final

Section 2 - Details of our Study We begin by reading our data: library(readxl) biomark <- read_excel( "Biomark.xls" ) voplus = biomark$voplus vominus = biomark$vominus oc = biomark$oc trap = biomark$trap knitr::kable(head(biomark)) voplus vominus oc loc trap ltrap lvoplus lvominus 1606 903 68.9 4.232656 19.4 2.965 7.382 6.806 2240 1761 56.3 4.030695 25.5 3.239 7.714 7.474 2221 1486 54.6 4.000034 19.0 2.944 7.706 7.304 896 1116 31.2 3.440418 9.0 2.197 6.798 7.018 2545 2236 36.4 3.594569 19.1 2.950 7.842 7.712 878 954 31.4 3.446808 14.6 2.681 6.778 6.861 Problem 11.36 a. #Numerical Analyses #VO+ Numerical Summary favstats(~ voplus, data= biomark) ## min Q1 median Q3 max mean sd n missing ## 285 542.5 870 1188.5 2545 985.8065 579.8581 31 0 #VO- Numerical Summary favstats(~ vominus, data= biomark) ## min Q1 median Q3 max mean sd n missing ## 254 554 903 1023 2236 889.1935 427.6161 31 0 #OC Numerical Summary favstats(~ oc, data= biomark) ## min Q1 median Q3 max mean sd n missing ## 8.1 18.6 30.2 46.05 77.9 33.41613 19.60974 31 0 #Trap Numerical Summary favstats(~ trap, data= biomark) ## min Q1 median Q3 max mean sd n missing ## 3.3 8.9 10.3 18.8 28.8 13.24839 6.52824 31 0 Now for the Graphical Analayses: hist(voplus) densityplot(~ voplus, data= biomark, main = "Density Plot of VO+" ) 1

Histogram of voplus voplus Frequency 0 1000 2000 3000 0 4 8 12 Density Plot of VO+ voplus Density 0e+00 2e-04 4e-04 6e-04 8e-04 0 1000 2000 3000 The plots of VO+ appear to be slightly right skewed as shown above hist(vominus) densityplot(~ vominus, data= biomark, main = "Density Plot of VO-" ) Histogram of vominus vominus Frequency 0 500 1500 2500 0 5 10 15 Density Plot of VO- vominus Density 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0 500 1500 2500 The plots of VO- appear to be fairly symmetric with a slight right skew as shown above hist(oc) densityplot(~ oc, data= biomark, main = "Density Plot of OC" ) 2

voplus 500 500 2000 5 20 500 2000 vominus oc 10 40 70 5 15 25 500 10 60 trap After numerical and graphical analysis of these relationships, we learn the following: VO+ and VO- have a strong positive relationship VO+ and OC have a moderately positive relationship VO+ and Trap have a strong positive relationship VO- and OC have a weak positive relationship VO- and Trap have a moderately positive relationship OC and Trap have a fairly strong positive relationship (numerically) Problem 11.37 a. simpleOC <- lm(voplus ~ oc, data= biomark) summary.lm(simpleOC) ## ## Call: ## lm(formula = voplus ~ oc, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -727.45 -234.43 -85.08 43.66 1500.99 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 334.034 159.241 2.098 0.0448 * ## oc 19.505 4.127 4.726 5.43e-05 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 443.3 on 29 degrees of freedom ## Multiple R-squared: 0.4351, Adjusted R-squared: 0.4156 ## F-statistic: 22.34 on 1 and 29 DF, p-value: 5.429e-05 plot(voplus ~ oc, data= biomark, main= "VO+ ~ OC" ) abline(simpleOC) 4

10 30 50 70 500 1500 VO+ ~ OC oc voplus Our linear regression equation is: [ VO+ = 334 . 034 + 19 . 505 OC H 0 : β 1 = 0 , there is no linear association between VO+ and OC. H a : β 1 ̸ = 0 , there is a substantial linear association between VO+ and OC. As shown in the summary, t = 4 . 726 and P = 5 . 43 ∗ 10 − 5 < 0 . 05 , so we reject the null hypothesis, concluding that there is enough evidence against the consistency to say that there is a statistically significant linear association between VO+ and OC. Now, to analyze the residuals: plot(simpleOC, which= 1 ) abline( h= 0 ) plot(simpleOC, which= 2 ) hist(residuals(simpleOC)) 600 1000 1600 -1000 500 Fitted values Residuals lm(voplus ~ oc) Residuals vs Fitted 5 32 -2 -1 0 1 2 -2 0 2 4 Theoretical Quantiles Standardized residuals lm(voplus ~ oc) Normal Q-Q 5 3 2 Histogram of residuals(simpleO residuals(simpleOC) Frequency -1000 0 1000 2000 0 5 10 As shown in the Residual plot, there appears to be a curve. Also, in the Q-Q Plot of the residuals, the data appears to curve off in the extremities. Normal Q-Q plots that exhibit this behavior usually mean the data has more extreme values than would be expected if they truly came from a Normal distribution. The histogram also appears to be slightly right skewed. 5

## -364.19 -158.57 -15.13 120.08 441.11 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -243.4877 94.2183 -2.584 0.01549 * ## oc 8.2349 2.8397 2.900 0.00733 ** ## trap 6.6071 10.3340 0.639 0.52797 ## vominus 0.9746 0.1211 8.048 1.2e-08 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 207.8 on 27 degrees of freedom ## Multiple R-squared: 0.8844, Adjusted R-squared: 0.8715 ## F-statistic: 68.84 on 3 and 27 DF, p-value: 9.031e-13 \ V O + = − 243 . 4877 + 8 . 2349 OC + 6 . 6071 TRAP + 0 . 9746 VO- H 0 : β 1 = β 2 = β 3 = 0 , there is no linear association between VO+ and OC/Trap/VO- (respectively). H a : β 1 ̸ = 0 , β 2 ̸ = 0 , β 3 ̸ = 0 , there is a substantial linear association between VO+ and OC/Trap/VO- (respectively). Since for OC, t = 2 . 900 and P = 0 . 00733 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying VO+ and OC have a statistically significant linear relationship. The coefficient of OC is then statistically significant to the model. Since for Trap, t = 0 . 639 and P = 0 . 52797 > 0 . 05 , we can conclude that there is not enough evidence against the consistency to reject the null hypothesis, saying VO+ and Trap do not have a statistically significant linear relationship. The coefficient of Trap is not statistically significant to the model. Since for VO-, t = 8 . 048 and P = 1 . 2 ∗ 10 − 8 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying VO+ and VO- have a statistically significant linear relationship. The coefficient of VO- is then statistically significant to the model. c. #VO+ using OC knitr::kable(coefficients(summary(simpleOC))) Estimate Std. Error t value Pr(>|t|) (Intercept) 334.03439 159.240942 2.097666 0.0447633 oc 19.50471 4.127054 4.726062 0.0000543 #VO+ using both OC and TRAP knitr::kable(coefficients(summary(OCTrapLM))) Estimate Std. Error t value Pr(>|t|) (Intercept) 57.704192 156.538826 0.3686254 0.7151796 oc 6.414658 5.124554 1.2517495 0.2210177 trap 53.874424 15.393305 3.4998607 0.0015770 #VO+ using OC, TRAP, and VO- knitr::kable(coefficients(summary(allLM))) 7

Estimate Std. Error t value Pr(>|t|) (Intercept) -243.4877089 94.2182880 -2.5842935 0.0154867 oc 8.2348785 2.8396522 2.8999603 0.0073318 trap 6.6071427 10.3339550 0.6393624 0.5279750 vominus 0.9745712 0.1210945 8.0480191 0.0000000 The P values in the first model were statistically significant, but in the third model, the P values for the same coefficients became no longer significant. In the third model, these coefficients (OC and Intercept) became statistically significant again. It is clear something is up with TRAP. The estimates of the coefficients change with every model. The intercept gets lower and lower going from models one to three. OC’s coefficient estimate goes down from the first to the second model, but increases going into the third model. TRAP’s coefficient estimate goes down from the second to the third model. d. For the each of the three models, the percent of variation in VO+ is 43 . 51% , 60 . 7% , and 88 . 44% , respectively. The residual standard errors are 443 . 3 on DF = 29 , 376 . 3 on DF = 28 , and 207 . 8 on DF = 27 , respectively. e. Model without TRAP suggested: noTrapLM <- lm(voplus ~ oc + vominus, data= biomark) summary.lm(noTrapLM) ## ## Call: ## lm(formula = voplus ~ oc + vominus, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -350.25 -153.94 -13.22 148.19 428.09 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -234.14400 92.09009 -2.543 0.016818 * ## oc 9.40388 2.14964 4.375 0.000153 *** ## vominus 1.01857 0.09858 10.333 4.65e-11 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 205.6 on 28 degrees of freedom ## Multiple R-squared: 0.8826, Adjusted R-squared: 0.8742 ## F-statistic: 105.3 on 2 and 28 DF, p-value: 9.418e-14 \ V O + = − 243 . 144 + 9 . 404 OC + 1 . 019 VO- H 0 : β 1 = β 2 = 0 , there is no linear association between VO+ and OC/VO- (respectively). H a : β 1 ̸ = 0 , β 2 ̸ = 0 , there is a substantial linear association between VO+ and OC/VO- (respectively). Since for OC, t = 4 . 375 and P = 0 . 000153 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying VO+ and OC have a statistically significant linear relationship. The coefficient of OC is then statistically significant to the model. Since for VO-, t = 10 . 333 and P = 4 . 65 ∗ 10 − 11 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying VO+ and VO- have a statistically significant 8

Histogram of lvoplus lvoplus Frequency 5.5 6.5 7.5 0 4 8 Density Plot of LVO+ lvoplus Density 0.0 0.2 0.4 0.6 5 6 7 8 The plots of LVO+ appear to be pretty symmetric as shown above hist(lvominus) densityplot(~ lvominus, data= biomark, main = "Density Plot of LVO-" ) Histogram of lvominus lvominus Frequency 5.5 6.5 7.5 0 5 10 15 Density Plot of LVO- lvominus Density 0.0 0.2 0.4 0.6 0.8 5 6 7 8 The plots of LVO- appear to be pretty symmetric (with a tiny right skew if anything) as shown above hist(loc) densityplot(~ loc, data= biomark, main = "Density Plot of LOC" ) 10

Histogram of loc loc Frequency 2.0 3.0 4.0 0 2 4 6 8 Density Plot of LOC loc Density 0.0 0.1 0.2 0.3 0.4 0.5 2 3 4 5 The density plot of LOC appears to be fairly symmetric, but the histogram reveals a slight left skew. hist(ltrap) densityplot(~ ltrap, data= biomark, main = "Density Plot of LTRAP" ) Histogram of ltrap ltrap Frequency 1.0 2.0 3.0 0 4 8 12 Density Plot of LTRAP ltrap Density 0.0 0.2 0.4 0.6 0.8 1 2 3 4 The plots of LTRAP appear to be fairly symmetric with a very slight left skew. Multivariate Analysis The potential correlation checked pair by pair: smallbio = subset(biomark, select= c( "lvoplus" , "lvominus" , "loc" , "ltrap" )) with(biomark, cor(smallbio)) ## lvoplus lvominus loc ltrap ## lvoplus 1.0000000 0.8396741 0.7735853 0.7549684 ## lvominus 0.8396741 1.0000000 0.5546070 0.6643005 ## loc 0.7735853 0.5546070 1.0000000 0.7953528 ## ltrap 0.7549684 0.6643005 0.7953528 1.0000000 We can also display this data graphically: pairs(smallbio, pch= "." ) 11

2.5 3.5 6.0 7.0 LVO+ ~ LOC loc lvoplus Our linear regression equation is: \ LVO+ = 4 . 3846 + 0 . 7062 LOC H 0 : β 1 = 0 , there is no linear association between LVO+ and LOC. H a : β 1 ̸ = 0 , there is a substantial linear association between LVO+ and LOC. Since t = 6 . 574 and P = 3 . 34 ∗ 10 − 7 < 0 . 05 , we reject the null hypothesis, concluding that there is enough evidence against the consistency to say that there is a statistically significant linear association between LVO+ and LOC. Now, to analyze the residuals: plot(simpleLOC, which= 1 ) abline( h= 0 ) plot(simpleLOC, which= 2 ) hist(residuals(simpleLOC)) 6.0 6.5 7.0 7.5 -0.5 0.5 Fitted values Residuals lm(lvoplus ~ loc) Residuals vs Fitted 5 17 28 -2 -1 0 1 2 -1 1 2 3 Theoretical Quantiles Standardized residuals lm(lvoplus ~ loc) Normal Q-Q 5 28 17 Histogram of residuals(simpleLO residuals(simpleLOC) Frequency -0.5 0.0 0.5 1.0 0 2 4 6 8 As shown in the Residual plot, there does appear to be a slight curve. The Q-Q Plot of the residuals looks pretty normal, just some curving off at the extremities still. The histogram appears to be right skewed. For our LVO+ ~ LOC + LTRAP Linear Regression Model: 13

LOCTrapLM <- lm(lvoplus ~ loc + ltrap, data= biomark) summary.lm(LOCTrapLM) ## ## Call: ## lm(formula = lvoplus ~ loc + ltrap, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.53385 -0.24194 -0.00337 0.18188 0.78501 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 4.2592 0.3506 12.147 1.12e-12 *** ## loc 0.4301 0.1680 2.560 0.0161 * ## ltrap 0.4243 0.2054 2.066 0.0482 * ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 0.3394 on 28 degrees of freedom ## Multiple R-squared: 0.6515, Adjusted R-squared: 0.6267 ## F-statistic: 26.18 on 2 and 28 DF, p-value: 3.89e-07 Our linear regression model is: \ LVO+ = 4 . 2592 + 0 . 4301 LOC + 0 . 4243 LTRAP H 0 : β 1 = β 2 = 0 , there is no linear association between LVO+ and LOC/LTRAP (respectively). H a : β 1 ̸ = 0 , β 2 ̸ = 0 , there is a substantial linear association between LVO+ and LOC/LTRAP (respectively). For the coefficient of LOC, we to reject the null hypothesis because t = 2 . 560 and P = 0 . 0161 < 0 . 05 . Thus, we conclude there is enough evidence against the consistency to say that there is a substantial linear association between LVO+ and LOC. For the coefficient of LTRAP, as shown in the summary, t = 2 . 066 and P = 0 . 0482 < 0 . 05 , so we reject the null hypothesis, concluding that there is enough evidence against the consistency to say that there is a statistically significant linear association between LVO+ and LTRAP. For our LVO+ ~ LOC + LTRAP + LVO- Linear Regression Model: allLM <- lm(lvoplus ~ loc + ltrap + lvominus, data= biomark) summary.lm(allLM) ## ## Call: ## lm(formula = lvoplus ~ loc + ltrap + lvominus, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.44029 -0.14718 -0.00694 0.16299 0.39917 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.87153 0.64015 1.361 0.18463 ## loc 0.39197 0.11535 3.398 0.00212 ** ## ltrap 0.02768 0.15697 0.176 0.86133 14

Once again, LTRAP seems to be the source of problems and is very statistically insignificant with a P value of 0.86 in the third model. So, we will create a model without it. For the each of the three models, the percent of variation in VO+ is 59 . 84% , 65 . 15% , and 84 . 2% , respectively. The residual standard errors are 0 . 358 on DF = 29 , 0 . 3394 on DF = 28 , and 0 . 2326 on DF = 27 , respectively. Model without LTRAP suggested: noLTrapLM <- lm(lvoplus ~ loc + lvominus, data= biomark) summary.lm(noLTrapLM) ## ## Call: ## lm(formula = lvoplus ~ loc + lvominus, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.44129 -0.14493 -0.00965 0.16497 0.40145 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.83180 0.58875 1.413 0.169 ## loc 0.40593 0.08242 4.925 3.40e-05 *** ## lvominus 0.68173 0.10379 6.569 4.02e-07 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 0.2286 on 28 degrees of freedom ## Multiple R-squared: 0.842, Adjusted R-squared: 0.8307 ## F-statistic: 74.59 on 2 and 28 DF, p-value: 6.061e-12 \ LV O + = 0 . 832 + 0 . 406 LOC + 0 . 682 LVO- H 0 : β 1 = β 2 = 0 , there is no linear association between LVO+ and LOC/LVO- (respectively). H a : β 1 ̸ = 0 , β 2 ̸ = 0 , there is a substantial linear association between LVO+ and LOC/LVO- (respectively). Since for LOC, t = 4 . 925 and P = 3 . 40 ∗ 10 − 5 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying LVO+ and LOC have a statistically significant linear relationship. The coefficient of LOC is then statistically significant to the model. Since for LVO-, t = 6 . 569 and P = 4 . 02 ∗ 10 − 7 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying LVO+ and LVO- have a statistically significant linear relationship. The coefficient of LVO- is then statistically significant to the model. This fourth model is clearly improved from our third model where we used LTRAP. Problem 11.40 For our VO- ~ OC Linear Regression Model: simpleOC <- lm(vominus ~ oc, data= biomark) summary.lm(simpleOC) ## ## Call: 16

## lm(formula = vominus ~ oc, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -510.08 -276.24 -81.27 177.19 1317.22 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 557.818 139.151 4.009 0.000391 *** ## oc 9.917 3.606 2.750 0.010161 * ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 387.4 on 29 degrees of freedom ## Multiple R-squared: 0.2068, Adjusted R-squared: 0.1795 ## F-statistic: 7.561 on 1 and 29 DF, p-value: 0.01016 plot(vominus ~ oc, data= biomark, main= "VO- ~ OC" ) abline(simpleOC) 10 30 50 70 500 1500 VO- ~ OC oc vominus Our linear regression equation is: eeeeeee VO- = 557 . 818 + 139 . 151 OC H 0 : β 1 = 0 , there is no linear association between VO- and OC. H a : β 1 ̸ = 0 , there is a substantial linear association between VO- and OC. Since t = 2 . 750 and P = 0 . 010161 < 0 . 05 , we reject the null hypothesis, concluding that there is enough evidence against the consistency to say that there is a statistically significant linear association between VO- and OC. Now, to analyze the residuals: plot(simpleOC, which= 1 ) abline( h= 0 ) plot(simpleOC, which= 2 ) 17

P = 0 . 67567 > 0 . 05 . Thus, we conclude there is not enough evidence against the consistency to say that there is a substantial linear association between VO- and OC. For our VO- ~ OC + TRAP Linear Regression Model: allLM <- lm(vominus ~ oc + trap + voplus, data= biomark) summary.lm(allLM) ## ## Call: ## lm(formula = vominus ~ oc + trap + voplus, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -346.99 -111.42 -4.38 118.33 317.70 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 267.26110 74.71782 3.577 0.00134 ** ## oc -6.51323 2.50744 -2.598 0.01502 * ## trap 9.48453 8.78782 1.079 0.29001 ## voplus 0.72420 0.08999 8.048 1.2e-08 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 179.2 on 27 degrees of freedom ## Multiple R-squared: 0.842, Adjusted R-squared: 0.8245 ## F-statistic: 47.97 on 3 and 27 DF, p-value: 5.974e-11 \ V O − = 267 . 26110 − 6 . 51323 OC + 9 . 48453 TRAP + 0 . 72420 VO+ H 0 : β 1 = β 2 = β 3 = 0 , there is no linear association between VO- and OC/TRAP/VO+ (respectively). H a : β 1 ̸ = 0 , β 2 ̸ = 0 , β 3 ̸ = 0 , there is a substantial linear association between VO- and OC/TRAP/VO+ (respectively). Since for OC, t = − 2 . 598 and P = 0 . 01502 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying VO- and OC have a statistically significant linear relationship. The coefficient of OC is then statistically significant to the model. Since for TRAP, t = 1 . 079 and P = 0 . 29001 > 0 . 05 , we can conclude that there is not enough evidence against the consistency to reject the null hypothesis, saying VO- and TRAP do not have a statistically significant linear relationship. The coefficient of TRAP is not statistically significant to the model. Since for VO+, t = 8 . 048 and P = 1 . 2 ∗ 10 − 8 < 0 . 05 , we can conclude that there is enough evidence against the consistency to reject the null hypothesis, saying VO- and VO+ have a statistically significant linear relationship. The coefficient of VO+ is then statistically significant to the model. #VO- using OC knitr::kable(coefficients(summary(simpleOC))) Estimate Std. Error t value Pr(>|t|) (Intercept) 557.817691 139.151268 4.008714 0.0003907 oc 9.916644 3.606389 2.749743 0.0101614 19

#VO- using both OC and TRAP knitr::kable(coefficients(summary(OCTrapLM))) Estimate Std. Error t value Pr(>|t|) (Intercept) 309.050693 134.941549 2.2902560 0.0297448 oc -1.867714 4.417532 -0.4227959 0.6756736 trap 48.500594 13.269529 3.6550351 0.0010509 #VO- using OC, TRAP, and VO+ knitr::kable(coefficients(summary(allLM))) Estimate Std. Error t value Pr(>|t|) (Intercept) 267.2610974 74.7178154 3.576939 0.0013396 oc -6.5132341 2.5074402 -2.597563 0.0150182 trap 9.4845308 8.7878244 1.079281 0.2900101 voplus 0.7242038 0.0899853 8.048019 0.0000000 Once again, TRAP seems to be the source of problems and is quite statistically insignificant with a P value of 0.29 in the third model. So, we will create a fourth model without it. For the each of the three models, the percent of variation in VO+ is 20 . 68% , 46 . 3% , and 84 . 2% , respectively. The residual standard errors are 387 . 4 on DF = 29 , 324 . 4 on DF = 28 , and 179 . 2 on DF = 27 , respectively. Model without LTRAP suggested: noTrapLM <- lm(vominus ~ oc + voplus, data= biomark) summary.lm(noTrapLM) ## ## Call: ## lm(formula = vominus ~ oc + voplus, data = biomark) ## ## Residuals: ## Min 1Q Median 3Q Max ## -334.86 -84.97 -6.61 135.47 285.01 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 298.01211 69.27509 4.302 0.000186 *** ## oc -5.25375 2.22586 -2.360 0.025459 * ## voplus 0.77778 0.07527 10.333 4.65e-11 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 179.7 on 28 degrees of freedom ## Multiple R-squared: 0.8352, Adjusted R-squared: 0.8234 ## F-statistic: 70.95 on 2 and 28 DF, p-value: 1.09e-11 \ V O − = 298 . 012 − 5 . 254 OC + 0 . 778 VO+ H 0 : β 1 = β 2 = 0 , there is no linear association between VO- and OC/VO+ (respectively). H a : β 1 ̸ = 0 , β 2 ̸ = 0 , there is a substantial linear association between VO- and OC/VO+ (respectively). 20