; variance analysis
Documents
User Generated
Resources
Learning Center

# variance analysis

VIEWS: 474 PAGES: 13

• pg 1
```									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, wed 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 Tukeys 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

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

```
To top
;