Looking for local water quality benefits of a small-scale cattle exclusion fencing project in Southern Alberta  

Statistical Methods

The comparison of a continuous response variable (E.coli concentration) between two treatments of a categorical independent variable (fenced vs. unfenced treatment) was completed using a version of the student's t test.  Individual t tests were made for each year: 2004, 2005, 2006, and 2007.

 

Assumptions

The use of a t test requires that the data, at each treatment level, meets several assumptions.  The data needs to be:
- Normally distributed
- Equal variances
- Equal sample size

Prior to begining analysis, the data were visually examined to confirm that they met these assumptions.  The assumption of equal sample size was met (n=12 for each treatment, each year).  Plotted histograms, however, presented non-normal distributions for E.coli concentrations (Figures 9 & 10).  Plots of residuals and Bartlett's test for homogeneity (p=0.6791) also suggested unequal variances between treatments for each year (Figure 11) . 

 

Figure 9. Histogram of E.coli concentrations from the fenced treatment, with a density estimate line and rug plot reflecting where the data actually lie along the x axis.

Figure 10. Histogram of E.coli concentrations from the unfenced/ background treatment, with a density estimate line and rug plot reflecting where the data actually lie along the x axis.

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Figure 11. Plot of residuals of E.coli concentrations by treatment and year Differences in variance are represented by the difference in vertical spread of the points at each treatment*year.  The presence of outliers creates unequal variances between treatments in two of the treatment*years.  The large limit on the y axis forces a clustering of data points, but variances appear to show some difference between treatments, as represented by the red line.

 

To allow the use of a parametric statistical test despite the known or potential presence of unequal variances, I selected a version of the t test, Welch's two sample t test, that handles inequality in variance by adjusting degrees of freedom (5).  Data distributions were normalized using a log-10 transformation of E.coli concentrations (Figures 12 & 13).  A comparison of pre- and post-transformed data is presented in the two probability plots below (Figure 14).

 

Figure 12. Histogram of log transformed E.coli concentrations from the fenced treatment.

Figure 13.  Histogram of the log transformed E.coli concentrations from the unfenced/background treatment.

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Figure 14.  Probability plots generated for untransformed and log-transformed E.coli concentration data.  With a normal distribution, the plotted points should lie on a straight line of unit slope from the origin; this is more closely approached in the log transformed data on the left. 

 

References cited:

(5)  Ruxton, GD. 2006. The unequal variance t test is an underused alternative to Student's t test and the Mann-Whitney U test. Behavioural Ecology 47: 688-690.