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SPSS 7.5 / 8.0 - A Survival Guide FACTORIAL ANALYSIS OF VARIANCE Balanced Two-Way ANOVA Unbalanced Two-Way ANOVA Three-Way ANOVA 69 SPSS 7.5 / 8.0 - A Survival Guide PROCEDURE MODEL: OVERVIEW Research Issue: Significance of Group Difference One Dependent Variable One Independent Variable Multiple Independent Variables No Covariate Covariate No Covariate Covariate Dependent (Within) Groups Independent (Between) Groups Two Groups Multiple Groups Two Groups Multiple Groups Dependent T-Test Repeated Measures ANOVA Independent T-Test One Way ANOVA ANCOVA Factorial ANOVA Factorial ANCOVA 70 SPSS 7.5 / 8.0 - A Survival Guide Factorial ANOVA Procedure Model Run Factorial ANOVA No Significant Interaction ? Yes Interpret Main Effect Run 1-Way ANOVA for Simple Main Effects Manually Calculate F using MSw from Factorial(1) Significant F More than 2 Levels ? No Stop No Significant F More than 2 Levels ? Yes Yes SPSS Post Hoc Analysis Manual Post Hoc Analysis using MSw from Factorial (2) (1) The MSw term for the main factorial ANOVA is more reflective of the population. The MSw term from the main factorial ANOVA should be used to manually calculate the F value. (2) The MSw term from the main factorial ANOVA is more reflective of the population variance. The MSw term from the main factorial ANOVA should be used to manually calculate observed values in the post hoc analysis. 71 SPSS 7.5 / 8.0 - A Survival Guide BALANCED TWO-WAY ANOVA Example Problem A marketing manager for a supermarket chain was interested in the effect of both retail price and display location on a new promotional line of cookies. A group of 24 stores with matching store volume, layout, and customer demographics was split at random into 6 groups of 4 stores each. One group was assigned to each of the 6 combinations of retail price (regular retail vs. discounted retail) and display location (entrance aisle, cookie aisle, and checkout). Below are the average weekly unit sales by store over a 13 week period. Table 1 Promotional Cookie: Average Weekly Unit Sales DISPLAY LOCATIONS 1 Entrance 1. Regular Retail 38 31 27 33 Price 2. Discount Retail 35 21 39 30 2 Cookies 28 25 23 20 3 Checkout 21 32 30 22 22 24 16 17 19 15 25 20 72 SPSS 7.5 / 8.0 - A Survival Guide BALANCED TWO-WAY ANOVA SPSS Procedures 1. Define and enter initial data. Define three variables as: Price, Location, and Sales. For the variable Price, assign the variable labels of Retail to the value of 1 and Discount to the value of 2. For the variable Location, assign the variable labels of Entrance to the value of 1, Aisle to the value 2, and 3 to the value of 3. Note: The entire data set is not displayed. 2. Implement the descriptive statistics Means procedure to obtain means for the variable SALES by PRICE by LOCATION. Choose Statistics, Compare Means, and Means from the SPSS main menu bar. The Means window will be displayed. Highlight and move the variable Sales to the Dependent List: text box. Highlight and move the variables Price and Location to the Independent List dialog box. In this example, it is necessary to obtain means and standard deviations for the two independent variables by subgrouping. This is done by opening the SPSS Syntax Editor. Using the mouse pointer, click the Paste button located on the right hand side of the Means dialog box. 73 SPSS 7.5 / 8.0 - A Survival Guide Clicking on the Paste button results in the Syntax! - SPSS Syntax Editor window appearing. The Syntax Editor displays the SPSS command scripting language statements generated by your menu selections in SPSS. In this case, the SPSS command line of Tables = sales BY price location must be modified in order to calculate means and standard deviations by price BY location. Position the blinking Windows cursor in front of the variable LOCATION and insert the word BY. The SPSS command statement should then read TABLES = sales BY price BY location. Click on the right arrow icon or select Run and All from the SPSS menu bar to execute the SPSS commands. 74 SPSS 7.5 / 8.0 - A Survival Guide Report SALES Retail Entrance Mean 32.25 N 4 Std. Deviation Mean 24.00 N 4 Std. Deviation Shown above left and left, is the results from executing the means procedure under SPSS. 4.57 Aisle 3.37 Checkout Mean 26.25 N 4 Std. Deviation Total Mean 27.50 N 12 Std. Deviation Mean 31.25 N 4 Std. Deviation Mean 19.75 N 4 Std. Deviation 5.56 Note: The apparent difference in appearance between the Case Processing Summary and Report is necessitated by the inability to display all of the information returned by SPSS on one screen. Rather than shrink the data to an unreadable size, it has been reformatted to an appearance similar to that presented by the SPSS Output Navigator window. TIP: The Results section of our report will contain a table of mean sales and standard deviations of unit sales as a function of price and display location. The table will also contain two Combined columns for summarization by levels of each independent variable. To obtain the means and standard deviations for completion of the table, it is necessary to re-execute the Means procedure, but with the order of the independent variables reversed. The SPSS command syntax line will read: TABLES = sales BY location BY price. The results of the second execution of the means procedure will contain much redundant data, however it will provide the data needed to complete the data table required for your report. 5.52 Discount Entrance 7.76 Aisle 3.86 Checkout Mean 19.75 N 4 Std. Deviation Total Mean 23.58 N 12 Std. Deviation Mean 31.75 N 8 Std. Deviation 4.11 7.56 Total Entrance 5.92 75 SPSS 7.5 / 8.0 - A Survival Guide Report SALES Entrance Retail Mean 32.25 N 4 Std. Deviation Mean 31.25 N 4 Std. Deviation Mean 31.75 N 8 Std. Deviation Mean 24.00 N 4 Std. Deviation Mean 19.75 N 4 Std. Deviation Mean 21.88 N 8 Std. Deviation Mean 26.25 N 4 Std. Deviation Mean 19.75 N 4 Std. Deviation Mean 23.00 N 8 Std. Deviation Mean 27.50 N 12 Std. Deviation Mean 23.58 N 12 Std. Deviation Mean 25.54 N 24 Std. Deviation 4.57 Discount 7.76 Total 5.92 Aisle Retail 3.37 Discount 3.86 Total 4.05 Checkout Retail 5.56 Discount 4.11 Total 5.71 Total Retail 5.52 Discount 7.56 Total 6.78 76 SPSS 7.5 / 8.0 - A Survival Guide Choose Statistics, General Linear Model, and GLM General Factorial. The general factorial ANOVA is referred to in Stevens (1990) as MANOVA. The GLM - General Factorial dialog box will be displayed. Move the variable SALES to the Dependent Variable: text box. Move the variables LOCATION and PRICE to the Fixed Factor(s): text box. Click on OK to continue. In SPSS 7.5 the GLM - General Factorial dialog box has been updated with a text box for Random Factors and Covariates. Additionally, this new version also automatically defines the range for your explanatory variables. 77 SPSS 7.5 / 8.0 - A Survival Guide BALANCED TWO-WAY ANOVA Selected Results from SPSS Execution Notes: [1] Initial interpretation. [a] The main effects are significant. The question is which main effect(s) is/are significant? [b] The LOCATION effect is significant, p = .002. [c] The PRICE effect is not significant, p = .076. [d] The LOCATION PRICE is called an interaction term. If the location differences were not the same for the two price factors, we would expect a significant interaction. For example, if the regular price did better in the entrance location, but the discount price did better at the checkout, wed have an interaction, i.e., the effect of price depends on location. In this example, the interaction is not significant, p = .564. 78 SPSS 7.5 / 8.0 - A Survival Guide [2] Additional steps: A post hoc analysis for main effects needs to be performed since significance was found in the LOCATION factor. A manual calculation must be performed. The formula for manually calculating the Tukey is contained in Stevens (1990) on page 52. Tukey calculation: (qα ,k , N −k )× MSW / n where q = table value and n = average sample size From table B.2 (Stevens (1990), p. 255), obtain the q value: .05,3,18 = 3.09 MSw (also known as error) from the ANOVA results = 25.875 N = 8 representing the average group size Therefore, the minimum difference which must exist between a pair wise group comparison is: q Comparison of Entrance and Cookie Aisle: 31.75 - 21.88 = 9.87 is greater than 6.4906; significantly different. Comparison of Cookie Aisle and Checkout: 21.88 - 23.00 = - 1.12; not significantly different. (Note: Compare the absolute difference between pair wise groups). Comparison of Entrance and Checkout: 31.75 - 23.00 = 8.75; significantly different. Note: 1 - SPSS does provide the Post Hoc Analysis option under the Factorial ANOVA model. However, although the program would provide a post hoc solution, the MSw error term is larger than the error term produced in the factorial analysis. The error term produced in the factorial ANOVA is a better estimate of the population variance than the MSw error term that is produced in a One-Way ANOVA; therefore the post hoc analysis should be calculated using the error term from the factorial analysis. 2 - In this case, the results of the post hoc do not differ, but this might be the case for other research problems. 79 SPSS 7.5 / 8.0 - A Survival Guide BALANCED TWO-WAY ANOVA Example Results and Discussion (Part 1) Results A two-way analysis of variance yielded no significant sales difference due to price and no significant interaction between price and display location. However, there was a significant difference among the display locations, F(.05,2,18) = 9.04, p =.002. A post hoc analysis using Tukeys procedure (α = .05) revealed the mean sales for the entrance location was significantly higher than both the cookie aisle and checkout location. There was no significant difference between the mean sales of cookie aisle and checkout location. Table 1 presents the means and standard deviations by group. Table 2 presents the analysis of variance summary. Table 1 Mean Sales and Standard Deviations of Unit Sales as a Function of Price and Display Location Display Location Retail Price Regular N M SD Discount N M SD Combined N M SD 8 31.75 5.92 8 21.88 4.05 8 23.00 5.71 24 25.54 6.78 Entrance Cookies Checkout Combined 4 32.25 4.57 4 24.00 3.37 4 26.25 5.56 12 27.50 5.52 4 31.25 7.76 4 19.75 3.86 4 19.75 4.11 12 23.58 7.56 80 SPSS 7.5 / 8.0 - A Survival Guide BALANCED TWO-WAY ANOVA Example Results and Discussion (Part 2) Table 2 Analysis of Variance Summary Source Price Location Price x Location Error *p = .002 DF 1 2 2 18 Sum of Squares 92.04 467.58 30.58 465.75 Mean Square 92.04 233.79 15.29 25.88 F Value 3.56 9.04* 0.59 Discussion Locating the new promotional cookie display in the entrance aisle appears to produce the most unit sales. It is interesting to note that there was no significant difference in sales based on retail price. The company can enjoy higher profits by displaying the cookies at regular price, regardless of the location. Notes: You will find various formats in the literature for presenting results in summary tables. Since the purpose of using tables is to assist the reader in interpreting your results, strive for a balance of simplicity and completeness. For example, including both the row and column means, as well as the cell means, in Table 1 allows the reader to clearly see the overall impact of each price and location factor on sales and the lack of interaction. Although the differences in sales by price were not statistically significant, you can see that the regular price outsold the discount price in every location. Given the effect of price on sales was close to significant (F = 3.56, p = .076), this might suggest some additional study on the price elasticity of cookies and similar products. 81

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