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

V. Inference from Samples
15
an estimate of the variance of the sampling distribution of
x− —that is, σˆ 2 Σ [x i Σ (x i /k)] 2 /(k 1), where σˆ denotes
estimate. Because σˆ 2 σˆ 2 /n, where n is the common sample
sum to the effective sample size for estimating the total
variability in the data ignoring group membership of the
observations [here kn 1 ( k 1) k ( n 1)].
size, inflation of σˆ 2 by n will produce a second estimate
of the variance of the underlying population assuming no
differences among the group means. These two estimates of Example 5
σ 2 should be about equal and their ratio close to 1.
If there is significant separation among some or all of the
group means, the second variance estimate, when computed
as described, will be larger than the first estimate, indicating
Suppose a study is designed to determine in vivo effects
of cytokines on in vitro clonogenicity assays (expansion of
CD34 progenitor cells). To this end, a total of 30 monkeys
(rhesus macaque) of the same age and gender were obtained
that a component of variance is being estimated beyond that
and assigned randomly as follows: 10 monkeys were administered
the traditional cytokine cocktail (TCC), 10 monkeys were
embodied only in an estimate of the within-group variability.
This additional variance component being estimated is
administered a conditioned media with anti-CD3 (ACD3S),
the variance among group means. ANOVA in this example
involves generating the two estimates of σ 2 and 10 monkeys served as controls and were administered
(called mean a placebo. Both cocktails were administered to induce production
of progenitor cells expressing antigen CD34. Bone
squares ) under the hypothesis that all of the group means
are equal. The hypothesis is then tested by forming the ratio
of the two mean squares (the second mean square divided by
the first mean square) called the F-statistic . The F-statistic
has a probability distribution called the F-distribution. If the
computed F -value is greater than the tabled distributional
marrow samples were taken from the monkeys 10 days after
administration, cultured for 14 days after which the number
of colonies was determined. (Expression of CD34 cells was
monitored by flow cytometry and the response is the number
of colonies detected in vitro. ) Figure 1-9 gives the counts
obtained for the three groups that were compared using
value, then the hypothesis is rejected and subsequently
ANOVA for the completely randomized design. The statistical
confidence intervals are constructed as described earlier to
software used was SPSS for Windows Release 11.
determine which group means are different. If the hypothesis
cannot be rejected, the process stops or perhaps, at most,
cocktail count
FIGURE 1-9 SPSS worksheet for
the data from all the groups might be pooled together and
comparing the responses of the three
1 control 45.00
the parameters of this single population estimated. Tables
groups of Example 6 using analysis
2 control 47.00
of the percentiles of the F -distributions can be found in all
of variance assuming a completely
3 control 39.00 randomized design. Source of data:
introductory texts on statistics ( Daniel, 2005 ; Dunn and
4 control 47.00 Mr. Nestor Montiel, California,
Clark, 2001 ; Schork and Remington, 2000 ), which also provide
instruction on how to read the tables. Also, these texts 6 control 39.00 University of California, Davis.
5 control 51.00 National Primate Research Center,
give the generalization of the among group mean square
when the sample size is not constant for all groups.
The results of an ANOVA are traditionally summarized
in a table called the ANOVA table, and Table 1-5 is such an
example. The first column of Table 1-5 shows the sources
7
8
9
10
11
control
control
control
control
tcc
40.00
42 00
35.00
38 00
52.00
12 tcc 52.00
of variability into which the total variability is decomposed.
13 tcc 53.00
In the present example, these sources are due to the variability
within groups and that which is reflective of variability
14 tcc 54.00
15 tcc 49.00
among group means. Column 4 gives the two independent 16 tcc 49.00
(under a hypothesis of no difference in response among 17 tcc 48.00
groups) estimates of σ 2 or mean squares, mean square
among group means ( MS A ), and mean square within groups
( MS W ). Columns 3 and 2 give, respectively, the numerator
(called the sum of squares) and the denominator (the effective
sample size or degrees of freedom) of the corresponding
18
19
20
21
22
tcc
tcc
tcc
acd3s
acd3s
51.00
52.00
51.00
66.00
64.00
mean square. The sums of squares provide a check on
23 acd3s 62.00
24 acd3s 61.00
calculations when generating an ANOVA table because
25 acd3s 58.00
the total of the sum of squares attributable to the various
26 acd3s 57.00
sources of variability is equal to the sum of squared deviations
of each observation across the k samples from the 28 acd3s 59.00
27 acd3s 56.00
grand mean of all the observations in the k samples, called 29 acd3s 59.00
the total sum of squares . In other words, the total sum of
squares is the numerator for estimating the total variability
in the data ignoring group membership. Similarly,
30 acd3s 60.00
the degrees of freedom for the sources of variability 3 3SPSS, Inc., 233 South Wacker Drive, Chicago, IL 16801-3008.