Two-way ANOVA with no replicates - Moodlerooms

1 Two-way ANOVA with no replicatesWhen experiments are conducted which involve two factors, and it is notpossible to obtain repeat readings for a given set of experimental conditions, atwo-way analysis of variance may be used. The following example assumes thatexperimental treatments are assigned at random. Note that if the factorsinvolved are each tested at only two levels, the full factorial analysis methoddescribed below could also be OF Two-way ANOVA with NOREPLICATESAn experiment was conducted to evaluate the effect of different detergentsand water temperatures on the cleanliness of ceramic substrates. The experi-menter selected three different detergents based on their pH levels, and con-ducted a series of experiments at four different water was quantified by measuring the contamination of a distilledwater beaker after rinsing the parts cleaned using each treatment coded data are shown in Table one of the Excel output (Table ) provides descriptive statistics on thedifferent treatment levels.

2 The ANOVA table is shown in part two. Note thatin the previously presented raw data table the rows represent the different tem-peratures and the columns the different detergents. Because there are no repli-cates, Excel is not able to provide an estimate of the interaction of detergentand water temperature. If you suspect that an interaction may be present, thenyou should try to replicate the experiment to estimate this effect. For this experi-ment, any P-value less than would indicate a significant effect. TheANOVA table indicates that there are significant differences between the dif-ferent detergents and the different water temperatures. To identifywhichdif-ferences are significant the experimenter can examine the means of thedifferent detergents and water temperatures usingt-tests. (Excel s data analysisExamples of applying common DOE methods using software617 Table experiment raw A DETERGENT B DETERGENT CCold151810 Cool12149 Warm10187 Hot6125tools add-in includes these tests.)

3 Be aware that the Type I error is affected byconducting multiplet-tests. If the Type I error on a singlet-test is , then theoverall Type I error forksuch tests is 1 1 k:For example, if 0:01and three pairs of means are examined,then the combined Type I error for allthreet-tests is 1 1 0:01 3 1 0:99 3 0:03. Statistical methods existthat guarantee an overall level of Type I error for simultaneous comparisons(Hicks, 1973, pp. 31^38). Two-way ANOVA with replicatesIf you are investigating two factors which might interact with one another,and you can obtain more than one resultfor each combination of experimentaltreatments, then Two-way analysis of variance with replicates may be used forthe analysis. Spreadsheets such as Microsoft Excel include functions that per-form this DESIGN OFEXPERIMENTST able experiment Two-way ANOVA output from Microsoft Excel.(Two-factor without replication.)SUMMARY OUTPUTC ountSum Average VarianceCold water335 water335 A443 B462 of variationSSdfMSFP-valueF critRows683 11 EXAMPLE OF Two-way ANOVA with REPLICATESAn investigator is interested in improving a process for bonding photoresistto copper clad printed circuit boards.

4 Two factors are to be evaluated: the pres-sure used to apply the photoresist material and the pre-heat temperature of thephotoresist. Three different pressures and three different temperatures are tobe evaluated; the number of levels need not be the same for each factor andthere is no restriction on the total number of levels. Each experimental com-bination of variables is repeated five times. Note that while Excel requiresequal numbers of replicates for each combination of treatments, most statisticalanalysis packages allow different sample sizes to be used. The experimenterrecorded the number of photoresist defects per batch of printed wiring coded data are shown in Table data were analyzed using Excel s Two-way ANOVA with replicatesfunction. The results are shown in Table before, part one of the Excel output provides descriptive statistics on thedifferent treatment levels. The ANOVA table is shown in part two.

5 Becausethere are now replicates , Excel is able to provide an estimate of the interactionof pressure and temperature. For this experiment, the experimenter decidedthat any P-value less than would indicate a significant effect. TheExamples of applying common DOE methods using software619 Table experiment raw data. ANOVA PRESSUREMED PRESSURELOW PRESSUREHigh temp393218303120352821432825252926 Med temp381022311528312529303126353620 Low temp302125352224362520372421392721620 DESIGN OFEXPERIMENTST able experiment Two-way ANOVA output from Microsoft Excel.(Two-factor with replication.)SUMMARY OUTPUTHigh pressureMed pressureLow pressureTotalHigh of variationSSdfMSFP-valueF table P-value of less than indicates that there are significant dif-ferences between the different columns (pressure), but the P-value of that there is not a significant difference between the rows (tempera-ture).

6 The interaction of pressure and temperature is also not significant, as indi-cated by the P-value of the P-value indicates that at least one difference is significant, we knowthat the largest difference of 23:06666667 11:2 is identifywhich otherdifferences are significantthe experimenter can exam-ine the means of the different pressures usingt-tests. (Excel s data analysistools add-in includes these tests.) Be aware that the Type I error is affected byconducting multiplet-tests. If the Type I error on a singlet-test is , then theoverall Type I error forksuchtestsis1 1 k. For example, if 0:01and three pairs of means are examined, then the combined Type I error for allthreet-tests is 1 1 0:01 3 1 0:99 3 0 and fractional factorialFull factorial experiments are those where at least one observation isobtained for every possible combinationof experimental variables. For exam-ple, if A has 2 levels, B has 3 levels and C has 5 levels, a full factorial experimentwouldhaveatleast2 3 5 30 factorialorfractional replicateare experiments where there aresome combinations of experimental variables where observations were notobtained.

7 Such experiments may not allow the estimation of every , when carefully planned, the experimenter can often obtain all of theinformation needed at a significant FACTORIAL EXPERIMENTSA simple method exists for analyzing the common 2nexperiment. Themethod, known as the Yates method, can be easily performed with a pocket cal-culator or programmed into a spreadsheet. It can be used with any properlydesigned 2nexperiment, regardless of the number of factors being use the Yates algorithm, the data are first arranged in standard order (ofcourse, the actual running order is random). The concept of standard order iseasier to understand if demonstrated. Assume that we have conducted an experi-ment with three factors, A, B, and C. Each of the three factors is evaluated attwo levels, which we will call low and high. A factor held at a low level will beidentified with a ^ sign, one held at a high level will be identified with a + sign.

8 The eight possible combinations of the three factors are identified usingthe scheme shown in the table of applying common DOE methods using software621