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Designs for estimating variability structure and implications for detecting watershed restoration effectiveness David P. Larsen –Western Ecology Division, NHEERL, USEPA –200 SW 35 th St. Corvallis, OR 97333 N. Scott Urquhart –Department of Statistics –Colorado State University –Ft. Collins, CO 80523

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Topics Linear trend detection –Applying the tools to restoration monitoring Organizing variability Expanding the linear trend detection model Variance summary Trend detection

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A 2% per Year Linear Trend (each point is a regional mean value) 2% / yr Increase ( Slope = 0 ?) For any patterned trend, there is an underlying linear component.

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A 2% per Year Linear Trend (each point is a regional mean value) Treatment Can we detect a difference is slope between treated and untreated systems? Reference

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Linear trend detection Hypothesis test: Slope = 0? Power: If a trend is present, what is the likelihood of detecting it? Hypothesis: Slope between treated and reference = 0 Power: likelihood of detecting if different?

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Linear trend detection Power depends on: - magnitude of the trend (slope), - variability of our measurements, - number of sites, - the duration of the study (how long we can wait for the information).

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Variance of a trend slope: How precisely can we estimate it?

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Organizing Variability Four major components: –Spatial Site-to-site –Temporal (year to year) Year Site x Year Interaction –Residual

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SITE VARIANCE: Persistent Site-to-Site Differences due to due to Different Landscape/Historical Contexts Different Levels of Human Disturbance ------ Gradient --------> ------ Gradient --------> ----------Stream Size ----------->.

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Year variation Concordant year-to-year variation across all sites Caused by regional phenomena such as: –Wet/Dry years –Ocean conditions –Major volcanic eruptions

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Interaction variation Independent year-to-year variation among sites Driven by local factors

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Residual variation The rest of it including: –Temporal or seasonal variation during sampling window –Fine scale spatial variation –Crew-to-crew differences in applying the protocol –Measurement error –…

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Design framework Multiple sites with revisits within and among years Need a sample size of 30-50 to get reasonable estimate of variance, i.e., 30 – 50 sites; 30-50 revisits within year; at least 5 years with some sites visited annually, or at least in pairs of adjacent years.

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AUGMENTED SERIALLY ALTERNATING SERIALLY ALTERNATING WITH CONSECUTIVE YEAR REVISITS

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Variance of a trend slope (New sites each year) site year interaction residual X i = Year ; N s = Number of sites in region; N v = Number of within-year revisits (Urquhart and Kincaid. 1999. J. Ag., Biol., and Env. Statistics 4:404-414)

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Variance of a trend slope (Revisiting the same sites each year) X i = Year ; N s = Number of sites in region; N v = Number of within-year revisits (See Urquhart & Kincaid, 1999) year interaction residual

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Implications Effect of site = 0 if sites are revisited across years Year is not sensitive to sample sizeand its effect can become dominant Residual is affected by within year revisits Interaction and residual are affected by number of sites in survey, therefore other factors being equal, better to add sites to the survey rather than revisit sites

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Some options (after adding sites doesnt help) Extend survey interval Focus on subpopulations to manage variance Monitor hypothesized covariates controlling year

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Adaptations for Effectiveness Monitoring Context –Comparing two watersheds

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Adaptations for Effectiveness Monitoring Context –Comparing multiple watersheds –Some treated ( ) –Some reference ( )

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Year Indicator Power to detect a 2% per year drift from reference?

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Variance of the difference in two trend slopes (New sites each Year) X i = Year ; N s = Number of sites in each region; N v = Number of within-year revisits site interaction residual

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Variance of the difference in two trend slopes (Revisiting the same sites Each Year) X i = Year ; N s = Number of sites in each region; N v = Number of within-year revisits interaction residual

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Duration (yrs) (X i – X) 2 960 1082.5 11110 12143 13182 14228 15280 Denominators effect

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Variance Summary (Large wood) Monitoring area SiteYearInteractionResidual North Coast 0.1310.0030.0090.033 Mid-Coast 0.08100.0030.014 Mid-South 0.2340.0070.0040.020 South Coast 0.166---0.0060.019 Umpqua 0.1380.00200.020

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Design for power curves Annual visits, # sites varies Serially alternating design, with annual panel Variance components values were selected as low and high for Log 10 (LW+0.1) Alpha = 0.1

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POWER CURVES FOR LOW VALUES OF VARIANCE COMPONENTS POWER YEAR

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POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS POWER YEAR

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POWER CURVES FOR HIGH VALUES OF VARIANCE COMPONENTS; AUGMENTED ROTATING PANEL DESIGN POWER YEAR

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Summary Characterization of spatial and temporal variation Design framework for estimating components of variation A framework for evaluating linear trend How variation affects trend detection Modifying the framework for evaluating restoration An example using large wood as an indicator

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