Page 1 of 8A Program to Compute Approximate ConfidenceIntervals on Certain Linear Combinations ofANOVA-based Mean SquaresBrandon L. ParisPage 2 of 8Overview of ProgramThis program was written to allow simple calculation of approximate confidence intervals for linearcombinations of ANOVA-based mean squares estimates. The linear combinations must follow one of the followingforms:∑==QiiiMSc1θ , or (1)∑∑+==−=QPjjjPiiiMScMSc11θ , (2)where all ci (i = 1, 2, . . . Q) are positive and the MSi are the ANOVA-based mean squares estimates. The methodsused in the program are from the text Confidence Intervals on Variance Components by Burdick and Graybill andare 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 tocompute the confidence intervals. The library, written by Brown, Levato, and Russell, provides routines that willmake various computations for a number of statistical distributions and is available in both C and Fortran on theWorld Wide Web at http://odin.mdacc.tmc.edu or http://www.netlib.no.Using the ProgramOnce linked and compiled, the program is invoked through the command-line as follows:> intervalInvoking the program in this manner will print output only to the display console. To also print the results to anoutput file, use the call:> interval outfilewhere 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 linearcombination 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-sidedconfidence intervalα (alpha) Proportion in (0,1) where the confidence level for the interval is (1–α) for one-sided intervalsor (1-2α) for two-sided intervals.ciConstant to be multiplied times the ith mean square included in the linear combination.MSiThe ith mean square included in the linear combination.dfiThe degrees of freedom associated with the ith mean square included in the linear combination.Page 3 of 8At times, the software may produce negative estimates of the lower and/or upper confidence limits. Thesoftware handles this differently depending on which form θ takes. If θ is of form (1), then it is constrained to benon-negative, and hence the confidence limits will follow this same constraint. If, however, θ is of form (2), thenthere are no general sign constraints on θ and the user must assess whether the final results make sense. Forinstance, a user may calculate a linear combination of form (2) in an attempt to estimate a variance, which must benon-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 confidencelimits will probably be acceptable.ExamplesExample One (Small Sample Sizes)Lorenzen and Anderson (1993) provide data for an experiment measuring the firing time of explosiveswitches where there were three factors of interest: the metal used in the switch, amount of primary initiator, and thepacking pressure of the explosive. A completely randomized design was used for the experiment. Because the metalused for the switches was made of recycled material, metal was considered to be a random effect while the remainingfactors were considered to be fixed. The experiment showed that only the metal used and amount of primaryinitiator were effecting firing time. As a result, the final ANOVA for the experiment included only these sources ofvariation and their interaction. This final ANOVA table, including the expected mean squares, is provided in Table1.As an example, consider that estimating the total variability of firing times is of interest. Simplemanipulation of the EMS from Table 1 shows that43129761181EMSEMSEMSTotal++=σ ,which can be estimated by43127778.01667.00556.0ˆMSMSMSTotal++≈σ .In this example, Q = P = 3. Using the software, 27.263ˆ2=Totalσ and the 95% two-sided confidence interval for2Totalσ 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 df SS MS EMSMetal 1 3136.00 3136.0022118MetalEMS σσ +=Initiator 2 513.72 256.86 )(126*22IEMSInitiatorMetalΦ++= σσMetal*Initiator 2 969.50 484.752*236InitiatorMetalEMS σσ +=Error 30 318.00 10.6024σ=EMSTotal 35 4937.22Page 4 of 8Table 2. ANOVA for Nested Experiment presented in Box, Hunter, and Hunter (1978).Source df SS MS EMSBatches 14 1,211.0 86.622242BatchesSamplesTestsσσσ ++Samples(Batches) 15 869.7 58.0222SamplesTestsσσ +Tests(Samples Batches) 30 27.5 0.92TestsσCorrected Total 59 2,108.2This 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 shouldconvince the reader that limitations do exist. Thus, the user should proceed with caution when interpreting theresults 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 pigmentpaste where the moisture content of the product is of interest. A nested experiment was conducted, where 15 batchesof pigment paste were sampled 2 times each, and each sample was tested for moisture content twice. Table 2 showsthe ANOVA for this study corrected for the mean. A goal of the study was to understand the variability in moisturecontent attributable to differences in batches as well as the variability caused by sampling technique. Based on theexpected mean squares, the variance components for Batches and Samples(Batches) can be estimated by4ˆ2SamplesBatchesBatchesMSMS −=σ , and2ˆ2TestsSamplesSamplesMSMS −=σ ,respectively. The program computed the 95% two-sided confidence interval for 2Batchesσ and 2Samplesσ 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 tosatisfy the