Clinical Biochemistry of Domestic Animals (Sixth Edition) - UMK ...

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Chapter | 1 Concepts of Normality in Clinical Biochemistry
differing significantly from 1 would indicate that MS d/a
was estimating something additional to σ e
2
, namely, n σ d/a 2 ,
where n is the number of replications per sample. Based
on the assumed data, the hypothesis σ d/a
2
0 is rejected.
σ d/a
2
in this design is the true intradog variability for this
response. The test result that the day effect is significant
means that this intradog variability is larger than zero.
The second test is that the variance component for
“ animals ” is equal to zero or σ a 2 0. It can be determined
by considering the expected mean squares of Table 1-14
that the test for no significant variability among animals
( H o : σ a
2
0) is made using MS d/a in the denominator of the
F -statistic. This test is also significant. This means that σ a
2
is
not equal to zero, indicating a significant source of variability
among dogs in the plasma cortisol levels recorded in the
experiment.
1 . Estimating Variance Components
Once significance has been established for one or more of
the variance components in an experimental design, interest
focuses on estimating the variance component(s). Estimates
are readily obtainable using the appropriate expected mean
squares in conjunction with the mean squares obtained
from the data. For example, Table 1-7 shows that an estimate
of σ e
2
is MS e (σˆ e
2
0.0096). An estimate of σ d/a
2
can
be obtained by noting that MS d/a estimates σ e
2
n σ d/a
2
;
solving for σ d/a 2 followed by substitution of MS e for σ e
2
yields ( MS d/a MS e )/n as the estimate. Based on the data
used in the example, σˆ d/a
2
(1.0958 0.0096)/2 0.5431.
An estimate of σ a
2
, obtained in a similar manner, is
( MS MS )/ nn
where n d is the number of days samples were taken (3).
Based on the data an estimate of σ a
2
is 0.4387. The term
σ a
2
is the estimate of the interanimal variability, whereas
σ d/a
2
estimates the intra-animal variability in plasma cortisol
level for the underlying population of dogs. Interval estimates
for these variance components can be obtained using
these point estimates by methods described elsewhere
( Harter and Lum, 1955 ; Mickey et al. , 2004 ; Satterthwaite,
1941).
d/a
2 . Estimating the Variance of the Grand
Mean Response
Another way to visualize the importance of these variance
components is to analyze their impact on the estimate
of the variance of interest. In some applications, there
would be interest in estimating the grand mean ( μ ) of the
response. In the present example, this would involve estimating
the mean plasma cortisol level taking into account
any random animal effect and day effect. The variance of
μˆ , Var( μˆ ), is given as σ a
2
/ n a σ d/a
2
/ n a n d σ e
2
/ n a n d n ( Little
d
et al. , 1991 ; Neter et al. , 1996 ). In the present example,
Var( μˆ ) is estimated as 0.4387/20 0.5431/60 0.0096/
120 0.0219 0.0091 0.0001 0.0311, and by far
the greatest contribution to this variance is that due to the
variability among animals in their response. The intradog
variance component, although slightly larger than the interdog
variance component, makes a considerably smaller
impact on Var( μˆ ).
3 . Estimating the Total Variability of a
Single Response
In other applications, interest centers about the total variability
( σ total 2 ) associated with a single response. A single
response is a linear combination of the terms in the response
model and, using the assumption of the independence of
terms in the model, has a variance equal simply to the sum of
the variance components. Specifically, σ total 2 σ a
2
σ d/a
2
σ e
2
( Kringle, 1994 ), which in this example is estimated as
0.4387 0.5431 0.0096 0.9914. Here the total variability
of a single response is divided nearly equally between
“animals ” and “days nested within animals. ”
Other possible designs could be considered. What has
been demonstrated is that the method of analysis of variance
in conjunction with experimental design can be useful
in answering a variety of questions.
Nested designs are frequently used to assess sources of
variability in an assay. For example, several laboratories
could be involved in doing a particular assay, with several
autoanalyzers in each laboratory and multiple technicians
running these autoanalyzers. Inference in this context centers
around being able to identify if there are significant
sources of variation among the laboratories, among autoanalyzers
within a given laboratory, and among technicians
operating a given autoanalyzer. The goal of analyses of
this sort is to identify large sources of variability. Once the
larger sources of variability have been identified, changes
are made in the system in an effort to reduce the variability
associated with each source. The long-term objective
is to have an assay with sources of variability that are as
small as possible. Clinical analysts conventionally divide
the square root of the estimates of the variance components
(the sample standard deviations) resulting from such
assay experiments by the grand mean to obtain coefficients
of variation for each source of variability ( Kringle, 1994 ).
These coefficients of variability should be much smaller
than those derived as intra-animal and interanimal variability.
Interested readers are strongly encouraged to consult
texts written on experimental design and ANOVA ( Mickey
et al. , 2004 ; Neter et al. , 1996 ).
REFERENCES
Björkhem , I. , Bergman , A. , and Falk , O. ( 1981 ). Clin. Chem. 27 ,
733 − 735 .