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2CHAPTER 6 
3MULTIVARIATE ANALYSIS OF VARIANCE 
4AND COVARIANCE 
5All of the statistical analysis techniques discussed to this point have involved only one depen- 
6dent variable. In this chapter, for the first time, we consider multivariate statistics—statistical proce- 
7dures that involve more than one dependent variable. The focus of this chapter is on two of the most 
8widely used multivariate procedures: the multivariate variations of analysis of variance and analysis of 
9covariance. These versions of analysis of variance and covariance are designed to handle two or more 
10dependent variables within the standard ANOVA/ANCOVA designs. We begin by discussing 
11multivariate analysis of variance in detail, followed by a discussion of the application Of covariance 
12analysis in the multivariate setting. 
131. MANOVA 
14Like ANOVA, multivariate analysis of variance (MANOVA) is designed to test the significance 
15of group differences. The only substantial difference between the two procedures is that MANOVA can 
16include several dependent variables, whereas ANOVA can handle only one DV. Oftentimes, these mul- 
17tiple dependent variables consist of different measures Of essentially the same thing (Aron & Aron, 
181999), but this need not always be the case. At a minimum, the DVs should have some degree of linear- 
19ity and share a common conceptual meaning (Stevens, 1992). They should "make sense" as a group of 
20variables. As you will soon see, the basic logic behind a MANOVA is essentially the same as in a uni- 
21variate analysis of variance. 
22SECTION 6.1 PRACTICAL VIEW 
23Purpose 
24The clear advantage of a multivariate analysis of variance over a univariate analysis of variance 
25is the inclusion of multiple dependent variables. Stevens (1992) provides two reasons why a researcher 
26should be interested in using more than one DV when comparing treatments or groups based on differ- 
27ing characteristics: 
28(1) Any worthwhile treatment or substantial characteristic will likely affect subjects in more than 
29one way; hence, the need for additional criterion (dependent) measures. 
30(2) The use of several criterion measures permits the researcher to obtain a more "holistic" picture, 
31and therefore a more detailed description, of the phenomenon under investigation (pp. 151-152). 
32This stems from the idea that it is extremely difficult to obtain a "good" measure of a trait (e.g., 
33Chapter 6 Multivariate Analysis of Variance and Covariance 
34math achievement, self-esteem, etc.) from one variable; multiple measures on variables repre- 
35senting a common characteristic are bound to be more representative of that characteristic. 
36ANOVA tests whether mean differences among k groups on a single DV are significant, or: 
37likely to have occurred by chance. However, when we move to the multivariate situation, the multiple 
38DVs are treated in combination. In other words, MANOVA tests whether mean differences among k 
39groups on a combination of Drs are likely to have occurred by chance. As palt of the actual analysis, a 
40"new" DV is created. This new DV is, in fact, a linear combination of the original measured DVs, com- 
41bined in such a way as to maximize the group differences (i.e., separate the k groups as much as possi- 
42ble). The new DV is created by developing a linear equation where each measured DV has an associ- 
43ated weight and, when combined and summed, creates maximum separation Of group means with re- 
44spect to the new DV: 
45at Yl + a2Y2 + any n, 
46(Equation 6. l) 
47where Yr, is an original DV, an is its associated weight, and n is the total number of original measured 
48DVs. An ANOVA is then conducted on this newly created variable. 
49Let us consider the following example: Assume we wanted to investigate the differences in 
50worker productivity, as measured by income level (DVI) and hours worked (DV2), for individuals of 
51different age categories (IV). Our analysis would involve the creation of a new DV, which would be a 
52linear combination (DVWV) of our subjects' income levels and numbers of hours worked that maximizes 
53the separation of our age category groups. Our new DV would then be subjected to a univariate 
54ANOVA by comparing variances on DVn„ for the various groups as defined by age category. 
55One could also have a factorial design that would involve multiple IVs as well as 
56multiple DVs. In this situation, a different linear combination of DVs is formed for each main effect 
57and each interaction (Tabachnick & Fidell, 1996). For example, we might consider investigating the 
58effects of gender (IV l) and job satisfaction (IV:) on employee income (DVI) and years of education 
59(DV2). Our analysis would actually provide three new DVs—the first linear combinåtion would maxi- 
60mize the separation between males and females (WI), the second linear combination would maximize 
61the separation among job satisfaction categories (IV2), and the third would maximize the separation 
62among the various cells of the interaction between gender and job satisfaction. 
63At this point, one might be inclined to question why a researcher would want to engage in a mul- 
64tivariate analysis of variance, as opposed to simply doing a couple of comparatively simple analyses of 
65variance. MANOVA has several advantages over its simpler univariate counterpart (Tabachnick & Fi- 
66dell, 1996). First, as previously mentioned, by measuring several DVs instead of only one, the chances 
67of discovering what actually changes as a result of the differing treatments or characteristics (and any 
68interactions) improves immensely. If we wanted to know what measures of work productivity are af- 
69fected by gender and age, we improve our chances of uncovering these effects by including hours 
70worked as well as income level. 
71There are also several statistical reasons for preferring a multivariate analysis over a univariate 
72one (Stevens, 1992). A second advantage is that, under certain conditions, MANOVA may reveal dif- 
73ferences not shown in separate ANOVAs (Tabachnick & Fidell, 1996; Stevens, 1992). Assume we have 
74a one-way design, with two levels on the IV and two DVs. If separate ANOVAs are conducted on two 
75DVs, the distributions for each of the two groups (and for each DV) might overlap sufficiently, such that 
76a mean difference probably would not be found. However, when the two DVs are considered in combi- 
77nation with each other, the two groups may differ substantially and could result in a statistically signifi- 
78Chapter 6 Multivariate Analysis Of Variance and Covariance 
79cant difference between groups. Therefore, a MANOVA may sometimes be more powerful than sepa- 
80rate ANOVAs. 
81Third, the use of several univariate analyses leads to a greatly inflated overall Type I error rate. 
82consider a simple design with one IV (with two levels) and five DVs. If we assume that we wanted to 
83test for group differences on each of the DVs (at a .05 level of significance), we would have to con- 
84duct five univariate tests. Recall that at an a-level of .05, we are assuming a 95% chance of no Type I 
85errors. Because of the assumption of independence, we can multiply the probabilities. The effect of 
86these error rates is compounded over all of the tests such that the overall probability of not making a 
87Type I error becomes: 
88- 77 
89In other words, the probability of at least one false rejection (i.e., Type I error) becomes 
90.23 
91.77 - 
92which, as we all know, is an unacceptably high rate of possible statistical decision error (Stevens, 1992). 
93Therefore, using this approach of fragmented univariate tests results in an overall error rate which is en- 
94tirely too risky. The use of MANOVA includes a condition that maintains the overall error rate at the 
95.05 level, or whatever a-level is pre-selected (Harris, 1998). 
96Finally, the use of several univariate tests ignores some very important information. Recall that 
97if several DVs are included in an analysis, they should be correlated to some degree. A multivariate 
98analysis incorporates the intercorrelations among DVs into the analysis (this is essentially the basis for 
99the linear combination ofDVs). 
100The reader should keep in mind, however, that there are disadvantages in the use of NIANOVA. 
101The main disadvantage is the fact that MANOVA is substantially more complicated than ANOVA (Ta- 
102bachnick & Fidell, 1996). In the use of MANOVA, there are several important assumptions that need to 
103be met. Furthermore, the results are sometimes ambiguous with respect to the effects of IVs on individ- 
104ual DVs. Finally, situations in which MANOVA is more powerful than ANOVA, as discussed a few 
105paragraphs ago, are quite limited; often the multivariate procedure is much less powerful than ANOVA 
106(Tabachnick & Fidell, 1996). It has been recommended that one carefully consider the need for addi- 
107tional DVs in an analysis in light of the added complexity (Tabachnick & Fidell, 1996). 
108In the unlvariate case, the null hypothesis stated that the population means are equal: 
109The calculations for MANOVA, however, are based on matrix algebra (as opposed to scalar algebra). 
110The null hypothesis in MANOVA states that the population mean vectors are equal: 
111For the univariate analysis of variance, recall that the F-statistic is used to test the tenability of 
112the null hypothesis. This test statistic is calculated by dividing the variance between the groups by the 
113variance within the groups. There are several available test statistics for multivariate analysis of vari- 
114ance, but the most commonly used criterion is Wilks' Lambda (A). (Other test statistics for MANOVA 
115include pillai's Trace, Hotelling's Trace, and Roy's Largest Root.) Without going into great detail, 
116Wilks' Lambda is obtained by calculating IWI (a measure of the within-groups sum-of-squares and 
117cross-products matrix—a multivariate generalization of the univariate sum-of-squares within tSSwl) and 
118Chapter 6 Multivariate Analysis Of Variance and Covariance 
119dividing it by ITI (a measure of the total sum-of-squares and cross-products matrix—also a multivariate 
120generalization, this time of the total sum-of-squares (SST)). The obtained value Of Wilks' A ranges from 
121zero to one. It is important to note that Wilks' A is an inverse criterion; i.e., the smaller the value of A, 
122the more evidence for treatment effects or group differences (Stevens, 1992). The reader should realize 
123that this is the opposite relationship that F has to the amount of treatment effect. 
124In conducting a MANOVA, one first tests the overall multivariate hypothesis (i.e., that all 
125groups are equal on the combination of DVs). This is accomplished by evaluating the significance of 
126the test associated with A. If the null hypothesis is retained, it is common practice to stop the interpreta- 
127tion of the analysis at this point and conclude that the treatments or conditions have no effect on the 
128DVs. However, if the overall multivariate test is significant, the researcher then would likely wish to 
129discover which of the DVs is being affected by the IV(s). To accomplish this, One conducts a series of 
130univariate analyses of variance on the individual DVs. This will undoubtedly result in multiple tests Of 
131significance, which will result in an inflated Type I error rate. 
132To counteract the potential of an inflated error rate due to multiple ANOVAs, an adjustment 
133must be made to the alpha level used for the tests. This Bonferroni-type adjustment involves setting a 
134more stringent alpha level for the test of each DV so that the alpha for the set of DVs does not exceed 
135some critical value (Tabachnick & Fidell, 1996). That critical value for testing each DV is usually the 
136overall a-level for the analysis (e.g., a .05) divided by the number ofDVs. For example, if one had 
137three DVs and wanted an overall a equal to .05, each univariate test could be conducted at a — .016, 
138since .05/3—.0167. One should note that rounding down is necessary to create an overall alpha less than 
139.05. The following equation may be used to check adjustment decisions: 
140where the overall error rate (a) is based on the error rate for testing the first DV (al), the second DV 
141and all others to the pth DV (ap). All alphas can be set at the same level, or more important DVs 
142can be given more liberal alphas (Tabachnick & Fidell, 1996). 
143Finally, for any univariate test of a DV that results in significance, one then conducts univariate 
144post hoc tests (as discussed in Chapter 4) in order to identify where specific differences lie (i.e., which 
145levels Of the IV are different from which other levels). TO summarize the analysis procedure for 
146MANOVA, a researcher would follow these steps: 
147(1) Examine the overall multivariate test of significance—if the results are significant, proceed to 
148the next step; if not, stop. 
149(2) Examine the univariate tests of individual D Vs—if any are significant, proceed to the next 
150step; if not, stop. 
151(3) Examine the post hoc tests for individual D Vs. 
152Sample Research Questions 
153In our first sample study in this chapter, we are concerned with investigating differences in 
154worker productivity, as measured by income level (DVI) and hours worked (DVD, for individuals of 
155different age categories one-way MANOVA design. Therefore, this study would address the 
156following research questions: 
157(l) Are there significant mean differences in worker productivity (as measured by the combination 
158of income and hours worked) for individuals of different ages? 
159(2) Are there significant mean differences in income levels for individuals of different ages? 
160122 
161Chapter 6 Multivariate Analysis Of Variance and Covariance 
162(2a) If so, which age categories differ? 
163(3) Are there significant mean differences in hours worked for individuals of different ages? 
164(3a) If so, which age categories differ? 
165Our second sample study will demonstrate a two-way MANOVA where we investigate the gen- 
166der (IVI) and job satisfaction (IV2) differences in income level (DVD and years of education (DVD. 
167One should note the following questions address the MANOVA analysis, univariate ANOVA analyses, 
168and post hoc analyses: 
169(1) 
170(2) 
171a. 
172b. 
173a. 
174b. 
175b. 
176C. 
177Are there significant mean differences in the combined DV of income and years Of educa- 
178tion for males and females? 
179Are there significant mean differences in the combined DV of income and years of educa- 
180tion for different levels of job satisfaction? Ifso, which job satisfaction categories differ? 
181Is there a significant interaction between gender and job satisfaction on the combined DV 
182Of income and years of education? 
183Are there significant mean differences on income between males and females? 
184Are there significant mean differences on income between different levels of job satisfac- 
185tion? If so, which job satisfaction categories differ? 
186Is there a significant interaction between gender and job satisfaction On income? 
187Are there significant mean differences in years of education between males and females? 
188Are there significant mean differences in years of education among different levels of job 
189satisfaction? If so, which job satisfaction categories differ? 
190Is there a significant interaction between gender and job satisfaction on years of educa- 
191tion ? 
192SECTION 6.2 ASSUMPTIONS AND LIMITATIONS 
193Since we are introducing our first truly multivariate technique in this chapter, we have a ftlew" 
194set of statistical assumptions to discuss. They are new in that they apply to the multivariate situation; 
195however, they are quite analogous to the assumptions for univariate analysis of variance, which we have 
196already examined (see Chapter 4). For multivariate analysis of variance, these assumptions are: 
197(l) The observations within each sample must be randomly sampled and must be independent Of 
198each other. 
199(2) The observations on all dependent variables must follow a multivariate normal distribution in 
200each group. 
201(3) The population covariance matrices for the dependent variables in each group must be equal 
202(this assumption is often referred to as the homogeneity of covariance matrices assumption or 
203the assumption of homoscedasticity). 
204(4) The relationships among all pairs ofDVs for each cell in the data matrix must be linear. 
205As a reminder to the reader, the assumption of independence is primarily a design issue, not a 
206statistical one. Provided the researcher has randomly sampled and assigned subjects to treatments, it is 
207usually safe to believe that this assumption has not been violated. We will focus our attention on the 
208assumptions of multivariate normality, homogeneity of covariance matrices, and linearity. 
209Chapter 6 Multivariate Analysis of Variance and Covariance 
210Methods of Testing Assumptions 
211As discussed in Chapter 3, multivariate normality implies that the sampling distribution of the 
212means of each DV in each cell and all linear combinations of DVs are normally distrlbuted (Tabachnick 
213& Fidell, 1996). Multivariate normality is a difficult entity to describe and even more difficult to assess. 
214Initial screening for multivariate normality consists of assessments for univariate normality (see Chapter 
2153) for all variables, as well as examinations of all bivariate scatter-plots (see Chapter 3) to check that 
216they are approximately elliptical (Stevens, 1992). Specific graphical tests for multivariate normality do 
217exist, but are not available in standard statistical software packages (Stevens, 1996) and will not be dis. 
218cussed here. 
219It is probably most important to remember that both ANOVA and MANOVA are robust to mod- 
220erate violations of normallty, provided the violation is created by skewness and not by outliers (Tabach- 
221nick & Fidell, 1996). With equal or unequal sample sizes and only a few DVs, a sample size of about 
22220 in the smallest cell should be sufficient to ensure robustness to violations of univariate and multivari- 
223ate normality. If it is determined that the data have substantially deviated from normal, transformations 
224of the original data should be considered. 
225Recall that the assumption of equal covariance matrices (i.e., homoscedasticity) is a necessary 
226condition for multivariate normality (Tabachnick & Fidel), 1996). The failure of the relationship be- 
227tween two variables to be homoscedastic is caused either by the nonnormality of One of the variables or 
228by the fact that one of the variables may have some sort of relationship to the transfortnation of the other 
229variable. Therefore, checking for univariate and multivariate norrnality is a good starting point for as- 
230sessrng possible violations of homoscedasticity. Specifically, possible violations of this assumption may 
231be assessed by intemreting the results of Box's Test. The reader should note that a violation of the as- 
232sumption of homoscedasticity, similar to a violation of homogeneity, will not prove, fatal to an analysis 
233(Tabachnick & Fidell, 1996; Kennedy & Bush, 1985). However, the results will be greatly' improved if 
234the heteroscedasticity is identified and corrected (Tabachnick & Fidell, 1996) by means of data trans- 
235formations. On the other hand, if homogeneity of variance-covariance is violated, a more robust multi- 
236variate test statistic, Pillai's Trace, can be selected when interpreting the multivariate results. 
237Linearity is best assessed through inspection of bivariate scatterplots. 
238If both variables in the 
239pair are normally distributed and linearly related, the shape of the scatterplot should be elliptical. If one 
240of the variables is not normally distributed, the relationship will not be linear and the scatter-plot between 
241the two variables will not appear oval shaped. As mentioned in Chapter 3, assessing linearity by means 
242of bivariate scattelplots is an extremely subjective procedure. In situations where nonlinearity between 
243variables is apparent, the data can once again be transformed in order to enhance the linear relationship. 
244SECTION 6.3 PROCESS AND LOGIC 
245The Logic Behind MANOVA 
246As previously mentioned, the calculations for MAN()VA somewhat parallel those for a univari- 
247ate ANOVA, although they exist in multivariate form (i.e., they rely on matrix algebra). Since several 
248variables are involved in this analysis, calculations are based on a matrix of values, as opposed to the 
249mathematical manipulations of a single value. Specifically, the matrix used in the calculations is the 
250sum-of-squares and cross-products (SSCP) matrix, which you will recall is the precursor to the variance- 
251covariance matrix (see Chapter l) 
252In univariate ANOVA, recollect that the calculations are based on a partitioning Of the total 
253sum-of-squares Into the sum-of-squares between the groups and the sum-of-squares within the groups: 
254Chapter 6 Multivariate Analysis Of Variance and Covariance 
255+ SSw,thi 
256In MANOVA. the calculations are based on the corresponding matrix analogue (Stevens, 1992), 
257in which the total sum-of-squares and cross-products matrix (T) is partitioned into a between sum-Of- 
258squares and cross-products matrix (B) and a within sum-of-squares and cross-products matrix (W): 
259SSCPBet..e„ + 
260(Equation 6.2) 
261Wilks' Lambda (A) is then calculated by using the Sort of generalized variance 
262for an entire set Of variables—of the SSCP matrices (Stevens, 1992). The resulting formula for A be. 
263comes: 
264ITI 
265(Equation 6.3) 
2661B WI 
267If there is no treatment effect Or group differences, then B O and A 1 indicating no differences be- 
268tween groups on the linear combination of DVs; whereas, if B were very large (i.e., substantially greater 
269than 0), then A would approach 0, indicating significant group differences on the combination of DVs. 
270As in all of our previously discussed ANOVA designs, we can again obtain a measure of 
271strength of association, or effect size. Recall that eta squared (i) is a measure of the magnitude of the 
272relationship between the independent and dependent variables and is interpreted as the proportion Of 
273variance in the dependent variable explained by the independent variable(s) in the sample. For 
274MANOVA, eta squared is obtained in the following manner: 
275In the multivariate situation, is interpreted as the variance accounted for in the best linear combina- 
276tion of DVs by the IV(s) and/or interactions of IVs. 
277Interpretation Of Results 
278The MANOVA procedure generates several test statistics to evaluate group differences on the 
279combined DV: Pillai's Trace, Wilks' Lambda, Hotelling's Trace, and Roy's Largest Root. When the 
280IV has only two categories, the F test for Pillars Trace, Wilks' Lambda, and Hotelling's Trace will be 
281identical. When the IV has three or more categories, the F test for these three statistics will differ 
282slightly but will maintain consistent significance or non-significance. Although these test statistics may -J 
283vary only slightly, Wilks' Lambda is the most commonly reported MANOVA statistic. Pillai's Trace is 
284used when homogeneity of variance-covariance is in question. If two or more IVs are included in the 
285analysis, factor interaction must be evaluated before main effects. 
286In addition to the multivariate tests, the output for MANOVA typically includes the test for ho- 
287mogeneity of variance-covariance (Box's Test), univariate ANOVAs, and univariate post hoc tests. 
288Since homogeneity of variance-covariance is a test assumption for MANOVA and has implications in 
289how to interpret the multivariate tests, the results Of Box's Test should be evaluated first. Highly sensi- 
290tive to the violation of normality, Box's Test should be interpreted with caution. Typically, if Box's 
291Test is significant at p•z.001 and group sample sizes are extremely unequal, then robustness cannot be 
292Chapter 6 Multivariate Analysis Of Variance and Covariance 
293assumed due to unequal variances among groups (Tabachnick & Fidell, 1996). In such a situation, a 
294more robust MANOVA test statistic, Pillai's Trace, is utilized when interpreting the MANOVA results, 
295If equal variances are assumed, Wilks' Lambda is commonly used as the MANOVA test statistic. Once 
296the test statistic has been determined, factor interaction (F ratio and p value) should be assessed if two Or 
297more IVs are included in the analysis. Like two-way ANOVA, if interaction is significant, then infer- 
298ences drawn from the main effects are limited. If factor interaction is not significant, then one should 
299proceed to examine the F ratios and p values for each main effect. Whemmul@yariate significance is 
300found, the univariate ANOVA results can indicate the degree to whigtvgroups@iffer for each DY2_A 
301Göieconservative alpha level should be applied using the Bonferroni adjustment. Post hoc results can 
302then indicate which groups are significantly different for the DV if univanate significance is found for 
303that particular DV. 
304In summary, the first step in interpreting the MANOVA results is to evaluate the Box's Test. If 
305homogeneity of variance-covariance is assumed, utilize the Wilks' Lambda statistic when interpreting 
306the multivariate tests. If the assumption of equal variances is violated, use Pillai's Trace. Once the mul- 
307tivariate test statistic has been identified, examine the significance (F ratios and p values) of factor inter- 
308action. This is necessary only if two or more IVs are included. Next evaluate the F ratios and p-VaÉes 
309for each factor's main effect. If multivariate significance is found, interpret the univariate ANOVA re- 
310suits to determine significant group differences for each DV. If univariate significance is revealed, ex- 
311amine the post hoc results to identify which groups are significantly different for each DV. 
312For our example that investigates age category (agecat4) differences in respondent's income 
313(rincom91) and hours worked per week (hrs l), data were screened for missing data and outliers and then 
314examined for fulfillment of test assumptions. Data screening led to the transformation of rincom91 to 
315rincom2 in order to eliminate all cases with income equal to zero and cases equal to or exceeding 22. 
316Hrs/ was also transformed to hrs2 as a means of reducing the number of outliers; those less than or 
317equal to 16 were recoded 17, and those greater than Or equal to 80 were recoded 79. Although normality 
318Of these transformed variables is still questionable, group sample sizes are quite large and fairly equiva- 
319lent. Therefore, normality will be assumed. Linearity of the two DVs was then tested by creating a 
320scatterplot and calculating the Pearson correlation coefficient. Results indicate a lineår relationship. 
321Although the corTelation coefficient is statistically significant, it is still quite low 
322last assumption, homogeneity Of variance-covariance, will be tested within MANOVA. Thus, 
323MANOVA was conducted utilizing the Multivariate procedure. The Box's Test (see Figure 6.1) 
324reveals that equal variances can be assumed, F(9, pe.648; therefore, Wilks' Lambda 
325will be used as the test statistic. Figure 6.2 presents the MANOVA results. The Wilks' Lambda criteria 
326indicates significant group differences in age category with respect to income and hours worked per 
327week, Wilks' A—.909, F.OOI, multivariate +.046. Univariate ANOVA results (see 
328Figure 6.3) were interpreted using a more conservative alpha level @.025). Results reveal that age 
329category significantly differs for only income (H3, pc.OOl, partial and not hours 
330worked per week (F(3, p—.919, partial 7/—.001). Examination of post hoc results reveal that 
331inconte of those 18-29 years of age significantly differs from all other age categories (see Figure 6.4). 
332In addition, income for individuals 30-39 years differ from those 40-49 years. 
333Chapter 6 Multivariate Analysis of Variance and Covariance 
334Figure 6.1 Box's Test for Homogeneity of Variance-Covariance. 
335Box's Test of Equality Of Covariance Matrices a 
3366.936 
337dfl 
3382886561 
339Tests the null hypothesis that the observed covariance 
340matrices of the dependent variables are equal across groups. 
341Design: Intercept*AGECAT4 
342gcx's test ig 
343Significant, 
344Lambda 
345u argd 
346Figure 6.2 Multivariate Tests for Income and Hours Worked by Age Category. 
347Multivariate TestsC 
348Hypothesi 
349ntercep 
350AGECAT4 
351Wilks' Lambda 
352Hotelling's Trace 
353ROY'S Largest Root 
354Pillars Trace 
355Wilks• Lambda 
356Hotelling•s Trace 
357Ray'S Largest Root 
358Value 
35922.080 
36022_080 
361Error df 
362000 
363680.000 
364Ego 
3651362.000 
3661360.000 
3671358000 
368681.000 
369750127? 
3702723 
3717507.27? 
37210.791 
37311 ,035a 
37411.279 
37522.457b 
3762,000 
3772000 
3782.000 
3792000 
3806000 
3816.000 
3826.000 
3833.000 
384a. Exact statistic 
385b, The statistic is an upper bound on F that yields a lower bound on the significance level. 
386c. Design: Intercept•AGECAT4 
387Indicates that age 
388Category SignifF 
389car-fly differs for 
390the cnmbhed DV. 
391Chapter 6 Multivariate Analysis of Variance and Covariance 
392Figure 6.3 Univariate ANOVA Summary Table. 
393Tests Of Between-Subjects Effects 
394S uarad 
3957864.965 
39610972.708 
39720.985 
398Intercept 
399AGEcnT4 
400C Total 
401R Squared 
402b. R Squared 
403Type Ill 
404ndent Variable uares 
405S are 
406343005 
40721427 
408128493.5 
4091410954 
410343005 
41121427 
41216.337 
413128,588 
414Indicates tat 
415category 
416effects 
417but NOT 
418HRS2 
419RINCOM2 
420HRS2 
421RINCOM2 
422HRS2 
423RINCOM2 
424HRS2 
425RINCOM2 
426HRS2 
427RINCOM2 
428HRS2 
4291029,otæ 
43064.281b 
4311284g3E 
4321410954 
4331029.016 
43464281 
43511125807 
43687568.119 
437149966.0 
4381575151 
43912154823 
4407632.400 
441_C85 (Adjusted R squared 
442.00 i (Adjusted R Squared 
443Writing Up Results 
444Once again, any data transformations utilized to increase the likelihood of fulfilling test assump- 
445tions should be reported in the summary of results. The summary should then report the results from the 
446multivariate tests by first indicating the test statistic utilized and its respective value and then reporting 
447the F ratio, degrees of freedom, p value, and effect size for each IV main effect. If follow-up analysis 
448was conducted using Univariate ANOVA, these results should be summarized next. Report the F ratio, 
449degrees of freedom, p value, and effect size for the main effect on each DV. Utilize the post hoc results 
450to indicate which groups were significantly different within each DV. Finally, you may want to create a 
451table of means and standard deviations for each DV by the IV categories. In summary, the MANOVA 
452results narrative should address the following: 
453(l) Subject elimination and/or variable transformation; 
454(2) MANOVA results (test statistic, F-ratio, degrees of freedom,p-value, and effect size); 
455(a) Main effects for each IV on the combined DV; 
456(b) Main effect for the interaction between IVs; 
457(3) Univariate ANOVA results (F-ratio, degrees of freedom, p-value, and effect size); 
458(a) Main effect for each IV and DV; 
459(b) Comparison of means to indicate which groups differ on each DV; 
460(4) Post hoc results (mean differences and levels of significance). 
461Utilizing our previous example, the following statement applies the results from Figures 6.1—6.4. 
462A one-way multivariate analysis of variance (MANOVA) was conducted to determine age cate- 
463gory differences in income and hours worked per week. Prior to the test, variables were trans- 
464formed to eliminate outliers. Cases with income equal to zero or equal to Or exceeding 22 were 
465eliminated. Hours worked per week was also transformed; those less than or equal to 16 were 
466Chapter 6 Multivariate Analysis OfVariance and Covariance 
467recoded 1 7 and those greater than or equal to 80 were recoded 79. MANOVA results revealed 
468significant differences among the age categories on the dependent variables, Wilks' 
469p€.001, multivariate Analysis of variance (ANOVA) was conducted 
470on each dependent variable as a follow-up test to MANOVA. Age category differences were 
471significant for income, F(3, pc-.001, partial Differences in hours worked 
472per week were not significant, F(3, p:.919, partial +.001. The Bonferroni post 
473hoc analysis revealed that income of those 18-29 years significantly differs from all other age 
474categories. In addition, income for individuals 30-39 years differs from those 40-49. Table I 
475presents means and standard deviations for income and hours worked per week by age category. 
476Table 1 Means and Standard Deviations for Income and Hours Worked per Week 
477by Age Category 
478Income 
479SD 
4804.14 
4813.88 
4823.87 
4834.42 
484Hours Worked per Week 
485Age 
48618-29 years 
48730-39 years 
48840-49 years 
48950+ years 
49011.87 
49114.03 
49215.32 
49314.96 
49446.32 
49547.03 
49646.49 
49746.33 
498SD 
49910.32 
50011.42 
50111.75 
50211.51 
503Figure 6.4 Post Hoc Results for Income and Hours Worked by Age Category. 
504u Rip 
505t Variable 
506'8-29 
507std, 
50812913 
5091.1145 
5101,221' 
51112913 
5121.1145 
51318-29 
5141 Bag 
5151B.2g 
5162 30-39 
5173 404 
5181 a.29 
5193 40-4g 
5202 30-3-9 
521aries 
522Wan 
523-2_1643• 
524-34570 • 
5252.1643' 
526-tn32• 
5273.0576 • 
5285.92* -03 
529.203 
530-3.3513 
531-4.6754 
53243960 
533-2.3443 
534-2.0776 
53522397 
536-l.ssog 
53740412 
538-3 SB61 
539-24010 
540-25258 
541•3.2473 
542-SMS'O 
543_&6sn 
544.3483-g 
545.2,2297 
546-1'846 
54734513 
548-2421 
5492.3443 
5501,550B 
5514360 
55220776 
553.etoa 
55426187 
5553 Q473 
5563 $572 
5574_0412 
55831570 
5591.2211 
560129 
561• is at .05 
562Chapter 6 Multivariate Analysis Of Variance and Covariance 
563SECTION 6.4 MANOVA SAMPLE STUDY AND ANALYSIS 
564This section provides a complete example that applies the entire process Of conducting 
565MANOVA: development of research questions and hypotheses, data screening methods, test methods, 
566interpretation of output, and presentation Of results. The SPSS data set gssft.sav is utilized. Our previ- 
567ous example demonstrates a one-way MANOVA, while this example will present a two-way 
568MANOVA. 
569Problem 
570This time, we are interested in determining the degree to which gender and job satisfaction af- 
571fects income and years Of education among employees. Since two IVs are tested in this analysis, ques- 
572tions must also take into account the possible interaction between factors. The following research ques- 
573tions and respective null hypotheses address the multivariate main effects for each IV and the possible 
574interaction between factors. 
575Research Questions 
576RQI: DO income and years of educa- 
577tion differ by gender among employees? 
578RQ2: Do income and years Of educa- 
579tion differ by job satisfaction among 
580employees? 
581RQ3: Do gender and job satisfaction 
582Null Hypotheses 
583HOI: Income and years of education 
584will not differ by gender among ern- 
585ployees. 
586H02: Income and years Of education 
587will not differ by job satisfaction among 
588employees. 
589H03: Gender and job satisfaction will 
590interact in the effect on income and not interact in the effect on income and 
591years of education? 
592years of education. 
593Both IVs are categorical and include gender (sex) and job satisfaction (satjob). One should note 
594that satjob represents four levels: very satisfied, moderately satisfied, a little dissatisfied, and very dis- 
595satisfied. The DVs are respondent's income (rincom2) and years of education (educ); both are quantita- 
596five. The variable, rincom2, is a transformation of rincom91 from the previous example. 
597Method 
598Data should first be examined for missing data, outliers and fulfillment oftest assumptions. The 
599Explore procedure was conducted to identify outliers and evaluate normality. Boxplots (see Figure 
6006.5) indicate extreme values in educ. Consequently, educ was transformed to educ2 in order to elimi- 
601nate subjects with 6 years Of education or less. Explore was conducted again to evaluate normality. 
602Tests indicate significant non-normality for both rincom2 and educ2 in maliy categories (see Figure 6.6). 
603Since MANOVA is fairly robust to non-normality, no further transformations will be performed. How- 
604ever, the significant non-normality coupled with the unequal group sample sizes, as in this example, 
605may lead to violation of homogeneity of variance-covariance. The next step in examining test assump- 
606tions was to determine linearity between the DVs. A scatterplot was created; Pearson correlation coeffl- 
607cients were calculated (see Figure 6.7). Both indicate a linear relationship. Although the correlation 
608130 
609Chapter 6 Multivariate Analysis of Variance and Covariance 
610coefficient is significant, it is still fairly weak ('—337, pq001). The final test assumption of homogene- 
611ity of variance-covariance will be tested with the MANOVA procedure. MANOVA was then conducted 
612using Mul tivariate. 
613Figure 6.5 Boxplots for Years of Education by Gender and Job Satisfaction. 
614-10 
615Male 
616Respondent's Sex 
617satisfied 
618Job Satisfaction 
619Fem 
620A little dissatisn•e 
621Very dissatisfied 
6223 
623.10 
624Chapter 6 Multivariate Analysis of Variance and Covariance 
625Figure 6.6 Tests of Normality of Income and Years of Education. 
626Tests of Normality 
627OrCWSmirn 
628Some distributions 
629by job 
630Wisf»ation are 
631Sign i fC"t'y 
632educate" by ice 
633satisfaction are 
634significanty rcn• 
635gender distrib'* 
636tiors for income 
637and education are 
638Correlation 
639cient L-dicates 
640Tlatdnship. 
641EDUC2 
642SATJOE Job satisfaction 
643T Very s 
6442 Mod 
6453 A little dissatisfied 
6464 Very dissatisfied 
647I Veri 
6482 Mod satisfied 
6493 A little dissatisfied 
6504 Very dissatisfied 
651Statistic 
652This a lower of the true significance. 
653Lillielors Significance Correction 
654SEX Re 
6552 Female 
656EOI_JC2 1 Male 
6572 Female 
658Tests Of Nor m 
659Kolmo 
660dent's Sex Statistic 
661a. Lilliefors Significance Correction 
662Figure 6.7 Correlation Coefficients for Income and Years of Education. 
663Correlations 
664RINCOM2 
665681 
6661.000 
667Pearson 
668re ation 
669Sig. (2-taited) 
670RINCOM2 Pearson Correlation 
671Sig. (2-tailed) 
672EDUC2 
673.337• 
674• Correlat/on is significant at the 0.01 level (2-tai'ed). 
675Chapter 6 Multivariate Analysis Of Variance and Covariance 
676Output and Interpretation of Results 
677Figures 6.8 —6.11 present some of the MANOVA output. The Box's Test (see Figure 6.8) is not 
678significant and indicates that homogeneity of variance-covariance is fulfilled, F(21, 
679"2.201, so Wilks' Lambda test statistic will be used in interpreting the MANOVA results. The multi- 
680variate tests are presented in Figure 6.9. Factor interaction was then examined and revealed nonsignifi- 
681cance, F(6, p:.610, The main effects of job satisfaction (F(6, 
682p—.OOl, 022.017) and gender (F(2, '/—.024) were both significant. However, multi- 
683variate effect sizes are very small. Prior to examining the univariate ANOVA results, the alpha level 
684was adjusted to a—.025 since two DVs were analyzed. Univariate ANOVA results (see Figure 6.10) 
685indicate that income significantly differs for job satisfaction (F(3, '122.031) and gen- 
686der (F(l, i —.023). Years of education do not significantly differ for job satisfac- 
687tion (F(3, p—.089, F.OIO) or gender (F(l, pz.310, '122.002). Scheffé post hoc 
688results (see Figure 6.11) for income and job satisfaction indicate that individuals very satisfied signifi- 
689cantly differ from those with only moderate satisfaction. Figures 6.12 and 6.13 present the unadjusted 
690and adjusted group means for income and years of education. 
691Presentation of Results 
692The following narrative summarizes the results for the two-way MANOVA example. 
693A two-way MANOVA was conducted to determine the effect Of job satisfaction and gender on 
694the two dependent variables of respondent's income and years of education. Data were first 
695transformed to eliminate outliers. Respondent's income was transformed to remove cases with 
696income of zero or equal to or exceeding 22. Years Of education was also transformed to elimi- 
697nate cases with 6 or fewer years. MANOVA results indicate that job satisfaction (Wilks' 
698A-.965, F(6, 1344)-3.98, p-.001, ,/-.017) and gender (Wilks' A-.976, F(2, 672)-8.14, 
699pc.001, +.024) significantly affect the combined DV of income and years of education. How- 
700ever, multivariate effect sizes are very small. Univariate ANOVA and Scheffé post hoc tests 
701were conducted as follow-up tests. ANOVA results indicate that income significantly differs for 
702job satisfaction (F(3, IN.OOI, '/—.031) and gender (F(l, 
703772—.023). Years of education does not significantly differ for job satisfaction (F(3, 
704p—.089, or gender (F(l, i 2.002). Scheffé post hoc results forin- 
705come and job satisfaction indicate that individuals very satisfied significantly differ from those 
706with only moderate satisfaction. Table 1 presents the adjusted and unadjusted group means for 
707income and years of education by job satisfaction and gender. 
708Chapter 6 Multivariate Analysis of Variance and Covariance 
709Table 1 Adjusted and Unadjusted Means for Income and Years Of Education by Job Satisfaction 
710and Gender 
711Gender 
712Male 
713Female 
714Job Satisfaction 
715Very Satisfied 
716Mod. Satisfied 
717Little Dissatisfied 
718Very Dissatisfied 
719Income 
720Years of Education 
721Adjusted M 
72214.37 
72314.04 
72414.33 
72513.84 
72613.99 
72714.68 
728Adjusted M 
72914.95 
73012.89 
73114.93 
73213.42 
73313.74 
73413.61 
735Unadjusted M 
73615.15 
73713.07 
73815.02 
73913.52 
74013.81 
74113.71 
742Unadjusted M 
74314.07 
74414.12 
74514.32 
74613.83 
74714.00 
74814.79 
749Figure 6.8 Box's Test for Homogeneity ofVariance-Covariance. 
750Box's Test Of Equality Of Covariance Matrices a 
751NOT 
7521245 
75321 
75420370 
755.201 
756Tests the null hypothesis that the observed covariance 
757matrices of dependent variables are equal across Voups. 
758a. Design: Intercept+SATJOB4SEX+SATJOB • SEX 
759SECTION" spss "How To"F0RMANOVA 
760This section presents the steps for conducting multivariate analysis of variance (MANOVA) us- 
761ing the Multivariate procedure for the preceding example, which utilizes the gssft.sav data set. To 
762open the Multivariate dialogue box as shown in Figure 6.14, select the following: 
763Analyze 
764General Linear Model 
765Mul tivqriage Dialogue Box (see Figure 6.14) 
766Once in this box, click the DVs (rincom2 and educ2) and move each to the Dependent Variables 
767box. Click the IVs (satjob and sex) and move each to the Fixed Factor(s) box. Then click Options. 
768Multivariate Options Dialogue Box (see Figure 6.15) 
769Move each IV to the Display Means box. Select Descriptive Statistics, Esti— 
770mates of Effect Size, and Homogeneity Tests under Display. These options are de- 
771scribed in Chapter 4. Click Continue. Back in the Dialogue Box, click post Hoc, 
772134 
773Chapter 6 Multivariate Analysis Of Variance and Covariance 
774Figure 6.9 MANOVA Summary Table. 
775cantly effects he 
776672000 
777672000 
778672_n 
779672 _ 000 
7801346.000 
78113u_w0 
7821342" 
783673,000 
784672W0 
7856720C0 
786672,000 
787672,000 
788'346 , 000 
7891342.000 
790673 An 
791.923 
79212.030 
7934042.1 loa 
7944042110• 
7954042 
7964042, 1 IC-• 
7973,967 
7983984B 
7997.2910 
8008.135' 
8011.051b 
8022000 
803zooc 
8042.000 
8056 000 
8066.000 
8073000 
8082.000 
8092.000 
8102.000 
8116.000 
8126.000 
8136.000 
814SAT.' 
815SAT-JOB 
816• SEX 
817Roys Largest Root 
818Pillai's Trace 
819WAS ' L amMa 
820Hotelling•s Trace 
821Roy's Largest Root 
822T race 
823Hotelling•s Trace 
824Roy's Larpst Root 
825Pinal'S 
826Wilks' Lambda 
827HoteUing•s Trace 
828a, Exact Statistic 
829b, The statste is an F that y•Ms a lower the significance 
830Design: Intercept.SATJoa.SEX.SATJOB • SEX 
831gender sgnificantty 
832be c.yn• 
833Significantly 
834com• 
835Chapter 6 Multivariate Analysis Of Variance and Covariance 
836Figure 6.10 Univariate ANOVA Summary Table. 
837Tests Of Effects 
8381,358 
8392949105 
8407,169 
8412178 
842'6.141 
8431030 
844tat 
845canny affects 
846but WT 
847years 
848s$nftant'y 
849effects 
850NOT 
851Intercept 
852SATJOB • SEX 
853Total 
854a- R Squared 
855RINC 
856EDUC2 
857RINCOM2 
858RINCOM2 
859EDUC2 
860EDUC2 
861dent V 
862Type 111 
863S uares 
86447955386 
86544.07B 
86615004 
86710943,317 
8684540639 
869139907.0 
870120446% 
871157,340 
8729.165 
8731 479553BE 
874116,576 
875262463 
8766,952 
8778,981 
8785.101 
879673 
88016.261 
8816.747 
882,091 (Adjusted R Squared .082) 
883014 R • W4) 
884Multivariate post HOC Dialogue Box (see Figyg 6.16 
885Under Factors, select the IVs (satjob) and move to Post Hoc Tests box. For our example, 
886only satjob was selected since gender has only two categories. Under Equal Variances Assumed, 
887select the desired post hoc test. We selected Scheffé. 
888Chapter 6 Multivariate Analysis Of Variance and Covariance 
889Figure 6.11 Post Hoc Tests for Income and Years of Education by Job Satisfaction. 
890Multip CompMisons 
891Variable 
892on 
893-lab 
8942 satisfied 
8953 A dissatisfied 
8964 Ve ry &SSatiSfiød 
8971 SetiSfwd 
8982 Mod "tisfied 
8993 A dissatisfied 
9004 dissatisfied 
901964 
902.997 
9031030 
904.866 
905.10826 
906.2,4351 
907-2.5895 
908-2.7450 
909-1.2443 
910-2.590B 
911-3.7127 
912-2.2107 
913-2.7861 
914-20150 
915-1.0923 
916-1.1582 
917-2.5095 
918-1.3050 
919-2.5234 
9202.4351 
9212.7450 
9223.7127 
9231.2443 
92422107 
9251.8184 
92627861 
92710738 
9281.1582 
9292.0192 
930Satisfaction 
931very satisfied 
9323 A little 
9334 Vvy 
934Very 
9354 di"atÉfied 
9361 "tisfied 
9372 "tisfied 
9384 
939"fish e d 
9403 A little 
9412 Mod "tisfed 
9424 disabsfied 
943a .tisfied 
9441.2174 
9451.31S1 
946•1.2174 
94797e4EaJ2 
948.1,3151 
949.9764E.02 
950,3211 
951.2139 
952.3511 
953,5179 
954.5516 
955• Tha mean diffg— is Significant at tha. 05 Oval. 
95611. MANCOVA 
957As with univariate ANCOVA, researchers often wish to control for the effects of concomitant 
958variables in a multivariate design. The appropriate analysis technique for this situation is a multivariate 
959analysis of covariance, or MANCOVA. Multivariate analysis of covariance is essentially a combination 
960of MANOVA and ANCOVA. MANCOVA asks if there are statistically significant mean differences 
961among groups after adjusting the newly created DV (a linear combination of all original DVs) for differ- 
962ences on one or more covariates. 
963SECTION 6.6 PRACTICAL VIEW' 
964Purpose 
965The main advantage of MANCOVA over MANOVA is the fact that the researcher can incorpo- 
966rate one Or more covariates into the analysis. The effects of these covariates are then removed from the 
967analysis, leaving the researcher with a clearer picture of the true effects of the IV(s) on the multiple 
968DVs. There are two main reasons for including several (i.e., more than one) covariates in the analysis 
969(Stevens, 1992). First, the inclusion of several covariates will result in a greater reduction in error vari- 
970ance than would result from incorporation of one covariate. Recall that in ANCOVA, the main reason 
971for including a covariate is to remove from the error term unwanted sources of variability (variance 
972within the groups), which could be attributed to the covariate. This ultimately results in a more sensitive 
973Chapter 6 Multivariate Analysis Of Variance and Covariance 
974F-test, which increases the likelihood of rejecting the null hypothesis. By including more covariates in a 
975MANCOVA analysis, we ,can reduce this unwanted error by an even greater amount, improving the 
976chances Of rejecting a null hypothesis that is really false, 
977Figure 6.12 Unadjusted Means for Income and Years of Education by Gender and Job Satisfaction. 
978R 
979OM2 
980Descriptive Statistics 
981SATJOa Job Satisfaction SEX Res ndenes Sex 
982Mean 
98315.81g3 
98414 0301 
98515.0234 
986148157 
98712.3182 
98813.5189 
98915.2571 
990122187 
99113.8060 
99214.2143 
993130000 
99413.7083 
99515_1524 
99613.0717 
99714.2144 
998142590 
99914.3gB5 
1000143211 
1001137484 
100213.g242 
100313.8282 
100414.1429 
100513.8438 
100614.0000 
100715.3571 
1008140000 
100914.7917 
101014.0722 
101114.1238 
101214.0954 
1013Deviation 
10143.738g 
10154.0093 
10163.g565 
10174.3237 
10184.0367 
10194.329B 
10204.1398 
10213.7566 
102242184 
10235.0563 
102428674 
10254.2475 
10264.1183 
10274.0376 
10284.2087 
10292.8856 
10302.2086 
103126030 
10322.6766 
10332.4762 
10342.5847 
10352_5541 
10362.662g 
10372.4685 
10382.1602 
10392.3953 
104027860 
10412.3635 
10422.6023 
10431 Very satisfied 
10442 Mod satisfied 
10453 A little dissatisfied 
10464 Very dissatisfied 
1047I Very satisfied 
10482 Mod satisfied 
10493 A little dissatisfied 
10504 Very dissatisfied 
10512 Female 
1052I Male 
10532 Female 
1054Total 
10551 Male 
10562 Female 
1057Total 
1058I Male 
10592 Female 
1060Total 
1061I Male 
1062Female 
1063Male 
10642 Female 
1065Male 
10662 Female 
1067Total 
10681 Male 
10692 Female 
10701 Male 
10712 Female 
1072Total 
1073I Male 
10742 Female 
1075Total 
1076EDUC2 
1077A second reason for including more than one covariate is that it becomes possible to make better 
1078adjustments for initial differences in situations where the research design includes the use of intact 
1079groups (Stevens, 1992). The researcher has even more information upon which to base the statistical 
1080matching procedure. In this case, the means of the linear combination of DVs for each group are ad- 
1081justed to what they would be if all groups had scored equally on the combination of covariates. 
1082Again, the researcher needs to be cognizant of the choice of covariates in a multivariate analysis. 
1083There should exist a significant relationship between the set of DVs and the covariate or set of covari- 
1084ates (Stevens, 1992). Similar to ANCOVA, if more than one covariate is being used, there should be 
1085relatively low intercorrelations among all covariates (roughly .40). In ANCOVA, the amount of error 
1086reduction was a result of the magnitude of the correlation between the DV and the covariate. In 
1087Chapter 6 Multivariate Analysis Of Variance and Covariance 
1088MANCOVA, if several covariates are being used, the amount of error reduction is determined by the 
1089magnitude of the multiple correlation (R2) between the newly created DV and the set of covariates (Ste- 
1090vens, 1992). A higher value for R2 is directly associated with low intercorrelations among covariates, 
1091which means a greater degree of error reduction. 
1092Figure 6.13 Adjusted Means for Income and Years of Education by Gender and Job Satisfaction. 
1093I. Respondent's Sex 
1094D endent Variable 
1095RI COM2 
1096EDUC2 
1097Oe ndent Variable 
1098COM2 
1099EDUC2 
1100Error 
1101386 
1102Error 
1103.31B 
1104Confidence Interval 
1105Lower Bound U er Bound 
1106Mean 
1107Res ndents Sex 
1108Std 
1109Male 
1110Female 
1111Male 
1112Female 
111314.952 
111412_eg2 
111514.377 
111614.042 
11172. Job Satisfaction 
111814.28B 
111912.135 
112013.950 
112113.554 
112215.615 
112313.649 
112414804 
112514 
1126Job Satisfaction 
1127Very satisfied 
1128Mod satisfied 
1129A little dissatisfied 
1130Very dissatisfied 
1131Very satisfied 
1132Mod satisfied 
1133A little dissatisfied 
1134V dissatisfied 
1135Mean 
113614Æ25 
113713417 
113813.738 
113913.607 
114014_32g 
114113.836 
114213.gg3 
114314 679 
1144Std. 
114595% Confidence Interval 
1146Lower Bound U er Bound 
114714464 
114812,951 
114912,770 
115011 
115114.032 
115213.536 
115313.370 
115413.623 
115515.385 
115613.883 
115714.706 
115815.246 
115914.626 
116014.137 
116114.617 
116215.734 
1163Figure 6.14 Multivariate Dialogue Box. 
1164@age 
1165e duc 
116610b? 
1167; m coral 
1168@educ2 
1169@satlOb 
1170Mcdei., 
1171pott 
1172Chapter 6 Multivanate Analysis Of Variance and Covariance 
1173Figure 6.15 Multivariate Options Dialogue Box. 
1174Multivariate: Optlons 
1175Estimated M Nginal Means 
1176Displ.yems For: 
1177Cprnpae mail effects 
1178Cr,afidence advustrrent 
1179(none/ 
1180Transformation 
1181Homogeneity tests 
1182r plc*' 
1183pHs 
1184Lack of fi test 
1185function 
1186satiob 
1187sex—lob 
1188Dtsplay 
118917 Qescripbve statistics 
1190Estimates ot size 
1191Observed 
1192Parameter estimates 
1193SSCP matrices 
1194Post fest for 
1195geroris not 
1196Significrvce Confidence intervals 
1197Ccntinue Cara 
1198Figure 6.16 Multivariate post Hoc Dialogue Box. 
1199rost Hoc compa,isons fo, 
1200T for. 
1201E quS A 
1202goriertmi 
1203SgWfe 
1204E qual N Ot Assumed 
1205r- r Dunnett•sc 
1206Chapter 6 Multivariate Analysis Of Variance and Covariance 
1207The null hypothesis being tested in MANCOVA is that the adjusted population mean vectors are 
1208equal: 
1209Wilks' Lambda (A) is again the most common test statistic used in MANCOVA. However, in this case, 
1210the sum-of-squares and cross-products (SSCP) matrices are first adjusted for the effects of the covari- 
1211ate(s). 
1212The procedure to be used in conducting MANCOVA mirrors that used in conducting 
1213MANC)VA. Following the statistical adjustment of newly created DV scores, the overall multivariate 
1214null hypothesis is evaluated using Wilks' A. If the null is retained, interpretation of the analysis ceases 
1215at this point. However, if the overall null hypothesis is rejected, the researcher then examines the results 
1216of univariate ANCOVAs in order to discover which DVs are being affected by the IV(s). A Bonferroni- 
1217type adjustment to protect from the potential of an inflated Type I error rate is again appropriate at this 
1218point, 
1219The reader may recall from Chapter 5 explicit mention of a specific application of MANCOVA 
1220that is used to assess the contribution of each individual DV to any significant differences in the IVs. 
1221This procedure is accomplished by removing the effects of all other DVs by treating them as covariates 
1222in the analysis. 
1223Sample Research Questions 
1224In the sample study presented earlier in this chapter, we investigated the differences in worker 
1225productivity (measured by income level, DVI, and hours worked, DV2) for Individuals in different age 
1226categories (IV). Assume that the variable of of education has been shown to relate to both DVs 
1227and we want to remove its effect from the analysis. Consequently, we decide to include years of educa- 
1228tion as a covariate in our analysis. Therefore, the design we have is now a one-way MANCOVA. Ac- 
1229cordingly, this study would address the following research questions: 
1230(l) Are there significant mean differences in worker productivity (as measured by the combination 
1231of income and hours worked) for individuals of different ages, after removing the effect of 
1232years of education? 
1233(2) Are there significant mean differences in income levels for individuals of different ages, after 
1234removing the effect of years of education? 
1235(2a) If so, which age categories differ? 
1236(3) Are there significant mean differences in hours worked for individuals of different ages, after 
1237removing the effect of years of education? 
1238(3a) If so, which age categories differ? 
1239For our second MANCOVA example, we will add a covariate to our two-factor design presented 
1240earlier. This two-way MANCOVA will investigate differences in the combined DV of income level 
1241(DVI,) and years of education (DV2) for individuals of different gender (IVt) and of different levels of 
1242job satisfaction (IV2), while controlling for age. Again, one should note that the following research 
1243questions address both the multivariate and univariate analyses within MANCOVA: 
1244(l) a. Are there significant mean differences in the combined DV of income and years of educa- 
1245tion between males and females, after removing the effect of age 0 
1246Chapter 6 Multivariate Analysis ofVariance and Covariance 
1247(2) 
1248(3) 
1249b. 
1250c. 
1251a. 
1252b. 
1253a. 
1254b. 
1255c. 
1256Are there significant mean differences in the combined DV of income and years of educa- 
1257tion for different levels of job satisfaction, after removing the effect of age? If so, which 
1258job satisfaction categories differ? 
1259Is there a significant interaction between gender and job satisfaction on the combined DV 
1260of income and years of education, after removing the effect of age? 
1261Are there significant mean differences on income between males and females, after remov- 
1262ing the effect of age? 
1263Are there significant mean differences on income among different levels of job satisfac- 
1264tion, after removing the effect of age? If so, which job satisfaction categories differ? 
1265Is there a significant interaction between gender and job satisfaction On income, after re- 
1266moving the effect of age? 
1267Are there significant mean differences in years Of education between males and females, 
1268after removing the effect of age? 
1269Are there significant mean differences in years of education among different levels of job 
1270satisfaction, after removing the effect Of age? If so, which job satisfaction categories dif- 
1271Is there a significant interaction between gender and job satisfaction on years of education, 
1272after removing the effect of age? 
1273SECTION 6.7 ASSUMPTIONS AND LIMITATIONS 
1274Multivariate analysis of covariance rests on the same basic assumptions as univariate ANCOVA. 
1275However, the assumptions for MANCOVA must accommodate multiple D Vs. The following list pre- 
1276sents the assumptions for MANCOVA, with an asterisk indicating modification from the ANCOVA as- 
1277sumption. 
1278(1) 
1279(3*) 
1280(5*) 
1281(6) 
1282The observations within each sample must be randomly sampled and must be independent Of 
1283each other. 
1284The distributions of scores on the dependent variables must be normal in the populations from 
1285which the data were sampled. 
1286The distributions of scores on the dependent variables must have equal variances. 
1287Linear relationships must exist between all pairs of DVs, all pairs of covariates, and all DV- 
1288covariate pairs in each cell. 
1289If two covariates are used, the regression planes for each group must be homogeneous or paral- 
1290lel. If more than two covariates are used, the regression hyperplanes must be homogeneous. 
1291The covariates are reliable and are measured without error. 
1292The first and sixth assumptions essentially remain unchanged. Assumptions #2 and #3 are sim- 
1293ply modified in order to include multiple DVs. Assumption #4 has a substantial modification in that we 
1294must now assume linear relationships not only between the DV and the covariate, but also among sev- 
1295eral other pairs of variables (Tabachnick & Fidel, 1996). There also exists an important modification to 
1296assumption number 5. Recall that if only one covariate is included in the analysis, there exists the as- 
1297sumption that covariate regression slopes for each group are homogeneous. However, if the 
1298MANCOVA analysis involves more than One covariate, the analogous assumption involves homogene- 
1299ity of regression planes (for 2 covariates) and hyperplanes (for 3 or more covariates). 
1300Chapter 6 Multivariate Analysis of Variance and Covariance 
1301Our discussion of assessing MANCOVA assumptions will center on the two substantially modi- 
1302fied assumptions (i.e., #4 and #5). Similar procedures, as have been discussed earlier. are used for test- 
1303ing the remaining assumptions. 
1304Methods of Testing Assumptions 
1305The assumption of norrnally distributed DVs is assessed in the usual manner. Initial assessment 
1306of normality is done through inspection of histograms, boxplots, and normal Q-Q plots. Statistical as- 
1307sessment of normality is accomplished by examining the values (and the associated significance tests) 
1308for skewness and kurtosis, and through the use of the Kolmogorov-Smirnov test. The assumption Of 
1309homoscedasticity is assessed primarily with Box's Test or using one of three different statistical tests 
1310discussed in previous chapters (i.e., Chapters 3 and 5), namely Hartley's F-max test, Cochran's test, or 
1311Levene's test. 
1312The assumption of linearity among all pairs Of DVs and covariates is crudely assessed by in- 
1313specting the within-cells bivariate scatterplots between all pairs of DVs, all pairs of covariates, and all 
1314DV-covariate pairs. This process is feasible if the analysis includes only a small number Of variables. 
1315However, the process becomes much more cumbersome (and potentially unmanageable!) with analyses 
1316involving the examination of numerous DVs and/or covariates—just imagine all of the possible bivari- 
1317ate pairings! If the researcher is involved in such an analysis, one recommendation is to engage in "spot 
1318checks" Of random bivariate relationships or bivariate relationships in which nonlinearity may be likely 
1319(Tabachnick & Fideli, 1996). 
1320Once again, if curvilinear relationships are indicated, they may be corrected by transforming 
1321some or all of the variables. Bear in mind that transforming the variables may create difficulty in inter- 
1322pretations. One possible solution might be to eliminate the covariate that appears to produce nonlinear- 
1323ity and replacing it with another appropriate covariate (Tabachnick & Fidell, 1996). 
1324The reader will remember that a violation of the assumption of homogeneity of regression slopes 
1325(as well as regression planes and hyperplanes) is an indication that there is a covariate by treatment (IV) 
1326interaction, meaning that the relationship between the covariate and the newly created DV is different at 
1327different levels of the IV(s). A preliminary or custom MANCOVA can be conducted to test the assump- 
1328tion of homogeneity of regression planes (in the case of two covariates) or regression hyperplanes (in 
1329the case of three or more covariates). If the analysis contains more than one covariate, there is an inter- 
1330action effect for each covariate. The effects are lumped together and tested as to whether the combined 
1331interactions are significant (Stevens, 1992). 
1332The null hypothesis being tested in these cases is that all regression planes/hyperplanes are equal 
1333and parallel. Rejecting this hypothesis means that there is a significant interaction between covariates 
1334and IVs and that the planes/hyperplanes are not equal. If the researcher is to continue in the use of mul- 
1335tivariate analysis Of covariance, one would hope to fail to reject this particular null hypothesis. In 
1336SPSS, this is determined by examining the results of the F-test for the interaction of the IV(s) by the co- 
1337variate(s). 
1338SECTION 6.8 PROCESS AND LOGIC 
1339The Logic Behind MANCOVA 
1340The calculations for MANCOVA are nearly identical to those for MANOVA, The Only substan- 
1341tial difference is that the sum-of-squares and cross-products (SSCP) matrices must first be adjusted for 
1342the effects Of the covariate(s). The adjusted matrices are symbolized by T • (adjusted total sum-Of- 
1343Chapter 6 Multivariate Analysis Of Variance and Covariance 
1344squares and cross-products matrix), W* (adjusted within sum-of-squares and cross-products matrix), 
1345and B* (adjusted between sum-of-squares and cross-products matrix). 
1346Wilks' A is again calculated by using the SSCP matrices (Stevens, 1992). We Can compare the 
1347MANOVA and MANCOVA formulas for A: 
1348ITI 
1349IWI 
1350IW*I 
1351(Equation 6.4) 
1352The interpretation of A remains as it was in MANOVA. If there is no treatment effect or group differ- 
1353ences, then B* — 0 and A* — I indicating no differences between groups On the linear combination of 
1354DVs after removing the effects of the covariate(s); whereas, if B* were very large, then A* would ap- 
1355proach 0, indicating significant group differences on the combination of DVs, after controlling for the 
1356co variate(s). 
1357As in MANOVA, eta squared for MANCOVA is obtained in the following manner: 
1358In the multivariate analysis Of covariance situation, 42 is interpreted as the variance accounted for in the 
1359best linear combination of DVs by the IV(s) and/or interactions Of IV(s), after removing the effects of 
1360any covariate(s). 
1361Interpretation of Results 
1362Interpretation of MANCOVA results is quite similar to that of MANOVA; however, with the 
1363inclusion of covariates, interpretation of a preliminary MANCOVA is necessary in order to test the as- 
1364sumption of homogeneity of regression slopes. Essentially, this analysis tests for the inteKiction be- 
1365tween the factors (IVs) and covariates. This preliminary or custom MANCOVA will also test homoge- 
1366neity of variance-covariance (Box's Test), which is actually interpreted first since it helps in identifying 
1367the appropriate test statistic to be utilized in examining the homogeneity of regression and the final 
1368MANCOVA results. If the Box's Test is significant at and group sample sizes are extremely 
1369unequal, then Pillai's Trace is utilized when interpreting the homogeneity of regression test and the 
1370MANOVA results. If equal variances are assumed, Wilks' Lambda should be used as the multivariate 
1371test statistic. Once the test statistic has been determined, then the homogeneity of regression slopes or 
1372planes results are interpreted by examining the F ratio and p value for the interaction. If factor-covariate 
1373interaction is significant, then MANCOVA is not an appropriate analysis technique. If interaction is not 
1374significant, then One can proceed with conducting the full MANCOVA analysis. Using the F ratio and p 
1375value for a test statistic that was identified in the preliminary analysis through the Box's Test, factor in- 
1376teraction should be examined iftwo or more IVs are utilized in the analysis. If factor interaction is sig- 
1377nificant, then main effects for each factor on the combined DV is not a valid indicator of effect. If factor 
1378interaction is not significant, the main effects for each IV can be accurately interpreted by examrning the 
1379F ratio,p value, and effect size for the appropriate test statistic. When main effects are significant, uni- 
1380144 
1381Chapter 6 Multivariate Analysis Of Variance and Covariance 
1382variate ANOVA results indicate group differences for each DV. Since MANCOVA does not provide 
1383post hoc analyses, examining group means (before and after covariate adjustment) for each DV can as- 
1384sist in determining how groups differed for each DV. 
1385In summary, the first step in interpreting the MANCOVA results is to evaluate the preliminary 
1386MANCOVA lesults that include the Box's Test and the test for homogeneity of regression slopes. If 
1387Box's Test is not significant, utilize the Wilks' Lambda statistic when interpreting the homogeneity of 
1388regression slopes and the subsequent multivariate tests. If Box's Test is significant, use pillai's Trace. 
1389Once the multivariate test statistic has been identified, examine the significance (F ratios and p values) 
1390of factor-covariate interaction (homogeneity of regression slopes). If factor-covariate interaction is not 
1391significant, then proceed with the full MANCOVA. TO interpret the full MANCOVA results, examine 
1392the significance (F ratios and p values) of factor interaction. This is necessao' only iftwo or more IVs 
1393are included. Next evaluate the F' ratio, p value, and effect size for each factor's main effect. If multi- 
1394variate significance is found, interpret the univariate ANOVA results to determine significant group dif- 
1395ferences for each DV. 
1396For our example that investigates age category (agecat4) differences in respondent's income 
1397(rincom91) and hours worked per week (hrsl) when controlling for education level (educ), the previ- 
1398ously transformed variables of rincom2 and hrs2 were utilized. These transformations are described in 
1399Section 6.3. Linearity of the two DVs and the covariate was then tested by creating a matrix scattemlot 
1400and calculating Pearson correlation coefficients. Results indicate linear relationships. Although the cor- 
1401relation coefflcients are statistically significant, all are quite low. The last assumption, homogeneity of 
1402variance-covariance, was tested within a preliminary MANCOVA analysis utilizing Multivariate. 
1403The Box's Test (see Figure 6.17) reveals that equal variances can be assumed, F(9, 
1404p—.769; therefore, Wilks' Lambda will be used as the multivariate statistic. Figure 6.18 presents the 
1405MANOVA results for the homogeneity of regression test. The interaction between agecat4 and educ2 is 
1406not significant, Wilks' A—.993, p—.558. A full MANCOVA was then conducted using 
1407Multivariate (see Figure 6.19). Wilks' Lambda criteria indicates significant groups differences in 
1408age category with respect to income and hours worked per week, Wilks' Ae.898, 
1409pe.001, multivariate Univariate ANOVA results (see Figure 6.20) reveal that age category 
1410significantly differs for only income (F(3, pc.001, partial +.097) and not hours worked 
1411per week (F(3, IF-.984, partial '72—.000). A comparison of adjusted means shows that indi- 
1412viduals 18-29 years of age have income that is more than 3 points lower than those 4049 and older than 
141350 (see Figure 6.21). 
1414Figure 6.17 Box's Test for Homogeneity of Variance-Covariance. 
1415Box's Test of Equality Of Covariance Matricesa 
1416Box's M 
1417dfl 
14185,740 
1419Box's Test is not 
1420significant use 
1421Wilks' Lamb:'a 
1422criteria 
14232827520 
1424Tests the null hypothesis that the observed covariance 
1425matrices of the dependent variables are equal across groups. 
1426a. Design: • 
1427EDUC2 
1428Clupter 6 Multivariate Analysis Of Variance and Covariance 
1429Figure 6.18 MANCOVA Summary Table: Test for Homogeneity of Regression Slopes. 
1430Hypt*hesi 
1431AGECA 
1432EDUC2 
1433Howling's Trace 
1434ROV S Rczt 
1435race 
1436WilkS' Lambda 
1437RCMS Lugest Root 
1438Pilafs Trace 
1439ROY'S Largegt Root 
1440Error df 
1441671,000 
1442674000 
1443674000 
1444674 _ OCX) 
1445674000 
1446674 
1447674000 
1448674.000 
1449674. cm 
14501350 000 
14511348,000 
14521346000 
1453675000 
1454agl 
14550397 
1456133 
1457133096' 
1458133096' 
1459133 096• 
14602 coo 
14612.000 
14622 
14632Coc 
1464AGECAT4•EDUC2 
1465Hotelling•s Trace 
1466a Exact statistic 
1467b. The statistic is an bound F that yields a on the 
1468"vecept+AGECAT4.EDUC2.AGECAT4 • EDUC2 
1469Figure 6.19 MANCOVA Summary Table. 
1470Multivariate Test' 
1471interaction is NOT 
1472Sig 
1473EndiCates hat 
1474influenæs 
1475hat 
1476cant'/ effects the 
1477AGECAT4 
1478WilkS' Lambda 
1479HoteUirWs Trace 
1480ROM S Largest Root 
1481Pinal's Trace 
1482WAS' 
1483Hotelling•s T race 
1484ROYS Largest 
1485Wilks• Larnt*'a 
1486Ha-telling'S Trace 
1487ROMs Largest RCK•t 
1488.424 
1489.874 
1490142.742' 
1491142.742' 
14924B„12g• 
149348428' 
149448.42B• 
14954BA28• 
149612 037 
149712356' 
149812673 
149925 
15002 
15012.000 
15022.000 
15032.000 
15042000 
15052,000 
15062 ooc 
15076.000 
15086.000 
15096. ooc 
15103.000 
1511a, Exact statistic 
1512b, statistic is upper F that —ds a bwer level. 
1513c. Design: 
1514146 
1515Chapter 6 Multivariate Analysis Of Variance and Covariance 
1516Figure 6.20 Univariate ANOVA Summary Table. 
1517Tests of Between-subjects Effects 
1518'2456 
15192840 
152064 g43 
152195.452 
15221 1086 
152324,177 
1524rected 
1525Intercept 
1526EDuc2 
1527AGECAT4 
1528Total 
1529Corrected Total 
1530a R Squared 
1531b. R Squared 
1532De Variable 
1533H RS2 
1534RINCOM2 
1535HRS2 
1536RINCOM2 
1537HRS? 
1538RINCOV2 
1539HRS2 
1540HRS2 
1541RINCOM2 
1542HRS2 
1543RINCOM2 
1544HRS2 
1545Type Ill 
1546Sum of 
154724M 930' 
1548922346 
154933502664 
15501439.92 
15511030099 
155219820 
15539586+57 
15548533 •,025 
15551491ggo 
15561%7n26 
15571 1998.587 
155886821 
1559Mean 
1560502 983 
1561372,613 
1562922 346 
156333502664 
15641355 E55 
15651439392 
1566343.366 
1567202 
1568hdigEteS that age 
1569cateø*y 
1570cantly effects 
1571NOT 
1572hours 
1573,2C1 ("jugted R -195) 
1574,017 (Adjusted R squared .011) 
1575Figure 6.21 Unadjusted and Adjusted Means for Income and Hours Worked per Week by 
1576Age Category. 
1577Descriptive Statistics 
1578194 
1579Figure 6.21 is continued on the next page. 
1580RINCO 
1581HRS2 
1582ACECAT4 4 
1583cat ories of a e 
15841 18-29 
15852 30-39 
15863 40-49 
1587Total 
15881 18-29 
15892 30-39 
15903 404 
15914 504 
1592Total 
1593Mean 
159411.8672 
159514.0315 
159615.3247 
159714.9574 
159814_1839 
159946.3203 
1600470315 
160146.48g7 
160246.3262 
160346 GOOC 
1604Deviation 
16054.1438 
160638810 
16073.8660 
16084.4173 
16094.2155 
1610100200 
161111.4182 
161211.7545 
161311.5149 
161411.3169 
1615Chapter 6 Multivariate Analysis Of Variance and Covariance 
1616Figure 6.21 Unadjusted and Adjusted Means for Income and Hours Worked per Week by 
1617Age Category. (Continued) 
16184 categories Of age 
161995% Confidence 
1620Interval 
1621Std. Error 
1622.253 
1623.272 
1624dent Variable 4 ca oriesof a e 
1625Mean 
162611.993B 
162713.88F 
162815.356a 
162915.165B 
163046.450a 
163146.882B 
163246.520 
163346 660B 
1634Lower 
1635Bound 
163611.33g 
163713,389 
163814.822 
163914.535 
164045.398 
164144.935 
164244.780 
1643upper 
1644Bound 
164512.54B 
164614.384 
164715.890 
164815.795 
164948.403 
165048.366 
165148.122 
165248.540 
1653NCOM2 
16541 18-29 
16552 30-39 
16563 40.49 
16572 30-39 
16583 40-49 
16594 50+ 
1660a. Evaluated at covariates appeared in the model: EDIJC2 14 0985 
1661Writing Up Results 
1662The process of summarizing MANCOVA results is almost identical to MANOVA; however, 
1663MANCOVA results will obviously include a statement of how the covariate influenced the DVs. One 
1664should note that although the preliminary MANCOVA results are quite important in the analysis proc- 
1665ess, these results are not reported slnce it is understood that if a full MANCOVA has been conducted, 
1666such assumptions have been fillfilled. Consequently, the MANCOVA results narrative should address 
1667the following: 
1668(1) Subject elimination and/or variable transformation; 
1669(2) Full MANCOVA results (test statistic, F ratio, degrees of freedom, p value, and effect size); 
1670(a) Main effects for each IV and covariate on the combined DV; 
1671(b) Main effect for the interaction between IVs; 
1672(3) Univariate ANOVA results (F ratio, degrees of freedom, p value, and effect size); 
1673(a) Main effect for each IV and DV; and 
1674(b) Comparison of means to indicate which groups differ on each DV. 
1675Often a table is created that compares the unadjusted and adjusted group means for each DV. For our 
1676example, the results statement includes all of these components with the exception of factor interaction 
1677since only one IV is utilized. The following results narrative applies the results from Figures 6.17 — 
16786.21. 
1679Multivariate analysis of covariance (MANCOVA) was conducted to determine the effect of age 
1680category on employee productivity as measured by income and hours worked per week while 
1681controlling for years of education. Prior to the test, variables were transformed to eliminate out- 
1682liers. Cases with income equal to zero and equal to or exceeding 22 were eliminated. Hours 
1683worked per week was transformed; those less than or equal to 16 were recoded 17 and those 
1684greater than or equal to 80 were recoded 79. Years Of education was also transformed to elimi- 
1685nate cases with 6 or fewer years. MANOVA results revealed significant differences among the 
1686age categories on the combined dependent variable, Wilks' A—.898, pc.001, 
1687multivariate pi—.052. The covariate (years of education) significantly influenced the comblned 
1688Chapter 6 Multivaroate Analysis Of Variance and Covariance 
1689dependent variable, Wilks' 74, multivariate 126. Analysis of 
1690covarlance (ANCOVA) was conducted on each dependent variable as a follow-up test to 
1691MANCOVA. Age category differences were significant for income, (F(3, pc.001, 
1692partial /12—.097) but not hours worked per week (F(3, partial A 
1693comparison of adjusted means revealed that income of those 18-29 years differs by more than 3 
1694points from those 40-49 years and those 50 years and older. Table I presents adjusted and unad- 
1695justed means for income and hours worked per week by age category. 
1696Table 1 Adjusted and Unadjusted Means for Income and Hours Worked per Week 
1697by Age Category 
1698Hours Worked per Week 
1699Age 
170018-29 years 
170130-39 years 
170240-49 years 
170350+ years 
1704Income 
1705Adjusted M 
170611.99 
170713.89 
170815.36 
170915.17 
1710Unadjusted M 
171146.32 
171247.03 
171346.49 
171446.33 
1715U nadjusted M Adjusted M 
171611.87 
171714.03 
171815.32 
171914.96 
172046.45 
172146.88 
172246.53 
172346.66 
1724SECTION 6.9 MANCOVA SAMPLE STUDY AND ANALYSIS 
1725This section provides a complete example that applies the entire process of conducting 
1726MANCOVA: development of research questions and hypotheses, data screening methods, test methods, 
1727interpretation of output, and presentation of results. The SPSS data set gssft.sav is utilized. Our previ- 
1728ous example demonstrates a one-way while this example will present a two-way 
1729MANCOVA. 
1730Problem 
1731Utilizing the two-way MANOVA example previously presented, in which we examined the de- 
1732gree to which gender and job satisfaction affects income and years of education among employees, we 
1733are now interested in adding the covariate of age. Since two IVs are tested in this analysis, questions 
1734must also take into account the possible interaction between factors. The following research questions 
1735and respective null hypotheses address the multivariate main effects ror each IV and the possible inter- 
1736action between factors. 
1737Research Questions 
1738RQI: Do income and years of educa- 
1739tion differ by gender among employees 
1740when controlling for age? 
1741RQ2: Do income and years of educa- 
1742tion differ by job satisfaction among 
1743employees when controlling for age? 
1744RQ3: DO gender and job satisfaction 
1745interact in the effect on income and 
1746years of education when controlling for 
1747age? 
1748Null Hypotheses 
1749Income and years of education 
1750HOI: 
1751will not differ by gender among em- 
1752ployees when controlling for age. 
1753H02: Income and years of education 
1754will not dlffer by job satisfaction among 
1755employees when controlling for age. 
1756HO: Gender and job satisfaction will 
1757not interact in the effect on income and 
1758years of education when controlling for 
1759age. 
1760Chapter 6 Multivariate Analysis Of Variance and Covariance 
1761Both IVs are categorical and include gender (sex) and job satisfaction (satjob). The DVs are 
1762respondent's income (rincom2) and years of education (educ2); both are quantitative. The covariate is 
1763years of age (age) and is quantitative. One should note that the variables rincom2 and educ2 are trans- 
1764formed variables of rincom91 and educ, respectively. Transfonnations of these variables are described 
1765in section 6.3 of this chapter. 
1766Method 
1767Since variables were previously transformed to eliminate outliers, data screening is complete. 
1768MANCOVA test assumptions should then be examined. Linearity between the DVs and covariate is 
1769first assessed by creating a scatterplot matrix and calculating Pearson correlation coefficients. Scatter- 
1770plots and correlation coefficients indicate linear relationships. Although three of the four correlation 
1771coefficients are significant (F.OOI), coefficients are still fairly weak. The final test assumptions of ho- 
1772mogeneity of variance-covariance and homogeneity of regression slopes will be tested in a preliminary 
1773MANCOVA using Multivariate. For our example, Box's Test (see Figure 6.22) indicates homo- 
1774geneity of variance-covariance, pz.204. Therefore, Wilks' Lambda will be utilized 
1775as the test statistic for all the multivariate tests. Figure 6.23 reveals that factor and covariate interaction 
1776is not significant, Wilks' A—.976, F(14, p—.315. Full MANCOVA was then conducted 
1777using Multivariate. 
1778Figure 6.22 Box's Test for Homogeneity of Variance-Covariance. 
1779Box's Test Of Equality of Covariance Matricesa 
1780ox's M 
178126.868 
178220374 
1783Sig. 
1784Box's Test nat 
1785significant use 
1786W i Iks• Lambda 
1787Tests the null hypothesis that the observed covariance 
1788matrices Of the dependent variables are equal across groups. 
1789a. Design: Intercept+SEX+SATJOB+AGE*SEX • SATJOB 
1790Output and Interpretation Of Results 
1791Figure 6.24 presents the unadjusted group means for each DV, while Figure 6.25 displays the 
1792adjusted means. MANCOVA results are presented in Figure 6.26 and indicate no significant interaction 
1793between the two factors of gender and job satisfaction, Wilks' AZ.993, F(6, p—.539. The 
1794main effects of gender (Wilks' A—.974, F(2, multivariate +.026) and job satisfac- 
1795tion (Wilks' A—.972, F(6, r.004, multivariate '122.014) indicate significant effect on the 
1796combined DV. However, one should note the extremely small effect sizes for each IV. The covariate 
1797significantly influenced the combined DV, Wilks' A—.908, F(2, multivariate 
1798Univariate ANOVA results (see Figure 6.27) indicate that only the DV of income was signifi- 
1799cantly effected by the IVs and covariate. 
1800150 
1801Chapter 6 Multivariate Analysis Of Variance and Covariance 
1802Figure 6.23 MANCOVA Summary Table: Test for Homogeneity of Regression Slopes. 
1803Multivariate Tests* 
1804Hypothesi 
1805Effect 
1806Intercept 
1807SATJOB 
1808AGE 
1809666.000 
18106EE.ooo 
18116E6.coa 
1812566000 
1813665000 
18141334 000 
18151332,aoo 
18161330000 
1817667 ooc 
1818666,000 
1819666.000 
18201334 000 
18211332100 
18221330000 
1823Value 
1824+63 
182585 
1826220 
1827220Bss•' 
1828220859 
1829U 100 
18301.100 
1831gg12a 
18329.9128 
1833gg12S 
18341,143 
18352000 
18362000 
18372.000 
18382000 
18392.000 
18402000 
1841scoo 
18426.000 
18436000 
18443.000 
18452.000 
18462000 
18472000 
184814.coc 
18491400t 
185014000 
18517000 
1852iliai'S race 
1853Wilks• Lambda 
1854Hclellirvs Trace 
1855ROWS L Mgest ROM 
1856pillars Trace 
1857Wilks• 
1858HatøV.ing•s Trace 
1859Ros's Largest Root 
1860Wilkg• Lambda 
1861Hotelling'S Trace 
1862ROY's Largest Root 
1863Pillai•s 
1864W Larnbda 
1865Hotellinos Trace 
1866ROY S Largest Root 
1867Trace 
1868Wilks' 
1869Hotelling's Trace 
1870Ray's La st Root 
1871Facer-wvariate 
1872NOT 
1873significant 
1874SEX • SATJOB • AGE 
1875Exact statistic 
1876b, Statistic an u 
1877bound an F that yields a lower bound on the 
1878Design: Intercept.SEx.SATJOB.AGE.SEX • SAT.10a • AGE 
1879Presentation of Results 
1880The following namtive summarizes the results from this two-way MANCOVA example. 
1881A two-way MANCOVA was conducted to determine the effect of gender and job satisfaction on 
1882income and years of education while controlling for years of age. Data were first transformed to 
1883eliminate outliers. Respondent's income was transformed to eliminate cases with income of 
1884zero and equal to or exceeding 22. Years of education was also transformed to eliminate cases 
1885with 6 or fewer years. The main effects of gender (Wilks' AZ.974, F(2, pc-.001, 
1886multivariate 022-026) and job satisfaction (Wilks' A—.972, F(6, multivari- 
1887ate 7/—.014) indicate significant effect on the combined DV. The covariate significantly influ- 
1888enced the combined DV, Wilks' A—.908, F(2, pc.001, multivariate '/—.092. Uni- 
1889variate ANOVA results (Figure 6.27) indicate that only the DV of income was significantly ef- 
1890fected by gender (F(l, partial job satisfaction (F(3, 
1891partlal ri—.025) and the covariate of age (F(l, pc-.001, partial //—.075). 
1892Table I presents the adjusted and unadjusted group means for income and years of education. 
1893Comparison of adjusted income means Indicates that those very satisfied have higher incomes 
1894than those less satisfied. 
1895Chapter 6 Multivariate Analysis of Variance and Covariance 
1896Table I Adjusted and Unadjusted Group Means for Income and Years ofEducation 
1897Gender 
1898Male 
1899Female 
1900Job Satisfaction 
1901Very Sat. 
1902Mod. Sat. 
1903A Little Dis. 
1904Very DIS. 
1905Income 
1906Unadjusted M 
190715.15 
190813.05 
190915.00 
191013.52 
191113.81 
19123.71 
1913Years of Education 
1914Adjusted M 
191515.00 
191612.92 
191714_so 
191813.51 
191913.73 
192013.80 
1921Adjusted M 
192214.37 
192314.04 
192414,35 
192513.83 
192613.99 
192714.67 
1928Unadjusted M 
192914.07 
193014.13 
193114.33 
193213.83 
193314.00 
193414.79 
1935Figure 6.24 Unadjusted Group Means for Years of Education and Income. 
1936RINCOM2 
1937SEX Res 
19381 Ma 
19392 Female 
1940Total 
19411 Male 
19422 Female 
1943Total 
1944ndent's Sex 
1945Descriptive Statistics 
1946SATJoa Job satisfaction 
1947I Very Satisfied 
19482 Mod satisfied 
19493 A little dissatisfied 
19504 Very dissatisfied 
1951Total 
1952I Very satisfied 
19532 Mod satisfied 
19543 A little dissatisfied 
19554 Very dissatisfied 
1956Total 
1957I Very satisfied 
19582 Mod satisfied 
19593 A little dissatisfied 
19604 Very dissatisfied 
1961Tot* 
1962I Very satisfied 
19632 Mod satisfied 
19643 A little dissatisfied 
19654 Very dissatisfied 
1966Total 
1967Very satisfied 
19682 Mod satisfied 
19693 A 'ittle dissatisfied 
19704 Very dissatisfied 
1971Total 
1972I Very satisfied 
19732 Mod satisfied 
19743 A little dissatisfied 
19754 Very dissatisfied 
1976Total 
1977Mean 
197814_2590 
197913, 7484 
1980141429 
198115.3571 
198214.0722 
198314.4167 
1984130242 
1985138438 
1986140000 
19871307 
1988143239 
1989138282 
199014.0000 
199114.7917 
199214.0985 
199315.8193 
1994145157 
199515.2571 
199614.2143 
199715.1524 
199813.9773 
199912.3182 
2000122187 
2001130000 
200213.045B 
200315.0034 
2004135189 
200513, 8060 
200613.7083 
200714.2044 
2008Deviation 
20092.8855 
201026766 
20112.7880 
20122.4585 
20132.7860 
20142.2070 
20152.4762 
20162.5541 
201721502 
20182.6039 
201925847 
20202G62g 
20212, 3953 
20222.6029 
20233_7389 
202443237 
20254.139B 
20265.0563 
20274.1183 
20283.9779 
20294.0367 
20302, 8674 
20314.0185 
20323,947 g 
20334.3298 
20344.2184 
20354.2475 
203642037 
2037Chapter 6 Multivariate Analysis of Variance and Covariance 
2038Figure 6.25 Adjusted Group Means for Years of Education and Income by Gender 
2039and Job Satisfaction. 
20401. Respondent's Sex 
204195% Confidence 
2042Error 
2043.218 
2044.249 
2045.325 
2046De endent Variable 
2047RINCOM2 
2048Res 
20492 Female 
2050I Male 
20512 Female 
2052ts Sex 
2053Mean Std, 
205414,374a 
205514.043B 
205614 _ 
205712.9218 
2058Bound 
205913,946 
206013.555 
206114.358 
206212.193 
2063upper 
2064Bound 
206514.802 
206614.532 
206715.633 
206813.649 
2069a- Evaluated at covariates appeared in the Il-yodel: AGE 
20702. Job Satisfaction 
2071Age Of Respondent 40.34 _ 
207295% Confidence 
2073Interval 
2074D 
2075ndent Variable 
2076Std 
2077Error 
2078.152 
2079.153 
2080.318 
2081.474 
2082.803 
2083Job Satisfaction 
2084Very satis 
20852 Mod satisfied 
20863 
2087A Ettie dissatisfied 
20884 
2089Very dissatisfied 
2090I Very satisfied 
20912 
2092Mod satisfied 
20933 A little dissatisfied 
20944 
2095Very dissatisfied 
2096Moan 
209714.345 
209813.830' 
209913_9g4a 
210014.666' 
210113.507' 
210213.727' 
210313,801" 
2104Sound 
210514046 
210613,530 
210713470 
210813.609 
210914.353 
211013_058 
211112.797 
211212_225 
2113upper 
2114Bound 
211514.643 
211614.131 
211714_618 
211815.723 
211915242 
212013 g55 
212114.658 
212215.378 
2123RINCOM2 
2124a. Evaluated at covariates appeared in the rnodel: AGE 
2125Age of ReswMent 40.34. 
2126Chapter 6 Multivariate Analysis of Variance and Covariance 
2127Figure 6.26 MANCOVA Summary Table. 
2128Multivariate Test* 
2129Hypothesi 
2130df 
2131670,000 
2132570,000 
2133570.000 
2134670.000 
2135670.000 
2136670.000 
2137670.oca 
2138670 000 
21396701300 
2140670.000 
2141670,000 
21421342000 
2143671 .ooo 
21441342.000 
21451340.000 
21461338.000 
2147671000 
2148The of 
2149infuerces the 
2150Gender 
2151cantly 
2152encas the 
2153Jab s absfadion 
2154significantly 
2155the 
2156ilterac• 
2157tan is NOT 
2158significant, 
2159Intercept 
2160SATJOB 
2161Wilkg• Lambda 
2162Hotel ling's Trace 
2163Roy'S L argest Root 
2164Pillars Trace 
2165Wilks' Lambda 
2166Hatelling's race 
2167Roys Largest Rent 
2168Pillars Trace 
2169Wilks' Lambda 
2170e g s race 
2171ROY' s Largest Rwt 
2172Pillars Trace 
2173Wilks' Lambda 
2174Hotel ling g race 
2175ROWS Largest Root 
2176Value 
2177.330 
21782.02B 
21792028 
2180-092 
2181. goa 
2182679424B 
2183679 424a 
2184679.424 a 
2185679.4248 
218633912* 
218733.912 
218833.912" 
21899.027•• 
2190g.027a 
21919.0278 
21923231 
21933.2428 
21945.8945 
2195IMaga 
21962.000 
21972.000 
21982.000 
21992.000 
22002 000 
22012.0cn 
22022.000 
22032,000 
22042.000 
22052.000 
22062.000 
22076.000 
22086.000 
22096.000 
22103.000 
22116.000 
22126.000 
22133.000 
2214SEX 
2215• SATu0B Pillars Trace 
2216Wilks• Lambda 
2217Ot g S 
2218Roy's La est Root 
2219a, Exact statistic 
2220b, The statistic is an upper bound on F that yieMs a Iowa: bound on the significance level. 
2221c. Design; • SATJOa 
2222154 
2223Chapter 6 Multivariate Analysis Of V ariance and Covariance 
2224Figure 6.27 Univariate ANOVA Summao' Table. 
2225SATJOB 
2226SEX • TJoe 
2227Correc:ted Total 
2228RINCOM2 
2229RINCOM2 
2230EOuC2 
2231EOUC2 
2232RINCOM2 
2233RINCOM2 
2234RINCOM2 
2235EDUC2 
2236Type 
2237'917923' 
22389098.497 
2239'331,310 
2240813.705 
224146.615 
22422S4.331 
224315551 
224429.773 
22454531.257 
224610080.664 
2247139763 0 
2248149199.0 
22498043 
2250239740 
22514331.310 
2252813705 
22536.75B 
225426629' 
225515,538 
225684.777 
22575.'84 
22589,924 
225915gsE 
2260'347029 
2261288.305 
2262S4_163 
22631.001 
226417.725 
22652.301 
2266NOT 
2267cmty e*ts 
2268,015 (Adjusted R Squared _003) 
2269SECTION SPSS "HOW TO" FOR MANCOVA 
2270This section describes the Steps for conducting both the preliminary MANCOVA and the full 
2271MANCOVA using the Multivariate procedure. Again, the preceding example from the gssft.sav 
2272data set is utilized in these steps. The first series Of steps describes the preliminary MANCOVA process 
2273for testing homogeneity Of variance-covariance and homogeneity Of regression slopes. To open the 
2274Multivariate dialogue box (see Figure 6.28), select the following: 
2275General 
2276Mu Itivari a te 
2277tivariate Dialo Box see Fi 6.28 
2278Once in this dialogue box, click each DV (rincom2 and educ2) and move to the Dependent 
2279Variables box. Click each IV (sex and satjob) and move to the Fixed Factor(s) box. Then click 
2280each covariate (age) and move to the Covariate box. Then click Model. 
2281Model Piglggue Box egg Figure 6.29) 
2282Under Specify Model, click Custom. Move each IV and covariate to the Model box. Then 
2283hold down the Ctrl key and highlight all IVs and covariate(s). Once highlighted, continue to hold down 
2284the shift key and move to the Model box. This should create the interaction between all IVs and covari- 
2285ate(s) (e.g., Also check to make sure that Interaction is specified in the Build Terms 
2286box. Click Continue. Back in the Multivariate Dialogue Box, click Options. 
2287Chapter 6 Multivariate Analysis Of Variance and Covariance 
2288Figure 6.28 Multivariate Dialogue Box. 
2289deg ee 
2290s atiDb2 
2291Mleft 
2292age 
2293eda:2 
229413 ' morn2 
2295Eiged 
2296CO" i&el 
2297@age 
2298Figure 6.29 Multivariate Model Dialogue Box. 
2299the 
2300factors and 
2301l',pelll 
2302Multivariate Options Dialogue Box (see Figure 6.30) 
2303Under Display, click Homogeneity Tests. Click Continue. Back in the Multivariate 
2304Dialogue Box, click OK. 
2305These steps will create the output to evaluate homogeneity of variance-covariance and homoge- 
2306neity of regression slopes. If interaction between the factors and covariates is not significant, then pro- 
2307ceed with the following steps for conducting the full MANCOVA. The same dialogue boxes are 
2308opened, but different commands will be used. Open the Multivariate Dialogue Box by selecting the fol- 
2309lowing: 
2310Analyze 
2311General Linear Model 
2312Mul tivariate 
2313Chapter 6 Multivariate Analysis of Variance and Covariance 
2314Figure 6.30 Multivariate Options Dialogue Box. 
2315SSP 
2316Mul tivariate Dialogue Box (see Fi41!e 6.u) 
2317If you have conducted the preliminaly MANCOVA, variables should already be identified. If 
2318not, proceed with the following. Click each DV (rincom2 and educ2) and move to the Dependent Vari- 
2319ables box. Click each IV (sex and satjob) and move to the Fixed Factor(s) box, Then click each covari- 
2320ate (age) and move to the Covariate box. Then click Model. 
2321Multivariate Model Dialogue Box (see Figure 6.31 ) 
2322Under specify model, click Full. Click Continue. 
2323Box, click 
2324Figure 6.31 Multivariate Model Dialogue Box, 
2325Back in the Multivariate Dialogue 
2326C Ontiræ 
2327Sse.'y — 
2328G Éii' 
2329157 
2330Chapter 6 Multivariate Analysis of Variance and Covariance 
2331Multivariate Options Dialogue Box (see Figugg 6.32) 
2332Under Factor(s) and Factor Interaction, click each IV and move to the Display Means box. 
2333der Display, click Descriptive Statistics and Estimates of EEEect Size. Click 
2334Continue. Back in the Multivariate Dialogue Box, click OK. 
2335Figure 6.32 Multivariate Options Dialogue Box. 
2336IOvERALL) 
2337v ed 
2338mere:es 
2339SUMMARY 
2340LSD .'r,oreq 
2341Spread plots 
2342estimable 
2343Multivariate analysis of variance (MANOVA) allows the researcher to examine group differ- 
2344ences within a set of dependent variables. Factorial MANOVA will test the main effect for each factor 
2345On the combined DV as well as the interaction among factors on the combined DV. Usually follow-up 
2346tests, such as Univariate ANOVA and post hoc tests, are conducted within MANOVA to determine the 
2347specificity Of group differences. Prior to conducting MANOVA, data should be screened for missing 
2348data and outliers. Data should also be examined for fulfillment Of test assumptions: normality, homo- 
2349geneity of vanance-covariance, and linearity of DVs. Box's Test for homogeneity Of variance- 
2350covariance will help determine which test statistic (e.g., Wilks' Lambda, Pillai's Trace) to utilize when 
2351interpreting the multivariate tests. The SPSS MANOVA table provides four different test statistics 
2352(Wilks' Lambda, Pillai's Trace, Hotelling's Trace, and Roy's Largest Root) with the F ratio, p value, 
2353and effect size that indicate the significance of factor main effects and interaction. Wilks' Lambda is the 
2354most commonly used criterion. If factor interaction is significant, then conclusions about main effects 
2355are limited. Univariate ANOVA and post hoc results determine group differences for each DV. Figure 
23566.33 provides a checklist for conducting MANOVA. 
2357Chapter 6 Multivariate Analysis Of Variance and Covariance 
2358Figure 6.33 Checklist for Conducting MANOVA. 
2359I. Screen D a t • 
2360Missing Data? 
2361Outliers? 
2362Run Outliers and review Stem—and-leaf plots Within or.. 
2363d. 
2364Eliminate or transform Outliers if necessary. 
2365Normality? 
2366Run Normality Plots With Tests Within Expl Ore . 
2367and histograms. 
2368Transform data if necessary _ 
2369Linearity Of D Vs? 
2370Create Scan erplots. 
2371Calculate Pearson correlation coefficients. 
2372Transform data if necessary. 
2373Of Variaxc-covariance? 
2374Run BOX 's Test With i iate. 
2375Analyze.. Model.. 
2376. Multivariate. 
237711. Conduct MANOVA 
2378Run MA A with post test. 
2379d. 
2380e. 
23816. 
23828. 
23839. 
2384Il. 
238512 _ 
238613. 
238714. 
2388Move DVs to Dependent Variable box. 
2389Move IVs to Fixed Inx. 
2390Model. 
2391Continue. 
2392Move each IV to the Display Means box. 
2393Check S s cs , 
2394Continue. 
2395Opost hoc. 
2396Move each IV to Test 
2397Select post hoc method. 
2398Continue . 
2399Estimates of Effect Size and 
2400H omogeneity Of e? 
2401Examine F•ratio and p•value for Box's Test. 
2402If significant at with extremely unequal group sample sizes, use Pillai's Trace for the test statistic. 
2403If NOT significant at with fairly equal grot& sample sizes. use Wilks• LamMa test statistic. 
2404Interpret factor interaction. 
2405If factor interaction is significant. main effects are erroneous. 
2406If factu interactim is NOT sgnificant. interpret main effects. 
2407Interpret main effects for each IV on the combined DV. 
2408Interpret Univariate ANOVA results. 
2409Interpret post hoc results. 
2410Ill. Summarize Results 
2411b. 
2412d. 
2413e. 
2414Describe any data elimination or transformation. 
2415Narrate Full MANOVA results. 
2416Main effects for each IV on the combined DV (test statistic, h ratio. p value, effect size). 
2417Main effect for factor interaction (test statistic, F•ratio, p•value, effect size). 
2418Narrate A results, 
2419Main e each I V and D V effect 
2420Narrate post hoc results. 
2421Draw conclusions. 
2422Chapter 6 Multivariate Analysis ofVariance and Covariance 
2423Multivariate analysis of covariance (MANCOVA) allows the researcher to examine group dif- 
2424ferences within a set of dependent variables while controlling for covariate(s). Essentially, the influence 
2425that the covariate(s) has on the combined DV is panitioned out before groups are compared, such that 
2426group means of the combined DV are adjusted to eliminate the effect of the covariate(s). One-way 
2427MANCOVA will test the main efTects for the factor on the combined DV while controlling for the co- 
2428variate(s). Factorial MANCOVA will do the same but will also test the interaction among factors on the 
2429combined DV while controlling for the covariate(s). Usually univariate ANCOVA is conducted within 
2430MANCOVA to determine the specificity of group differences. Prior to conducting MANCOVA, data 
2431should be screened for missing data and outliers. Data should also be examined for fulfillment of test 
2432assumptions: normality, homogeneity of variance-covariance, homogeneity of regression slopes, and 
2433linearity of DVs and covariates. A prelimina1V or custom MANCOVA must be conducted to test the 
2434assumptions of homogeneity of variance-covariance and homogeneity of regression slopes. Box's Test 
2435for homogeneity of variance-covariance will help determine which test statistic (e.g., Wilks' Lambda, 
2436Pillai's Trace) to utilize when interpreting the test for homogeneity of regression slopes and the full 
2437MANCOVA analyses. The test for homogeneity of regression slopes will indicate the degree to which 
2438the factors and covariate(s) interact to effect the combined DV. If interaction is significant, as indicated 
2439by the F ratio and p value for the appropriate test statistic, then the full MANCOVA should NOT be 
2440conducted. If interaction is not significant, then the full MANCOVA can be conducted. Once the full 
2441MANCOVA has been completed, factor interaction should be examined when two or more IVs are util- 
2442ized. If factor interaction is significant, then conclusions about main effects are limited. Interpretation 
2443of the multivariate main effects and interaction is similar to MANOVA. Univariate ANOVA results 
2444determine the significance of group differences for each DV. Figure 6.34 provides a checklist for con- 
2445ducting MANCOVA. 
2446Chapter 6 Multivariate Analysis Of Variance and Covariance 
2447Figure 6—34 Checklist for Conducting MANCOVA. 
2448Screen Data 
2449d. 
2450e. 
2451Missing Data? 
2452outliers? 
2453Run Outliers and stem-and-leaf plots Within Explore . 
2454Eliminate or transtarm it' necessary 
2455Normality' 
2456Run Plots Tests within 
2457Review boxplots and histograms. 
2458Transform dala if neeessary, 
2459Linearity ofDVs and covariate(s)? 
2460Create Scatterplots. 
2461Calculate Pearson correlation coefficients. 
2462Transform data if necessary. 
2463Test remaining assumptions hy conducting preliminary MANCOVA. 
2464Conduct preliminary (Custom) MANCOVA 
2465Run custom MANCOVA_ 
2466-'t General Linear Model ...C Multivariate. 
2467Move DVs to Dependent Variable box. 
2468M0'.e IVs to Fixed Factor box, 
2469Move covariate(s) to Covanate box. 
2470s -'t Model. 
2471Move IV and covariate the Model 
2472Hold down Ctrl key and highlight all IVs and covariate(s), while still holding dmvn the Ctrl key in order 
2473to interaction to Model box. 
2474Continue _ 
247510. 
2476Options. 
2477Check Homogeneity Tests _ 
2478Continue. 
247912. 
248013. 
2481Homogeneity 
2482Examine F-rutio and p-value for Box's Test. 
2483If significant at pz.001 with exlremely unequal group sample sizes, use Pillars Trace for the test statistic. 
2484If NOT significant at with fairly equal group sample sizes, use Wilks• Lambda for the test statistic. 
2485Homogeneity of Regression Slopes? 
2486Using the appropriate test statistic. examine F-ratio and p-value for the interaction among IVs and covariates. 
2487If is significant, do not proceed with Full MANCOVA. 
2488If interaction is NOT significant, proceed with Full MANCOVA. 
2489111. conduct MANCOVA 
2490Run Full MANCOVA. 
2491Linear 
2492Move DVs to Dependent Variable 
2493Move IVs to Fixed Factor box. 
2494Move covariate(s) Covariate box. 
24956 Full. 
2496Option s. 
2497Move each IV to lhe Display Means box- 
2498Check Descriptive Statistics and of Effect Size. 
249910 
2500Il. 
250112, 
2502Continue _ 
2503Interpret factor interaction. 
2504If factor interaction is significant. main effects are erroneous. 
2505If factor interaction is NOT significant. interpret main effects. 
2506Interpret main effects cach IV on the ccnnbined DVs. 
2507Interpret Cni'.ariate ANOVA results. 
2508Chapter 6 Multivariate Analysis Of Variance and Covariance 
2509Figure Checklist for Conducting MANCOVA (Continued) 
2510IV. Summarize Results 
2511a. 
2512C. 
2513e. 
2514Describe any data elimination Ot transformation. 
2515Narrate Full MANCOVA results. 
2516Main effects each V on DV statistic. F—ratiO. c Size). 
2517Main effect for factor interaction (test statistic, F-ratio, p-value, effect size). 
2518Narrate Univariate ANOVA results. 
2519Main effects for each IV and DV (F-ratio. "-value. effect size). 
2520Compare groLQ means to indicate which groups differ on each DV 
2521Draw conclusions. 
2522Exercises for Chapter 6 
2523The two exercises below utilize the data set gssft.sav, which can be downloaded from this Web site: 
2524http:/_ledhd.bgsu.edu/amm/datasets.html 
2525l. You are interested in evaluating the effect of job satisfaction (satjob2) and age category (agecat4) 
2526on the combined DV of hours worked per week (hrs I) and years of education (educ). 
2527b. 
2528C. 
2529d. 
2530Develop the appropriate research questions and/or hypotheses for main effects and interaction. 
2531Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing 
2532data and outliers? 
2533Test the assumptions Of normality and linearity of DVs. 
2534i, What steps, if any, are necessary for increasing normality? 
2535ii. Are DVs linearly related? 
2536Conduct MANOVA with post hoc (be sure to test for homogeneity Of variance-covariance). 
2537Can you conclude homogeneity ofvariance-covariance? Which test statistic is most 
2538appropriate for interpretation of multivariate results? 
2539Is factor interaction significant? Explain. 
2540162 
2541Chapter 6 Multivariate Analysis Of Variance and Covariance 
2542Are main effects significant? Explain. 
2543iv. What can you conclude from univariate ANOVA and post hoc results? 
2544e. Write a results statement. 
25452. Building on the previous problem in which you investigated the effects of job satisfaction (satjoh2) 
2546and age category (agecat4) on the combined dependent variable of hours worked per week (hrsl) 
2547and of education (educ), you are now interested in controlling for respondent's income such 
2548that rincom91 will be used as a covariate. Complete the following. 
2549a. Develop the appropriate research questions and/or hypotheses for main effects and interaction. 
2550b. Screen data for missing data and outliers. What steps, if any, are necessary for reducing missing 
2551data and outliers? 
2552c. Test the assumptions of normality and linearity of DVs and covariate. 
2553i. What steps, if any, are necessary for increasing normality? 
2554ii. Are DVs and covariate linearly related? 
2555d. Conduct a preliminary MANCOVA to test the assumptions of homogeneity of variance- 
2556covariance and homogeneity of regression slopes/planes_ 
2557i. Can you conclude homogeneity of variance-covariance? Which test statistic is most ap- 
2558propriate for interpretation of multivariate results? 
2559ii. Do factors and covariate significantly interact? Explain. 
2560e. Conduct MANCOVA. 
2561n. 
2562Is factor interaction significant? Explain. 
2563Are main effects significant? Explain. 
2564Chapter 6 Multivariate Analysis of Variance and Covariance 
2565iii. What Can you conclude from univariate ANOVA results? 
2566f. Write a results statement. 
2567Compare the results from problems number I and number 2. Explain the differences in main 
2568effects. 
2569164