of ŷ from the regression line for x = 6.5, we would expect a blood alcohol content of mg/ml. yˆ = * yˆ = = mg / ml 8

9 Coefficient of Determination (R 2 ) Measures usefulness of regression prediction R 2 (or r 2, the square of the correlation): measures the percentage of the variation in the values of the response variable (y) that is explained by the regression line r=1: R 2 =1: regression line explains all (100%) of the variation in y r=.7: R 2 =.49: regression line explains almost half (50%) of the variation in y r = 1 r 2 = 1 Changes in x explain 100% of the variations in y. y can be entirely predicted for any given value of x. r = 0.87 r 2 = 0.76 r = 0 r 2 = 0 Changes in x explain 0% of the variations in y. The value(s) y takes is (are) entirely independent of what value x takes. Here the change in x only explains 76% of the change in y. The rest of the change in y (the vertical scatter, shown as red arrows) must be explained by something other than x.!!! Extrapolation is the use of a regression line for predictions outside the range of x values used to obtain the line. This can be a very stupid thing to do, as seen here. Height in Inches Height in Inches!!! 9

10 Correlation Does Not Imply Causation Even very strong correlations may not correspond to a real causal relationship. Evidence of Causation A properly conducted experiment establishes the connection Other considerations: A reasonable explanation for a cause and effect exists The connection happens in repeated trials The connection happens under varying conditions Potential confounding factors are ruled out Alleged cause precedes the effect in time Reasons for relationships between variables 1. Explanatory variable is the direct cause of the response variable 2. The response variable is causing a change in the explanatory variable 3. The explanatory variable is contributing to but not the sole cause of change in the response variable 4. Confounders may exist 5. Both variables result from a common cause 6. Both variables are changing over time 7. The association is coincidence 10

11 Association and causation It appears that lung cancer is associated with smoking. How do we know that both of these variables are not being affected by an unobserved third (lurking) variable? For instance, what if there is a genetic predisposition that causes people to both get lung cancer and become addicted to smoking, but the smoking itself doesn t CAUSE lung cancer? We can evaluate the association using the following criteria: 1) The association is strong. 2) The association is consistent. 3) Higher doses are associated with stronger responses. 4) The alleged cause precedes the effect. 5) The alleged cause is plausible. Ch 14 & 15 concepts Statistical vs. Deterministic Relationships Statistical Significance Correlation Coefficient Problems with Correlations LS Regression Equation R 2 Correlation does not imply causation! 11