The importance of graphing the data: Anscombe s regression examples


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1 The importance of graphing the data: Anscombe s regression examples Bruce Weaver Northern Health Research Conference Nipissing University, North Bay May 3031, 2008 B. Weaver, NHRC
2 The Objective To demonstrate that good graphs are an essential part of linear regression analysis. B. Weaver, NHRC
3 Not this kind of regression analysis B. Weaver, NHRC
4 This kind of regression analysis B. Weaver, NHRC
5 A very brief primer on simple linear regression B. Weaver, NHRC
6 Simple linear regression A model in which X is used to predict Y. Y is a continuous variable with interval scale properties. In the prototypical case, X is also a continuous variable with intervalscale properties. Example: Y = distance in a 6minute walk test X = FEV1 B. Weaver, NHRC
7 Back to high school Equation for a straight line Y = bx + a SLOPE INTERCEPT b = slope of the line = the rise over the run a = the value of Y when X = 0 B. Weaver, NHRC
8 Example of a straight line Gym membership Annual fee = $100 Fee per visit = $2 Let X = the number of visits to the gym Let Y = the total cost Y = 2X Let X = 200 visits to the gym Total cost = 2(200) = $500 B. Weaver, NHRC
9 What if the relationship is imperfect? Straight line for a perfect relationship: Y = bx + a Straight line for an imperfect relationship: Y = bx + a Y = bx + a Two different symbols for the predicted value of Y B. Weaver, NHRC
10 Rsquared Rsquared = the proportion of variability in Y that is accounted for by explanatory variables in the model. For a simple linear regression model (i.e., one predictor variable), Rsquared = the proportion of the variability in Y that can be accounted for by the linear relationship between X and Y The adjusted Rsquared corrects for upward bias in Rsquared B. Weaver, NHRC
11 Anscombe s examples (1973) Frank Anscombe devised 4 sets of XY pairs He performed simple linear regression for each data set Here are the results B. Weaver, NHRC
12 Means & Standard Deviations X Y Data Set N Mean SD Mean SD The means and SDs for the 4 data sets are identical to two decimals. B. Weaver, NHRC
13 Correlations between X and Y Data Set Pearson r Rsquared Adj. Rsq SE Correlations, Rsquared, adjusted R squared, and standard errors are all identical to two decimals. B. Weaver, NHRC
14 ANOVA Summary Tables Data Set Source SS df MS F p Regression Residual Total Regression Residual Total Regression Residual Total Regression Residual Total B. Weaver, NHRC
15 The Regression Coefficients Data Set B SE t p 95% CI Lower Upper Constant X Constant X Constant X Constant X For all 4 models, Y = 0.5(X) + 3 B. Weaver, NHRC
16 Which Model is Best? Judging by everything we ve just seen, it appears that the models are all equally good But if that were true, I wouldn t be doing this talk! It is well known that good graphs are an essential part of data analysis (Tukey, 1977; Tufte, 1997) Let s look at some graphs that show the relationship between X and Y B. Weaver, NHRC
17 Scatterplot for Data Set 1 10 data points Influential point Not a good model B. Weaver, NHRC
18 Scatterplot for Data Set 2 Perfect linear relationship except for one outlier Better model than for Data Set 1, but still not great. B. Weaver, NHRC
19 Scatterplot for Data Set 3 Wrong model! The relationship between X and Y is curvilinear, not linear! The model should include both X and X 2 as predictors. B. Weaver, NHRC
20 Scatterplot for Data Set 4 This is a good looking plot. No influential points; straight line provides a good fit. B. Weaver, NHRC
21 Summary The usual summary statistics for the 4 regression models were virtually identical Scatterplots revealed that only one of the 4 data sets gave us a good model Appropriate graphs are an essential part of data analysis B. Weaver, NHRC
22 What about multivariable models? Scatterplots are useful for simple linear regression models (i.e., only one predictor variable) But often, we have multiple, or multivariable regression models (i.e., 2 or more predictor variables) In that case, it is more common to assess the fit of the model by looking at residual plots B. Weaver, NHRC
23 What is a residual? In linear regression, a residual is an error in prediction Residual = (Y Y ) = (actual score predicted score) B. Weaver, NHRC
24 Set 1: Scatterplot vs. Residual Plot Scatterplot Residual Plot Y Residual X Predicted value of Y B. Weaver, NHRC
25 Set 2: Scatterplot vs. Residual Plot Scatterplot Residual Plot Residual Predicted value of Y B. Weaver, NHRC
26 Set 3: Scatterplot vs. Residual Plot Scatterplot Residual Plot Residual Predicted value of Y Runs of samesign residuals B. Weaver, NHRC
27 Set 4: Scatterplot vs. Residual Plot Scatterplot Residual Plot Residual Predicted value of Y B. Weaver, NHRC
28 Summary The usual summary statistics for the 4 regression models were virtually identical Scatterplots revealed that only one of the 4 data sets gave us a good model Residual plots reveal the same thing, and have the advantage of being applicable to multivariable regression models Appropriate graphs are an essential part of data analysis B. Weaver, NHRC
29 Questions? I think you should be more explicit here in step 2. B. Weaver, NHRC
30 References Anscombe FJ. (1973). Graphs in statistical analysis. The American Statistician, 27, Tufte ER. (1997). Visual Explanations, Images and Quantities, Evidence and Narrative (3rd Ed.). Graphics Press: Cheshire. Tukey JW. (1977). Exploratory data analysis. AddisonWesley: Reading, Mass. B. Weaver, NHRC
31 Extra Slides B. Weaver, NHRC
32 Just as one would expect! The experimentalist comes running excitedly into the theorist's office, waving a graph taken off his latest experiment. "Hmmm," says the theorist, "That's exactly where you'd expect to see that peak. Here's the reason (long logical explanation follows)." In the middle of it, the experimentalist says "Wait a minute", studies the chart for a second, and says, "Oops, this is upside down." He fixes it. "Hmmm," says the theorist, "you'd expect to see a dip in exactly that position. Here's the reason...". B. Weaver, NHRC
33 Bestfitting line: Least squares criterion Many lines could be placed on the scatterplot, but only one of them is considered the bestfitting line. The most common criterion for bestfitting is that the sum of the squared errors in prediction is minimized. This is called the leastsquares criterion. B. Weaver, NHRC
34 Illustration of Least Squares Error in prediction B. Weaver, NHRC
35 Illustration of Least Squares Squared error in prediction Error = 0 for this point, so no square Squared error in prediction B. Weaver, NHRC
36 Illustration of Least Squares Sum of squared errors = the sum of the areas of all these squares For any other regression line, the sum of the squared errors would be greater. B. Weaver, NHRC
37 What is a residual plot? Scatterplot with: X = the fitted (or predicted) value of Y Y = the residual (i.e., the error in prediction) Residuals should be independent of the fitted value of Y There should be no serial correlation in the residuals (e.g., long runs of samesign residuals) Both of these problems (plus some others) can be detected via residual plots Advantage of residual plots: they can be used in multivariable (i.e., multipredictor) regression models B. Weaver, NHRC
38 Examples of residual plots Curvilinear relationship Residual Predicted Y Outlier Heteroscedasticity B. Weaver, NHRC
39 Example of a good residual plot B. Weaver, NHRC
40 Example of a zigzag pattern You do not want to see this kind of zigzag pattern in the residual plot. B. Weaver, NHRC
41 Simple linear regression & correlation Pearson r = the correlation It measures of the direction and strength of the linear association between X and Y It ranges from 1 to +1 B. Weaver, NHRC
42 Direction of the linear relationship Positive relationship Negative relationship As X increases, Y increases As X increases, Y decreases B. Weaver, NHRC
43 Perfect vs. Imperfect Relationship Perfect relationship Imperfect relationship B. Weaver, NHRC
44 rsquared The square of Pearson r is a measure of how well the regression model fits the observed data It gives the proportion of variability in Y that is accounted for the linear relationship between X and Y. E.g., let r = 0.6 (or 0.6) r 2 = 0.36 So 36% of the variability in the Yscores is accounted for by the linear relationship between X and Y B. Weaver, NHRC
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