Logging and Water Quality

Description

Data from an observational study of nitrate levels measured at three week intervals for five years in two watersheds. One of the watersheds was undisturbed and the other had been logged with a patchwork pattern.

Usage

case1501

Format

A data frame with 88 observations on the following 3 variables.

Week

week after the start of the study

Patch

natural logarithm of nitrate level (ppm) in the logged watershed (ppm)

NoCut

natural logarithm of nitrate level in the undisturbed watershed (ppm)

Source

Ramsey, F.L. and Schafer, D.W. (2013). The Statistical Sleuth: A Course in Methods of Data Analysis (3rd ed), Cengage Learnings.

References

Harr, R.D., Friderksen, R.L., and Rothacher, J. (1979). Changes in Streamflow Following Timber Harvests in Southwestern Oregon, USDA/USFS Research Paper PNW-249, Pacific NW Forest and Range Experiment Station, Portland, Oregon.

Examples

str(case1501)
attach(case1501)

## EXPLORATION
opar <- par(no.readonly=TRUE)  # Store current graphics parameters settings
par(mfrow=c(2,1))   # Set graphics parameters: 2 row, 1 column layout
plot(NoCut ~ Week,  type="b", ylab="Log of Nitrate Concentration; NoCut")
abline(h=mean(NoCut))  # Horizontal line at the mean
plot(Patch ~ Week,  type="b", ylab="Log of Nitrate Concentration; Patch Cut")         
abline(h=mean(Patch))
 
par(opar) # Restore previous graphics settings
lag.plot(NoCut,do.lines=FALSE)  # Lag plot for NoCut
lag.plot(Patch,do.lines=FALSE)  # Lag plot for Patch
pacf(NoCut)  # partial autocorrelation function plot; noCut
pacf(Patch)  # partial autocorrelation function plot; Patch

## INFERENCE  (2-sample comparison, accounting for first serial correlation)
diff     <- mean(Patch) - mean(NoCut)
nPatch   <- length(Patch)  # length of Patch series
nNoCut   <- length(NoCut)   # length of NoCut series
acfPatch <- acf(Patch, type="covariance")  # auto covariances for Patch series
c0Patch  <- acfPatch$acf[1]*nPatch/(nPatch-1) # variance; n-1 divisor (Patch) 
c1Patch  <- acfPatch$acf[2]*nPatch/(nPatch-1) # autocov; n-1 divisor (Patch)  
acfNoCut <- acf(NoCut, type="covariance") # auto covariances for NoCut series
c0NoCut  <- acfNoCut$acf[1]*nNoCut/(nNoCut - 1) # variance; n-1 divisor (NoCut)
c1NoCut  <- acfNoCut$acf[2]*nNoCut/(nNoCut - 1) # autocov; n-1 divisor (NoCut) 
dfPatch  <- nPatch - 1     # DF (n-1); Patch
dfNoCut  <- nNoCut - 1     # DF (n-1); NoCut

c0Pooled   <- (dfPatch*c0Patch + dfNoCut*c0NoCut)/(dfPatch + dfNoCut)
c0Pooled   #[1] 1.413295  = pooled estimate of variance
c1Pooled   <- (dfPatch*c1Patch + dfNoCut*c1NoCut)/(dfPatch + dfNoCut)
c1Pooled   #[1] 0.9103366 = pooled estimate of lag 1 covariance

# Pooled estimate of first serial correlation coefficient:
r1 <- c1Pooled/c0Pooled                  #[1] 0.6441233
SEdiff  <- sqrt((1 + r1)/(1-r1))*sqrt(c0Pooled*(1/nPatch + 1/nNoCut))    

# t-test and confidence interval
tStat      <- diff/SEdiff #[1] 0.2713923
pValue     <- 1 - pt(tStat,dfPatch + dfNoCut)     # One-sided p-value   
halfWidth  <- qt(.975,dfPatch + dfNoCut)*SEdiff   # half width of 95% CI
diff + c(-1,1)*halfWidth  #95% CI -0.6557578  0.8648487

## GRAPHICAL DISPLAY FOR PRESENTATION  
par(mfrow=c(1,1))                   # Reset mfrow to a single plot per page
plot(exp(Patch) ~ Week, # Use exp(Patch) to show results in original units
  log="y", type="b", xlab="Weeks After Logging",
  ylab="Nitrate Concentration in Watershed Runoff (ppm)",
  main="Nitrate Series in Patch-Cut and Undisturbed Watersheds",
  pch=21, col="dark green", lwd=3, bg="green", cex=1.3 ) 
points(exp(NoCut) ~ Week, pch=24, col="dark blue", lwd=3, bg="orange",cex=1.3)
lines(exp(NoCut) ~ Week, lwd=3, col="dark blue",lty=3)            
abline(h=exp(mean(Patch)),col="dark green",lwd=2)
abline(h=exp(mean(NoCut)),col="dark blue", lwd=2,lty=2)
legend(205,100,legend=c("Patch Cut", "Undisturbed"),
  pch=c(21,24), col=c("dark green","dark blue"), pt.bg = c("green","orange"),
  pt.cex=c(1.3,1.3), lty=c(1,3), lwd=c(3,3))
text(-1, 8.5, "Mean",col="dark green")
text(-1,6.3,"Mean", col="dark blue")


detach(case1501)

Results