Stat 501, F. Chiaromonte. Lecture #8

Transcription

1 Stat 501, F. Chiaromonte Lecture #8 Data set: BEARS.MTW In the minitab example data sets (for description, get into the help option and search for "Data Set Description"). Wild bears were anesthetized, and their bodies were measured and weighed. One goal of the study was to make a table (or perhaps a set of tables) for hunters, so they could estimate the weight of a bear based on other measurements (this would be used because in the forest it is easier to measure the length of a bear, for example, than it is to weigh it). Variable Count (N) Description Age 143 Bear's age, in months (can be derived). Sex = male 2 = female Head.L 143 Length of the head, in inches Head.W 143 Width of the head, in inches Neck.G 143 Girth (distance around) the neck, in inches Length 143 Body length, in inches Chest.G 143 Girth (distance around) the chest, in inches Weight 143 Weight of the bear, in pounds Unit=observation and measurement of a beard (but some animals were re-captured and re-measured several times ) (this is not all the data set) 1

2 Well suited example for a regression application: 1. Construct and estimate a regression model for Y = Weight on one or more of the measurements. 2. Use the model to estimate the mean-weight, or predict a new weight, given the measurement(s) --point estimate, confidence interval, prediction interval. Descriptive Statistics Variable N Mean Median TrMean StDev Age Head.L Head.W Neck.G Length Chest.G Weight Variable SE Mean Minimum Maximum Q1 Q3 Age Head.L Head.W Neck.G Length Chest.G Weight

3 Correlations (Pearson) Ag Hd.L Hd.W Nk.G Lgt Ct.G We Ag Hd.L Hd.W Nk.G Lgt Ct.G We Age Head.L Head.W Neck.G Length Chest.G Weight

4 As to be expected, measurements on a body are very much related to one another, with strong linear components in the relationships (although, as the scatter plot matrix shows, the relationships are not necessarily purely or mainly linear ) Since we are still doing simple regression, we need to costruct a model that uses only one measurement. As a try, let us use the one that has the strongest linear relationship with Weight, i.e. Chest.G Regression Analysis The regression equation is Weight = Chest.G Predictor Coef StDev T P Constant Chest.G S = R-Sq = 93.3% R-Sq(adj) = 93.3% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

5 The regression line parameters estimates can be derived from the descriptive statistics in the previous outputs: b 1 = (s Y /s X ) r XY = b 0 = Av(Y) - b 1 Av(X) = Note that b 0 is negative, and anyway Chest.G=0 is not "reasonable". In order to give an interpretation of the "location" of our line (given the slope), we could interpret: A Weight value: The fitted value on min(chest.g)=19 A Chest.G value: The Chest.G value at which the estimated line crosses the abscisa axis 5

6 Regression Plot Y = X R-Sq = 93.3 % Weight Chest.G 6

7 Take Chest.G = 40. From the regression output, the descriptive statistics, and tables for the t-distribution, we can calculate Confidence int. for the mean-weight (Chest.G=40) Prediction int. for a new weight (Chest.G=40) This is very close to the sample mean of Chest.G, and well within the sample range. Now do the same at Chest.G = 80 (far out) Confidence int. for the mean-weight (Chest.G=80) Prediction int. for a new weight (Chest.G=80) With Minitab we can get the bands about the estimated regression line (in "Fitted line plot", check out the "Options") 7

8 Regression Plot Y = X R-Sq = 93.3 % Weight Regression 95% CI 95% PI Chest.G 8

9 This could be a good summary to give to a hunter: Measure the Chest.G, and you have A point-estimate of the corresponding mean-weight An interval about it, which is guaranteed to contain the unknown meanweight with probability.95 Another interval about it (wider), which is guaranteed to contain a new weight observation with probability.95 But we are in for a surprise! Although the estimated regression line captured a lot of the Weight-variability, maybe a line in Chest.G is not the most appropriate regression model to use! 9

10 100 RESI Chest.G Normal Probability Plot for RESI1 ML Estimates Percent Mean: StDev: Data 10

11 The residuals show quite a bit of curvature along the predictor. The coefficient of determination is quite high (the linear component of the relationship strong) But the simple regression is not performing well because it leaves residuals containing a clear patern along the predictor (a curve here) (If we take the residuals as the sample-equivalent of the errors, we have graphical evidence against the assumption: E(ε) = 0 at every level of X) We will get into diagnostics next time; for now, intuitively: This ough to suggest that the mean-relationship between Y and X is not linear Maybe we should change our model and consider a parabolic relationship (second order polynomial): Weight = β 0 + β 1 Chest.G + β 2 Chest.G 2 + ε with ε indep. X, E(ε) = 0, var(ε) = σ 2 This model, although it still refers to one variable, contains two terms (linear in the parameters) so it is technically a multiple regression. (create the variable (Chest.G)^2 in Minitab through "Calc", and then regress Weight on both Chest.G and (Chest.G)^2) 11

12 Regression Analysis The regression equation is Weight = Chest.G (Chest.G)^2 Predictor Coef StDev T P Constant Chest.G (Chest.G)^ S = R-Sq = 95.6% R-Sq(adj) = 95.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Regression Plot Y = X X**2 R-Sq = 95.6 % Weight Regression 95% CI 95% PI Chest.G 12

13 RESI Chest.G Normal Probability Plot for RESI2 ML Estimates Percent Mean: StDev: Data 13

14 Now we have eliminated the curved pattern in the residual plot quite efficiently But now we can see a clear non-constant variance: Residuals seem to be larger at larger predictor levels. (we have graphical evidence against the assumption: var(ε) = σ 2 at every level of X) One way to eliminate or weaken this kind of phenomenon is to transform the response via a variance stabilizing transformation. A typical one is the log(.). Maybe we should change our model again and consider: log(weight) = β 0 + β 1 Chest.G + β 2 Chest.G 2 + ε with ε indep. X, E(ε) = 0, var(ε) = σ 2 (again, create the variable log(weight) in Minitab through "Calc") Why should it "stabilize" the variability? It tends to "blow up" very small values, and "shrink" very large ones --in comparison. 14

15 Regression Analysis The regression equation is log(weight)= Chest.G (Chest.G)^2 Predictor Coef StDev T P Constant Chest.G Chest.G^ S = R-Sq = 94.3% R-Sq(adj) = 94.3% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Regression Plot Y = E-02X E-04X**2 R-Sq = 94.3 % 2.5 log(weight) 2.0 Regression % CI 95% PI Chest.G 15

16 RESI Chest.G Normal Probability Plot for RESI3 ML Estimates Percent Mean: StDev: Data 16

17 Sign of the quadratic term in the parabola has changed passing to log of the Weight. Coeff. of determination has decreased slightly, but the non-constant variance in the residual plot has almost completely vanished. Notice there seems to be an outlier (a point far out from the bulk of the observations, that seems to "come from another population" and not to satisfy the relationship between response and predictor that is suggested by the other points). We could repeat the analysis eliminating the far out point. Also, we could separate females and males: We have not used the qualitative variable "sex" yet. Construct and estimate two separate regression models for males and females: If one obtains very similar models, based on the sample data the relationship between Weigt and Chest.G appears to be the same in the two sex-subpopulations. Might as well have one common model for both.. If one obtains very different models, based on the sample data the relationship between Weight ad Chest.G is sex-specific. Two separate models will fit the data much better than a common one. (in other words: is the gain in explanatory power and precision large enough to justify the fact that the hunter will carry along two tables/models, one for male and one for female bears?) Notice we are reasoning on how to introduce a qualitative variable in a (multiple) regression. 17