hw4.sol - Page 1 of 6 Stat209/Ed260 D Rogosa 1/31/09...

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Stat209/Ed260 D Rogosa 1/31/09 Solutions Assignment 4. Multilevel data, ecological fallacy Problem 1 Following p.750 Greenland and Robins American Journal of Epidemiology Vol. 139, No. 8: 747-760 Reading through their exposition probably is helpful here's my attempt (it can be done more elegantly) #make the region indices > r = c(0:40) > r [1] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 [26] 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 #now radon concentrations > regradon = .1 + .3*sqrt(r) > regradon [1] 0.1000000 0.4000000 0.5242641 0.6196152 0.7000000 0.7708204 0.8348469 [8] 0.8937254 0.9485281 1.0000000 1.0486833 1.0949874 1.1392305 1.1816654 [15] 1.2224972 1.2618950 1.3000000 1.3369317 1.3727922 1.4076697 1.4416408 [22] 1.4747727 1.5071247 1.5387495 1.5696938 1.6000000 1.6297059 1.6588457 [29] 1.6874508 1.7155494 1.7431677 1.7703293 1.7970563 1.8233688 1.8492856 [36] 1.8748239 1.9000000 1.9248288 1.9493242 1.9734994 1.9973666 > # set up smoking variable (counterbalanced with radon) > u = rnorm(41) > var(u) [1] 0.9883518 > mean(u) [1] 0.1323434 > #that's a good enough u > s = .2*(34 + .4*r -u) + .4*(13 - .2*r -.5*u) > s [1] 11.65626 12.02922 11.45928 11.86984 11.96606 12.05849 11.54374 11.99844 [9] 11.42576 12.42389 11.88704 12.42439 11.93378 12.39780 11.55852 11.78481 [17] 11.69073 11.74597 12.69344 12.01483 11.84285 10.74535 11.70950 12.54065 [25] 12.25184 12.75854 12.22308 12.12930 11.85289 12.27538 11.76754 11.37832 [33] 12.27040 12.22711 11.75117 11.99551 12.49188 12.12246 11.60682 11.72744 [41] 11.59924 #smoking proportions > p0 = 53 - .2*r + 1.5*u > p1 = 34 + .4*r - u > p2 = 13 - .2*r - .5*u #eq2 p.751 > lungc = .40*(1 + .2*regradon)*exp(s/10)*(p0 + 2.7183*2.7183*p1 + exp(4)*p2) #resulting data > cor(lungc, s) [1] 0.8518681 > cor(lungc, regradon) [1] -0.2588493 > cor(s, regradon) [1] 0.1254401 > plot( regradon, lungc) #my plot is at http://www-stat.stanford.edu/~rag/stat209/hw4p1.pdf #run the regression > luncreg = lm(lungc ~ regradon + s) > summary(luncreg) Call: lm(formula = lungc ~ regradon + s) Residuals: Min 1Q Median 3Q Max -165.490 -28.550 9.463 38.072 77.763 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2012.52 239.73 -8.395 3.48e-10 *** regradon -102.60 16.80 -6.107 4.06e-07 *** s 298.39 20.21 14.768 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 50.42 on 38 degrees of freedom Multiple R-Squared: 0.8616, Adjusted R-squared: 0.8543 F-statistic: 118.2 on 2 and 38 DF, p-value: < 2.2e-16 even with confounder (smoking) "controlled for" [laughter supressed] radon concentration negatively associated with lung cancer in this regional-level regression. Causal conclusion--start pumping radioctive gas into everyone's basements?
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hw4.sol - Page 1 of 6 Stat209/Ed260 D Rogosa 1/31/09...

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