muss<-read.csv(url("http://cknudson.com/data/mussels.csv"))
head(muss)## GroupID dry.mass count attached lipid protein carbo ash Kcal ammonia
## 1 1 0.55 20 Rock 8.14 47.43 21.59 5.51 3.61 0.07
## 2 2 0.45 19 Rock 9.34 53.89 23.41 6.34 4.06 0.07
## 3 3 0.37 20 Rock 9.12 49.01 21.10 5.63 3.74 0.07
## 4 4 0.63 20 Rock 10.32 49.25 16.55 5.41 3.66 0.11
## 5 5 0.57 20 Rock 10.08 50.17 17.51 6.10 3.72 0.11
## 6 6 0.57 22 Rock 10.83 53.84 19.97 6.36 4.04 0.11
## O2 AvgAmmonia AvgO2 AvgMass
## 1 0.82 0.00350000 0.04100 0.027500
## 2 0.70 0.00368421 0.03684 0.023684
## 3 0.62 0.00350000 0.03100 0.018500
## 4 0.89 0.00550000 0.04450 0.031500
## 5 1.09 0.00550000 0.05450 0.028500
## 6 1.00 0.00500000 0.04545 0.025909attach(muss)
summary(muss)## GroupID dry.mass count attached
## Min. : 1.00 Min. :0.3000 Min. :19.00 Amblema:13
## 1st Qu.: 7.75 1st Qu.:0.4375 1st Qu.:20.00 Rock :15
## Median :14.50 Median :0.5450 Median :21.50
## Mean :14.93 Mean :0.5836 Mean :27.79
## 3rd Qu.:22.25 3rd Qu.:0.6425 3rd Qu.:31.00
## Max. :30.00 Max. :1.1600 Max. :72.00
##
## lipid protein carbo ash
## Min. : 6.500 Min. :43.49 Min. : 9.97 Min. :4.620
## 1st Qu.: 8.250 1st Qu.:50.05 1st Qu.:17.11 1st Qu.:5.395
## Median : 9.405 Median :53.94 Median :20.09 Median :5.550
## Mean : 9.210 Mean :54.91 Mean :19.96 Mean :5.663
## 3rd Qu.:10.195 3rd Qu.:58.90 3rd Qu.:21.96 3rd Qu.:6.075
## Max. :10.830 Max. :71.94 Max. :29.89 Max. :6.570
## NA's :1 NA's :1 NA's :2
## Kcal ammonia O2 AvgAmmonia
## Min. :3.270 Min. :0.0500 Min. :0.3800 Min. :0.002500
## 1st Qu.:3.743 1st Qu.:0.0875 1st Qu.:0.7975 1st Qu.:0.003500
## Median :4.005 Median :0.1100 Median :0.9250 Median :0.004749
## Mean :3.955 Mean :0.1443 Mean :1.2157 Mean :0.005296
## 3rd Qu.:4.110 3rd Qu.:0.1875 3rd Qu.:1.4125 3rd Qu.:0.005647
## Max. :4.610 Max. :0.4100 Max. :2.7700 Max. :0.012963
## NA's :2
## AvgO2 AvgMass
## Min. :0.01458 Min. :0.006389
## 1st Qu.:0.03100 1st Qu.:0.016885
## Median :0.04439 Median :0.024048
## Mean :0.04564 Mean :0.023107
## 3rd Qu.:0.04939 3rd Qu.:0.027500
## Max. :0.09571 Max. :0.047826
## We create a linear model to find the relationship that average mass and whether or not the mussel is attached to a rock have with average ammonia.
(mod <- lm(AvgAmmonia ~ AvgMass + attached))##
## Call:
## lm(formula = AvgAmmonia ~ AvgMass + attached)
##
## Coefficients:
## (Intercept) AvgMass attachedRock
## 0.001140 0.239279 -0.002563Then, our regression equation is: $$hat{AvgAmmonia}=0.001140
We can conduct a hypothesis test to determine whether average mass and average ammonia have a linear relatiohsip, regardless of if the mussel is attached to a rock or not.
summary(mod)##
## Call:
## lm(formula = AvgAmmonia ~ AvgMass + attached)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.019e-03 -5.240e-04 -5.959e-05 3.429e-04 2.526e-03
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0011398 0.0005533 2.060 0.05 *
## AvgMass 0.2392793 0.0215863 11.085 3.86e-11 ***
## attachedRock -0.0025629 0.0003931 -6.519 7.91e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.00103 on 25 degrees of freedom
## Multiple R-squared: 0.8574, Adjusted R-squared: 0.846
## F-statistic: 75.18 on 2 and 25 DF, p-value: 2.66e-11confint(mod)## 2.5 % 97.5 %
## (Intercept) 1.999745e-07 0.002279427
## AvgMass 1.948215e-01 0.283737200
## attachedRock -3.372584e-03 -0.001753235We are 95% confident that given a constant average mass, we predict the average ammonia will increase between \(-3.372584\times 10^{-3}\) and \(-0.001753235\) if the mussel is attached to a rock.