# SPSS INSTRUCTION CHAPTER 9

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1 SPSS INSTRUCTION CHAPTER 9 Chapter 9 does no more than introduce the repeated-measures ANOVA, the MANOVA, and the ANCOVA, and discriminant analysis. But, you can likely envision how complicated it can be to obtain calculated values for these tests. Calculations for any of these tests may cause anxiety for those uncomfortable with math. So, the possibility of needing to combine these operations for tests such as a repeated-measures ANCOVA or a multiple discriminant analysis may seem utterly overwhelming! Luckily, SPSS provides an option for those who wish to avoid the time-consuming and labor-intensive calculations. Each of the following sections provides instructions for using SPSS to perform its respective test as well as for interpreting the test s output. Repeated-Measures ANOVA with SPSS Your first order of business when conducting a repeated-measures ANOVA in SPSS is to organize your data correctly on the Data View screen. Unlike the setup for a betweensubjects ANOVA, you cannot use dummy variables to distinguish between groups for a repeated measures ANOVA. Dummy variables do not indicate the associations between subjects. To specify these associations for a repeated-measures ANOVA, you must assign a row to each group. The procedure to enter data for the paired-subjects t-test can serve as a guide. The arrangement of data for a repeated-measures ANOVA differs only in that it uses more than two rows because it compares more than two categories. Example SPSS Data View Screen for Repeated Measures ANOVA A partial display of the imaginary data used to create the tables in Example 9.10 shows three separate rows, each pertaining to one of the conditions in which subjects complete crossword puzzles. TABLE 9.9 SPSS REPEATED MEASURES ANOVA DATA ARRANGEMENT Placing data points from the samples side by side in the SPSS data view screen indicates the links between scores. The user inserts column headings, which describe the conditions for each sample, in the variable view screen.

2 For this study, the same people complete a puzzle from a newspaper, a puzzle from a magazine, and a puzzle from a crossword puzzle book. Each row contains the times that it took a particular subject to complete the puzzles in the three conditions. Reading across each row, SPSS knows that the values pertain to the same subjects or to subjects who have some connection with each other. The program looks for this association when you instruct it to perform a repeated measures ANOVA. SPSS regards this association as an additional factor in the analysis. Even a oneway repeated-measures ANOVA requires attention to this additional factor. As a result, you must use SPSS s General Linear Model function to perform the test. This function, also used for the multi-way ANOVA described in Chapter 7, suits situations involving a comparison of at least three groups that have relationships among themselves or with other variables. Its wide applicability makes it appropriate for some of the other tests described in Chapter 9 as well. SPSS, however, requires more input for the repeated-measures ANOVA than Chapter 7 s multi-way ANOVA. The necessary steps for a one-way repeated measures ANOVA are as follows. 1. Choose the General Linear Model option in SPSS Analyze pull-down menu. 2. Choose Repeated Measures from the prompts given. A window entitled Repeated Measures Define Factor(s) should appear. FIGURE 9.6 SPSS REPEATED MEASURES DEFINE FACTORS WINDOW

3 The user inputs preliminary information about the repeated measures test in this box. For a one-way repeated-measures ANOVA, attention should focus upon the top half of the window. The Within-Subject Factor Name refers to a term that describes the condition that distinguishs between groups. The Number of Levels refers to the number of groups involved in the comparison. 3. You have two main tasks in the Repeated Measures Define Factor(s) window. a. SPSS asks you to create a name that describes the overall comparison factor. This name, commonly terms such as condition, or time, distinguishes between the sets of data that you wish to compare. You should type this term into the box marked Within-Subject Factor Name at the top of the window. b. The number that you type into the Number of Levels box tells SPSS how many data sets you wish to compare. Your analysis can include all of your data sets or only some of them. In the next window, you can specify which data sets to include in the analysis. 4. Click Define. A window entitled Repeated Measures appears. FIGURE 9.7 SPSS REPEATED MEASURES WINDOW The Within-Subjects Variables box contains spaces for the number of variables specified as the Number of Levels when defining factors. This particular window would suit a comparison of four means, hence the four spaces in the Within-Subjects Variables box. 5. A box on the left side of the window contains the names of all variables for which you have entered data. For each variable that you would like to include in the analysis, click on its name and, then, on the arrow pointing to the Within-Subjects Variables box. Doing so should move the variable name.

4 6. To include descriptive statistics for the groups in the output, click on the window s Options button. a. Move the name of the analysis within subjects factor to the box labeled Display Means for box. b. Mark Descriptive Statistics in Display box. c. Click Continue to return to the Repeated Measures box. 7. Click OK. These steps create many output tables. Not all of these tables provide new information or values that help you to determine whether category means differ significantly. The one of primary interest to you should be the Tests of Within-Subjects Effects Table, which contains significance values for the repeated-measures ANOVA, itself. In the upper portion of this table, labeled with the independent variable s name, four F and four p values appear. These values are usually the same or almost the same. However, for a standard repeatedmeasures ANOVA the row labeled Sphericity Assumed provides the F and p values that you need. If you requested that SPSS provide you with descriptive statistics (Step #6 in the process) output also includes a table entitled Descriptive Statistics. Values in this table become especially useful when you reject the null hypothesis. In this situation, you must refer to the descriptive statitics to determine what category or categories means differ from the other(s) and the direction of the difference. This information can also help you determine how to begin post-hoc tests. Example 9.16 Selected SPSS Output for One-Way Repeated Measures ANOVA A oneway analysis performed using an expansion of the mock data set shown in Table 9.9 produces the following descriptive statistics and within-subject effects values. Descriptive Statistics Mean Std. Deviation N Newspaper Magazine Book Measure:MEASURE_1 Tests of Within-Subjects Effects

5 Type III Sum of Source Squares Df Mean Square F Sig. Publication Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound Error(publication) Sphericity Assumed Greenhouse-Geisser Huynh-Feldt Lower-bound AND TABLE 9.11 SELECTED SPSS OUTPUT FOR ONEWAY REPEATED-MEASURES ANOVA According to the means listed in the Descriptive Statistics Table (Table 9.10), subjects spent almost the same amount of time completing crossword puzzles from newspapers and from magazines. Puzzles from crossword puzzle books, however, took more time than did those from either or the other two publications. The Sphericity Assumed Row in the section of Table 9.11, entitled Tests of Within-Subjects Effects, contains the F and p values that indicate whether these means differ significantly. The researcher must strongly contemplate the decision about accepting or rejecting the null hypothesis for this analysis. The p value of.088 exceeds that standard α of.05, suggesting that no significant differences exist between the means listed in Table Raising the α level to.10, however, would allow the researcher to reject the null hypothesis of equality between means. He or she should weigh the importance of finding significant differences against the increased chance of making a Type I error when deciding whether to change the α value. If the repeated-measures ANOVA indicates significant differences between category means, you must conduct post-hoc tests. These tests search for sources of the significant omnibus results by comparing two groups or two combinations of groups using t-tests. The strategy for determining which groups or combinations of groups to compare follows that explained for the ANOVAs in Chapter 7. However, rather than using independent-samples t tests for the post-hoc tests, as explained in Chapter 7, you must use paired-samples t tests for a repeated-measures ANOVA s post hoc tests. By using the paired-samples t-tests, you continue to acknowledge the one-to-one relationships between subjects in the independent-variable categories that made the repeated-measures ANOVA necessary in the first place. MANOVA with SPSS

6 If you instruct SPSS to perform a MANOVA, it automatically arranges your dependent variables into a canonical variate. The program, then, compares the mean canonical variate values for each independent variable group. You can include as many independent variables as you wish in the analysis by entering their names as fixed factors. For a oneway MANOVA, though, you should identify only one fixed factor, as explained in the following steps. 1. Choose the General Linear Model option in SPSS Analyze pull-down menu. 2. Choose Multivariate from the prompts given. A window entitled Multivariate should appear. FIGURE 9.8 SPSS MULTIVARIATE WINDOW The user performs a MANOVA in SPSS by moving the names of relevant variables from the box on the left side of the window to the Dependent Variables and Fixed Factor(s) boxes in the center of the window. Because the MANOVA involves multiple dependent variables, the Dependent Variables box should contain at least two variable names. The number of variable names moved to the Fixed Factor(s) box depends upon the number of independent variables involved in the analysis. 3. Identify the variables involved in the analysis. a. Move the names of dependent variables from the box on the left side of the window to the box labeled Dependent Variables. b. Move the name of the independent variable from the box on the left side of the window to the box labeled Fixed Factor(s). 4. To include descriptive statistics for the groups in the output, click on the window s Options button. a. Move the name of the independent variable to the box labeled Display Means for box. b. Mark Descriptive Statistics in the Display box. c. Click Continue to return to the Multivariate window.

7 5. Click OK. As with almost all SPSS output, the first table shown simply identifies the categories and the number of subjects in each one. Of more interest that this information, however, is likely the Descriptive Statistics output table, which appears only if you included Step #4 in the process of requesting the MANOVA. This table contains group means and standard deviations for each individual dependent variable. To assess the significance of differences between the mean values, you must evaluate values in the Multivariate Tests table and, in some cases, the Tests of Between-Subjects Effects table. The first of these tables contains F and p values for the MANOVA analysis comparing groups canonical variate means. The Tests of Between Subject Effects table provides data for ANOVAs performed using each individual dependent variable. Example 9.17 Selected SPSS Output for Oneway MANOVA Tables 9.12, 9.13, and 9.14 show sample MANOVA output based upon imaginary data for the scenario described in Example 9.4. SPSS output for the MANOVA contains other tables as well. However, these three tables provide the information needed to address the omnibus hypothesis and the role of the dependent variables in determining whether canonical variate means differ significantly. Descriptive Statistics Genre Mean Std. Deviation N Setting Written Film Musical Total Characters Written Film Musical Total Plot Written Film Musical Total

8 Multivariate Tests c Effect Value F Hypothesis df Error df Sig. Intercept Pillai's Trace a Wilks' Lambda a Hotelling's Trace a Roy's Largest Root a Genre Pillai's Trace Wilks' Lambda a Hotelling's Trace Roy's Largest Root b Tests of Between-Subjects Effects Source Dependent Variable Type III Sum of Squares df Mean Square F Sig. Corrected Model Setting a Characters b Plot c Intercept Setting Characters Plot Genre Setting Characters Plot Error Setting Characters Plot Total Setting Characters Plot Corrected Total Setting Characters

9 Plot TABLE 9.12, TABLE 9.13, AND TABLE 9.14 SELECTED SPSS OUTPUT FOR ONEWAY MANOVA Category means and standard deviations for the canonical variate appear in Table 9.12, entitled Descriptive Statistics. Values in the Multivariate Tests table (Table 9.13) indicate whether these means differ significantly. In this table, the row labeled Wilks Lambda contains the values pertaining to the MANOVA procedure described in Chapter 9. To further understand the p value included in this table, the researcher might find values in Table 9.14, Test of Between-Subjects Effects, useful. This table provides p values for oneway ANOVAs comparing category means for each of the dependent variables that compose the canonical variate. Values in lower portion of the Multivariate Tests table, labeled genre, indicate whether canonical variate means differ significantly for those who experienced the story by reading it, watching it as a film, and watching it as a Broadway musical. In this table, SPSS presents the results from four possible techniques of obtaining F for the MANOVA. For an analysis using Section s method involving Λ, values in the Wilks Lambda row of the table should be examined. The F of and the p of.000) indicate a significant difference between the mean canonical variate values for each genre. The presence of a significant difference in canonical variate means, however, does not imply significant differences in the means for each dependent variable. The results of ANOVAs that compare the mean setting, characters, and plot scores for each category appear in Table According to the values in the genre row of this table and based upon the standard α of.05, subjects in the three independent-variable categories do not have significantly different recall of characters (F=1.715, p=.182). They do, however, have significantly different recall of the story s setting (F=14.932, p=.000) and plot (F=7.355, p=.001). The differences in these dependent variable scores, provide a mathematical explanation for the differences in canonical variate scores. Although not an issue for this analysis, values from the Table 9.14 can also provide a behind the scenes look when you have insignificant results. One cannot assume that accepting the MANOVA s null hypothesis implies that the independent variable groups have equal scores on each dependent variable. Scores for one or more dependent variables may differ significantly among groups. But, a majority of dependent variables with similar scores may mute these differences in the canonical variate. The ANOVA results presented in the Tests of Between Subjects Effects table identify any individual dependent variables with significantly different group means. Had results from Example 9.17 s analysis led to an accepted null hypothesis, you could end your analysis by stating that no significant differences between mean canonical variate

10 values exist. However, with a rejected null hypothesis, you must continue the analysis with post-hoc comparisons to find at least one reason for the significant difference The same technique for performing post-hoc analyses for the ANOVA applies to the MANOVA. However, rather than comparing category means for individual dependent variables, the MANOVA s post-hoc analyses compare category means for canonical variates. So, you should begin by identifying a category or categories with combinations of dependent-variable scores that you believe differ from the others. The total values in the Descriptive Statistics table can help you to determine which category or categories you should contrast from the others. Performing the post-hoc comparisons of canonical variates requires more MANOVAs. These MANOVAs, however, compare only two independent variable categories. SPSS s Select Cases function allows you to specify the categories that you wish to include in the analysis. When necessary, you can also combine categories by recoding them. (See Chapter 2 for instructions about selecting cases and recoding categories.) As with any post-hoc exercise, you must continue making comparisons until you find at least one disparity that produces a p<α. The distinction between the categories that produces these results helps to explain the significant omnibus results. You can obtain very specific information about the source of significant omnibus MANOVA results by determining whether you can associate these differences with particular dependent variables. To do so, you need to compare the means for a particular dependent variable across categories or combinations of categories that your original post-hoc tests identified as different. This investigation uses values in the Tests of Between-Subjects Effects table. The rows labeled with independent variable names contain results from ANOVAs that compare dependent variable means. (Note that, in Example 9.17, these values and those that appear in the Corrected Model row are the same. The two rows contain identical values only for a oneway test.) The values that appear to the right of each dependent variable name indicate whether category means for that dependent variable, alone, differ significantly. If a dependent variable s scores don t differ significantly among groups (p>α) then that dependent variable doesn t contribute to the difference in canonical variate values. But, you may wish to give some attention to dependent variables with scores that do differ significantly (p<α). Post-hoc comparisons of these dependent variable s means amount to nothing more than the t-tests used for post-hoc analyses of ANOVA results, described in Chapter 7. When results of these tests indicate significant differences between means, you know that that scores for this component of the canonical variate helps to account for its significantly different canonical variate means.

11 ANCOVA and MANCOVA with SPSS If you know how to use SPSS s Univariate window to perform a multi-way ANOVA, then you simply need to add a step to the process for an ANCOVA. Similarly, performing a MANCOVA requires just one more step than performing a MANOVA using SPSS s Multivariate window. In both cases, this step involves the identification of covariates. Both the Univariate and the Multivariate windows contain a box labeled Covariate(s). The entire process for performing an ANCOVA in SPSS, then, requires six steps. 1. Choose Compare Means from the Analyze pull-down menu. 2. Choose General Linear Model from the options provided. A new menu should appear to the right of the pull-down menu. Select Univariate from the new menu. A Univariate window should appear on the screen. FIGURE 9.9 SPSS UNIVARIATE WINDOW The user performs an ANCOVA by selecting the appropriate variable names from those listed in the box on the left side of the window. The names of the independent variables should be moved to the fixed factor(s) box. The name of the Dependent Variable and covariate(s) should also be moved to the appropriate areas in the center of the window. 3. Highlight the name of the dependent variable from the list appearing in the upper left corner of the window. Click on the arrow to the left of the Dependent Variable box. The name of the variable should move to this box. 4. Highlight the name of one independent variable from the list appearing in the upper left corner of the window. Click on the arrow to the left of the Fixed Factor(s) box. The

12 name of the variable should move to this box. Continue this process with each independent variable name until they all appear as fixed factors. 5. Highlight the name of one covariate from the list appearing in the upper left corner of the window. Click on the arrow to the left of the Covariate(s) box. The name of the variable should move to this box. Continue this process with each independent variable name until they all appear as covariates. 6. If you would like your output to include descriptive statistics, select the Options button, located on the right side of the window. A new window, entitled Univariate: Options should appear. Select Descriptive Statistics from the Display portion of this window. Then, click Continue to return to the One-Way ANOVA window. Failing to complete this step will still produce valid ANCOVA results. 7. Click OK. Assuming you performed Step #6, above, the SPSS output for an ANCOVA begins with descriptive statistics for each independent-variable category. The results of the significance test appear in the table entitled Tests of Between-Subjects Effects. The Corrected Model values in this table provide the ANCOVA s adjusted sum of squares and the resulting F and significance (p) values. Example 9.18 Selected SPSS Output for Oneway ANCOVA. SPSS output for an analysis of sample data from the situation presented in Example 9.6, which addresses the effectiveness of relaxation techniques, appears as follows. Descriptive Statistics Dependent Variable:change technique Mean Std. Deviation N yoga meditation biofeedback Total Dependent Variable:change Tests of Between-Subjects Effects Source Type III Sum of Squares Df Mean Square F Sig.

13 Corrected Model a Intercept health technique Error Total Corrected Total a. R Squared =.033 (Adjusted R Squared =.013) TABLE 9.15 AND TABLE 9.16 SELECTED SPSS OUTPUT FOR ONEWAY ANCOVA Descriptive statistics for each category of the independent variable appear in Table 9.15, labeled Descriptive Statistics. The Tests of Between-Subjects Effects table (Table 9.16) lists both the independent variable, in this case, technique, and the covariate, in this case health, as predictors of the dependent variable. The values used to determine whether changes in heart rate differ significantly with respect to the independent variable and considering the possible effects of the covariate appear in the top row of this table. According to results of this analysis, those exposed each of the three relaxation techniques did not experience significantly different changes in heart rate. The p value of.183 lies above the standard α of.05 as well as above an elevated α of.10, indicating that one would accept the null hypothesis of equality at these levels of significance. The analysis considered differences in the overall health of patients in the three independent-variable conditions when calculating these results, hence the designation of a Type III Sum of Squares value in the Tests of Between-Subjects Effects table. The process used to request and analyze SPSS results of an ANCOVA translate easily into a MANCOVA. Performing a MANCOVA in SPSS requires the same steps, only you would need to use SPSS s Multivariate, rather than Univariate window. In the Multivariate window, you can identify as many dependent variables as needed for the analysis. SPSS assembles the values for the dependent variables into canonical variate scores. By inputting names of covariates into the Covariate(s) box, you tell SPSS to consider the roles of these covariates upon the relationship between the independent variables and the canonical variate. The MANCOVA output that results contains a Multivariate Tests table. This table resembles the Multivariate Tests table produced for a MANCOVA, however, it also includes the names of covariates. Assuming you wish to consider results based upon the Wilks Lambda procedure for obtaining F, you should focus upon values in this row of the table. A p-value that exceeds α indicates significant differences between mean canonical variate values for the covariate-biased independent-variable categories.

14 Discriminant Analysis with SPSS Rather than working with pre-existing classifications of subjects, as the other tests in Chapter 9 do, a discriminant analysis attempts to create classifications. To conduct a discriminant analysis in SPSS, therefore, you cannot use the General Linear Model function. The following process allows you to use continuous values to predict subjects group placements. 1. Choose the Classify option in SPSS Analyze pull-down menu. 2. Identify your desired type of classification as Discriminant. Choose Discriminant from the prompts given. A window entitled a window entitled Discriminant Analysis should appear. FIGURE 9.9 SPSS DISCRIMINANT ANALYSIS WINDOW The user identifies the variables involved in a one-way discriminant analysis by selecting their names from those listed on the left side of the Discriminant Analysis window. SPSS performs the test using variables with names placed into the Independents and variables with names placed into the Grouping Variables box. 3. In this window, you can define the variables involved in the analysis as follows a. Move the name of the categorical dependent variable from the box on the left to the Grouping Variable box. You must also click on the Define Range button below this box and type the values for the lowest and highest dummy-variable values used to identify groups. b. Identify the continuous measure(s) used to predict subjects categories by moving the names of the predictor(s) to the Independents box. 4. Click OK. The Discriminant Analysis Independents Variable box allows you to identify more than one predictor of subjects categories. Inputting more than one independent variable leads to a multiple discriminant analysis. The analysis presented in Chapter 9 s examples, though, use a single independent variable.

15 Example SPSS Output for Discriminant Analysis Tables 9.18 through 9.21 show the some of the output from applying these steps to imaginary data for the acreage and fencing style example first presented in Example 9.9. As with the output for most tests of significance, SPSS first presents descriptive statistics and then follows with values that indicate predictability. Among these values is a measure of significance based upon the conversion of Wilks Lambda into F, as described in Section 9.5. Group Statistics Valid N (listwise) Fence Unweighted Weighted chain link acreage wrought iron acreage wood acreage vinyl acreage Total acreage Eigenvalues Functio n Eigenvalue % of Variance Cumulative % Canonical Correlation a a. First 1 canonical discriminant functions were used in the analysis. Wilks' Lambda Test of Functio n(s) Wilks' Lambda Chi-square Df Sig Standardized Canonical Discriminant Function Coefficients Function 1

16 Acreage 1 TABLE 9.18, TABLE 9.19, TABLE 9.20, AND TABLE 9.21 SPSS OUTPUT FOR DISCRIMINANT ANALYSIS The number of subjects in each grouping variable category appear as Group Statistics in Table 9.18 The remainder of the tables provide information regarding the predictability of these groups from continuous predictor variable values. The Eigenvalues table (Table 9.19) contains a correlation coefficient (See Chapter 8) representing the linear relationship between the predictor variable and the grouping variable. With the significance value in the Wilks Lambda table (Table 9.20) and the coefficient in the Standardized Canonical Discriminant Function Coefficients table (Table 9.21), the user can determine the strength of the relationship between variables. Of course, given the fact that this analysis involves only one independent variable, the output is relatively simplistic compared to the output for a multiple discriminant analysis. The canonical correlation shown in Table 9.19 amounts to the pairwise correlation between the two variables. For a multiple discriminant analysis, it would describe the linear relationship between the canonical variate (a combination of independent variables) and the grouping variable. Also, the coefficient of 1, shown in Table 9.21, implies a discriminating function of G=x. This equation suggests that all of the responsibility for predicting fencing style lies with acreage. Still, you can easily see, based upon the significance value in Table 9.20, that acreage sufficiently predicts the type of fencing used to enclose property. The p value of.004 indicates a significant relationship between acreage and fencing type at both α=.05 and α=.01. For evaluations that involve more predictors than that in Example 9.20 does, you can use output values in a variety of ways. In particular, researchers often use values in the Standardized Canonical Discriminant Function Coefficient table for more than just identifying the discriminating function. These values can signify the importance of each predictor variable in the relationship with the grouping variable. Because predictor variables with very small coefficients have weak linear relationships with the grouping variable, they likely add little to the predictability of the model. You may wish to perform another discriminant analysis, omitting the predictor variables with low coefficients, to determine whether you really need them to help classify subjects. If results of this analysis also indicate significance, then you know that their presence makes little difference in the ability to classify subjects. So, you do not have to regard them as contributors to the overall canonical predictor. This process allows you to limit your grouping variables to only those that truly help to predict subjects categories.

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