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2 Announcement Student Opinion of Courses and Teaching (SOCT) forms will be available for you during the dates indicated: STT : 4/14/2014-5/2/2014(SOCT only) STT : 4/14/2014-5/2/2014 (SOCT only) STT : 4/14/2014-5/2/2014(SOCT only)

3 Chapter7 Scatterplots, Association, and Correlation

4 Looking at Scatterplots Scatterplotsmay be the most common and most effective display for data. In a scatterplot, you can see patterns, trends, relationships, and even the occasional extraordinary value sitting apart from the others. Scatterplots are the best way to start observing the relationship and the ideal way to picture associationsbetween two quantitative variables.

5 Looking at Scatterplots (cont.) When looking at scatterplots, we will look for direction,form, strength, and unusual features. Direction: A pattern that runs from the upper left to the lower right is said to have a negativedirection. A trend running the other way has a positive direction.

6 Looking at Scatterplots (cont.) The figure shows a negative direction between the year since 1970 and the and the prediction errors made by NOAA. As the years have passed, the predictions have improved (errors have decreased).

7 Looking at Scatterplots (cont.) As the central pressure increases, the maximum wind speed decreases.

8 Looking at Scatterplots (cont.) Form: If there is a straight line (linear) relationship, it will appear as a cloud or swarm of points stretched out in a generally consistent, straight form.

9 Looking at Scatterplots (cont.) Form: Ifthe relationship isn t straight, but curves gently, while still increasing or decreasing steadily,

10 Looking at Scatterplots (cont.) Form: If the relationship curves sharply,

11 Looking at Scatterplots (cont.) Strength: At one extreme, the points appear to follow a single stream (whether straight, curved, or bending all over the place).

12 Looking at Scatterplots (cont.) Strength: At the other extreme, the points appear as a vague cloud with no discernible trend or pattern:

13 Looking at Scatterplots (cont.) Unusual features: Look for the unexpected. One example of such a surprise is an outlier standing away from the overall pattern of the scatter plot. Clusters or subgroups should also raise questions.

14 Roles for Variables It is important to determine which of the two quantitative variables goes on the x-axis and which on the y-axis. This determination is made based on the roles played by the variables. When the roles are clear, the explanatoryor predictor variablegoes on the x-axis, and the response variable goes on the y-axis.

15 Roles for Variables (cont.) Do baseballteamsthatscoremorerunssell more tickets to their games? Do studentswhoscorehigheron theirsat testshavehidhergradepointaveragesin college? Do olderhousessellforlessthannewerones of comparable size and quality?

16 Intheseexamplesthetwovariablesplay different roles. Wecallthevariableof interesttheresponse variable And the other the explanatory variable or predictor variable.

17 Correlation Data collected from students in Statistics classes included their heights (in inches) and weights (in pounds): Here we see a positive association and a fairly straight form, although there seems to be a high outlier.

18 Correlation (cont.) How strong is the association between weight and height of Statistics students? If we had to put a number on the strength, we would not want it to depend on the units we used. A scatterplot of heights (in centimeters) and weights (in kilograms) doesn t change the shape of the pattern:

19 Correlation (cont.) The correlation coefficient (r)gives us a numerical measurement of the strength of the linear relationship between the explanatory and response variables.

20 Correlation Conditions Correlationmeasures the strength of the linearassociation between two quantitative variables. Before you use correlation, you must check several conditions: Quantitative Variables Condition Straight Enough Condition Outlier Condition

21 Correlation Conditions (cont.) Outlier Condition: Outliers can distort the correlation dramatically. An outlier can make an otherwise small correlation look big, or hide a large correlation. It can even give an otherwise positive association a negative correlation coefficient (and vice versa). When you see an outlier, it s often a good idea to report the correlations with and without that point.

22 Correlation Properties The sign of a correlation coefficient gives the direction of the association. Correlation is always between -1 and +1. Correlation canbe exactly equal to -1 or +1, but these values are unusual in real data because they mean that all the data points fall exactlyon a single straight line. A correlation near zero corresponds to a weak linear association.

23 Correlation Properties (cont.) Correlation treats xand ysymmetrically: The correlation of xwith yis the same as the correlation of ywith x. Correlation has no units. Correlation is not affected by changes in the center or scale of either variable.

24 Correlation Properties (cont.) Correlation measures the strength of the linearassociation between the two variables. Variables can have a strong association but still have a small correlation if the association isn t linear. Correlation is sensitive to outliers. A single outlying value can make a small correlation large or make a large one small.

25 Correlation Causa on Whenever we have a strong correlation, it is tempting to explain it by imagining that the predictor variable has caused the response to help. Scatterplotsand correlation coefficients neverprove causation.

26 Correlation Tables It is common in some fields to compute the correlations between every pair of variables in a collection of variables and arrange these correlations in a table.

27 Becareful!!! Don t correlate categorical variables. Be sure to check the Quantitative Variables Condition. Be sure the association is linear. There may be a strong association between two variables that have a nonlinear association. Slide 1-27

28 Becareful!!! Don t assume the relationship is linear just because the correlation coefficient is high. Here the correlation is 0.979, but the relationship is actually bent.

29 Becareful!!! Beware of outliers. Even a single outlier can dominate the correlation value. Make sure to check the Outlier Condition.

30 Chapter8 LinearRegression

31 Fat Versus Protein: An Example The following is a scatterplot of total fat versus proteinfor 30 items on the Burger King menu:

32 Ifwewant25 gramsof protein ourlunch, how muchfatshouldweexpecttoconsumeat Burger King? The correlation between Fat and protein is 0.83 The strength of the linear relationship between fat and protein is fairly strong. But correlation does not tell us anything about howmuchfatweshouldconsumetohave25 grams of protein. Wewillusea linearmodel toanswerthis question.

33 The Linear Model Remember from Algebra that a straight line can be written as: y= mx+ b In Statistics we use a slightly different notation: ŷ= b0 + b1 x We write ŷ to emphasize that the points that satisfy this equation are just our predicted values, not the actual data values.

34 The Linear Model (cont.) We write b 1 and b 0 for the slope and intercept of the line. The b sare called the coefficients of the linear model. The coefficient b 1is the slope, which tells us how rapidly ŷ changes with respect to x. It showshowmuchchangeoccursat y in average withoneunitincreasein x. The coefficient b 0 is the intercept, which tells where the line hits (intercepts) the y-axis.

35 Residuals The model won t be perfect, regardless of the line we draw. Some points will be above the line and some will be below. The estimate made from a model is the predicted value(denoted as ). ŷ

36 Residuals (cont.) The difference between the observed value and its associated predicted value is called the residual. To find the residuals, we always subtract the predicted value from the observed one: residual= observed predicted = y yˆ

37 Residuals (cont.) A negative residual means the predicted value s too big (an overestimate). A positive residual means the predicted value s too small (an underestimate).

38 Best Fit Means Least Squares Howdo wefindtheactualvaluesof slopeand intercept? Weneedtobuildthemodel thatfitsthedata best. Andthelineshouldgothroughthemeanof y-mean of x point. Sowemayjusttrytobuilda model byminimizing thedistancebetweenthelineandobserveddata values. The distance between the line and observed values are residuals. Instead of minimizing each residual value, we can try to minize their total.

39 Best Fit Means Least Squares(cont.) Some residuals are positive, others are negative, and, on average, they cancel each other out. So, we can t assess how well the line fits by adding up all the residuals. Similar to what we did with Standard deviations, we square the residuals and add the squares. The smaller the sum, the better the fit. The line of best fit is the line for which the sum of the squared residuals is smallest, the least squares line.

40 The Least Squares Line In our model, we have a slope (b 1 ): The slope is built from the correlation and the standard deviations: b 1 = s r s y x

41 The Least Squares Line (cont.) In our model, we also have an intercept (b 0 ). The intercept is built from the means and the slope: b = y b x 0 1

42 Fat Versus Protein: An Example The regression line for the Burger King data fits the data well: The equation is Onegram increasein protein will cause 0.97 grams increase in Fat in average. Lessformally, wemightsay that Burger King foods pack about0.97 gramsof fatper gram of protein. The predicted fatcontent for a BK Broiler chicken sandwich is (30) = 35.9 grams of fat.

43 The Least Squares Line (cont.) Since regression and correlation are closely related, we need to check the same conditions for regressions as we did for correlations: Quantitative Variables Condition Straight Enough Condition Outlier Condition

44 Exercise7.33 Doeshowlongchildrenremainat thelunchtable help predict how much they eat? Thedata is for20 toddlersobservedoverseveral months at a nursery school. Time is theaveragenumberof minutesa child spentat thetablewhenlunchwasserved. Calories is the average number of calories the child consumed during lunch, calculated from careful observation of what the child ate each day.

45 Exercise7.33 Time Calories

46 Exercise7.33 The relation ship between time and calories seems linear. (Read the exercise for scatter plot) Correlationbetweentime andcaloriesis r = It seems we have negative relationship. Ifwehad recordedtime in hoursinsteadof minutes how would r have changed? Itwouldnot havechanged. Which one of these variables is response variable? Calories

47 Exercise7.33 The standard deviation for calories=29.94 The standard deviation for time=6.32 Average for calories=456 Average for time=34.01 Build the linear regression model.

48 Exercise7.33 Lettingchildstayat thetableonemoreminute will cause 3.07 unit decrease for the calories that child consumes in average.

49 Exercise8.12 x_bar S x y_bar s y r Regressionmodel ? ?? ? y_hat=-10+15x

50 Residuals Revisited The linear model assumes that the relationship between the two variables is a perfect straight line. The residuals are the part of the data that hasn t been modeled. or (equivalently) Or, in symbols, Data = Model + Residual Residual = Data Model e= y yˆ

51 Residuals Revisited (cont.) Residuals help us to see whether the model makes sense. When a regression model is appropriate, nothing interesting should be left behind. After we fit a regression model, we usually plot the residuals in the hope of finding nothing. A scatterplotof theresidualsversusthex-valuesshouldbe the most boring scatter plot you ve ever seen. It shouldn t have any interesting features like a direction or shape. It should stretch horizantally, with about the same amount of scatter throughout. Itshouldhaveno bends, andit shouldhaveno outliers.

52 Residuals Revisited (cont.) The residuals for the BurgerKingmenu regression look appropriately boring:

53 R 2 The Variation Accounted For The variation in the residuals is the key to assessing how well the model fits. In the BK menu items example, total fathas a standard deviation of 16.4 grams. The standard deviation of the residuals is 9.2 grams.

54 R 2 The Variation Accounted For (cont.) If the correlation were 1.0 and the model predicted the fatvalues perfectly, the residuals would all be zero and have no variation. As it is, the correlation is 0.83 not perfection. However, we did see that the model residuals had less variation than total fat alone. We can determine how much of the variation is accounted for by the model and how much is left in the residuals.

55 R 2 The Variation Accounted For (cont.) The squared correlation, r 2, gives the fraction of the data s variance accounted for by the model. Thus, 1 r 2 is the fraction of the original variance left in the residuals. For the BK model, r 2 = = 0.69, so 31% of the variability in total fathas been left in the residuals.

56 R 2 The Variation Accounted For (cont.) All regression analyses include this statistic, although by tradition, it is written R 2 (pronounced R-squared ). An R 2 of 0 means that none of the variance in the data is in the model; all of it is still in the residuals. When interpreting a regression model you need to Tellwhat R 2 means. In the BK example, according to our linear model, 69% of the variation in total fatis accounted for by variation in the protein content.

57 How Big Should R 2 Be? R 2 is always between 0% and 100%. What makes a good R 2 value depends on the kind of data you are analyzing and on what you want to do with it.

58 How Big Should R 2 Be (cont)? Along with the slope and intercept for a regression, you should always report R 2 so that readers can judge for themselves how successful the regression is at fitting the data. Statistics is about variation, and R 2 measures the success of the regression model in terms of the fraction of the variation of yaccounted for by the regression.

59 What Can Go Wrong? Don t fit a straight line to a nonlinear relationship. Beware of extraordinary points (y-values that stand off from the linear pattern or extreme x-values). Don t invert the regression. To swap the predictor-response roles of the variables, we must fit a new regression equation. Don t extrapolate beyond the data the linear model may no longer hold outside of the range of the data. Don t infer that xcauses yjust because there is a good linear model for their relationship association is not causation. Don t choose a model based on R 2 alone.

60 Exercises8.30 A) Yes, therelationshipis not verystrong, but it is reasonably straight. B) Thelinearmodel on numberof winsaccountsfor 48.5% of the variaiton in Attendance. C) The residuals spread out. There is more variaiton in Attendanceas thenumberof winsincreases. Thisplotshowsthatthereis someproblem with the model. So it needs further examination. D) The Yankees attandence was about 13,000 fans morethanwemightexpectgiventhenumberof wins.

61 Exercise8.34 A) B) 12,581 people C) Everywinaddsan average peoplein attendance. D) Itmeansthattheteam saverageattendanceis lowerthantheaveragefora teamwithas many wins. E) 12, attendees. This means that the Cardinals averaged over 12,000 more attendees thanonewouldpredictfora teamwith83 wins.