A Program to Compute Approximate Confidence
Intervals on Certain Linear Combinations of
ANOVA-based Mean Squares
Brandon L. Paris
Page 1 of 8
Overview of Program
This program was written to allow simple calculation of approximate confidence intervals for linear
combinations of ANOVA-based mean squares estimates. The linear combinations must follow one of the following
forms:
Q
θ =
∑ c MS
i
i
,
or
(1)
i =1
Q
P
θ =
∑
ci MS i −
i =1
∑ c MS
j
j
,
(2)
j = P +1
where all ci (i = 1, 2, . . . Q) are positive and the MSi are the ANOVA-based mean squares estimates. The methods
used in the program are from the text Confidence Intervals on Variance Components by Burdick and Graybill and
are based on large-sample approximations. The user is directed to this source for further information.
The software makes use of several routines from DCDFLIB to obtain the F and χ2 quantiles necessary to
compute the confidence intervals. The library, written by Brown, Levato, and Russell, provides routines that will
make various computations for a number of statistical distributions and is available in both C and Fortran on the
World Wide Web at http://odin.mdacc.tmc.edu or http://www.netlib.no.
Using the Program
Once linked and compiled, the program is invoked through the command-line as follows:
> interval
Invoking the program in this manner will print output only to the display console. To also print the results to an
output file, use the call:
> interval outfile
where outfile is the name of the file the results will be placed.
The program will prompt the user for several inputs.
Q
The total number of sources of variation (SOV) in the linear combination.
P
The number of SOV to be included in the first summation. When Q = P, the linear
combination is like (1), otherwise, the linear combination is like (2).
type
The type of confidence interval: one-sided upper CI, one-sided lower CI, or two-sided
confidence interval
α (alpha)
Proportion in (0,1) where the confidence level for the interval is (1–α) for one-sided intervals
or (1-2α) for two-sided intervals.
ci
Constant to be multiplied times the ith mean square included in the linear combination.
MSi
The ith mean square included in the linear combination.
dfi
The degrees of freedom associated with the ith mean square included in the linear combination.
Page 2 of 8
At times, the software may produce negative estimates of the lower and/or upper confidence limits. The
software handles this differently depending on which form θ takes. If θ is of form (1), then it is constrained to be
non-negative, and hence the confidence limits will follow this same constraint. If, however, θ is of form (2), then
there are no general sign constraints on θ and the user must assess whether the final results make sense. For
instance, a user may calculate a linear combination of form (2) in an attempt to estimate a variance, which must be
non-negative. In this situation, negative confidence limits should be interpreted as zero by the user. At other times,
the goal may be to determine whether two variances are statistically different. In this instance, negative confidence
limits will probably be acceptable.
Examples
Example One (Small Sample Sizes)
Lorenzen and Anderson (1993) provide data for an experiment measuring the firing time of explosive
switches where there were three factors of interest: the metal used in the switch, amount of primary initiator, and the
packing pressure of the explosive. A completely randomized design was used for the experiment. Because the metal
used for the switches was made of recycled material, metal was considered to be a random effect while the remaining
factors were considered to be fixed. The experiment showed that only the metal used and amount of primary
initiator were effecting firing time. As a result, the final ANOVA for the experiment included only these sources of
variation and their interaction. This final ANOVA table, including the expected mean squares, is provided in Table
1.
As an example, consider that estimating the total variability of firing times is of interest. Simple
manipulation of the EMS from Table 1 shows that
2
σ Total
=
1
1
7
EMS1 + EMS 3 + EMS 4 ,
18
6
9
which can be estimated by
2
σˆ Total
≈ 0.0556MS1 + 0.1667MS 3 + 0.7778MS 4 .
2
In this example, Q = P = 3. Using the software, σˆ Total
= 263.27 and the 95% two-sided confidence interval for
2
σ Total
is (111.8, 1.78x105). A specific listing of both the inputs and program output is provided in Figure 1.
Table 1. Final ANOVA for Experiment of Explosive Switches from Lorenzen and Anderson (1993).
Source
Metal
df
1
SS
3136.00
MS
3136.00
Initiator
2
513.72
256.86
EMS 2 = σ 2 + 6σ Metal *Initiator + 12Φ ( I )
Metal*Initiator
2
969.50
484.75
2
EMS 3 = σ 2 + 6σ Metal
*Initiator
30
35
318.00
4937.22
10.60
Error
Total
EMS
2
EMS1 = σ + 18σ Metal
2
EMS4 = σ 2
Page 3 of 8
Table 2. ANOVA for Nested Experiment presented in Box, Hunter, and Hunter (1978).
Source
df
SS
MS
EMS
Batches
14
1,211.0
86.6
2
σ Tests
2
+ 2σ Samples
Samples(Batches)
15
869.7
58.0
2
2
σ Tests
+ 2σ Samples
Tests(Samples Batches)
Corrected Total
30
59
27.5
2,108.2
0.9
2
+ 4σ Batches
2
σ Tests
This example is presented to show the limitations of the software for use with data with small sample sizes
(small degrees of freedom for metals, initiators, and their interaction), and the fairly wide confidence intervals should
convince the reader that limitations do exist. Thus, the user should proceed with caution when interpreting the
results in these situations, as the calculated interval could be too wide to be of any use.
Example Two (Large Sample Sizes)
Box, Hunter, and Hunter (1978) present data obtained from a batch process for manufacturing a pigment
paste where the moisture content of the product is of interest. A nested experiment was conducted, where 15 batches
of pigment paste were sampled 2 times each, and each sample was tested for moisture content twice. Table 2 shows
the ANOVA for this study corrected for the mean. A goal of the study was to understand the variability in moisture
content attributable to differences in batches as well as the variability caused by sampling technique. Based on the
expected mean squares, the variance components for Batches and Samples(Batches) can be estimated by
2
σˆ Batches
=
2
σˆ Samples
=
MS Batches − MS Samples
4
MS Samples − MS Tests
2
, and
,
2
2
respectively. The program computed the 95% two-sided confidence interval for σ Batches
and σ Samples
as (-15.0,
39.7) and (15.4, 69.0), respectively. Note that the user would interpret the first interval as (0, 39.7) in order to
satisfy the constraint of non-negative variances. Figures 2a and 2b show the program output for these confidence
intervals.
Page 4 of 8
Figure 1. Listing of Inputs and Results for Example 1.
Confidence Intervals on Linear Combinations of Expected Mean Squares
Enter Q . . . 3
Enter P . . . 3
1) One-Sided Lower Confidence Interval (Upper Bound),
2) One-Sided Upper Confidence Interval (Lower Bound), or
3) Two-Sided Confidence Interval
Type of Confidence Interval to compute . . . 3
Enter alpha such that (1-2*alpha)*100% is the confidence level
0.025
Enter
Enter
Enter
Enter
Enter
Enter
Enter
Enter
Enter
c[1]
MS[1]
df[1]
c[2]
MS[2]
df[2]
c[3]
MS[3]
df[3]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.05556
3136
1
0.16667
484.75
2
0.7778
10.60
30
Confidence Intervals on Linear Combinations of Expected Mean Squares
Q = 3
P = 3
Interval Type:
i
--1
2
3
c[i]
------0.056
0.167
0.778
Point Estimate:
Two-sided CI
MS[i]
------3136.0
484.8
10.6
df[i]
------1.0
2.0
30.0
theta_HAT = 263.274
95.0% Confidence Interval
(111.773, 177533.738)
Page 5 of 8
Figure 2a. Listing of Inputs and Results for Example 2 (Batches Source of Variation).
Confidence Intervals on Linear Combinations of Expected Mean Squares
Enter Q . . . 2
Enter P . . . 1
1) One-Sided Lower Confidence Interval (Upper Bound),
2) One-Sided Upper Confidence Interval (Lower Bound), or
3) Two-Sided Confidence Interval
Type of Confidence Interval to compute . . . 3
Enter alpha such that (1-2*alpha)*100% is the confidence level
0.025
Enter
Enter
Enter
Enter
Enter
Enter
c[1]
MS[1]
df[1]
c[2]
MS[2]
df[2]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.25
86.6
14
0.25
58.0
15
Confidence Intervals on Linear Combinations of Expected Mean Squares
Q = 2
P = 1
Interval Type:
i
--1
2
c[i]
------0.250
0.250
Point Estimate:
Two-sided CI
MS[i]
------86.6
58.0
df[i]
------14.0
15.0
theta_HAT = 7.150
95.0% Confidence Interval
(-15.029, 39.678)
Page 6 of 8
Figure 2b. Listing of Inputs and Results for Example 2 (Samples within a batch Source of Variation).
Confidence Intervals on Linear Combinations of Expected Mean Squares
Enter Q . . . 2
Enter P . . . 1
1) One-Sided Lower Confidence Interval (Upper Bound),
2) One-Sided Upper Confidence Interval (Lower Bound), or
3) Two-Sided Confidence Interval
Type of Confidence Interval to compute . . . 3
Enter alpha such that (1-2*alpha)*100% is the confidence level
0.025
Enter
Enter
Enter
Enter
Enter
Enter
c[1]
MS[1]
df[1]
c[2]
MS[2]
df[2]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.50
58
15
0.5
0.9
30
Confidence Intervals on Linear Combinations of Expected Mean Squares
Q = 2
P = 1
Interval Type:
i
--1
2
c[i]
------0.500
0.500
Point Estimate:
Two-sided CI
MS[i]
------58.0
0.9
df[i]
------15.0
30.0
theta_HAT = 28.550
95.0% Confidence Interval
(15.372, 69.006)
Page 7 of 8
References
Brown, Barry W., James Lovato, and Kathy Russell. DCDFLIB: Library of C Routines for Cumulative Distribution
Functions, Inverses, and Other Parameters. Houston, TX: The University of Texas, M.D. Anderson Cancer
Center, 1994.
Burdick, Richard K. and Franklin A. Graybill. Confidence Intervals on Variance Components. New York, NY:
Marcel Dekker, Inc., 1992.
Lorenzen, Thomas J. and Virgil L. Anderson. Design of Experiments: A No-Name Approach. New York, NY:
Marcel Dekker, Inc., 1993.
Box, George E. P., William G. Hunter, and J. Stuart Hunter. Statistics for Experimenters: An Introduction to
Design, Data Analysis, and Model Building. New York, NY: John Wiley & Sons, Inc., 1978.
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