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Example 1.0 Indirect Effects and the Test of Mediation using the winBUGS software

Example 1.0 Indirect Effects and the Test of Mediation using the winBUGS software. Use this tutorial along with the generic example 1.0. One of the many findings: vegetation recovery was a function of the age of the stand that burned. r = -0.35. cover is in proportions. age in years. age of

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Example 1.0 Indirect Effects and the Test of Mediation using the winBUGS software

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  1. Example 1.0Indirect Effects and the Test of Mediation using the winBUGS software Use this tutorial along with the generic example 1.0.

  2. One of the many findings: vegetation recovery was a function of the age of the stand that burned. r = -0.35 cover is in proportions. age in years.

  3. age of stand that burned e1 post-fire vegetation cover Indirect Effects as Causal Tests: Step 1 Model A

  4. winBUGS code for Model A # Title: Simple Regression of cov (cover) on age MODEL simple { # Priors b ~ dnorm(0,0.00001) a ~ dnorm(0,0.00001) tau ~ dgamma(0.5,1) # Likelihoods for(i in 1:N) { covhat[i] <- a + b*age[i] cov[i] ~ dnorm(covhat[i],tau) } # Calculate standardized coefficients covstd <- sd(cov[]) covvar <- covstd*covstd agestd <- sd(age[]) bstd <- b*agestd/covstd # Calculate Prediction Efficiency (Rsqr) covmu <- mean(cov[]) covhatmu <- mean(covhat[]) for (i in 1:N) { covdev[i] <- cov[i]-covmu covhatdev[i] <- covhat[i]-covhatmu } covar <- inprod(covdev[] , covhatdev[])/(N-1) bpred <- covar/(sd(cov[])*sd(covhat[])) #bpred is correlation between predicted and observed Rsqr <- bpred*bpred } CONTINUED ON NEXT PAGE

  5. winBUGS code for Model A (cont.) Initial Values list(a=0, b=1, tau=0.01) Data list(N=90,age=c( 40,25,15,15,23,24,35,14,45,35,45,35,35,30,15,40,21,39,39,29,12,19,20,21,3, 17,40,40,57,52,35,40,5,5,28,33,31,48,55,22,21,13,13,25,15,12,28,16,25,28, 38,10,26,23,16,19,15,27,30,30,3,6,25,28,27,19,25,11,11,15,36,24,24,26,26, 6,31,20,15,15,15,16,20,33,13,20,48,35,60,36), cov=c(1.03879744,0.47759241,0.94893572,1.19490019,1.29818904,1.17348657,0.86158595,0.41906623,0.12851131,0.3062645,0.68247702,0.5310083,0.2957547,0.98468564,0.69869688,1.16663393,0.84068273,1.11949189,0.41204136,0.84594715,1.17590928,0.66100507,0.6517726,0.41172415,0.53282988,0.49739296,0.20203482,0.05557657,0.12124606,0.08297883,0.54920017,0.54646221,1.27310233,0.84805791,0.37941846,0.35841475,0.63674533,0.290822,0.50717877,0.63749067,1.06473979,0.76755405,1.32067202,1.07900425,0.9377264,0.69915774,0.30582847,0.55437438,0.83649961,0.8507124,0.74700881,0.51896384,1.12002246,0.56923477,0.79578273,0.9167156,0.57708428,0.67706467,0.61410302,0.62828578,1.1074659,0.84054394,0.90855822,0.73840393,0.27628054,0.77771328,0.59338111,0.58821905,0.51846397,0.34806766,1.09761107,0.54498063,0.67605775,0.28858364,0.68021606,1.12022933,0.44701614,0.48445559,1.535408,0.92860808,0.46919542,0.45107384,0.51761156,0.5572632,0.5144298,1.16590727,1.08351251,0.56744869,0.5808883,0.37861547))

  6. Indirect Effects as Causal Tests: Step 1 – Results for Model A Statistics: node mean sd MC error 2.5% median 97.5% start sample Rsqr 0.1228 0.0 8.238E-13 0.1228 0.1228 0.1228 1000 19001 a 0.9175 0.08044 5.282E-4 0.7604 0.9169 1.078 1000 19001 b -0.008861 0.002836 1.967E-5 -0.01452 -0.008858 -0.003299 1000 19001 bstd -0.351 0.1123 7.792E-4 -0.5753 -0.3509 -0.1307 1000 19001 0

  7. e1 e2 age of stand that burned fire severity Model B post-fire vegetation cover Indirect Effects as Causal Tests: Step 2

  8. winBUGS code for Model B # Title: Example 1.1 Model B, with cov, fire (fire severity), and age MODEL simple { # Priors b1 ~ dnorm(0,0.00001); b2 ~ dnorm(0,0.00001) a1 ~ dnorm(0,0.00001); a2 ~ dnorm(0,0.00001) tau1 ~ dgamma(0.5,1); tau2 ~ dgamma(0.5,1) # Likelihoods for(i in 1:N) { firehat[i] <- a1 + b1*age[i] fire[i] ~ dnorm(firehat[i],tau1) covhat[i] <- a2 + b2*fire[i] cov[i] ~ dnorm(covhat[i],tau2) } # Calculate standardized coefficients firestd <- sd(fire[]); agestd <- sd(age[]); b1std <- b1*agestd/firestd covstd <- sd(cov[]); b2std <- b2*firestd/covstd CODE CONTINUED ON NEXT PAGE

  9. winBUGS code for Model B # Calculate Prediction Efficiency (Rsqr) firemu <- mean(fire[]) firehatmu <- mean(firehat[]) covmu <- mean(cov[]) covhatmu <- mean(covhat[]) for (i in 1:N) { firedev[i] <- fire[i]-firemu firehatdev[i] <- firehat[i]-firehatmu covdev[i] <- cov[i]-covmu covhatdev[i] <- covhat[i]-covhatmu } covar1 <- inprod(firedev[] , firehatdev[])/(N-1) b1pred <- covar1/(sd(fire[])*sd(firehat[])) #b1pred is cor between predicted and observed Rsqr1 <- b1pred*b1pred covar2 <- inprod(covdev[] , covhatdev[])/(N-1) b2pred <- covar2/(sd(cov[])*sd(covhat[])) #b2pred is cor between predicted and observed Rsqr2 <- b2pred*b2pred } DATA CONTINUED ON NEXT PAGE

  10. winBUGS code for Model B (cont.) Initial Values list(a1=0, a2=0, b1=1, b2=1, tau1=0.01, tau2=0.01) Data list(N=90,age=c( 40,25,15,15,23,24,35,14,45,35,45,35,35,30,15,40,21,39,39,29,12,19,20,21,3, 17,40,40,57,52,35,40,5,5,28,33,31,48,55,22,21,13,13,25,15,12,28,16,25,28, 38,10,26,23,16,19,15,27,30,30,3,6,25,28,27,19,25,11,11,15,36,24,24,26,26, 6,31,20,15,15,15,16,20,33,13,20,48,35,60,36), cov=c(1.03879744,0.47759241,0.94893572,1.19490019,1.29818904,1.17348657,0.86158595,0.41906623,0.12851131,0.3062645,0.68247702,0.5310083,0.2957547,0.98468564,0.69869688,1.16663393,0.84068273,1.11949189,0.41204136,0.84594715,1.17590928,0.66100507,0.6517726,0.41172415,0.53282988,0.49739296,0.20203482,0.05557657,0.12124606,0.08297883,0.54920017,0.54646221,1.27310233,0.84805791,0.37941846,0.35841475,0.63674533,0.290822,0.50717877,0.63749067,1.06473979,0.76755405,1.32067202,1.07900425,0.9377264,0.69915774,0.30582847,0.55437438,0.83649961,0.8507124,0.74700881,0.51896384,1.12002246,0.56923477,0.79578273,0.9167156,0.57708428,0.67706467,0.61410302,0.62828578,1.1074659,0.84054394,0.90855822,0.73840393,0.27628054,0.77771328,0.59338111,0.58821905,0.51846397,0.34806766,1.09761107,0.54498063,0.67605775,0.28858364,0.68021606,1.12022933,0.44701614,0.48445559,1.535408,0.92860808,0.46919542,0.45107384,0.51761156,0.5572632,0.5144298,1.16590727,1.08351251,0.56744869,0.5808883,0.37861547), fire=c(3.5,4.05,2.6,2.9,4.3,4,4.8,4.8,7.25,6.2,8.05,7.55,7.25, 7.3,3.8,5.7,4.3,4.2,4.25,5.1,1.95,4.4,4.45,3,1.2,6.4,8.2,6.85,6.15,7.7,6.1,4.3,3.75,3.7,5.7,5.4,4.85,4.85,4.3,3.25,2.5,3.85,2.55,3.3,4,3.4,3.9,4,7.5,4.9,4.7,1.7,3,6,6.7,2,6.05,3.9,2.9,3.7,1.2,1.3,3.6,4.4,3.7,4.85,4.85,5.6,4.1,5.7,4.2,4.8,7.2,9.2,6.2,2.1,3.85,3.9,4.85,5.3,4.6,3.3,5.4,4.3,3.6,3,3.8,4.5,3.9,4.6))

  11. Indirect Effects as Causal Tests: Step 2 – Results for Model B Statistics: node mean sd MC error 2.5% median 97.5% start sample Rsqr1 0.206 0.0 4.517E-13 0.206 0.206 0.206 1000 49001 Rsqr2 0.1911 0.0 4.517E-13 0.1911 0.1911 0.1911 1000 49001 a1 3.043 0.3595 0.001567 2.338 3.042 3.752 1000 49001 a2 1.074 0.1017 4.759E-4 0.873 1.074 1.271 1000 49001 b1 0.05955 0.01263 5.623E-5 0.03447 0.05955 0.08426 1000 49001 b1std 0.4529 0.09608 4.276E-4 0.2622 0.4529 0.6408 1000 49001 b2 -0.08385 0.02095 9.758E-5 -0.1246 -0.08386 -0.04258 1000 49001 b2std -0.4367 0.1091 5.082E-4 -0.6491 -0.4368 -0.2218 1000 49001

  12. e1 e2 age of stand that burned fire severity post-fire vegetation cover Indirect Effects as Causal Tests: Step 3 Model C

  13. winBUGS code for Model C # Title: Example 1.1 Model C, with cov, fire (fire severity), and age MODEL simple { # Priors b1 ~ dnorm(0,0.00001); b2 ~ dnorm(0,0.00001); b3 ~ dnorm(0,00001) a1 ~ dnorm(0,0.00001); a2 ~ dnorm(0,0.00001) tau1 ~ dgamma(0.5,1); tau2 ~ dgamma(0.5,1) # Likelihoods for(i in 1:N) { firehat[i] <- a1 + b1*age[i] fire[i] ~ dnorm(firehat[i],tau1) covhat[i] <- a2 + b2*fire[i] + b3*age[i] cov[i] ~ dnorm(covhat[i],tau2) } # Calculate standardized coefficients firestd <- sd(fire[]); agestd <- sd(age[]); b1std <- b1*agestd/firestd covstd <- sd(cov[]); b2std <- b2*firestd/covstd b3std <- b3*agestd/covstd THE REST OF THE CODE AND THE DATA ARE THE SAME AS FOR MODEL B

  14. Indirect Effects as Causal Tests: Step 3 – Results for Model C Statistics: node mean sd MC error 2.5% median 97.5% start sample Rsqr1 0.206 0.0 4.517E-13 0.206 0.206 0.206 1000 49001 Rsqr2 0.2084 0.0171 8.172E-5 0.1597 0.215 0.2202 1000 49001 a1 3.039 0.3575 0.001604 2.337 3.041 3.737 1000 49001 a2 1.121 0.1052 4.759E-4 0.9138 1.122 1.327 1000 49001 b1 0.0597 0.01259 5.691E-5 0.03501 0.05964 0.08446 1000 49001 b1std 0.454 0.09578 4.328E-4 0.2663 0.4536 0.6423 1000 49001 b2 -0.06713 0.02323 1.113E-4 -0.1122 -0.06716 -0.02123 1000 49001 b2std -0.3497 0.121 5.796E-4 -0.5842 -0.3498 -0.1106 1000 49001 b3 -0.00485 0.003076 1.287E-5 -0.01085 -0.004854 0.001184 1000 49001 b3std -0.1921 0.1218 5.099E-4 -0.4298 -0.1923 0.04689 1000 49001 DENSITY FUNCTIONS ON NEXT PAGE

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