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and the overall model for diffQ is, therefore, known as
<br />a mixed model. (See, for example, Snedecor and
<br />Cochran (1967) for a more detailed discussion of the
<br />distinction between fixed and random effects.) The
<br />random effects associated with site and date each have
<br />a variance (known as variance components), and the
<br />variance of diffQ thus is the sum of three constituent
<br />terms: the site variance, the date variance, and an error
<br />variance, which represents variability (such as
<br />measurement error) that is not accounted for by any
<br />known factors.
<br />Therefore, a mixed analysis of variance model
<br />with both fixed and random effects was applied as
<br />follows: The three fixed (nonrandom) effects of
<br />interest were: (1) method, with levels P, C, and M;
<br />(2) make, with levels M, S, X, and B; and (3) type,
<br />with levels O, L, S, and C. The eight values for
<br />make R were not included in the analysis because
<br />the differences in instantaneous discharge were so
<br />much greater in magnitude than all the other values.
<br />Boxplots for all the discharge data pooled and for each
<br />level of the three fixed effects are shown in figure 3.
<br />More than 80 percent of the differences in the paired
<br />discharge measurements for the entire network of
<br />wells were less than 10 percent, more than 50 percent
<br />of the differences were less than 5 percent, and the
<br />median difference was less than 1 percent (fig. 3A).
<br />The distribution of the differences varied among the
<br />three fixed effects (method, make, and type) (figs. 3B,
<br />3C, and 3D).
<br />In addition to the fixed effects, two random
<br />effects were included in the analysis: (4) site and (5)
<br />date. The sites were classified as to make and type; for
<br />example, each site was associated with one and only
<br />one make and type. Thus, random factor site (4) is said
<br />to be nested under fixed effects make (2) and type (3).
<br />Likewise, random factor date (5) was nested under
<br />fixed factor site (4). The portable flowmeter methods
<br />[factor (1)] were applied at all sites, and often two or
<br />more methods were applied at the same well on the
<br />same date, so there was no nesting used for this factor.
<br />This analysis of variance design is referred to as a
<br />split -plot design, with "plots" corresponding to a given
<br />site on a given day. Snedecor and Cochran (1967) and
<br />Helsel and Hirsch (1992) provide more in -depth
<br />discussion of fixed and random effects and of nested
<br />(or hierarchical) designs.
<br />The mathematical model for diffQ may be
<br />written as
<br />dlffQijkmn = It + ai + Pi +Yk (5)
<br />+ 'Sjkm + Cjkmn + eijkmn
<br />where
<br />µ is the intercept term,
<br />ai is the effect (fixed) for the portable flowmeter
<br />method i,
<br />R j is the effect (fixed) for totalizing flowmeter
<br />make j,
<br />7 k is the effect (fixed) for distribution system
<br />type k,
<br />Sjkm is the effect (random) for site m of wells with
<br />make j and type k,
<br />C jkmn is the effect (random) for make j and type k
<br />on day n at site m, and
<br />eijkmn is a random error term.
<br />In this model, the random terms S, C, and e are
<br />assumed to be independent and normally distributed
<br />with mean 0 and variances 6s , 62 , and a2 , respec-
<br />tively. The analysis of variance provides estimates of
<br />the fixed effects and of the magnitudes of these three
<br />variances (known as "variance components" because
<br />they constitute a partitioning of the random variability
<br />of diffQ) as well.
<br />The three fixed effects were included in order to
<br />determine if average values of diffQ tend to change
<br />systematically with method, make, or type. The
<br />random effects for site and date were included to
<br />account for the correlation among measurements taken
<br />at the same site and on the same day. In most cases,
<br />more than one portable flowmeter method was used at
<br />a given site on the same day. In many cases, the well
<br />discharge measurements made at the same site on the
<br />same day by portable flowmeters clustered together
<br />and exhibited similar deviation from the TFM
<br />discharge. This clustering tendency is shown in
<br />figure 4, which shows how diffQ varies with site.
<br />The magnitude of the tendency for differences to
<br />cluster is evaluated by the site -and date - variance
<br />components. The site variance as is a measure of the
<br />tendency for all the measurements made at a well to
<br />exhibit a systematic discrepancy between portable
<br />14 Comparison of Two Approaches for Determining Ground -Water Discharge and Pumpage in the
<br />Lower Arkansas River Basin, Colorado, 1997 -98
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