6. ANÁLISIS Y PROPUESTA DE MEJORAMIENTO DE LOS PROCESOS DE SERVICIO POSTVENTA QUE IMPACTAN EN LA SATISFACCIÓN DEL
6.2. MANEJO DE NOVEDADES Y ATENCIONES DE LOS CLIENTES (SOPORTE TÉCNICO)
Fisher’s pioneering work on DOE involved using an analysis technique called ANOVA. As men-tioned earlier, ANOVA techniques study the variation between the total responses compared to the variation of responses within each factor. The ANOVA studies are augmented with results attained from applying multiple regression techniques to the yield (responses) data. Using the regression, we can form prediction equations that model the study responses obtained.
All of the ANOVA experimental design methods are essentially tests of hypothesis. A hypothesis test is used to determine the equivalence of the multiple means (average of each level) for each fac-tor. The general null and alternate hypotheses statements for each factor are of the form:
H0:mI= mII= mIII= … = mk and H1: At least one mean is different
The ANOVA study results in a test statistic per factor analyzed with a hypothesis test to deter-mine the significance of each factor.
4.7.1 Single-Factor Design or Completely Randomized Design
In the classical design all factors are fixed except the one under investigation. The one factor could be fertilizer from our earlier analogy with three brands as the factor levels or treatments. Thus a total of nine tests could be run: three tests for each of the three brands. The rainfall, time of harvest, tem-perature, and all other factors are held equivalent (controlled) for each treatment. The major draw-back to this method is that the conclusions about the brands would apply only to the specific conditions run in the experiment. The table shown below is a way to visualize this design where the numbers are not the response values but a simple numbering of the total number of tests that are to be performed.
The single factor design is also known as the completely randomized design. This naming is because the nine tests that are to be completed are performed in a completely random fashion. This will randomize any random variation in fixed factors (e.g., water, sunshine, and temperature at the time of each test).
In ANOVA terminology, this would be called a one-way analysis of variance since all of the stud-ied variation in responses is contained only in the columns (treatments).
I II III
3 8 1
6 2 9
4 7 5
I II III
1 2 3
4 5 6
7 8 9
4.7.2 Calculations for Single-Factor ANOVA Tables
Single Factor
Treatment 1 Treatment 2 … Treatment k
y1,1 y1,2 … y1,k
Some sources give and use a different set of equations for balanced (same number of observations in each treatment) and unbalanced (different number of observations in each treatment) designs. We will use the one set of equations given below for both designs. As the yields (responses) are captured from the tests, they are recorded in a matrix. The equations reference elements of the yield matrix are shown below. When there are a variable number of responses (y’s) in any treatment (column), this is the unbalanced design. The balanced design is simply when every treatment has the same number of responses (i.e., the same number of rows per treatment such that n1= n2= … = nk).
Constants from the inputs of the design:
k= number of treatments
nj= number of samples in the jth treatment N= total sample size = n1+ n2+ … + nk yi,j= yield in row i and column j Calculations made from yield matrix:
Sum of square calculations:
4.7.3 Single-Factor ANOVA Table
Values of k, N, SST, SSE and TSS from the above equations are used to complete the ANOVA table shown below. In this table, the sums of square (SS) terms are converted to variances (MS terms)
SST sum of squares for treatments
TSS total sum of square
SSE sum of squares for error TSS SST SSB
= = −
sample total of th treatment
overall sample total or
sum of squares of all samples
,
and the F-test is the ratio of the MS terms. Some minor computations are necessary to complete the
Using a hypothesis test, the F-test value of the treatment determines the significance of the factor (i.e., whether the treatment’s means are equivalent to each other).
4.7.4 Two-Factor Design or Randomized Block Design
The next design recognizes a second factor, called blocks (e.g., crops A, B, and C, where A is acorn squash, B is beans, and C is corn). Both the original factor with its treatments and the added factor, the blocks, are studied. Again, data for each response must be collected in a completely randomized fashion.
In the randomized block design each block (row) is a crop (acorn squash, beans, corn) and the fertilizer brands are the treatments (columns). Each brand, crop combination is tested in random order. This guards against any possible bias due to the order in which the brands and crops are used.
Fertilizer
The randomized block design has advantages in the subsequent data analysis and conclusions.
First, from the same nine observations, a hypothesis test can be run to compare brands and a sepa-rate hypothesis test run to compare crops. Second, the conclusions concerning brands apply for the three crops and vice versa, thus providing conclusions over a wider range of conditions.
The ANOVA terminology calls the randomized block design two-way analysis of variance since the studied variations in responses are contained both in the columns (treatments) and in the rows (blocks).
4.7.5 Calculations for Two-Factor ANOVA Tables
The generalized matrix for two-factor solutions is shown below. As with the one-factor equations, the two way matrix and elements are referenced by the two-way equations.
Treatment 1 Treatment 2 … Treatment k Totals
1 y1,1 y1,2 … y1,k
The two-factor equations are shown below. In addition to calculating the total sum of squares (TSS) for all yields (responses) and the sum of squares for all treatments (SST), the sum of squares for all blocks (SSB) must be calculated. This time the sum of squares of the error (SSE) is the dif-ference between TSS and both SST and SSB.
Constants from the design:
In the two factor ANOVA table, there is an additional row calculation added for the sum of squares for the blocks (SSB). As with single factors, the SS terms are converted to variances (MST, MSB, and MSE terms). Values of k, b, N, SST, SSB, SSE and TSS from the above equations are used to complete the ANOVA table shown below. This time two hypothesis tests are required: one for treat-ments and one for blocks.
SST sum of squares for treatments
SSB sum of squares for blocks
TSS total sum of square
SSE sum of squares for error TSS SST SSB
= = −
4.7.7 Two Factor with Interaction Design
The last issue to consider is interaction between the two factors. A new term, interaction, is defined as the effect of mixing, in this case, of specific fertilizer brands with specific crops. There may be a possibility that under a given set of conditions something “strange” happens when there are interac-tions among the factors. The two factor with interaction design investigates not only the two main factors but the possible interaction between them. In this design each test is repeated (replicated) for every combination of the main factors. In our agricultural example with three replications using the three brands of fertilizer and three crops, we have 3 × 3 × 3 or 27 possibilities (responses). Separate tests of hypothesis can be run to evaluate the main factors and the possible interaction.
The two-factor design with interaction is also known as two-way analysis of variance with replications.
4.7.8 Calculations for Two Factor with Interaction ANOVA Tables
To show the yield matrix for this situation, the term replication must be defined. Replication is where yields for all treatments and blocks are observed for “r” times. The generalized matrix is shown on the below:
I I II III
Hi --- --- ---
---Med --- --- ---
---Lo --- --- ---
---Factor A
Factor B
1
T1,1 T1,2 T1,3 T1,a B1 1
2
3
b Totals
2 3 a Totals
T2,1 T2,2 T2,3 T2,a B2
T3,1 T3,2 T3,3 T3,a B3
Tb,1
A1 A2 A3 Aa ∑y
Tb,2 Tb,3 Tb,a Bb
The associated calculations for the above matrix are shown below:
Input from the Design:
Calculations from the yield matrix:
Sum of Squares Calculations:
4.7.9 Two Factor with Interaction ANOVA Table
The ANOVA table for two factors with interaction is shown below. This time hypothesis tests are required for A, B, and AB to determine significance.
Source df SS MS F-test
SST Sum of Squares for all Treatments
SS(A) Sum of Squares for Factor A
SS(B) Sum of Squares for Factor B
SS(A B) Sum of Squares for Interaction A B SST SS(A) SS(B) TSS Total Sum of Square
SSE Sum of Squares for Error TSS SST SS(A)
= = − ∑
( )
Replication Total of , th Treatment
Sample Total of th Block
4.7.10 Factorial Base Experimental Designs
Factorial solutions use a matrix approach that can study multiple factors, interactions, and levels. A full factorial design gets responses at every level for each factor. The following table shows a full factorial design for studying four factors—tire compounds, road temperatures, tire pressures, and vehicle types.
The design tests all four factors at each of the two levels, in every possible combination.
Factor Levels
Compounding formula X Y
Road temperature 75 80
Tire pressure 28 34
Vehicle type I II
This is called a 24design, since there are two levels and four factors. The “2” is for the two levels and the “4” is for the factors. The general form is 2k, where the k is the number of factors. To test this model using a full factorial design, a minimum of sixteen (24) different experimental runs are needed.
4.7.11 Full Factorial of Two Factors at Two Levels (22) Design
This table shows the complete layout of a two factor with two levels (22) experiment. Factor A could be road temperature and factor B could be tire pressure. The yields of interest could be tire wear.
Run (Trial) Factors
1 75 28
2 80 28
3 75 34
4 80 34
4.7.12 Full Factorial 22Design with Two Replications
When only one trial is performed for each combination, statistical analysis is difficult and may be misleading. When the experiment is replicated it is possible to perform a test of hypothesis on the effects to determine which, if any, are statistically significant. In this example we will use the twice-replicated runs (r= 2) for the experiment.
When there is more than one replication, it is necessary to compute the mean of each run. The variance will also be useful.
Run Factor A Factor B Y1 Y2 Average s2
1 75 28 55 56 55.5 0.5
2 80 28 70 69 69.5 0.5
3 75 34 65 71 68.0 18.0
4 80 34 45 47 46.0 2.0
The results indicate that run 2 gets the best average yield and has a low variance. Run 3 has a slightly lower average but a larger variance. Common sense would indicate that the experiments should use the settings of run 2 for the best results.
Taking a graphical approach, look first at factor A. When low settings are used the average yield is 61.75—the two averaged yields where A is set low are 55.5 and 68.0, which is an average of 61.75.
Similarly for the high setting A the average yield is 57.75. That means that as factor A increases––
goes from the low setting to the high setting––then the magnitude changes (decreases) by a mag-nitude of −4.0. This is called the effect of factor A. See factor A chart below. A similar graph is plot-ted for the factor B effect, which is −5.5.
The figure below shows the interaction of the factors. If the high setting for factor A is chosen, and factor B is set low, the least fire wear is achived. Note that varying only one factor at a time would have missed the large interaction between A and B.
4.7.13 Full Factorial 22Design—Linear Equation Model
Graphical analysis does not provide a complete picture. We now need to investigate a statistical method of analysis. The graphical analysis, especially when coupled with a statistical analysis, allows the experimenter to get reasonable ideas as to the behavior of the process.
Returning to the two-factor (22) example we will assume a linear modeling equation of:
ˆy= b0+ b1A+ b2B.
The b1A and b2B terms are called the main effects. The coefficients b1and b2are the slopes of factors A and B respectively. The term b3AB is the interaction effect and coefficient b3is the slope of the interaction AB.
As an aid in the statistical analysis we will use codes instead of the words low and high. A stan-dard coding terminology substitutes a “−1” for each low setting. A “+1” is used for a high setting.
For the factor “tire pressure” in the table, the 28 would be the low setting and the 34 would be the high setting. In the “coded” design they would be specified −1 and +1, respectively. Note that in each coded column there is the same number of pluses as there are minuses for each run. This means that all factors at all levels are sampled equally, and that the design is balanced. This order is sometimes called the standard order.
4.7.14 Full Factorial 22Design Calculations
Returning to the previous example, the statistical calculations are initiated by adding the “codes,”
columns, and rows to the matrix array. The first step is to rewrite using the codes for high and low values.
Run A B Y1 Y2 Mean s2
1 −1 −1 55 56 55.5 0.5
2 +1 −1 70 69 69.5 0.5
3 −1 +1 65 71 68.0 18.0
4 +1 +1 45 47 46.0 2.0
A column for interaction or the AB column is added next. The coded value for AB is determined by multiplying the A times B codes for each run.
Run A B AB Y1 Y2 Mean s2
1 −1 −1 +1 55 56 55.5 0.5
2 +1 −1 −1 70 69 69.5 0.5
3 −1 +1 −1 65 71 68.0 18.0
4 +1 +1 +1 45 47 46.0 2.0
Then the mean yield times the codes of A, B, and AB, are multiplied, respectively.
Run A B AB Y1 Y2 Mean s2 A B AB
1 −1 −1 +1 55 56 55.5 0.5 −55.5 −55.5 +55.5
2 +1 −1 −1 70 69 69.5 0.5 +69.5 −69.5 −69.5
3 −1 +1 −1 65 71 68.0 18.0 −68.0 +68.0 −68.0
4 +1 +1 +1 45 47 46.0 2.0 +46.0 +46.0 +46.0
Next the contrasts are calculated by summing each run value in columns.
Run A B AB Y1 Y2 Mean s2 A B AB
1 −1 −1 +1 55 56 55.5 0.5 −55.5 −55.5 +55.5
2 +1 −1 −1 70 69 69.5 0.5 +69.5 −69.5 −69.5
3 −1 +1 −1 65 71 68.0 18.0 −68.0 +68.0 −68.0
4 +1 +1 +1 45 47 46.0 2.0 +46.0 +46.0 +46.0
Contrasts 239 21.0 −8.0 −11.0 −36.0
The effects are the contrasts divided by 2k− 1or 22− 1= 2 in this case. Note that the means and variance columns do not have effects.
Run A B AB Y1 Y2 Mean s2 A B AB
1 −1 −1 +1 55 56 55.5 0.5 −55.5 −55.5 +55.5
2 +1 −1 −1 70 69 69.5 0.5 +69.5 −69.5 −69.5
3 −1 +1 −1 65 71 68.0 18.0 −68.0 +68.0 −68.0
4 +1 +1 +1 45 47 46.0 2.0 +46.0 +46.0 +46.0
Contrasts 239 21.0 −8.0 −11.0 −36.0
Effects N/A N/A −4.0 −5.5 −18.0
The calculations in the row labeled “Effects” are nothing more than what we saw in the effect plots earlier. Again, the largest effect is the interaction effect, as shown by the −18 in the AB column, which is much more significant than the effect of A or B.
The final addition to the table is the coefficients. The effects sum for A, B, and AB divided by two. For the means, it is a divided by 2k= 4. Nothing applies to the variance column.
Run A B AB Y1 Y2 Mean s2 A B AB
1 −1 −1 +1 55 56 55.5 0.5 −55.5 −55.5 +55.5
2 +1 −1 −1 70 69 69.5 0.5 +69.5 −69.5 −69.5
3 −1 +1 −1 65 71 68.0 18.0 −68.0 +68.0 −68.0
4 +1 +1 +1 45 47 46.0 2.0 +46.0 +46.0 +46.0
Contrasts 239 21.0 −8.0 −11.0 −36.0
Effects N/A N/A −4.0 −5.5 −18.0
Coefficients 59.75 N/A −2.0 −2.75 −9.0
The coefficients are used to predict the yield for any setting of the factors. However, only signif-icant factors are used in this equation. The purpose of this example is to conduct a “t-test” on each of the coefficients. First, the standard errors Seand Sbare found as follows:
Pooled variance = SSE = sp2= (sum of the run variances) = 21 Standard error = se= = 2.291
Standard error of the coefficients = sb=
(Where k= number of factors = 2, r = number replications = 2) Then the hypothesis test for each coefficient is stated as:
H0A: coefficient of A = 0 H1A: It is significant H0B: coefficient of B = 0 H1B: It is significant H0AB: coefficient of AB = 0 H1AB: It is significant In each case the test statistic is calculated using the relationship:
ttest= [coefficient/sb]
The three test statistics are calculated.
tA= [−2.0/0.81] = −2.47 tB= [−2.75/0.81] = −3.40 tAB= [−9.0/0.81] = −11.11
The degrees of freedom are determined from (r− 1) × 2k= (2 − 1) × 22= 1 × 4 = 4
The “t” table two-sided critical value, when a= .05 and df = 4, is 2.776. Comparing the calculated t values with the table values, the coefficients for factor B and the interaction AB are significant.
The resulting linear predicting relationship for yields is:
ˆy= 59.75 −2.75B −9.0AB
4.7.15 Observations
• The full factorial experiment can be extended for 2 levels of three factors (23) and larger experiments.
The number of interactions grows dramatically as the size of the experiment increases, though.
• Experiments are not really done in the order listed. They are performed in a random fashion that causes resetting of the variables and thus reduces bias in the results. Remember the way they are listed is called the standard order.
4.7.16 Full Factorial Three Factor with Two Levels (23) Designs
Instead of explaining a build up of steps starting with graphical methods and then going onto statis-tical analysis of a 23design, this section combines the steps. The layout of 23design for a milling machine’s power consumption (wattage) for three factors [speed (A), pressure (B), and angle (C)] is as follows:
Factor Levels
A 200 600
B Low High
C 20 28
The null hypotheses are that there is no significant difference in any factor, regardless of the set-ting, and that there is no significant interaction. Appropriate alternative hypotheses are that there is a significant difference in a factor based on the setting and that there is significant interaction.
4.7.17 Graphical and Statistical Analysis of a Full Factorial 23Design
Testing the above model using a full factorial design will require eight separate runs or trials.
Performing two replications (r= 2) of the above design and adding the coding resulted in the fol-lowing matrix showing the two sets of yields (watts used) from the runs.
Run Factors A B C Y1 Y2
1 200 Low 20 −1 −1 −1 221 311
2 600 Low 20 1 −1 −1 325 435
3 200 High 20 −1 1 −1 354 348
4 600 High 20 1 1 −1 552 472
5 200 Low 28 −1 −1 1 440 453
6 600 Low 28 1 −1 1 406 377
7 200 High 28 −1 1 1 605 500
8 600 High 28 1 1 1 392 419
Beginning below the additional columns and rows will be added to the table so that the results can be analyzed.
Run Average S2 A B C AB AC BC ABC
1 266 4050 −266 −266 −266 266 266 266 −266
2 380 6050 380 −380 −380 −380 −380 380 380
3 351 18 −351 351 −351 −351 351 −351 351
4 512 3200 512 512 −512 512 −512 −512 −512
5 446.5 84.5 −446.5 −446.5 446.5 446.5 −446.5 −446.5 446.5 6 391.5 420.5 391.5 −391.5 391.5 −391.5 391.5 −391.5 −391.5 7 552.5 5512.5 −552.5 552.5 552.5 −552.5 −552.5 552.5 −552.5
8 405.5 364.5 405.5 405.5 406 405.5 405.5 405.5 405.5
Columns of the table were removed for space considerations.
The contrasts, effects and coefficients are summed. The contrasts, except the one for yield, are divided by 2k− 1to calculate the effects. Coefficients are calculated by dividing the effects by two, except the coefficient for yield is its contrast divided by 23or 8.
Run Average S2 A B C AB AC BC ABC
1 266 4050 −266 −266 −266 266 266 266 −266
2 380 6050 380 −380 −380 −380 −380 380 380
3 351 18 −351 351 −351 −351 351 −351 351
4 512 3200 512 512 −512 512 −512 −512 −512
5 446.5 84.5 −446.5 −446.5 446.5 446.5 −446.5 −446.5 446.5
6 391.5 420.5 391.5 −391.5 391.5 −391.5 391.5 −391.5 −391.5
7 552.5 5512.5 −552.5 552.5 552.5 −552.5 −552.5 552.5 −552.5
8 405.5 364.5 405.5 405.5 405.5 405.5 405.5 405.5 405.5
3305 19700 Contrasts 73 337 287 −45 −477 −97 −139
N/A sP= SSE Effects 18.25 84.25 71.75 −11.25 −119.25 −24.25 −34.75 Bo 413.125 N/A Coefficients 9.125 42.125 35.875 −5.625 −59.625 −12.125 −17.375 TSS 134587.8 SSi 1332.25 28392.25 20592.25 506.25 56882.25 2352.25 4830.25 sigma= se= 49.624 ttest 0.73554 3.395563 2.89177 −0.453 −4.80618 −0.9774 −1.4005
Sbeta 12.406 B1 B2 B3 B4 B5 B6 B7
Run Factors A B C AB AC BC ABC Y1 Y2
1 200 Low 20 −1 −1 −1 1 1 1 −1 221 311
2 600 Low 20 1 −1 −1 −1 −1 1 1 325 435
3 200 High 20 −1 1 −1 −1 1 −1 1 354 348
4 600 High 20 1 1 −1 1 −1 −1 −1 552 472
5 200 Low 28 −1 −1 1 1 −1 −1 1 440 453
6 600 Low 28 1 −1 1 −1 1 −1 −1 406 377
7 200 High 28 −1 1 1 −1 −1 1 −1 605 500
8 600 High 28 1 1 1 1 1 1 1 392 419
The same equations listed earlier for 22apply and they are:
Pooled variance = SSE = sp2= (sum of the run variances) = 19700 Standard error = se= = 49.6236
Standard error of the coefficients = sb=
(Where k= number of factors = 2, r = number replications = 2)
Sum square of each effect = SSi= for example SSA= 1332.25 The test values for the t’s are calculated from: ttest= Coefficienti/sbfor example the B1test value is calculated as follows = 9.125/12.4059 = 0.73554.
The critical value for a t-test at the .05 level with 8 df is 2.306, comparing each of the test values to the critical results in significant coefficients for main effect B, main effect C, and interaction effect AC.
The ANOVA table can be constructed from the sum squares calculated above.
r ×73 =
Source df SS MS F sig F Regression Significant?
Regression 7 114887.75 16412.536 6.664989 0.0079 yes
Error 8 19700 2462.5
Total 15 134587.75
Remember that sum square of the regression is calculated by adding up the seven main and inter-action effects sum squares.
SSR= SSA + SSB + SSC + SSAB + SSAC + SSBC + SSABC
Total sum square, or TSS, is the numerator term for the variance of all of the yields and is equal to sum square of regression plus sum square of error
(TSS= SSR + SSE)
Another statistics to look at is the coefficient of correlation R. Its value is simply calculated from R= SSR/TSS = 0.9239 and so R2= 0.8536
4.7.18 Additional Thoughts on Factorial Designs
Higher order full factorial designs, such as 24could be run having four factors at two levels each. It would require 16 trials to collect all the data. For 2kdesigns there are 2k− 1columns of main and inter-action effects to code and calculate. This means for four factors a total of 15 columns are needed to account for the main, 2nd, 3rd, and 4th order effects.
Higher order interactions, such as ABC, ABCD, or ABCDE, are rarely important. As a general rule, avoid assuming that all 2nd order interactions have no meaning in order to perform a statistical analysis. While higher order interactions are rarely important or significant, second order interac-tions are frequently significant.
Analysis techniques are available that consider only part of the full factorial runs and they are called fractional factorials. Also available are 2k− pfactorial designs which use fractional designs combined with placing additional factors in the third order and above interactions. The 2k− pis a way to add factors without requiring the addition runs necessary in full factorial designs. Whole books are devoted to such designs.
Situations calling for three levels call for a 3kdesign. The calculations change from the above 2k technique and require an additional centering run. A three factor with 3 levels would require 3 or 27 runs plus the centering run.
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