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Microsoft Excel. For detailed mathematical background on these methods, the reader is referred to any of the many fine books on the subject (e.g. Box et al., 1978; Hicks, 1993; Montgomery, 1996). DOE PC, a full-featured commercial software for design and analy­sis of experiments is available from http://www.qualityamerica.com. A statistical anal­ysis shareware package for Windows operating systems can be downloaded from http://www.dagonet.com/scalc.htm. MINITAB includes DOE capabilities.

Two-Way AN OVA with No Replicates When experiments are conducted which involve two factors, and it is not possible to obtain repeat readings for a given set of experimental conditions, a two-way analysis of variance may be used. The following example assumes that experimental treatments are assigned at random. Note that if the factors involved are each tested at only two levels, the full factorial analysis method described below could also be used.

Example of Two-Way ANOVA with No Replicates An experiment was conducted to evaluate the effect of different detergents and water temperatures on the cleanliness of ceramic substrates. The experimenter selected three different detergents based on their pH levels, and conducted a series of experiments at four different water temperatures. Cleanliness was quantified by measuring the con­tamination of a distilled water beaker after rinsing the parts cleaned using each treat­ment combination. The coded data are shown in Table 10.5.

Part one of the Excel output (Table 10.6) provides descriptive statistics on the differ­ent treatment levels. The ANOVA table is shown in part two. Note that in the previously presented raw data table the rows represent the different temperatures and the columns the different detergents. Because there are no replicates, Excel is not able to provide an estimate of the interaction of detergent and water temperature. If you suspect that an interaction may be present, then you should try to replicate the experiment to estimate this effect. For this experiment, any P-value less than 0.05 would indicate a significant effect. The ANOVA table indicates that there are significant differences between the dif­ferent detergents and the different water temperatures. To identify which differences are significant the experimenter can examine the means of the different detergents and water temperatures using t-tests. (Excel's data analysis tools add-in includes these tests.) Be aware that the Type I error is affected by conducting multiple t-tests. If the Type I error on a single t-test is a, then the overall Type I error for k such tests is 1- (1 - at For example, if a = 0.01 and three pairs of means are examined, then the combined Type I error for all three t-tests is 1 - (1 - 0.01)3 = 1 - (0.99)3 = 0.03. Statistical methods exist that guarantee an overall level of Type I error for simultaneous comparisons (Hicks, 1973, pp. 31-38).

I Detergent A Detergent B Detergent C

Cold 15 18 10

Cool 12 14 9

Warm 10 18 7

Hot 6 12 5

TABLE 10.5 Cleaning Experiment Raw Data

360 C hap te r Ten

SUMMARY OUTPUT

Count Sum Average Variance

Cold water 3 43 4.333333 16.33333

Cool water 3 35 11.666667 6.333333

Warm water 3 35 11.666667 32.33333

Hot water 3 23 7.6666667 14.33333

Detergent A 4 43 10.75 14.25

Detergent B 4 62 15.5 9

Detergent C 4 31 7.75 4.916667

ANOVA

Source of

IF crit variation SS df MS F P-value

Rows 68 3 22.666667 8.242424 0.015043179 4.757055

Columns 122.1666667 2 61.083333 22.21212 0.001684751 5.143249

Error 16.5 6 2.75

Total 206.6666667 11

TABLE 10.6 Cleaning Experiment Two-Way ANOVA Output from Microsoft Excel (Two-Factor without Replication)

Two-Way AN OVA with Replicates If you are investigating two factors which might interact with one another, and you can obtain more than one result for each combination of experimental treatments, then two­way analysis of variance with replicates may be used for the analysis. Spreadsheets such as Microsoft Excel include functions that perform this analysis.

Example of Two-Way ANOVA with Replicates An investigator is interested in improving a process for bonding photoresist to copper clad printed circuit boards. Two factors are to be evaluated: the pressure used to apply the photoresist material and the preheat temperature of the photoresist. Three different pressures and three different temperatures are to be evaluated; the number of levels need not be the same for each factor and there is no restriction on the total number of levels. Each experimental combination of variables is repeated 5 times. Note that while Excel requires equal numbers of replicates for each combination of treatments, most statistical analysis packages allow different sample sizes to be used. The experimenter recorded the number of photoresist defects per batch of printed wiring boards. The coded data are shown in Table 10.7.

These data were analyzed using Excel's two-way ANOVA with replicates function. The results are shown in Table 10.8.

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High Pressure Med Pressure Low Pressure

High temp 39 32 18

30 31 20

35 28 21

43 28 25

25 29 26

Med temp 38 10 22

31 15 28

31 25 29

30 31 26

35 36 20

Low temp 30 21 25

35 22 24

36 25 20

37 24 21

39 27 21

TABLE 10.7 Photoresist Experiment Raw Data ANOVA Results

As before, part one of the Excel output provides descriptive statistics on the differ­ent treatment levels. The ANOVA table is shown in part two. Because there are now replicates, Excel is able to provide an estimate of the interaction of pressure and tem­perature. For this experiment, the experimenter decided that any P-value less than 0.05 would indicate a significant effect. The ANOVA table P-value of less than 0.001 indi­cates that there are significant differences between the different columns (pressure), but the P-value of 0.6363 indicates that there is not a significant difference between the rows (temperature). The interaction of pressure and temperature is also not significant, as indicated by the P-value of 0.267501.

Since the P-value indicates that at least one difference is significant, we know that the largest difference of 34.26666667 - 23.06666667 = 11.2 is significant. To identify which other differences are significant the experimenter can examine the means of the different pressures using t-tests. (Excel's data analysis tools add-in includes these tests.) Be aware that the Type I error is affected by conducting multiple t-tests. If the Type I error on a single t-test is a, then the overall Type I error for k such tests is 1 - (1 - at For example, if a = 0.01 and three pairs of means are examined, then the combined Type I error for all three t-tests is 1 - (1 - 0.01)3 = 1 - (0.99)3 = 0.03.

Full and Fractional Factorial Full factorial experiments are those where at least one observation is obtained for every pos­sible combination of experimental variables. For example, if Ahas 2 levels, B has 3 levels and C has 5 levels, a full factorial experiment would have at least 2 x 3 x 5 = 30 observations.