Statistics for Managers 2
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Factorial ANOVA: More Than One Independent Variable
Learning Objectives
After reading this chapter, you should be able to:
• Contrast one-way ANOVA with factorial ANOVA.
• Explain a statistical interaction.
• Demonstrate the impact on error of adding independent variables to the analysis.
• Calculate the effect size for the different elements of the factorial ANOVA.
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CHAPTER 6Section 6.1 Extending the Independent Samples t-Test and One-Way ANOVA
Chapter Outline
6.1 Extending the Independent Samples t-Test and the One-Way ANOVA
6.2 The Factorial ANOVA Multiple Independent Variables and Multiple Questions Calculating the Factorial ANOVA Partitioning the SSbet Graphing the Interaction
6.3 Using Excel to Complete a Factorial ANOVA Describing the Factorial ANOVA The Effect Sizes
6.4 Another ANOVA Problem
6.5 A 2 3 2 ANOVA Problem
Chapter Summary
Introduction
In the day-to-day world of business, there are precious few problems for which one fac-tor provides an adequate explanation. Although one-way ANOVA represents an impor- tant advancement in statistical analysis from the independent samples t-test, both those tests impose a substantial limitation because neither allows for more than one indepen- dent variable at a time to be elements of the analysis. The one-way procedure allows any number of categories, but they must be the categories of just one IV. As vital as they are to understanding the evolution of modern statistical analysis, the independent samples t-test and the one-way ANOVA make it easy to oversimplify complex problems, focusing on the most obvious variable and ignoring other important factors that might also explain the dependent variable.
6.1 Extending the Independent Samples t-Test and the One- Way ANOVA
Thinking back to the oil change example in Chapter 5, recall that the price charged was the IV, and the number of oil changes sold during the pricing period was the DV. What if besides the price of the oil change, the service manager wished to assess the sales impact of including a complimentary car wash? Both an independent samples t-test and an ANOVA can accommodate whether a car wash was included or not (the IV), and then analyze for significant differences in the number of oil changes sold (the DV). However, neither procedure can accommodate both the price of the oil change (IV #1) and whether a complimentary car wash was included (IV #2) in the same analysis.
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CHAPTER 6Section 6.2 The Factorial ANOVA
Explaining the number of oil changes a service center sells in a particular period as a function of price presumes that only price is relevant, or at least that other variables are comparatively unimportant. Examples of other variables that might impact the number of oil changes sold may include:
• The speed with which the service is provided. • The cleanliness of the waiting area. • The courteousness of the employees. • Whether free snacks are provided to customers. • The area of the city in which the service center is located. • Whether the weather was more adverse during one particular pricing period.
While price may be the most important factor in the number of oil changes a service center sells, it is unlikely to be the only factor. If some of the variables in the questions above also affect the number of oil changes sold, but only the price is considered, what happens to the variability in the number of oil changes stemming from those other variables?
In any ANOVA, the total sum of squares (SStot) measures all variability from all sources. The approach in Chapter 5 was to calculate the total variability and then partition that number into two components, the sum of squares between (SSbet), which measures the impact of the independent variable, and the sum of squares within (SSwith), which mea- sures data variability from all other sources.
Because the one-way ANOVA can accommodate just a single independent variable, any differences in the dependent variable that do not stem from that variable show up in the calculations in the SSwith. The SSwith measures the error variance in an analysis. Recall that error variance is just uncontrolled variability; it is the total effect of all the other factors apart from the IV that explain why the dependent variable has the values that it has. In an ANOVA problem on the number of oil changes performed by a service center when prices change, changes in the number of oil changes due to everything except the price constitute error variance.
6.2 The Factorial ANOVA
In the language of statistics, a factor is an independent variable in an analysis. Factorial ANOVA describes an analysis of variance with more than one IV. By extending much of the logic we used in Chapter 5 with one-way ANOVA, we can extend the analysis to mul- tiple variables. Although there are practical limitations, related primarily to sample size, there are no theoretical limits to the number of variables that can be included in a factorial ANOVA. Moreover, the higher the relevance of the variables introduced in an analysis, the more complete the explanation of the dependent variable will be.
Let’s illustrate with an example. Three importers of sports apparel, Swoosh, Matrix, and Impact, report sales by their top eight sales associates for a particular period. The data are as follows in $10,000 dollars:
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CHAPTER 6Section 6.2 The Factorial ANOVA
Swoosh: 4, 5, 4.5, 2, 7, 5.5, 4.5, 6 Matrix: 3, 3.5, 1, 4, 4.5, 4, 2, 5.5 Impact: 4.5, 5, 5.5, 3, 7, 4.5, 3.75, 4.5
The initial question is whether there are significant differences in total sales for the top sales associates among the three companies over this period. This question can be answered with a one-way ANOVA, with the company as the IV and the value of sales as the DV. The results of figuring the problem with Excel are shown in Table 6.1.
Table 6.1: A one-way ANOVA of sales data by company
ANOVA: Single Factor
SUMMARY
Groups Count Sum Average Variance
Swoosh 8 38.5 4.8125 2.209821
Matrix 8 27.5 3.4375 2.03125
Impact 8 37.75 4.71875 1.418527
ANOVA
Source of Variation
SS df MS F P-value Fcrit
Between Groups
9.442708 2 4.721354 2.502662 0.105962 3.4668
Within Groups
39.61719 21 1.886533
Total 49.0599 23
With F 5 2.503 and a probability of that value of F occurring by chance p 5 .106, the result is not statistically significant. It appears that in terms of sales over the period, the top sales associates in the three companies are performing similarly.
Multiple Independent Variables and Multiple Questions
One-way ANOVA works well as long as the only question is whether there are significant differences in sales between companies. That query involves just a single independent variable and just a single question to answer. However, suppose that in each of the three companies, half the top performers have sales experience in the sports apparel industry and the other half do not. Recall that by definition, any source of data variability not explained by the analysis is error variance. If some of the variability in sales is related to the industry experience of the sales associates, and that source of variability is not
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introduced into the analysis, it emerges as error variance. That means that in the one-way analysis, any difference in sales related to industry experience shows up in the error term, SSwith.
Factorial ANOVA can account for both IVs simultaneously. To analyze this data using factorial ANOVA, it will have to be treated differently. The sales of each of the companies must be further separated into whether they were made by associates with or without past industry experience. By arranging the sales figures to fit the new problem into a table, the analysis is easier to visualize. The rows in Table 6.2 indicate the particular company. The columns indicate the sales associate’s experience category. Because the means will be needed for the longhand calculations, they are also indicated.
Table 6.2: The data for an ANOVA with two independent variables
Company No Industry Experience
Industry Experience Row Means
Swoosh 4, 4.5, 2, 7 Ma 5 4.375 a
5, 5.5, 4.5, 6 Mb 5 5.25 b
Mab 5 4.813
Matrix 4, 4.5, 4, 5.5 Mc 5 4.5 c
3, 3.5, 1, 2 Md 5 2.375 d
Mcd 5 3.438
Impact 3, 3.75, 4.5, 4.5 Me 5 3.938 e
4.5, 5, 5.5, 7 Mf 5 5.5 f
Mef 5 4.719
Column Means Mace 5 4.271 Mbdf 5 4.375 MG 5 4.323
In addition to questions about differences from company to company, and about indus- try experience, there is an additional question. Any time there is more than one indepen- dent variable, there is a possibility that the variables will have a combined effect. When it occurs, it is called a statistical interaction of the independent variables. It is the possi- bility of an interaction that makes drug manufacturers warn against using sedatives when the user is also consuming alcohol. The sedative has one effect and the alcohol has another effect, but the two in combination can have third effect that is more profound than either of their effects alone.
The way the data are arranged in Table 6.2 makes it clear that there are two IVs at work. The data in each box reflects the combination of one of the levels of each IV. The top left box contains the sales at Swoosh by sales associates with no past industry experience. The bottom right box is sales at Impact by sales asso- ciates with past industry experience.
The one-way ANOVA indicated that differences in sales by the three companies’ top sales associates were not statistically significant. This analysis will revisit that question and in addition indicate whether there are differences in sales that can be accounted for by the industry experience of the sales associates, and whether the combined effect of experience and company is the same in all cases.
Key Terms: When independent variables have a different effect in combination than they have independently, there is a statis- tical interaction.
Section 6.2 The Factorial ANOVA
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With the separation within company according to the two categories of sales associates’ experience, there are now six groups. For ease of navigation, the top left box of Table 6.2 is labeled “cell a,” the top right “cell b,” and so on down to the bottom right box, which is “cell f.” Note that both of the IVs are nominal (categorical) variables. This is always the case for ANOVA problems.
Calculating the Factorial ANOVA
The initial steps for calculating the ANOVA longhand are the same for a factorial problem as they were for a one-way ANOVA. The three components of SStot, SSbet, and SSwith are still the basic components of the analysis. What is unique to factorial problems is that the SSbet is further divided, or partitioned, into the variability explained by each IV and the variability stemming from the interaction of IVs.
The sum of squares total is calculated the same way it was calculated for the one-way ANOVA:
Formula 5.1 SStot 5 Σ(x 2 MG) 2
Recall that SStot is the sum of the squares of each individual sales value subtracted from the mean of all the data (MG), beginning with the first value in cell a, and continuing to the last value in cell f. With MG 5 4.323 the result is:
(4.0 2 4.323)2 1 . . . 1 (7.0 2 4.323)2
The subtractions and squaring for the 24 sales values aren’t repeated here but verify that the result is:
SStot 5 49.060 (or thereabouts, depending on round-off differences).
Although sales are divided into six groups in the factorial problem rather than three, the number and value of the sales total have not changed from the one-way problem, Conse- quently, SStot for the factorial problem has the same value as SStot in the one-way problem.
The SSbet has changed, however. With the data in six groups rather than three, the number and the values of the mean scores change. For the factorial problem, Ma indicates the mean of sales for cell a, which would be Swoosh sales by those without past industry experience. The formula takes the form below, which is the same as Formula 5.3, made to accommodate 6 groups.
SSbet 5 (Ma 2 MG) 2na 1 (Mb 2 MG)
2nb 1 . . . 1 (Mf 2 MG) 2nf
Referring to the cell means in Table 6.2, and adjusting the number of scores in each cell n 5 4 provides this result:
Review Question A: If a manager’s effec- tiveness is explained by 1) past experience, and 2) education, and no data are collected on education, where does the variability in effectiveness due to education emerge in the analysis?
Section 6.2 The Factorial ANOVA
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SSbet 5 (4.375 2 4.323) 24 1 (5.25 2 4.323)24 1 . . . 1 (5.5 2 4.323)24
Verify that SSbet 5 24.888
As with the one-way ANOVA, the SSwith is determined by:
• subtracting from each individual score in each group the mean of that group, • squaring the difference, • summing the squared differences for each group, and • summing the totals across the (now) six groups.
For the factorial problem the adaptation of Formula 5.4 is:
SSwith 5 Σ[(xa1 2 Ma) 2 1 (xa2 2 Ma)
2 1 . . . 1 (xa4 2 Ma) 2] 1 Σ[(xb1 2 Mb)
2 1 (xb2 2 Mb) 2 1
. . . 1 (xb4 2 Mb) 2] 1 . . . 1 Σ[(xf 1 2 Mf )
2 1 (xf 2 2 Mf ) 2 1 . . . 1 (xf 4 2 Mf )
2]
Using Excel for the repetitive subtraction and squaring will speed up the calculations. The result will be approximately:
SSwith 5 24.172
Recall that the accuracy of the SSwith result can be confirmed by adding SSwith to SSbet to make sure that they total SStot. That relationship holds with any ANOVA problem.
SSwith 1 SSbet 5 SStot
24.172 1 24.888 5 49.060, which is the same value earlier determined for SStot.
It is important to note that since factorial ANOVA allows for introducing a second variable into the analysis (the industry experience variable, in this example), if this additional vari- able explains something that was unexplained in the previous one-way ANOVA, the term that measures unexplained variability, the SSwith, will shrink. In the one-way ANOVA, SSwith 5 39.617. For this factorial ANOVA, SSwith 5 24.172. Unexplained variability is indeed reduced in the factorial ANOVA. It is not that the variability is eliminated; SStot is still the same, so the variability is still there. The difference is that if either of the new terms in the analysis, the second independent variable and the interaction of the variables, is significant, they will explain what was previously unexplained and was, therefore, error variance in the one-way ANOVA.
Partitioning the SSbet The SSbet value in any ANOVA problem contains the variability related to all independent variables in the analysis, as well as the combined effects of IVs called the interaction. Because in this problem there is a second independent variable, part of the anal-
ysis involves determining the portion of variance contained in the SSbet that is related to the first IV, the variance related to the second IV, and the variance prompted by the
Key Terms: In contrast to interactions, independent vari- ables in an analysis of variance are also called main effects.
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Note that the SSIV1 involves the same values organized the same way as they were in SSbet in the one-way ANOVA. The results should confirm that SSIV1 from the factorial problem 5 SSbet from the one-way problem.
The SS for the second IV is:
• The comparison of cells a 1 c 1 e (the first column) to cells b 1 d 1 f (in the second column).
• MG is subtracted from the mean of the data from cells a 1 c 1 e (Mace) • the difference is squared. • the result is multiplied by the number of sales figures in cells a, c, and e • this process is repeated for cells b, d, and f. • the results are summed.
The associated formula is:
interaction. Any of the three, in addition to the model as a whole, which is what the SSbet represents, could be statistically significant. In this particular problem, then, there will be four F ratios calculated, but the sums of squares must be calculated first. In the present problem where the first IV (company) has three categories, and the second IV (past sales experience) has two categories, the analysis is a “3 by 2” ANOVA. In factorial ANOVA problems, to help distinguish the independent variables from the interaction, the various independent variables are often referred to as main effects. The way they are listed here makes company the first main effect and experience the second main effect.
The SS for the first IV is:
• the comparison of cells a 1 b (Swoosh), to c 1 d (Matrix), to e 1 f (Impact) • MG is subtracted from the mean of the data from cells a 1 b (Mab), • the difference is squared, and • the result is multiplied by the number of scores in cells a and b • this process is repeated for cells c and d, and then for cells e and f • the results are summed
The associated formula is:
Formula 6.1 SSIV1 5 (Mab 2 MG) 2 (nab) 1 (Mcd 2 MG)
2 (ncd) 1 (Mef 2 MG) 2 (nef )
SScomp 5 (4.813 2 4.323) 2 (8) 1 (3.813 2 4.323)2 (8) 1 (4.719 2 4.323)2 (8)
5 9.443
Section 6.2 The Factorial ANOVA
Formula 6.2 SSIV2 5 (Mace 2 MG) 2 (nace) 1 (Mbdf 2 MG)
2 (nbdf )
SSexp 5 (4.271 2 4.323) 2 (12) 1 (4.375 2 4.323)2 (12)
5 .065
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The data variability from the interaction is greater than either of the independent variables in isolation. Although sums of squares values can be difficult to interpret by themselves, these results indicate that the combined effect of the IVs may be greater than either of the IVs by themselves. This is something we can only know for sure, however, by examining the MS values and then the F ratios, as we did in the one-way ANOVA. The results for this problem are displayed in Table 6.3. It looks very much like the table for the one-way ANOVA, but there are some important differences.
Formula 6.3 SSinteraction 5 SSbet 2 SSIV1 2 SSIV2
SScomp 3 exp 5 SSbet 2 SScomp 2 SSexp
5 24.888 2 9.443 2 .065
5 15.380
Because results were not organized according to the sale associates’ industry experience in the one-way ANOVA, any variability from that source emerged as error variance in the one-way ANOVA. It was part of the SSwith. In fact, that is the case with any source of vari- ance not controlled (accounted for) by introducing the factor responsible. If the gender of the sales associate is a source of variability in sales—if women and men have different sales totals—and it isn’t controlled (accounted for) because the sales associates are not analyzed according to whether they are male or female—that source of variance is still present in the analysis, but it emerges as part of the SSwith, the error term.
The data variability stemming from industry experience was not analyzed in the one- way problem. With SSexp 5 .065 the associated variability appears to be comparatively minor—certainly less than the SSwithin, which is the error variance, for example. It is a source of variance that went unanalyzed in the earlier problem. As a result, any variability in sales data related to industry experience would have emerged in the error term in the first problem. Although there is little reason to be concerned about differences in industry experience by themselves, the analysis of the interaction will indicate that although they appear innocuous independently, their influence emerges another way.
Any variance in the SSbet that is not related to one of the independent variables is attrib- uted to the interaction of the IVs and calculated as the interaction SS. This value is deter- mined by subtraction. Once the sums of squares for all independent variables have been calculated, those values are subtracted from the SSbet, and what remains is the interaction sum of squares. For a problem with 2 IVs, that makes the formula:
Section 6.2 The Factorial ANOVA
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Table 6.3: Factorial ANOVA table
Source SS df MS F Fcrit
Total 49.060 23
Between 24.888 5 4.978 3.707 F.05(5,18) 5 2.77
Company 9.443 2 4.722 3.516 F.05(2,18) 5 3.55
Experience .065 1 .065 .048 F.05(1,18) 5 4.41
Co. 3 Exp. 15.380 2 7.690 5.726 F.05(2,18) 5 3.55
Within 24.172 18 1.343
Because there are multiple IVs or main effects, there are lines beneath SSbet for each of them as well as for the interaction of the IVs. In any factorial ANOVA there are several tests, each with its own F. The various F ratios are calculated using the same procedures as the one-way ANOVA. First the SS for the particular source of data variance is divided by its associated degrees of freedom to compute the mean square (MS) value. The different mean squares each have their own degrees of freedom. They are as follows:
• The df for SStot remain N 2 1, the total of all values, minus one. Here that value is 24 2 1 5 23, just as it was in the one-way ANOVA.
• The df for SSbet remain at k 2 1 with but with k now the number of cells, 6 2 1 5 5.
• The df for each IV are the number of categories of that particular IV, 21. For the analysis of company differences, SScomp, df 5 3 2 1 5 2. For the analysis of age differences, SSage, df 5 2 2 1 5 1.
• The df for SSinteraction is the product of the df for the IVs. With df 5 2 for the company IV and df 5 1 for the age IV, dfinteraction 5 2 × 1 5 2.
• The df for SSwith remains N 2 K, but with k now 5 6 dfwith 5 24 2 6 5 18.
Having determined the various MS values, they can then be used to determine the multiple F values that every factorial ANOVA produces. For every MS except the MSwith there will be an associated F test. In a factorial ANOVA with two main effects, there will be MS values for the model as a whole (the MSbet), for each of the main effects, and for the interaction. Each of the respective MS values is divided by the MSwith to determine the particular F values. Every time the combination of degrees of freedom involve change for the several F values, there is a different critical value of F to look up to determine statistical significance.
Section 6.2 The Factorial ANOVA
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The first F value is the ratio of the MSbet 4 MSwith, which in a factorial analysis is for the model as a whole. This F provides an aggregate of both IVs plus the interaction and indi- cates whether any of the possible F values are statistically significant. In Table 6.3, the calculated F values and their associated critical values from the table of critical values for F indicate that:
• The F 5 3.707 for between is significant; it exceeds the critical value of F.05(5,18) 5 2.77.
• The F 5 3.516 for company is not significant; it is less than F.05(2,18) 5 3.55. • The F 5 .048 for experience is not significant; it is less than the critical value of
F.05(1,18) 5 4.41. • The F 5 5.726 for the interaction is significant; it exceeds the critical value of
F.05(2,18) 5 3.55.
The significant F for between indicates that at least one of the other F tests analyzed in this model is also statistically significant. Neither of the main effects is significant, but their combined effects are; there is a statistically significant interaction of the main effects.
Graphing the Interaction
Statistically significant interactions can be difficult to “wrap your head around,” and it is usually easier to understand them if they are represented in a diagram or graph. They are particularly easy to construct for a two-way ANOVA—a factorial ANOVA with two inde- pendent variables. The interaction for the company by industry experience in the above example is graphed in Figure 6.1. In the graph:
• The vertical (y) axis represents the range of values for the dependent variable (sales).
• The horizontal (x) axis is used to represent one of the independent variables. It does not matter which one, although the graph will be visually simpler if the variable has only two categories, such as the industry experience versus no industry experience example above.
• The other independent variable (company) is represented in the graph by drawing a line for each category of that variable. The lines are anchored by the mean for the combination of that variable with one level of the other vari- able—the mean sales at Swoosh by associates with no industry experience (M 5 4.375), for example—to the mean for the combination of that variable with the other level of the second variable, sales at Swoosh by associates with industry experience (M 5 5.25).
Section 6.2 The Factorial ANOVA
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Figure 6.1: The interaction of company and the age of the top sellers
A statistically significant interaction indicates that the com- bined effects of the independent variables have different effects in different situations. The combined effect of company and experience on sales is different in some situations than in others. In this problem it is manifested by the fact that for the sales people at Impact and Swoosh, sales associates with industry experience sold more. At Matrix, it is the opposite.
Because they are the combined effect of multiple IVs, interactions can’t emerge when one variable is studied at a time, as they are in a one-way ANOVA. By extension, that means that if IVs that interact are studied independently, any conclusions that are drawn are likely to be incorrect. In the problem here, company-to-company differences appeared to be just random differences; they were not statistically significant. However, when com- pany differences are combined with experience, the differences due to the combination of those variables are not random. For that reason, some statisticians maintain that if the interaction in a factorial ANOVA is statistically significant, the analyst should explain the interaction and be relatively unconcerned with main effects that may or may not be statis- tically significant since explaining their influence in isolation may be deceptive.
The significance tests provide definitive answers about statistical significance that the graph does not provide, but the appearance and orientation of the lines can provide a good deal of supplemental information. For the company-by-experience example:
• Substantial differences in the point at which each of the three lines begins would have suggested differences from company to company between sales associates with no industry experience. As it was, the lines all began with quite similar means.
No Experience Experience
S a le
s in
$ k
7
6
5
4
3
2
1
(2.375) Matrix
(5.25) Swoosh (5.5) Impact
(4.50)
(4.375) (3.938)
Review Question B: If none of the main effects in a factorial ANOVA is significant, can there be a signifi- cant interaction?
Section 6.2 The Factorial ANOVA
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CHAPTER 6Section 6.3 Using Excel to Complete a Factorial ANOVA
• The positions of the line ends on the right-hand side of the graph suggest how much difference there was in sales from company to company between sales associates with industry experience. There is a good deal more data vari- ance here than there was in the sales associates with no industry experience category. Any company-to-company differences are largely a function of what the sales associates with industry experience did, but these differences are not great enough for the overall company result to be statistically significant.
• When the lines steeply incline or decline, it indicates differences within the particular company. There are differences between the experience groups in all the companies, but they are most pronounced in Matrix because its line has the steepest slope.
• The fact that the lines are not parallel (or reasonably close to being parallel) suggests the presence of the interaction, although it is the F value from the table that will indicate whether it is actually statistically significant. The lines could all decline or incline dramatically, but as long as they move in basically the same direction, there is no interaction. The significant interaction in this problem is suggested by the very different orientation of the Matrix line compared to the other two. The lines for Swoosh and Impact have similar trajectories in spite of the fact that they cross.
There might be several ways to explain this interaction. Perhaps there is a policy at Matrix to provide sales associates with no industry experience with more intensive train- ing, which causes them to outperform their experienced col- leagues. As an alternative, maybe the experienced sales asso- ciates at Matrix are tasked with mentoring and coaching their less experienced colleagues, and as a result have less time to sell than their similarly experienced counterparts at the other two companies. For whatever reason, experience and com- pany together result in different sales results at Matrix than at Swoosh or Impact.
6.3 Using Excel to Complete a Factorial ANOVA Excel makes it very easy to complete a factorial ANOVA, although it is limited to two-way analyses such as the one completed here. The data from the problem just completed are arranged just as the data table was arranged above, with the rows used for one indepen- dent variable and the columns for the other. The difference is that each individual score must be in a separate cell. If we use Row 1 and Column A for data labels, the sales total values are entered as follows:
• In cell B1 enter the label No Experience. • In cell C1 enter the label Experience. • In cells A2 to A5 enter the label Swoosh. • In cells A6 to A9 enter the label Matrix. • In cells A10 to A13 enter the label Impact. • In cells B2 to B5 enter the four sales totals for the Swoosh sales associates with
no industry experience.
Review Question C: When graphing an interaction, what does the vertical (y) axis gauge?
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• In cells B6 to B9 enter the four sales totals for the Matrix sales associates with no industry experience.
• In cells B10 to B13 enter the four sales totals for the Impact sales associates with no industry experience.
• In cells C2 to C5 enter the four sales totals for the Swoosh sales associates with industry experience.
• In cells C6 to C9 enter the four sales totals for the Matrix sales associates with industry experience.
• In cells C10 to C13 enter the four sales totals for the Impact sales associates with industry experience.
With the data entered, the commands for the analysis are as follows:
• Select the Data tab at the top of the screen. • Select Data Analysis. • Select ANOVA Two Factor With Replication—click OK. • Indicate the input range which will be A1:C13. • In the rows per sample box indicate 4, which indicates that there are 4 scores
for each combination of variables. • For the Output Range option, indicate E1 so that the output doesn’t overwrite
the data set.
The result is provided in Figure 6.2.
Figure 6.2: The factorial ANOVA with Excel
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Note that the Excel output includes the descriptive data. They include the means, sums, and variance figures for each of the companies and for the two experience categories. In the ANOVA table, the labels differ from the longhand calculations but the figures agree, although Excel does not provide the figures for “between.” Recall, however, that between can be determined by adding together the sums of squares for the main effects and the interaction.
Describing the Factorial ANOVA
Factorial ANOVA problems are referred to by the number of categories in each indepen- dent variable. The order of the IVs usually is not important.
• With three categories of one IV and two levels of the other, the model is a 3 3 2, said “3 by 2” (or “2 by 3” ANOVA).
• If there were four companies involved and three categories of sales associate experience (e.g., Industry Sales Experience, Unrelated Sales Experience, No Sales Experience), the problem becomes a 4 3 3 ANOVA.
• Regardless of the number of categories there are in each independent vari- able, all ANOVA procedures with two IVs are called “two-way ANOVAs.”
• If there is a third variable, the example becomes a three-way ANOVA. In the example above, if the analyst decided to factor in the gender of the sales asso- ciate, this would become a 3 3 2 3 2 ANOVA.
The Number of Independent Variables and the Sample Size
While there are no theoretical limits to the number of independent variables that can be included in a factorial ANOVA, there are some important practical limits that ought to be considered. As the number of variables increases so, logically, must the number of catego- ries for which there must be data. With the one-way problem, there had to be data in just the three categories that represented the three companies. When the age of the sales staff was added, the number of categories became 6, cells a through f in Table 6.2. If whether the representatives have earned college degrees is made part of the analysis, the number of cells becomes 12, and the table might look something like Figure 6.3.
Section 6.3 Using Excel to Complete a Factorial ANOVA
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CHAPTER 6
Figure 6.3: A “3 3 2 3 2” ANOVA
There are no hard-and-fast rules about how many observations are required in each cell to complete an ANOVA, but more are usually better. Furthermore, it is a good idea, although not essential, to have the same number of entries in each cell, particularly when the num- bers are comparatively low. With 8 scores in each company, there were 24 total scores in the one-way analysis of just company-to-company differences. If the analyst wanted to main- tain 8 per cell when the industry experience categories were added, the number necessary becomes 48. If whether the top sales associates had earned college degrees was analyzed, the number necessary for 8 values in each cell becomes 96, and if sales are also analyzed by day of week for a 6-day week, the result is 576, for a minimum of 8 measures in each cell. This last combination would result in a 3 × 2 × 2 × 6 ANOVA. Data demands rapidly increase when additional variables are added, particularly when there are many categories of the variable.
The Post Hoc Test
Recall that for the one-way ANOVA, a significant F prompted a post hoc test to determine which groups were significantly different from which. Excel has no option for a post hoc test, but if it is requested with one of the major statistical packages, SPSS (Statistical Pack- age for the Social Sciences) for example, there will be post hoc tests for each of the IVs, but not for the interaction. This appears to be guided by thinking such as that by Glass and Hopkins (1978) who said of factorial ANOVA analyses that post hoc tests are “ordinarily not used in interactions—interactions are a generic phenomenon best understood by studying the interaction graph” (p. 384), which is the reason for our discussion of Figure 6.1 above.
Had the company-to-company differences produced a significant F, Tukey’s HSD could have been calculated to determine which pair(s) of companies were different. Had the F for industry experience been significant, no post hoc test would have been needed, of course, since there could have been only one possibility—a significant difference between the sales for the two categories.
The three companies are represented in the “rows” from top to bottom.
The columns back to front indicate whether the sales representatives have earned college degrees.
The columns left to right indicate whether sales representatives have past industry experience.
Section 6.3 Using Excel to Complete a Factorial ANOVA
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CHAPTER 6
The Effect Sizes
With the one-way ANOVA in Chapter 5, we were careful to distinguish between statistical and practical significance and noted that a nonrandom (statistically significant) outcome is not necessarily important in the everyday world. We used the partial eta-squared value as a measure of practical importance. For the one-way problem in this chapter, no effect size was necessary since the result was not statistically significant. However, in the fac- torial problem where there were multiple F’s, two were statistically significant, that for between, and that for the interaction. They each require a corresponding effect size calcu- lation. The eta-squared formula used in Chapter 5 will be adjusted for the particular effect size to be calculated. For between, recall that Formula 5.7 for partial eta-squared was:
hp 2 5
SSbet SSbet 1 SSwith
Inserting the values from the ANOVA table for the factorial problem provides:
hp 2 5
24.888 24.888 1 24.172
5 .507
The result indicates that about 51% of the variance in sales data can be explained by the entire model. That is, the company, industry experience, and the interaction of those 2 IVs explains 51% of the variability in sales. To look specifically at just the proportion of vari- ance explained by the interaction, the SSco×exp (the sum of squares for the interaction) is substituted for the SSbet in both the numerator and denominator of the partial eta formula, the SSwith is retained, and the calculations are completed as follows:
hp 2 5
SSco 3 age SSco 3 age 1 SSwith
With the relevant values from the ANOVA table inserted:
hp 2 5
15.380 15.380 1 24.172
5 .388
Note that the SSwith is constant in the calculations; it is the error term regardless of which component of variability we’re trying to determine. It is the SS for the particular IV or for the interaction of the IVs that changes. The partial eta for the interaction indicates that about 39% of the variance in sales data can be explained by the way company and indus- try experience interact.
Recognize that the various partial eta values are not directly additive. Had the main effects (IVs) both been statistically significant, the sum of the partial eta values for the two main effects plus the partial eta value for the interaction would not have the same value as the par- tial eta for the between component, even though the sum of squares between is the total of the sums of squares for the independent variables plus that for the interaction. Although they each provide a helpful indicator of the importance of the particular component of the analysis, the various partial eta values overlap each other. Their values are not entirely independent.
Section 6.3 Using Excel to Complete a Factorial ANOVA
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CHAPTER 6Section 6.4 Another ANOVA Problem
6.4 Another ANOVA Problem
In the sports apparel company problem, the SSbet in the one-way analysis that measured company-to-company differences became one of two independent variables in the fac- torial problem. What was initially the SSbet became the SScom, but with the same value since the same scores were involved in either case, just with different labels. That is not always the case, as can be demonstrated with a second problem.
A company uses three different shifts of workers at an assembly plant—day, swing, and night. The production manager is interested in the comparative productivity of the three shifts and so tracks the number of units assembled over an eight-day period for all three shifts. The results are in hundreds of components assembled.
1 2 3 4 5 6 7 8
Day 4 5 4.5 2 7 5.5 4.5 6
Swing 3 3.5 4 4 4.5 4 2 5.5
Night 4.5 5 5.5 3 7 4.5 3.75 4.5
If a one-way ANOVA is completed in Excel to check for significant differences in produc- tivity, the results are those in Table 6.4.
Table 6.4: A one-way ANOVA of shift differences in productivity
ANOVA: Single Factor
SUMMARY
Groups Count Sum Average Variance
Days 8 38.5 4.8125 2.209821
Swing 8 30.5 3.8125 1.066964
Nights 8 37.75 4.71875 1.418527
ANOVA
Source of Variation
SS df MS F P-value Fcrit
Between Groups
4.880208 2 2.440104 1.559068 0.233719 3.4668
Within Groups
32.86719 21 1.565104
Total 37.7474 23
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CHAPTER 6Section 6.4 Another ANOVA Problem
With F 5 1.559 and a probability of that value of F occurring by chance p 5 .234, the result is not statistically significant. In terms of productivity, the differences in the three shifts are minimal enough that they can be attributed to random differences.
If the productivity values for each day are divided into the four days before a pay raise and the four days after a pay raise, the problem becomes a factorial ANOVA:
• The shift-to-shift differences would be the first IV. • The before/after pay raise would be the second IV. • Then there would be the interaction of the shift variable (day/swing/night)
with the pay increase variable (before/after) to analyze.
Even with the division of the four days’ productivity into before and after the pay raise, the SSshift in the factorial problem would have the same value that SSbet had in the one-way analysis. That can happen because the SSshift calculation for the factorial problem and the SSbet calculation for the one-way problem involve the same 24 scores and the same mean values organized the same way.
If individual scores are altered, however, this changes. To illustrate, perhaps the data for each day’s production figures are reduced to the production in hundreds of units before break, and the production after break for each shift. Although the production totals for the eight days will remain the same, the individual values change. Table 6.5 indicates what the before-and-after break production values are for each of the 8 days. Because each of the former scores has been reduced to two parts, neither IV will match the SSbet from the one-way analysis.
Table 6.5: The data for analyzing the productivity of shift, by time of day
Company Before Break After Break Row Means
Day 1, 2, 1, .5, 2, 2, 2, 1.75 Ma 5 1.531
a
3, 3, 3.5, 1.5, 5, 3.5, 2.5, 4.25 Mb 5 3.281
b
Mab 5 2.406
Swing 2, 2.75, 2.5, 3, 2.5, 2.5, 1.5, 3.5 Mc 5 2.531
c
1, .75, 1.5, 1, 2, 1.5, .5, 2 Md 5 1.281
d
Mcd 5 1.906
Night 1.5, 1.5, 2, 1, 2.5, 1.75, 1.25, 1.5 Me 5 1.625
e
3, 3.5, 3.5, 2, 4.5, 2.75, 2.5, 3 Mf 5 3.094
f
Mef 5 2.360
Column Means Mace 5 1.896 Mbdf 5 2.552 MG 5 2.224
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CHAPTER 6
The way the calculations are completed are the same as before, but there are more scores involved. The SStot remains the square of the difference between each individual value and the mean of all the data. The difference is squared, and the squared differences summed across all (now) 48 scores beginning with the first value in box “a” and continuing to the last value in box “f.”
SStot 5 Σ(x1 2 MG) 2 1 Σ(x2 2 MG)
2 1 . . . 1 Σ(x48 2 MG) 2
Substituting the appropriate values the result will be:
SStot 5 (1.0 2 2.224) 2 1 (2.0 2 2.224)2 . . . 1 (3.0 2 2.224)2 5 50.280
The calculation for the SSbet was according to this formula in the one-way analysis:
SSbet 5 (Ma 2 MG ) 2na 1 (Mb 2 MG )
2nb 1 (Mc 2 MG ) 2nc
The difference is that now there are six groups rather than the three from the one-way ANOVA. Substituting the new group mean and grand mean values results in the following:
SSbet 5 (1.531 2 2.224) 2 8 1 (3.281 2 2.224)2 8 1 . . . 1 (3.094 2 2.224)2 8 5 29.569
Although the production figures for each shift have the same total, there are two produc- tion figures for each shift in the factorial ANOVA problem—one for before break and one for after. Recall that one of the differences between sums of squares and other measures of data variability such as the standard deviation is that more measures translate inevitably into larger variability measures. That happened here. The SSbet 5 4.880 value for the one- way analysis grew to SSbet 5 29.569 for the two-way analysis, in spite of the fact that total production figures from both analyses have the same total.
For the SSwith recall that from each individual in each group is subtracted the mean for that group, the difference is squared, the squared differences are summed for each group, and then the totals are summed across groups, as follows:
SSwith 5 Σ[(xa1 2 Ma) 2 1 (xa2 2 Ma)
2 1 . . . 1 xa8 2 Ma) 2] 1 Σ[(xb1 2 Mb)
2 1 (xb2 2 Mb ) 2 1
. . . 1 xb8 2 Mb ) 2] 1 . . . 1 Σ[(xf 1 2 Mf )
2 1 (xf 2 2 Mf ) 2 1 . . . 1 xf 8 2 Mf )
2] 5 20.711
Remembering that SStot is the sum of SSbet 1 SSwith, the accuracy of that value can be con- firmed, once again, by subtracting SSbet from SStot to make sure that the difference is the same value as was calculated for the SSwith. If there is no error in the calculations, and we allow for whatever minor rounding differences there may be, the result should be 50.280, which the subtraction below confirms.
SSwith 1 SSbet 5 SStot; 20.711 1 29.569 5 50.280
Section 6.4 Another ANOVA Problem
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CHAPTER 6
The shift-to-shift differences, SSshift, are represented in the differences between the rows in Table 6.5. The row data compare the values in cell a plus cell b, to the values in cell c plus cell d, compared to the values in cell e plus cell f.
SSshift 5 (Mab 2 MG) 2 (nab) 1 (Mcd 2 MG)
2 (ncd) 1 (Mef 2 MG) 2 (nef )
5 (2.406 2 2.224)2 (16) 1 (1.906 2 2.224)2 (16) 1 (2.360 2 2.224)2 (16)
5 2.444
The sum of squares for the time of day (SStime) will have the same form the SSexp calcula- tions took in the first factorial problem we did in this chapter. The data from the cells that make up the first column (before break: a 1 c 1 e) will be analyzed with the data in the cells in the second column (after break: b 1 d 1 f ) .
SStime 5 (Mace 2 MG) 2 (nace) 1 (Mbdf 2 MG)
2 (nbdf )
5 (1.896 2 2.224)2 (24) 1 (2.552 2 2.224)2 (24)
5 5.164
The sum of squares for the interaction remains the sum of squares for between minus the sums of squares for the independent variables SSbet 2 SSshift 2 SStime. The SSshift × time vari- ability is:
SSshift 3 time 5 SSbet 2 SSshift 2 SStime
29.569 2 2.444 2 5.164 5 21.961
In the one-way analysis, no distinction could be made between the shift-to-shift differences and any differences there might have been from comparing the before-to-after break produc- tion. Both of those shift and orientation-to-break variables, as well as any interaction there might have been between the two variables were the SSbet in the one-way analysis. Once they’re separated out as they are here, and with the effect that twice the number of scores has on the amount of variability measured, note that the sum of the three components is much greater than the SSbet was in the one-way ANOVA.
The sums of squares values can each be divided by their respective degrees of freedom to determine the mean square values, after which the F values can be calculated. As with any two-way ANOVA, there are four F ratios:
• MSbet /M3Swith • MSshift /M3Swith • MStime /M3Swith • MSshift 3 time /MSwith
Review Question D: What is the relation- ship between the number of measures and the magnitude of sums of squares values?
Section 6.4 Another ANOVA Problem
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CHAPTER 6
The results are in Table 6.6.
Table 6.6: A second factorial ANOVA: Productivity of shift by time of day
Source SS df MS F Fcrit
Total 50.280 47
Between 29.569 5 5.914 11.996 F.05(5,42) 5 2.44
Shift 2.444 2 1.222 2.479 F.05(2,42) 5 3.22
Time 5.164 1 5.164 10.475 F.05(1,42) 5 4.07
Shift 3 Time 21.961 2 10.981 22.274 F.05(2,42) 5 3.22
Within 20.711 42 .493
• The F for the model as a whole (MSbet/MSwith) is significant. • The F for shift is not significant. • The F for time (before/after break) is significant. • The interaction of shift and time is significant.
A statistically significant interaction of main effects should be graphed. Figure 6.4 shows that productivity is higher for the swing shift workers before break. It is higher for work- ers on the other two shifts after break.
In terms of effect sizes, the variance explained is calculated for all significant F tests. With h2p 5 SS for the particular component divided by the SS for that same component plus the SSwith
, we have the following:
• Between h2p 5 29.569/(29.569 1 20.711) 5 .588 • Time: 5.164/(5.164 1 20.711) 5 .20 • The interaction: 21.961/(21.961 1 20.711) 5 .515
The combination of the main effects and the interaction (between) explain about 59% of the variability in scoring. Whether the workers are pre- or post-break explains about 20% of the variance, and the interaction of the time relative to break and the shift explains about 52% of the production variance.
Section 6.4 Another ANOVA Problem
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CHAPTER 6Section 6.5 A 2 3 2 ANOVA Problem
Figure 6.4: Production by shift and break
6.5 A 2 3 2 ANOVA Problem
Factorial analysis of variance problems are described by the number of independent variables included, and the number of levels of each independent variable. The prob- lem represented in Table 6.5 is a “3 × 2” problem since there are 2 independent variables, 1 with 3 levels, and the other with 2. For the sake of one more example, perhaps the man- ager of a pastry shop is interested in whether sales are significantly different on Saturdays versus weekdays, and mornings versus afternoons. The manager compares the sales on each Wednesday in a particular month, morning and afternoon, to the Saturdays of the same month, mornings and afternoons. Representing the sales data in a 2 × 2 table pro- duces the following:
AM PM
Wed 430, 410, 390, 415 310, 320, 300, 315
Sat 475, 490, 500, 515 380, 390, 350, 365
The relevant mean scores are as follows:
MWed, AM 5 411.250 MWed, PM 5 311.250 MSat, AM 5 495.000 MSat, PM 5 371.250 MG 5 397.188
SStot 5 Σ(x1 2 MG) 2 1 Σ(x2 2 MG)
2 1 . . . 1 Σ(x16 2 MG) 2
Before Break After Break
U n
its o
f P
ro d
u ct
io n
3.5
3.0
2.5
2.0
1.5
1.0
.5
(1.281) Swing
(3.094) Night (3.281) Day
(2.531)
(1.531)
(1.625)(1.625)
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CHAPTER 6Section 6.5 A 2 3 2 ANOVA Problem
For SStot with the relevant values entered:
SStot 5 (430 2 397.188) 2 1 (410 2 397.188)2 . . . 1 (365 2 397.188)2 5 74098.437
SSbet 5 (Ma 2 MG) 2na 1 (Mb 2 MG)
2nb 1 . . . 1 (Md 2 MG) 2nc
Inserting the relevant mean values:
SSbet 5 (411.250 2 397.188) 24 1 (411.250 2 397.188)24 1 (411.250 2 397.188)24 1
(411.250 2 397.188)24 5 71292.187
SSday 5 (Mab 2 MG) 2nab1 (Mcd 2 MG)
2ncd
Inserting the relevant mean values:
SSday 5 (361.250 2 397.188) 28 1 (433.125 2 397.188)28 5 20664.063
SStime 5 (Mac 2 MG) 2nac 1 (Mbd 2 MG)
2nbd
Inserting the relevant mean values:
SStime 5 (453.125 2 397.188) 28 1 (341.250 2 397.188)28 5 50064.063
SSday 3 time 5 SSbet 2 SSday 2 SStime
5 71292.187 2 20664.063 2 50064.063 5 564.061
SSwith 5 SStot 2 SSbet 5 74098.437 2 71292.187 5 2806.250
Completing the ANOVA table results in the following:
Source SS df MS F Fcrit h 2
p
Total 74098.437 15
Between 71292.187 3 23764.062 101.619 F.05(3,12) 5 3.49 .962
Days 20664.063 1 20664.063 88.363 F.05(1,12) 5 4.75 .880
Time 50064.063 1 50064.063 214.082 F.05(1,12) 5 4.75 .947
Days 3 Time 564.061 1 564.061 2.412 F.05(1,12) 5 4.75
Within 2806.250 12 233.854
The model as a whole, the “between,” is statistically significant. The sales related to the day of the week and the time of day are statistically significant. The partial eta value indi- cates that those two variables explain 96.2% of the variance in sales. The value is unreal- istically high, of course, which can happen when the values of the dependent variable are contrived. Days and time are also independently statistically significant, explaining 88% and 94.7% respectively. The interaction of the main effects (the independent variables) is not statistically significant, so there is no interaction to graph.
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CHAPTER 6Chapter Summary
Since the problem is a 2 × 2 ANOVA, there is also no need for a post hoc test. Since in each case there is only one possibility, the morning sales must be significantly different from the afternoon sales, and the Wednesday sales must be significantly different from the Sat- urday sales.
Chapter Summary
Factorial ANOVA makes it possible to reflect some of the real-world complexity that is inherent in most business problems. The t-tests and one-way ANOVA only accommo- date one independent variable. Most business decisions are too involved for one-variable explanations. Factorial ANOVA allows for multiple categorical independent variables (Objective 1).
As soon as more than one variable is involved in an explanation, there must be allowance for the possibility that the variables interact. Sometimes, one variable conditions, or modi- fies, the way another variable in the same analysis affects the result. This is the concept behind a statistical interaction (Objective 2).
Since there are several sources of data variance in any factorial ANOVA problem, there are several F values to be calculated. There is an F for the overall variance between groups that is an overall picture of the analysis, and there are F values calculated for each of the main effects (IVs) and for all possible interactions. In the simplest of factorial ANOVAs, the two-way ANOVA, there will be an F calculated for between, for both of the two indepen- dent variables, and for the interaction of the IVs.
Besides the advantage of being able to examine specific variables, the advantage of intro- ducing multiple independent variables is that variability that would have been consid- ered random becomes explained. When random variability is explained, the amount of error variance is reduced. Since the F values all have the MSwith, the error term in ANOVA, as the denominator, reductions in error make it more likely that a particular F will be sta- tistically significant (Objective 3).
Like the results of any statistical test, when the F values in a factorial ANOVA are statisti- cally significant it means only that they are unlikely to have occurred by chance. For a gauge of the practical importance of significant outcomes, the partial eta-squared values are calculated (Objective 4). The resulting values are proportions ranging from 0 to 1.0, which, when multiplied by 100, will indicate the percentage of the variance in the depen- dent variable that can be explained by a particular main effect, or by an interaction of main effects.
Answers to Review Questions
A. Score variability stemming from a variable not included in the analysis emerges as error variance in the SSwith.
B. Yes, sometimes despite the fact that none of the main effects is statistically significant, the interaction of the main effects can be significant.
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CHAPTER 6Management Application Exercises
C. The vertical axis in an interaction graph indicates the scores on the dependent variable.
D. The greater the number of values, the larger the sums of squares values become. This is in contrast to other measures of variability, such as the standard devia- tion, where more values can often mean a shrinking variability estimate.
Chapter Formulas
Formula 6.1 SSIV1 5 (Mab 2 MG) 2nab 1 (Mcd 1 MG)
2 ncd 1 (Mef 2 MG) 2nef
In a factorial ANOVA the variability between groups is partitioned into the SS due to each of the IVs and the interaction. In a table with rows designating IV1 (three groups) and columns designating IV2 (two groups), the levels of IV1 will be cells a plus b, cells c plus d, and cells e plus f.
Formula 6.2 SSIV2 5 (Mace 2 MG) 2nace 1 (Mbdf 2 MG)
2nbdf
The variability between two groups on the second IV in a factorial ANOVA, and attribut- able to that variable. In a table with rows for IV1 (three groups) and columns for IV2 (two groups), the cells related to this variable will be a, c, and e for one level, and b, d, and f for the other.
Formula 6.3 SSinteraction 5 SSbet 2 SSIV1 2 SSIV2
The variance in the DV due to the interaction is whatever is left over once the variability due to the main effects is removed from the between sum of squares.
Management Application Exercises
Unless otherwise stated, use p 5 .05 in all your answers.
1. For a particular ANOVA problem,
SStot 5 450
SSbet 5 255
SS for the first IV 5 85
SS for the second IV 5 110
a. What is the SSwith? b. What is the SSinteraction?
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CHAPTER 6Management Application Exercises
2. A produce supplier to a grocery chain wishes to determine whether the gen- der and marital status of the purchaser of produce are factors in how much is spent. The dependent variable is the amount spent in dollars on fresh fruit and vegetables.
Married, female consumers: 2.9, 3.0, 3.3, 3.5, 3.7 Single female consumers: 2.2, 2.6, 2.8, 2.8, 2.9 Married male consumers: 2.8, 2.9, 3.0, 3.1, 3.4 Single male consumers: 3.3, 3.4, 3.4, 3.5, 3.7 Divorced female consumers: 2.8, 3.4, 3.7, 3.7, 3.8 Divorced male consumers: 2.4, 2.4, 2.6, 2.7, 2.8
Describe the kind of a Factorial ANOVA this is (2 3 2, etc.).
3. How many F values will be calculated in the course of completing in item 2? a. Are there significant differences related to marital status? b. Are there significant differences related to gender? c. Is the interaction of marital status and gender significant? d. How much of the consumption variance can be explained by a total of
marital status, gender, and the interaction of the two?
4. The CEO at a sheet metal manufacturing plant is analyzing the company’s day versus night shift production during a two-week period before and after workers received a raise. Data are as follows: cells a and b represent level 1 of main effect 1 and:
Day workers before the raise (in tons): 7.5, 7.7, 7.7, 8.9, 8.2 Day workers after the raise: 7.9, 8.5, 8.8, 8.9, 9.0 Night workers before the raise: 6.6, 6.3, 6.9, 7.1, 7.0 Night workers after the raise: 6.5, 6.1, 6.0, 6.0, 5.9
a. How much of the variability is explained by shift? b. Why is no post hoc test necessary in problems of this type?
5. For item 4, if there is variability in sheet metal production related to the weather, where will it emerge?
6. Regarding problem 4, how should a partial eta for between be interpreted?
7. An extermination company uses summer college interns and regular year-round staff to sell service contracts to homeowners. The sales areas are divided into lower- and middle-class. Write a narrative that explains what a statistically signifi- cant interaction of the main effects would indicate.
8. If, for item 7, salespeople were divided by gender and by marital status as well as by whether they are full-time or students, and whether they sell in lower- or mid- dle-class regions, how many independent variables would there be for a related factorial ANOVA?
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CHAPTER 6Key Terms
9. Three growers of organic vegetables create a co-op to market their crops more eco- nomically. If data are gathered on the value of sales over a five-day period, what test will indicate whether there are significant differences between growers?
10. With reference to item 9, if produce is further divided into vegetables and fruits, and sales are calculated accordingly, the 5-day sales results in dollars are as follows:
Vegetables Fruit
Grower 1 64, 58, 72, 66, 53 34, 37, 42, 45, 51
Grower 2 55, 51, 49, 43, 58 65, 61, 58, 55, 59
Grower 3 38, 33, 36, 44, 40 45, 44, 38, 49, 44
a. Which variables/combinations of variables are statistically significant? b. What partial eta explains the most variance?
Key Terms
• A statistical interaction indicates that independent variables have a different effect in combination than they have independently.
• Main effects is another term for independent variables in an analysis of variance problem.
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