Stat 112: Lecture 9 Notes Homework 3: Due next Thursday Prediction Intervals for Multiple Regression (Chapter 4.5) Multicollinearity (Chapter 4.6). Summary of F tests Partial F tests are used to test whether a subset of the slopes in multiple regression are zero. The whole model F test (test of the useful of the model) tests whether the slopes on all variables in multiple regression are zero, i.e., it tests whether the multiple regression is more useful for prediction than just ignoring the Xs and using to predict Y. Y For testing whether one slope in multiple regression is zero, we can use the t-test. But in fact, the partial F test for one slope being zero is equivalent to the t-test (it gives the same p-values and the same decisions). Why use the F test to test whether two or more slopes are not both equal to zero rather than two t-tests? The F test is more powerful. This will be illustrated later in the lecture. Prediction in Automobile
Example The design team is planning a new car with the following characteristics: horsepower = 200, weight = 4000 lb, cargo = 18 ft3, seating = 5 adults. What is a 95% prediction interval for the GPM1000 of this car? Prediction with Multiple Regression Equation Prediction interval for individual with x1,,xK: y p b0 b1 x1 bK xK CI :y p t / 2,n K 1 * s p s p SE ( y p y p ) Finding Prediction Interval in JMP Enter a line with the independent variables x1,,xK for the new individual. Do not enter a y for the new individual. Fit the model. Because the new individual does not have a y, JMP will not include the new individual when calculating the least squares fit.
Click red triangle next to response, click Save Columns: To find y p , click Predicted Values. Creates column with y p To find 95% PI, click Indiv Confid Interval. Creates column with lower and upper endpoints of 95% PI. Prediction in Automobile Example The design team is planning a new car with the following characteristics: horsepower = 200, weight = 4000 lb, cargo = 18 ft3, seating = 5 adults. From JMP, y p 45.08 95% prediction interval: (37.86, 52.31) Multicollinearity DATA: A real estate agents wants to develop a model to predict the selling price of a home. The agent takes a random sample of 100 homes that were recently sold and records the selling price (y), the
number of bedrooms (x1), the size in square feet (x2) and the lot size in square feet (x 3). Data is in houseprice.JMP. Scatterplot Matrix 200000 Price 100000 5.0 4.0 3.0 2.0 3000 2500 2000 1500 Bedrooms
House Size 8000 6000 Lot Size 4000 100000 2.0 3.5 5.0 1500 250040007000 Response Price Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 0.559998
0.546248 25022.71 154066 100 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 3 7.65017e10 2.5501e10 40.7269 Prob > F Error 96 6.0109e+10 626135896 C. Total 99 1.36611e11
<.0001 Parameter Estimates Term Intercept Bedrooms House Size Lot Size Estimate Std Error t Ratio Prob>|t| 37717.595 14176.74 2.66 0.0091 2306.0808 6994.192 0.33 0.7423 74.296806 52.97858 1.40 0.1640 -4.363783 17.024 -0.26 0.7982
There is strong evidence that predictors are useful, p-value for F-test <.0001 and R 2 .560 , but the t-tests for each coefficient are not significant. Indicative of multicollinearity. Note: These results illustrate how the F test is more powerful for testing whether a group of slopes in multiple regression are all zero than individual t tests. Multicollinearity Multicollinearity: Explanatory variables are highly correlated with each other. It is often hard to determine their individual regression coefficients. Multivariate Correlations Bedrooms Bedrooms House Size There
is Lot Size 1.0000 0.8465 very little 0.8374 House Size Lot Size 0.8465 0.8374 1.0000 0.9936 information in the 0.9936 1.0000
data set to find out what would happen if we fix house size and change lot size. Since house size and lot size are highly correlated, for fixed house size, lot sizes do not change much. The standard error for estimating the coefficient of lot sizes is large. Consequently the coefficient may not be significant. Similarly for the coefficient of house size. So, while it seems that at least one of the coefficients is significant (See ANOVA) you can not tell which one is the useful one. Consequences of Multicollinearity Standard errors of regression coefficients are large. As a result t statistics for testing the population regression coefficients are small. Regression coefficient estimates are unstable. Signs of coefficients may be opposite of what is intuitively reasonable (e.g., negative sign on lot size). Dropping or adding one variable in the
regression causes large change in estimates of coefficients of other variables. Detecting Multicollinearity 1. Pairwise correlations between explanatory variables are high. 2. Large overall F-statistic for testing usefulness of predictors but small t statistics. 3. Variance inflation factors Variance Inflation Factors Variance inflation factor (VIF): Let R 2j denote the R2 for the multiple regression of xj on the other x-variables. Then 1 VIF j . 2 1 Rj Fact: MSE
SD j VIF n 1 S x2 j j 2 VIFj for variable xj: Measure of the increase in the variance of the coefficient on xj due to the correlation among the explanatory variables compared to what the variance of the coefficient on xj would be if xj were independent of the other explanatory variables. Using VIFs To obtain VIFs, after Fit Model, go to Parameter Estimates, right click, click Columns and click VIFs. Detecting multicollinearity with VIFs: Any individual VIF greater than 10 indicates multicollinearity.
Average of all VIFs considerably greater than 1 also indicates multicollinearity. Summary of Fit RSquare 0.559998 Parameter Estimates Term Intercept Bedrooms House Size Lot Size Estimate Std Error t Ratio
Prob>|t| VIF 37717.595 14176.74 2.66 0.0091 . 2306.0808 6994.192 0.33 0.7423 3.5399784 74.296806 52.97858 1.40 0.1640 83.066839 -4.363783 17.024 -0.26 0.7982 78.841292 Multicollinearity and Prediction If interest is in predicting y, as long as pattern of multicollinearity continues for those observations where forecasts are desired (e.g., house size and lot size are either both high, both medium or both small), multicollinearity is not particularly problematic. If interest is in predicting y for observations where pattern of multicollinearity is different than that in sample (e.g., large house size, small lot size), no good solution (this would be extrapolation).
Problems caused by multicollinearity If interest is in predicting y, as long as pattern of multicollinearity continues for those observations where forecasts are desired (e.g., house size and lot size are either both high, both medium or both small), multicollinearity is not particularly problematic. If interest is in obtaining individual regression coefficients, there is no good solution in face of multicollinearity. If interest is in predicting y for observations where pattern of multicollinearity is different than that in sample (e.g., large house size, small lot size), no good solution (this would be extrapolation). Dealing with Multicollinearity Suffer: If prediction within the range of the data is the only goal, not the interpretation of the coefficients, then leave the multicollinearity alone. Combine: In some cases, it may be possible to combine variables to reduce multicollinearity (see next slide) Omit a variable. Multicollinearity can be reduced by removing one of the highly correlated variables.
However, if one wants to estimate the partial slope of one variable holding fixed the other variables, omitting a variable is not an option, as it changes the interpretation of the slope. Combining horsepower and weight in cars data Response GP1000MHwy Parameter Estimates Term Intercept Weight(lb) Cargo Seating Horsepower Estimate 19.100521 0.0040877 0.0533 0.0268912 0.0426999
Std Error 2.098478 0.001203 0.013787 0.428283 0.01567 t Ratio 9.10 3.40 3.87 0.06 2.73 VIF . 3.8589527 1.3173318 1.9717046 3.4149672
Combining Horsepower and Weight into Horsepower/Weight Parameter Estimates Term Intercept Cargo Seating Horsepower//Weight Estimate 15.021983 0.0544811 1.5680411 302.16217 Std Error 3.699961 0.017328 0.470098 51.41088 t Ratio 4.06
3.14 3.34 5.88 VIF . 1.3150096 1.5011661 1.1905447 Multiple Regression Example: California Test Score Data The California Standardized Testing and Reporting (STAR) data set californiastar.JMP contains data on test performance, school characteristics and student demographic backgrounds from 1998-1999. Average Test Score is the average of the reading and math scores for a standardized test administered to 5th grade students. One interesting question: What would be the causal effect of decreasing the student-teacher ratio by one student per teacher?
Multiple Regression and Causal Inference Goal: Figure out what the causal effect on average test score would be of decreasing student-teacher ratio and keeping everything else in the world fixed. Lurking variable: A variable that is associated with both average test score and student-teacher ratio. In order to figure out whether a drop in student-teacher ratio causes higher test scores, we want to compare mean test scores among schools with different student-teacher ratios but the same values of the lurking variables, i.e. we want to hold the value of the lurking variable fixed. If we include all of the lurking variables in the multiple regression model, the coefficient on student-teacher ratio represents the change in the mean of test scores that is caused by a one unit increase in studentteacher ratio. Omitted Variables Bias Response Average Test Score Parameter Estimates Term Intercept Student Teacher Ratio
Estimate 698.93295 -2.279808 Std Error 9.467491 0.479826 t Ratio 73.82 -4.75 Prob>|t| <.0001 <.0001 Response Average Test Score Parameter Estimates Term Intercept Student Teacher Ratio
Percent of English Learners Estimate 686.03225 -1.101296 -0.649777 Std Error 7.411312 0.380278 0.039343 t Ratio 92.57 -2.90 -16.52 Prob>|t| <.0001 0.0040 <.0001
Schools with many English learners tend to have worst resources. The multiple regression that shows how mean test score changes when student teacher ratio changes but percent of English learners is held fixed gives a better idea of the causal effect of the student-teacher ratio than the simple linear regression that does not hold percent of English learners fixed. Omitted variables bias: bias in estimating the causal effect of a variable from omitting a lurking variable from the multiple regression. Omitted variables bias of omitting percentage of English learners = -2.28-(-1.10)=-1.28. Key Warning About Multiple Regression Even if we have included many lurking variables in the multiple regression, we may have failed to include one or not have enough data to include one. There will then be omitted variables bias. The best way to study causal effects is to do a randomized experiment.
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