CHAPTER 4

RESULTS AND DATA ANALYSIS

Loss of Participants and Missing Data

            One-hundred-eight persons responded to the newspaper ads for this study. Of these 108, 93 made it to the stage of random assignment, with 31 designated for each of the three groups. Out of these 93 people, 81 completed the pretest surveys, and 62 completed the entire study. For the 81 who completed the pretest, 26 were in the maha mantra group, 27 were in the alternate mantra group, and 28 were in the control group, and for the 62 who completed the experiment, 24 were in the maha mantra group, 19 in the alternate mantra group, and 19 in the control group. Reasons for subjects leaving the study were described in the procedures section of the methodology chapter (chapter 3).

            Cohen and Cohen (1983) write "Because Y represents the outcome or effect of the IVs, when the Y value for a subject is not known, there is little that can be done in MRC but drop that subject......because the research is focally concerned with Y, we find unattractive, in general, attempts to make up Y values so as to avoid the loss of information in the IVs and the reduction in n” (p. 276). Thus, they recommend that data from dropout subjects be dropped from the statistical analysis. Still, for the sake of completeness, means and standard deviations for subjects who did not complete the study were calculated, and t tests were performed to determine if dropout scores at pretest (n=19) were significantly different from non-dropout scores at pretest (n=62). Specifically, pretest scores for dropouts for each group for each of the five dependent variables, as well as for gender and age, were compared with pretest scores for non-dropouts. Altogether 21 t tests were conducted (3 groups x 7 variables), and none of them were significant at the .05 level. This indicates that dropouts from the survey were random with respect to the variables studied, rather than due to some systematic factor.

            There were very few missing data points in the surveys that were completed. There were no missing data for any of the independent variables, and none of the dependent variables had more than 1.1% of its scores missing. Cohen and Cohen (1983) describe a process for creating a missing data variable in order to positively utilize missing data as valuable information. However, they explain that when missing values are very few, such a variable is unnecessary. Therefore, the pairwise deletion method was used for missing values in the surveys. Nunnally and Bernstein (1994) write that pairwise deletion is preferable to listwise deletion when there are only a small number of omissions.

Demographics

            Tables 3 to 5 show demographic statistics for age, gender, and chanting frequency.

            Demographic statistics for age are shown in Table 3. The average age of participants was 24.63 years, ranging from 18 to 49 years. For the maha mantra group the average age was 22.46 years, ranging from 18 to 48 years, for the alternate mantra group the average age was 24 years, ranging from 19 to 39 years, and for the control group the average age was 28 years, ranging from 19 to 49 years. An ANOVA comparing average age for each group resulted in a non-significant F statistic at the .05 level.

Table 3

Demographic Statistics for Age

Variable     Mean       Std Dev      Minimum   Maximum   N

AGE          24.63       7.71        18.00     49.00     62

AGEMAHA      22.46       5.90        18.00     48.00     24

AGEALT       24.00       5.76        19.00     39.00     19

AGECON       28.00      10.24        19.00     49.00     19

     Demographics for gender are shown in Table 4. Among subjects who completed the study, 31 were female and 31 were male. In the maha mantra group there were 9 males and 15 females, in the alternate mantra group there were 13 males and 6 females and in the control group there were 9 males and 10 females. A chi-square statistic comparing the gender distribution between groups resulted in a non-significant p value at the .05 level.

Table 4

Demographic Statistics for Gender

Variable       Females     Males    N

GENDER        31        31        62

GENMAHA         15         9       24

GENALT           6        13        19

GENCON          10         9        19

            Table 5 shows average chanting frequency for the maha mantra group and alternate mantra group. A t test comparing the differences between these two groups resulted in a nonsignificant p value at the .05 level.

Table 5

Chanting Frequency Means and Standard Deviations

N = 24 for Maha Group; N = 19 for Alternate Group         

Scores on Dependent Measures by Group

Stress

            Table 6 shows stress scores, as measured by the ICS, for each group at pretest, posttest and followup. An ANOVA comparing mean ICS scores for each group at pretest resulted in a non-significant F statistic at the .05 level.

Table 6

Stress Scores by Group at Pretest, Posttest and Followup

N = 24 for Maha Group; N = 19 for Alternate Group; N = 19 for Control Group

Pretest                       Mean ICS Score           Std Dev

     Maha Group                    33.43                11.13    

     Alternate Group               36.59                19.32

     Control Group                 27.11                13.97

Posttest

     Maha Group                    22.11                6.80

     Alternate Group               40.39                23.16

     Control Group                 28.06                16.52

Followup

     Maha Group                    26.76                 9.77

     Alternate Group               35.87                22.11

     Control Group                 28.28                14.25

Depression

            Table 7 shows depression scores, as measured by the GCS, for each group at pretest, posttest and followup. An ANOVA comparing mean GCS scores for each group found a significant difference at pretest (F= 5.56; p= .006).

Table 7

Depression Scores by Group at Pretest, Posttest and Followup

N = 24 for Maha Group; N = 19 for Alternate Group; N = 19 for Control Group

Pretest                  Mean GCS Score                Std Dev

     Maha Group             29.75                       11.23

     Alternate Group        32.02                       12.25   

     Control Group          21.07                        8.28

Posttest

     Maha Group             21.48                        6.76

     Alternate Group        30.41                       15.59

     Control Group          20.34                        8.91

Followup

     Maha Group             25.18                        11.27   

     Alternate Group        32.30                        15.02

     Control Group          21.49                        9.10

Sattva

            Table 8 shows sattva scores, as measured by the sattva subscale of the VPI, for each group at pretest, posttest and followup. An ANOVA comparing mean pretest sattva scores between groups found a significant difference at pretest (F= 3.30; p= .044).

Table 8

Sattva Scores by Group at Pretest, Posttest and Followup

N = 24 for Maha Group; N = 19 for Alternate Group; N = 19 for Control Group

Pretest                       Mean Sattva Score            Std Dev

     Maha Group                    71.38                   7.71  

     Alternate Group               67.30                   8.28      

     Control Group                 73.71                  7.43

Posttest

     Maha Group                    76.61                   5.07

     Alternate Group               66.45                   9.08 

     Control Group                 72.92                  7.00  

Followup

     Maha Group                    71.95                   6.78

     Alternate Group               65.74                   9.99

     Control Group                 73.83                   6.87

Rajas

            Table 9 shows rajas scores, as measured by the rajas subscale of the VPI, for each group at pretest, posttest and followup. An ANCOVA comparing mean rajas scores at pretest for each group found a significant difference at pretest (F= 8.75; p= .001).

Table 9

Rajas Scores by Group at Pretest, Posttest and Followup

N = 24 for Maha Group; N = 19 for Alternate Group; N = 19 for Control Group

Pretest                  Mean Rajas Score     Std Dev

     Maha Group               52.44            9.12

     Alternate Group          56.21            7.67

     Control Group            44.42            9.73

Posttest

     Maha Group               50.79            6.60

     Alternate Group          51.40            9.27

     Control Group            46.10            9.77

Followup

     Maha Group               53.75           10.56

     Alternate Group          52.02            9.46

     Control Group            44.96            9.88

Tamas

                Table 10 shows tamas scores, as measured by the tamas subscale of the VPI, for each group at pretest, posttest and followup. An ANOVA comparing pretest tamas scores between groups found a significant difference at pretest (F= 5.70; p= .010).

Table 10

Tamas Scores by Group at Pretest, Posttest and Followup

N = 24 for Maha Group; N = 19 for Alternate Group; N = 19 for Control Group

Pretest                  Mean Tamas Score          Std Dev

     Maha Group               49.97                   10.22

     Alternate Group          52.31                   10.62

     Control Group            40.64                   13.35

Posttest

     Maha Group               43.61                    7.08

     Alternate Group          49.12                   10.68

     Control Group            41.53                   13.03

Followup

     Maha Group               46.19                    7.51

Table 10- Continued      Mean Tamas Score     Std Dev

     Alternate Group          50.52              9.74  

     Control Group              40.27           13.08

Correlations of Covariates

            A large correlation matrix was generated, with computations of all correlations between all variables in the study. This included correlations involving dependent and independent variables, as well as all possible combinations of test time (pre, post, and followup) and group (maha, alternate, and control). This section discusses correlations involving the covariates in this study, namely gender, age, and chanting frequency, which were assessed as covariates of the primary independent variable, group status.

Gender

            Correlations between gender and the other variables in the study were calculated using Pearson r. Specifically, correlations for gender in each group- maha, alternate and control- were calculated for each variable in that group. For instance, correlations for gender in the alternate group were calculated for pretest, posttest, and followup scores for the alternate group for each of the five dependent variables. Also, correlations between gender and age, and gender and chanting frequency, were computed.

            For the maha mantra group there were 18 correlations computed (3 test times x 5 dependent variables + age + chanting frequency + gender). Of these 18 calculations, none were significant at the .05 level, except of course for gender with itself.

            For the alternate mantra group there were also 18 calculations, and of these the only correlation significant at the .05 level, other than gender with itself, was gender alternate group with age alternate group (r=-.63; p=.004). This indicates that for the alternate mantra group the 13 males were significantly younger than the 6 females. For the gender variable females were coded as “1” and males were coded as “2”.

            For the control group there were 17 calculations, because the control group had no chanting frequency values. Of these 17 correlations, 6 were significant at the .05 level. Gender for the control group was significantly correlated with control group scores for Pretest Sattva (r=-.46; p=.048), Followup Sattva (r=-.50; p=.029), Pretest Tamas (r=.50; p=.028), Posttest Tamas (r=.46; p=.047), Followup Tamas (r=.50; p=.029), and Gender (r=1.00; p=.000).

            These correlations indicate that for the control group males had higher tamas scores, at all three measurement times, than females, and that females had higher sattva scores, at pretest and followup, than males. 

            For gender, 6 out of 50 (12%) of the correlations with other variables were significant at the .05 level. Gender was hypothesized to not be significantly correlated with the other variables.  

Age

            Correlations between age and the other variables in the study were calculated using Pearson r. Specifically, correlations for age in each group- maha, alternate and control- were calculated for each variable in that group. For instance, correlations for age in the alternate group were calculated for pretest, posttest, and followup scores for the Alternate Group for each of the five dependent variables. Also, correlations between age and gender, and age and chanting frequency, were computed.

            For the maha mantra group there were 18 correlations computed (3 test times x 5 dependent variables + gender + chanting frequency + age). Of these 18 calculations, none were significant at the .05 level, except for the correlation of age with itself.

            For the alternate mantra group there were also 18 calculations, and of these three were significant at the .05 level. The correlation with Gender was significant, as described above, and the correlation between Age Alternate Group and Chanting Frequency Alternate Group was also significant (r--.47; p=.041), as was the correlation of age with itself (r=1.00; p=.000). This  shows that for the Alternate Group younger subjects had a higher chanting frequency.       

            For the control group there were 17 calculations, since the control group had no chanting frequency values. None of these were significant at the .05 level, except for the correlation of age with itself.

            For age, 2 out of 50 correlations (4%) with other variables were significant at the .05 level. Age was hypothesized to not be correlated with the other variables.

Chanting Frequency

            Chanting frequency was hypothesized for the maha mantra group to correlate positively with sattva, and negatively with stress, depression, rajas and tamas. For the alternate mantra group chanting frequency was hypothesized to have no correlation with the other variables.

            For the maha mantra and alternate groups correlations between chanting frequency and all other variables in that group were computed using Pearson r. For each of these groups there were 18 correlations calculated, as described above for the gender and age variables.

            For the maha mantra group none of the correlations were significant, except for chanting frequency with itself.

            For the alternate mantra group 11 of 17 correlations were significant at the .05 level. These correlations are presented in Table 11.

Table 11

Significant Chanting Frequency Correlations

                                                                                    Pearson r                    p value

Pretest Stress Alternate Group                                      -.54                              .017

Pretest Sattva Alternate Group                                      -.60                              .007

Pretest Rajas Alternate Group                                       -.65                              .003

Posttest Stress Alternate Group                                     -.55                              .015

Posttest Rajas Alternate Group                                      -.56                              .013

Posttest Depression Alternate Group                             -.47                              .043

Followup Stress Alternate Group                                   -.64                              .003

Followup Rajas Alternate Group                                    -.55                              .016

Followup Depression Alt. Group                                    -.48                              .038

Chanting Frequency                                                       1.00                             .000

            Also, as explained in the above section on Age, the correlation between age alternate group and chanting frequency alternate group was significant (r=-.47; p=.041). These data indicate that those who chanted more, compared with those who chanted less, scored lower on stress at all three measurement periods. For depression, increased chanting was associated with lower depression scores at posttest and followup. Also, increased chanting correlated with lower rajas scores at all measurement periods.

            Contrary to the secondary hypotheses, chanting frequency had no correlation with the maha mantra group variables, and it did significantly correlate with several of the alternate mantra group variables. This will be discussed in the next chapter.

General Covariate Correlations with Other Covariates

            Without specifying group status, age and chanting frequency were significantly correlated (r=-.31; p=.015). That is, chanting frequency is significantly correlated with age, without regard for whether subjects were in the maha mantra group or the alternate mantra group. Other intercovariate correlations, without specifying group status, were not significant. This information is useful in assessing the effects of multicollinearity on the ANCOVA models that will be presented later in this chapter. Specifically, age and chanting frequency were significantly correlated at the .05 level, and therefore it is possible that their effects on dependent variables were somewhat overlapping. Concerning multicollinearity, Cohen and Cohen (1983) explain that a hierarchical approach to ANCOVA serves to separate effects of the different independent variables. This hierarchical approach was used in this study.

Pretest Group Comparisons

            In order to evaluate the effectiveness of random assignment in this experimental design, pretest scores for each group were compared using ANOVA tests. At the .05 level there were significant differences between at least two groups at pretest for depression (p=.006), sattva (p=.044), rajas (p=.001), and tamas (p=.010). Therefore, for these four variables random assignment was apparently not successful.

            For depression, the control group differed significantly at pretest from both the maha mantra and alternate mantra groups. For sattva, the difference in pretest scores between the alternate and control groups was significant. For rajas and tamas, the control group differed significantly at pretest from both the maha mantra and alternate mantra groups.

Reliability Analyses for Dependent Variable Measures

            For all five measures, observed Cronbach’s alpha was less than that reported in the literature, and as described in the methodology chapter of this dissertation. Still, all of the scores were in the acceptable range, as Nunnally and Bernstein (1994) explain that, for group comparisons, an alpha of .70 is satisfactory (see Table 12). Therefore, each of the scale or subscale scores was retained for subsequent analyses.

Table 12

Alpha for Dependent Measures

                                                Alpha

Sattva (VPI):                            .86          

Rajas (VPI):                             .82          

Tamas (VPI):                            .87          

Stress (ICS):                             .94

Depression (GCS):                    .90

Analysis of Effects of Group Status and
 Covariates on the Dependent Variables

            ANCOVA was used to assess the effects of group status on the dependent variables. Gender, age and chanting frequency served as covariates in the ANCOVA analyses. These analyses were performed hierarchically, with age and gender assessed first, then chanting frequency, and then group status. This order of variable analysis was selected in accord with the principles of causal priority and removal of confounding variables, as described in Cohen and Cohen (1983). According to the principle of causal priority, variables that are temporally prior and unlikely to be affected by other variables should be analyzed first. Since age and gender fit this description, they were the first variables analyzed in the hierarchical ANCOVAs. The principle of removal of confounding variables dictates that variables other than the primary variable(s) being studied should be assessed prior to the main independent variables, in order to remove the effects of secondary variables when evaluating the effects of the main variables (Cohen & Cohen, 1983). Therefore, chanting frequency was analyzed prior to group status in the hierarchical analysis, so that the effects of all covariates were removed when assessing group status.

            ANCOVA is a combination of analysis of variance with a categorical level independent variable, and standard regression analysis with an interval level independent variable. This is an effective method for modeling an interval dependent variable in terms of both interval and categorical independent variables (Agresti and Finlay, 1986). This type of analysis is relevant to this study, because all the dependent variables are interval level, and the independent variables are both categorical, such as group status and gender, and interval, such as chanting frequency and age.

            The main hypotheses of this experiment, stated in the hypotheses section of the methodology chapter (chapter 3), were that chanting the maha mantra will decrease stress, depression, rajas and tamas, and will increase sattva, from pretest to posttest. These changes were additionally predicted to be significantly greater, at the alpha = .05 level, than any similar changes in the dependent variables occurring in the alternate mantra or control groups. Secondary hypotheses are that chanting frequency will effect the dependent variables in the same direction as the maha mantra group. That is, the greater the chanting frequency of the maha mantra, the greater was the expected decrease in stress, depression, rajas and tamas, and the greater was the expected increase in sattva. Chanting frequency of the alternate mantra was predicted to not have an effect on the dependent variables. Also, gender and age were predicted to not have an effect on the dependent variables. Changes in pretest-followup scores were expected to be in the same direction as for pretest-posttest for each variable, though the changes for pretest-followup were predicted to be smaller than the changes from pretest-posttest.

            Since the hypotheses described above involve comparisons of means while controlling for covariates, the ANCOVA statistic was chosen for the purpose. Although MANCOVA could have also been used in this analysis, with difference scores of pretest-posttest and pretest-followup serving as the simultaneous dependent variables, the sample size of the study did not provide sufficient power for efficacious use of the MANCOVA (Montgomery, 1997). Also, although repeated measures ANCOVA could be used for this experimental design, repeated measures ANCOVA is more typically used for time series analysis and trend studies with many more than three data points per subject. Therefore, ANCOVA was selected for this analysis (McNeil, Newman, & Kelly, 1996).

            Since group equivalence of means was not achieved for some of the dependent variables, it was especially important that the statistical procedure incorporated the difference in pretest scores. Calculating difference scores (e.g., from pretest to posttest) and then analyzing these difference scores with ANCOVA is one method of doing this. However, Cohen and Cohen (1983) note that the reliability of a difference score is likely to be lower than the variables being differenced, and that change scores tend to be correlated with pretest scores. Therefore, they are not an objective measure because the difference scores contain some variance that is due to the pretest score, and thus change scores do not actually remove the effect of the pretest. Cohen and Cohen recommend partialling the pretest score from the observed difference score, thus creating a regressed change score. Nunnally and Bernstein (1994), however, write “...the observed difference score...is the simplest and most direct definition of change, despite its problems. Also, remember that standardizing the elements of a difference score may produce spurious results” (p. 245). Nunnally and Bernstein critique the partiallizing method, pointing out shortcomings such as treating the pretest score as if it were an error-free true score. They conclude, “Consequently, observed differences need not be as fatally flawed as was once thought” (p. 246).

            To compensate for pretest inequalities, ANCOVA was first performed using the partialling method. To check for substantial differences in the results of the two statistical approaches, ANCOVA using observed difference scores was also performed for each dependent variable. Regressed change scores, as described in Cohen and Cohen (1983, p. 416), were calculated according to the following formula:

a-b(rab*sda / sdb)

a: posttest score

b: pretest score

rab: correlation of pretest scores with posttest scores

sda: standard deviation of posttest scores

sdb: standard deviation of posttest scores

            Before running ANCOVA, the data for each dependent variable was evaluated for adherence to assumptions for the ANCOVA statistic. Bartlett’s Box statistic was used to assess homoscedasticity, with a .05 significance level chosen to determine whether there is adequate homoscedasticity for a distribution. Dowdy and Wearden (1991) compare the homoscedasticity tests of Cochran, Hartley and Bartlett, and conclude that Bartlett’s test is the most powerful of the three. Also, they explain that the F test is robust with respect to departures from homogeneity, and that a significance level of p = .05 in the Bartlett statistic is an adequate measure of homoscedasticity for evaluating the homoscedasticity assumption for using the F distribution. The method of weighted least squares is a viable option for using regression techniques when there is severe violation of the homoscedasticity assumption. Sen and Srivastava (1990) explain that the weighted least squares approach ascribes smaller weights to larger errors, and that the method is a special case of generalized least squares. They caution that weighted least squares should not be utilized unless deviation from homoscedasticity is extreme.

            For each ANCOVA a test comparing a linear explanation for the data with a nonlinear relationship was performed. In some cases neither a linear nor a nonlinear model produced a significant p value (at  p = .05). This indicates that the relationship between the dependent variable and the independent variable being tested for linearity was so weak that no model explained a statistically significant portion of the variance. In the case of dependent variables regressed on the age variable, a non-significant p value was predicted by the hypotheses of this study, namely that age will not have a significant effect on the dependent variables. Still, when neither the p value for a linear or nonlinear explanation was significant, the lower p value indicated which type of model best explained the relationship between the two variables (Ryan, 1997). That is, the type of model with the lower p value explained more variance than the other type of model. Besides age, chanting frequency was the other interval-level independent variable, and therefore chanting frequency was also tested for a linear relationship with the dependent variables. According to the hypotheses of this study chanting frequency was predicted to have a relationship with the dependent variables, and therefore the p values for chanting frequency in the linearity tests are expected to be lower than the p values for age, indicating that a linear model explains more of the variance for chanting frequency than for age.

            Slopes for the dependent variables for each group of the age variable were calculated to estimate similarity of slopes and effects of interactions. Rigdon, Shumacker, and Wothke (1996) explain that markedly different slopes across groups between two variables indicate that an interaction term is necessary to explain the relationship between the three variables. According to Rigdon et al., similarity of slopes is also support for the fulfillment of the linearity assumption. Specific to japa studies, if the slope of age regressed on the dependent variable is the same for the maha mantra group, the alternate group, and the control group, then it means that there is no significant interaction between age and group status in relation to the dependent variable. Sheskin (1997) suggests that a difference in slopes between groups of more than .50 may be considered large, and this standard was used in the following analyses.

            If the total explained variance for an ANCOVA was significant at the .05 level, then t tests were performed to determine which of the three group comparisons had significant differences at the .05 level. Also, 95% confidence intervals (CI) were calculated for each t test. For each ANCOVA a model with a group status-gender interaction term was computed. In no case did this term have a statistically significant effect on the dependent variable, and therefore the term was dropped from the model in all cases.

            After the ANCOVA, t tests for the main effects of the different groups on the dependent variable were performed if the F test for the explained variance of the entire model was significant at the .05 level. In addition, effect sizes are presented with the eta squared statistic. Nunnally and Bernstein (1994) describe eta as a universal measure of relationship that can be used regardless of the form of the relationship. Eta is a correlation ratio that is calculated by computing the variance in the dependent variable about any curve of the relationship. Further, they state that eta applies equally well to categorical or continuous variables, which is relevant to this study because there are independent variables that are categorical and continuous. They write “Although F is basic to statistical inferences about group mean differences in the population, eta indicates how strong the relationship is, thus describing the independent variable’s explanatory power.” (p. 138).

Difference Scores

            The ANCOVAs in the following analyses used difference scores. Therefore, difference scores for the dependent variables, which can also be calculated from Tables 6-10, are presented below in Table 13.

Table 13

Difference Scores for Dependent Variables by Group from Pretest to Posttest and from Pretest to Followup

                                                                                                                                    Mean Difference

Stress Scores

     Maha Group From Pretest To Posttest                    -11.32

     Alternate Group From Pretest To Posttest                 3.80

     Control Group From Pretest To Posttest                    .95

     Maha Group From Pretest To Followup                     -6.67

     Alternate Group From Pretest To Followup                 -.72

     Control Group From Pretest To Followup                   1.17

Depression Scores

     Maha Group From Pretest To Posttest                     -8.27

     Alternate Group From Pretest To Posttest                -1.61

     Control Group From Pretest To Posttest                   -.73

     Maha Group From Pretest To Followup                     -4.57

     Alternate Group From Pretest To Followup                  .28

     Control Group From Pretest To Followup                    .42

Sattva Scores

     Maha Group From Pretest To Posttest                      5.23

     Alternate Group From Pretest To Posttest                 -.85

     Control Group From Pretest To Posttest                   -.79

     Maha Group From Pretest To Followup                       .57

     Alternate Group From Pretest To Followup                -1.56

     Control Group From Pretest To Followup                    .12

Rajas Scores

     Maha Group From Pretest To Posttest                     -1.65

     Alternate Group From Pretest To Posttest                -4.81

     Control Group From Pretest To Posttest                   1.68

     Maha Group From Pretest To Followup                      1.31

Table 13- continued                                                                                              Mean Difference

     Alternate Group From Pretest To Followup                -4.19

     Control Group From Pretest To Followup                    .54

Tamas Scores

     Maha Group From Pretest To Posttest                     -6.36

     Alternate Group From Pretest To Posttest                -3.19

     Control Group From Pretest To Posttest                    .89

     Maha Group From Pretest To Followup                     -3.78

     Alternate Group From Pretest To Followup                -1.79

     Control Group From Pretest To Followup                   -.37

            The difference scores shown in Table 13 reflect the change in average group scores from pretest to posttest and pretest to followup for each dependent variable. ANCOVAs tested the difference in these change scores between groups, controlling for age, gender, and chanting frequency. Table 14 shows the difference in change scores between groups, which are the values that were actually evaluated for statistical significance. The values in Table 14 can be calculated from Table 13. For calculation of partiallized change scores, the formula in Cohen and Cohen (1983, p. 46), which was described in the preceding section, was applied to the mean difference scores in Table 14.

Table 14

Difference in Change Scores for Dependent Variables by Group from Pretest to Posttest and from Pretest to Followup

                                                                                                                        Mean Difference

Stress Scores                                                  

            Maha/Alternate From Pretest to Posttest                                                -15.12*

            Maha/Control From Pretest to Posttest                                                  -12.27*

            Alternate/Control From Pretest to Posttest                                                2.85

            Maha/Alternate From Pretest to Followup                                              -5.95

            Maha/Control From Pretest to Followup                                                -7.84

            Alternate/Control From Pretest to Followup                                            -1.89

Depression Scores

            Maha/Alternate From Pretest to Posttest                                                -6.66*

            Maha/Control From Pretest to Posttest                                                  -7.54*

            Alternate/Control From Pretest to Posttest                                               -.88

Depression Scores 

            Maha/Alternate From Pretest to Followup                                             -4.85

Table 14- continued                                                                                              Mean Difference

            Maha/Control From Pretest to Followup                                                        -4.99*

            Alternate/Control From Pretest to Followup                                                     -.14

Sattva Scores

            Maha/Alternate From Pretest to Posttest                                                         6.08*

            Maha/Control From Pretest to Posttest                                                           6.02*

            Alternate/Control From Pretest to Posttest                                                      -.06

            Maha/Alternate From Pretest to Followup                                                     2.13

            Maha/Control From Pretest to Followup                                                          .45

            Alternate/Control From Pretest to Followup                                                 -1.68

Rajas Scores

            Maha/Alternate From Pretest to Posttest                                                         3.16

            Maha/Control From Pretest to Posttest                                                         -3.33

            Alternate/Control From Pretest to Posttest                                                    -6.49

            Maha/Alternate From Pretest to Followup                                                       5.50

            Maha/Control From Pretest to Followup                                                           .77

            Alternate/Control From Pretest to Followup                                                  -4.73

Tamas Scores

            Maha/Alternate From Pretest to Posttest                                                       -3.17

            Maha/Control From Pretest to Posttest                                                         -7.25*

            Alternate/Control From Pretest to Posttest                                                   -4.08

            Maha/Alternate From Pretest to Followup                                                   -1.99

            Maha/Control From Pretest to Followup                                                      -3.41*

            Alternate/Control From Pretest to Followup                                                 -1.42

*Indicates a statistically significant value at alpha = .05.

Statistical Analysis of the Stress Variable

Pretest-Posttest Analysis of Stress Variable

Tests of Assumptions, Interaction, and Outliers for Pretest-Posttest Analysis of the Stress Variable.

            Analysis of residuals was performed for partiallized difference scores and observed difference scores for the pretest-posttest analysis of the stress variable. For partiallized scores, Bartlett’s Box statistic for homogeneity, F (2, 7601), had a p value of .166, and for observed scores, Bartlett’s Box statistic for homogeneity, F (2, 7601), had a p value of .125. These non-significant p values indicate that the data possess adequate homogeneity of variance for the ANCOVA.

            According to Nunnally and Bernstein (1994), about 5% of standardized residuals can be expected to have a value greater than 2.  Cohen and Cohen (1983) state “When residuals are standardized by dividing them by their standard deviation, a residual that is as much as three (or, certainly, four) of these units in absolute size is reasonably considered an outlier” (p. 48).  They further state that outliers are particularly bothersome when they are predominantly of the same sign.

            For partiallized scores, 4 of 62 standardized residual scores had values greater than 2. These four values were -3.64, -3.29, 2.97, and 2.04. The raw data for these scores were examined for correct entry, and no mistakes were found. Since posttest roughly coincided with finals week, the end of the semester, and the start of the December holiday season, the author conjectured that these factors combined to produce substantially increased or decreased amounts of stress on the outliers. These influences are expected, especially with a student population. For this reason, as well as the fact that two outliers were positive and two were negative, thus reducing their negative impact on the statistical calculation, these outlying scores were retained for the analysis. Also, 4 outliers out of 62 scores is only slightly above 5% (6.45%), and Cohen and Cohen (1983) caution that the decision to discard data from outliers “should not be taken lightly” (p. 128).

            For observed difference scores, only 3 (4.84%) of the data points had a standardized residual greater than 2, and one of these outlying scores was of a different sign than the other two. The same analysis as provided for the partiallized residuals applies, and therefore the data for the outliers was retained for the ANCOVA analysis.

            To test for adherence of the data to the linearity assumption of the ANCOVA, F tests were conducted to determine whether a linear or non-linear model best explained the relationship between pretest-posttest stress scores and chanting frequency scores. The p value of the F statistic for a linear explanation was .087, and the p value of the F statistic for a non-linear explanation was .477. This indicates that a linear model is a better fit for the data. The same statistical procedure was used to assess the form of the relationship between pretest-posttest stress scores and age values. The p value of the F statistic for a linear explanation was .037, and for a non-linear explanation the p value was .320. This indicates that a linear explanation for the relationship between age and pretest-posttest difference scores for stress explained more variance than a non-linear explanation, and was therefore a better fit for the data.

            As an additional test for linearity, observed difference scores for each group for pretest-posttest were regressed on the age variable. For the alternate mantra group the slope was -.03, for the maha mantra group the slope was .03, and for the control group the slope was .04. These similar slopes across groups suggest a linear relationship.

ANCOVA Using Partiallized Pretest-Posttest Stress Difference Scores.

            Hypothesis 1 in the methodology chapter stated that the maha mantra group will show significantly decreased stress, at the .05 level, from pretest to posttest compared with the alternate group and the control group. Table 15 shows the results of a hierarchical ANCOVA, using partiallized difference scores, assessing the effects of group status on stress, with gender, age and chanting frequency as covariates. Effects of gender and age were calculated first, then chanting frequency was assessed, and then the effects of group status were analyzed:

Table 15

Results of ANCOVA Using Partiallized Pretest-Posttest Stress Difference Scores

     Pearson R for this ANCOVA was .62 (Multiple R2= .39). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and alternate comparison (p= .000; CI for difference= [8.26, 24.19]) and the maha and control comparison (p= .001; CI for difference= [-15.84, -4.30]).            

     The result of the F test of the ANCOVA for group status (p= .000) indicates that group status had an effect on stress. The t tests and confidence intervals for maha-control and maha-alternate comparisons show that the decrease in the maha group's stress score was significant at the .05 level, compared to the change in score of either of the other groups. This was shown by the significance levels as well as the confidence intervals, both of which indicate that such results can be expected by chance less than 1 % of the time. The Multiple R2 value of .39 means that 39% of the variance in stress difference scores was accounted for by the complete model. The partial eta squared values for the four variables show the proportion of variance explained by each variable, controlling for all of the other variables in the model. From the F tests, age was the only covariate with a significant effect on stress, though the effect size for age was only 6.0%.

ANCOVA Using Observed Pretest-Posttest Stress Difference Scores.

            Table 16 shows the results of a hierarchical ANCOVA, using observed difference scores, assessing the effects of group status on stress, with gender, age and chanting frequency as covariates. Effects of gender and age were calculated first, then chanting frequency was assessed, and then the effects of group status were analyzed.

Table 16

Results of ANCOVA Using Observed Pretest-Posttest Stress Difference Scores

     Pearson R for this ANCOVA was .56 (Multiple R2= .31). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and alternate comparison (p= .001; CI for difference= [-23.14, -7.11]) and the maha and control comparison (p= .000; CI for difference= [6.23, 18.32]).

     Using observed difference scores instead of partialled difference scores, the results were essentially unchanged. The lower value for Multiple R2 indicates that the model explained less of the variance using observed scores, and the lower partial eta squared score for group status shows that group status explained less of the variance in the stress variable, compared with the partialled score calculation. However, the effects of group status were still statistically significant for maha-alternate and maha-control comparisons. Effect size measures and significance tests reveal that age was less of an influence on stress using observed difference scores. Chanting frequency and gender remained non-significant factors.

Pretest-Followup Analysis of the Stress Variable

Tests of Assumptions for Pretest-Followup Analysis of the Stress Variable.

            Analysis of residuals was performed for partiallized difference scores and observed difference scores for the pretest-followup analysis of the stress variable. For partiallized scores, Bartlett’s Box statistic for homogeneity, F (2, 7601), had a p value of .108, and for observed scores, Bartlett’s Box statistic for homogeneity, F (2, 7601), had a p value of .104. These non-significant p values indicate that the data possess adequate homogeneity of variance for the ANCOVA.

            For partiallized scores, only 1 of 62 (1.61%) standardized residual values were greater than 2. This outlying score, after confirming that data entry for the score is correct, was retained for the statistical analysis. For observed difference scores, only 2 of 62 (3.23%) scores had a standardized residual value greater than 2. Because they represent less than 5% of the scores, and because the two outliers are of opposite sign, they were retained for the analysis.

            To test for adherence of the data to the linearity assumption of the ANCOVA, F tests were conducted to determine whether a linear or non-linear model best explains the relationship between pretest-followup stress scores and chanting frequency scores. The p value of the F statistic for a linear explanation was .0056, and the p value of the F statistic for a non-linear explanation was .130. This indicates that a linear model is a better fit for the data. The same statistical procedure was used to assess the form of the relationship between pretest-posttest stress scores and age values. The p value of the F statistic for a linear explanation was .078, and for a non-linear explanation the p value was .303. This indicates that a linear explanation for the relationship between age and pretest-followup difference scores for stress explains more variance than a non-linear explanation, and is therefore a better fit for the data.

            As an additional test for linearity, observed difference scores for each group for pretest-followup were regressed on the age variable. For the maha mantra group the slope was -.155, for the alternate mantra group the slope was .199, and for the control group the slope was .069. These similar slopes across groups suggest a linear relationship, and thus the linearity assumption for the ANCOVA was adequately satisfied.

ANCOVA Using Partiallized Pretest-Followup Stress Difference Scores.

            Hypothesis 6 in the Methodology Chapter stated that the maha mantra group will show significantly decreased stress, at the .05 level, from pretest to followup compared with the alternate group and the control group, though this decrease was hypothesized to be less than the decrease from pretest to posttest. Using a hierarchical ANCOVA with partiallized difference scores, assessing the effects of group status on stress from pretest to followup, with gender, age and chanting frequency as covariates, the p value for the F statistic for the variance explained for the entire model was .096, which is not significant at the .05 level, and the p value for the effects of group status was .724. Partial eta2 for group status was .011, and none of the covariates had an eta squared greater than .02, nor did any of the covariates have a significant t value at the .05 level. These results show that the model as a whole does not explain variance in the dependent variable at a significant level. Group status t tests were not performed because the F test was not significant.

ANCOVA Using Observed Difference Scores for Pretest-Followup Stress Scores.

            Using a hierarchical ANCOVA with observed difference scores, assessing the effects of group status on stress from pretest to followup, with gender, age and chanting frequency as covariates, the p value for the F statistic for the variance explained for the entire model was .056, and the p value for the effects of group status was .082. Eta squared for group status using observed differences was .12. Chanting frequency had a significant p value for the F test (.004), indicating that, controlling for the other variables in the model, chanting frequency had a significant effect on stress from pretest to followup. Chanting frequency had an eta2 of .155. Group comparison t tests for group status were not conducted because the F test for total explained variance was not significant at the .05 level.

Statistical Analysis of the Depression Variable

Pretest-Posttest Analysis of Depression

Tests of Assumptions, Interaction and Outliers for Pretest-Posttest Depression Scores.

            Residual analysis for partiallized difference pretest-posttest depression scores revealed five standardized residual values greater than 2. These values were 2.31, 2.01, -2.18, -2.201, and -2.26. Since these values contain two positive numbers and three negative numbers, and because none of them exceed 2 by more than .308 in absolute value, all these scores were retained for the ANCOVA analysis. Cohen and Cohen (1983) explain that “a residual that is as much as three (or, certainly, four) of these units in absolute size is reasonably considered an outlier” (p. 48). Bartlett’s Box statistic for homoscedasticity was .130, indicating that the data was adequately homoscedastistic for the ANCOVA. Residual analysis for observed difference pretest-posttest depression scores also showed five standardized residual values greater than 2. These values were 2.09, 2.067, 2.13, -2.09, and -2.37. For the same reasons applied in the case of partiallized pretest-posttest depression scores, all five outlying scores were retained for the ANCOVA. Bartlett’s Box statistic for observed scores was .240, and thus the distribution was adequately homoscedastistic for the ANCOVA.

            To test for linearity, F tests were run to compare the appropriateness of a linear explanation for the depression scores charted against chanting frequency values with a non-linear explanation. The p value for the F statistic for a linear explanation was .004, and the p value for a non-linear explanation was .011, indicating that a linear model is a better fit. For depression scores charted against age, the p value for the F statistic for a linear explanation was .052, and the p value for a non-linear explanation was .557, again providing statistical evidence for the linear model.

            Observed difference scores for each group for pretest-posttest were regressed on the age variable. For the maha mantra group the slope was -.191, for the alternate mantra group the slope was -.250, and for the control group the slope was -.034. These similar slopes across groups suggest a linear relationship and lack of interaction.

ANCOVA for Pretest-Posttest Depression Scores Using Partiallized Differences.

            Hypothesis 2 in the Methodology Chapter stated that the maha mantra group will show significantly decreased depression, at the .05 level, from pretest to posttest compared with the alternate group and the control group. Table 17 shows the results of a hierarchical ANCOVA, using partiallized difference scores, analyzing the effects of group status on depression, with gender, age and chanting frequency as covariates. Effects of gender and age were calculated first, then chanting frequency was evaluated, and then the effects of group status were assessed.

Table 17

Results of ANCOVA Using Partiallized Pretest-Posttest Depression Difference Scores

     Pearson R for this ANCOVA was .57 (Multiple R2= .33). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and alternate comparison (p= .002; CI for difference= [1.03, 11.88]) and the maha and control comparison (p= .010; CI for difference= [-12.58, -4.06]).

            The result of the F test of the ANCOVA (p = .001) for group status indicates that group status had an effect on depression, and the p value for overall explained variance (p= .000) shows that the model as a whole had a significant effect on pretest-posttest depression scores. Significance levels of t tests and t test confidence intervals for maha-alternate and maha-control show that the decrease in the maha group’s depression score was significant at the .05 level, compared to the change in score of either of the other groups, controlling for the covariates. This was demonstrated by the p values as well as the confidence intervals, both of which indicate that such results can be expected by chance less than 1% of the time. The Multiple R2 value of .33 means that 33% of the variance in depression difference scores is accounted for by the complete model. The partial eta2 values for the four variables show the proportion of variance explained by each variable, controlling for the other variables in the model. Although the p value for the F test for age was significant at the .05 level, age explained less than 1% of the variance, while chanting frequency, which also had a significant p value for the F statistic, explained 15% of the variance.

ANCOVA Using Observed Difference Scores for Depression Pretest-Posttest.

            Table 18 shows the results of a hierarchical ANCOVA, using observed difference scores, evaluating the effects of group status on pretest-posttest depression scores, with gender, age and chanting frequency as covariates. Effects of gender and age were computed first, then chanting frequency was assessed, and then the effects of group status were analyzed:

Table 18

Results of ANCOVA Using Observed Pretest-Posttest Depression Difference Scores

     Pearson R for this ANCOVA was .59 (Multiple R2= .35). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and alternate comparison (p= .013; CI for difference= [-11.85, -1.47]) and the maha and control comparison (p= .000; CI for difference= [3.62, 11.46]).

     Using observed difference scores instead of partiallized difference scores, the results were basically the same. Effects of group status were significant, as are effects of chanting frequency and age. However, age explained only a very small percentage of the variance (1%). Significance levels of t tests illustrate that maha-alternate and maha-control differences were significant.

Pretest-Followup Analysis of the Depression Variable

Tests of Assumptions, Interaction and Outliers for Pretest-Followup Depression Scores.

            Analysis of residuals was performed for partiallized difference scores and observed difference scores for the pretest-followup analysis of the Depression variable. Partiallized and observed scores each had 3 values with a standardized residual greater than 2. Also, for each method two of the outliers were of one sign, and the third was of the opposite sign. Therefore all outlying scores were retained for the ANCOVAs.

            The p value for Bartlett’s Box statistic for homogeneity, F (2, 7601), for partiallized scores was .109, and for observed scores the p value was .103, indicating that the data possess sufficient homogeneity of variance for the ANCOVA.

            F tests were performed to determine whether a linear or non-linear model best explained the relationship between pretest-followup depression scores and chanting frequency scores. For a linear explanation the p value was .004, and for a non-linear explanation the p value was .054. Comparing the dependent variable with age, the linearity p value was .025, and the non-linear value was .3480. Therefore, these relationships adequately satisfied the linearity assumption of the ANCOVA.

            Observed difference scores for each group for pretest-followup were regressed on the age variable. For the maha mantra group the slope was -.191, for the alternate mantra group the slope was -.1433, and for the control group the slope was -.0245. These similar slopes indicate that the linearity assumption was satisfied for the pretest-followup depression data.

ANCOVA Using Partiallized Pretest-Followup Depression Difference Scores.

            Hypothesis 7 in the Methodology Chapter stated that the maha mantra group will show significantly decreased depression, at the .05 level, from pretest to followup compared with the alternate group and the control group, though this decrease was hypothesized to be less than the decrease from pretest to posttest. Table 19 shows the results of a hierarchical ANCOVA, using partiallized difference scores, evaluating the effects of group status on pretest-followup depression scores, with gender, age and chanting frequency as covariates. Effects of gender and age were computed first, then chanting frequency was assessed, and then the effects of group status were analyzed.

Table 19

Results of ANCOVA Using Partiallized Pretest-Followup Depression Difference Scores

     Pearson R for this ANCOVA was .46 (Multiple R2= .22). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and control comparison (p= .004; CI for difference= [-12.23, -2.54]).

            The p value for group status (.024) was statistically significant at the .05 level, and shows that group status had an effect on depression from pretest to followup. Although the overall effect of the three covariates was not statistically significantly at the .05 level, chanting frequency did have a significant effect (p = .012), controlling for other variables. Group status and chanting frequency explained 12% and 11%, respectively, of the variance. Gender and age did not have significant p values. Results of the t tests show that the maha-control comparison was the only significant difference at the .05 level for pretest-followup depression data.

ANCOVA Using Observed Difference Scores for Pretest-Followup Depression Values.

            Table 20 shows the results of a hierarchical ANCOVA, using observed difference scores, evaluating the effects of group status on pretest-followup depression scores, with gender, age and chanting frequency as covariates. Effects of gender and age were computed first, then chanting frequency was assessed, and then the effects of group status were analyzed.

Table 20

Results of ANCOVA Using Observed Pretest-Followup Depression Difference Scores

     Pearson R for this ANCOVA was .49 (Multiple R2= .24). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and control comparison (p= .016; CI for difference= [.88, 9.10]).

     Using observed difference scores instead of partiallized difference scores for pretest-followup depression, there were minor differences in the results. Most notably, chanting frequency had a significant effect (.012) on depression difference scores using partiallized differences, and the effect of chanting frequency was non-significant using observed differences.

Statistical Analysis of the Sattva Variable

Pretest-Posttest Analysis of the Sattva Variable

Tests of Assumptions, Interaction and Outliers for Pretest-Posttest Analysis of the Sattva Variable.

            Analysis of residuals was performed for partiallized difference scores and observed difference scores for the pretest-posttest analysis of the sattva variable. For partiallized scores there were 2 outliers, with values of -2.02 and -2.25. Since these two scores represented only 3.23% of the values, and 5% of the values can be expected by chance to have standardized residuals greater than 2, these scores were retained for the ANCOVA. For standardized scores there were 3 outliers (4.84%), none of which had an absolute value greater than 2.152, and therefore these scores were also retained for the ANCOVA.

            The p value for Bartlett’s Box statistic for the partiallized distribution was .065, and for the observed distribution the p value was .169. These values, which were not significant at the .05 level, indicate that the distributions are adequately homosecedastistic for the ANCOVA.

            To test for the linearity assumption, F tests were performed to determine whether a linear or non-linear model best explains the relationship between pretest-posttest sattva scores and chanting frequency scores. The p value of the F statistic for a linear explanation was .040, and for a non-linear explanation the p value was .169, indicating that a linear approximation is a better fit for the relationship than a non-linear approximation. The same procedure was applied to the relationship between pretest-posttest sattva scores and age scores. For this relationship, the p value of the F statistic for a linear explanation was .088, and for a non-linear explanation the p value was .234, suggesting that a linear approximation is a better fit for the data.

            As an additional test for linearity, observed difference scores for each group for pretest-posttest were regressed on the age variable. For the maha mantra group the slope was -.189, for the alternate mantra group the slope was .033, and for the control group the slope was .057. These similar slopes across groups suggest a linear relationship and lack of interaction.

ANCOVA Using Partiallized Pretest-Posttest Sattva Difference Scores.

            Hypothesis 3 in the Methodology Chapter stated that the maha mantra group will show significantly increased sattva, at the .05 level, from pretest to posttest compared with the alternate group and the control group. Table 21 shows the results of a hierarchical ANCOVA, using partiallized difference scores, assessing the effects of group status on sattva, with gender, age and chanting frequency as covariates. Effects of gender and age were calculated first, then chanting frequency was assessed, and then the effects of group status were analyzed.

Table 21

Results of ANCOVA Using Partiallized Pretest-Posttest Sattva Difference Scores

     Pearson R for this ANCOVA was .52 (Multiple R2= .27). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and alternate comparison (p= .011; CI for difference= [1.47, 10.68]) and the maha and control comparison (p= .001; CI for difference= [-9.43, -2.60]).

     These statistics show that group status had a statistically significant effect on sattva from pretest to posttest at the .05 level, accounting for 23% of the variance. As a whole (p = .259) and individually the covariates did not have a statistically significant effect on the dependent variable. Significance levels of the t tests provide evidence that the maha mantra group differed significantly from both the alternate mantra group (p = .011) and the control group (p = .001), and that the Alternate Mantra and control groups did not significantly differ from each other. 

Results of ANCOVA Using Observed Pretest-Posttest Sattva Difference Scores.

            Table 22 shows the results of a hierarchical ANCOVA, using observed difference scores, assessing the effects of group status on sattva, with gender, age and chanting frequency as covariates. Effects of gender and age were calculated first, then chanting frequency was assessed, and then the effects of group status were analyzed.

Table 22

Results of ANCOVA Using Observed Pretest-Posttest Sattva Difference Scores

     Pearson R for this ANCOVA was .49 (Multiple R2= .24). Three t tests were performed to identify significant comparisons. Significant p values were found for the maha and alternate comparison (p= .014; CI for difference= [1.49, 10.70]) and the maha and control comparison (p= .002; CI for difference= [-9.70, -2.32]).

     Using observed difference scores instead of partiallized difference scores, the results were essentially unchanged. With observed scores group status explained more of the variance, and chanting frequency explained less of the variance, compared with partiallized scores. Still, covariates, as a whole and individually were not significant at the .05 level, and group status was significant, with both the maha-alternate and maha-control comparisons showing statistically significant comparisons.

Pretest-Followup Analysis of the Sattva Variable

Tests of Assumptions, Interaction and Outliers for Pretest-Followup Sattva Scores.

            Analysis of residuals was performed for partiallized difference scores and observed difference scores for the pretest-followup analysis of the sattva variable. For both observed and partiallized scores there were 3 outliers, representing 4.84% of the 62 scores in each data set. Since this was less than the 5% of scores that would be expected by chance to have a standardized residual greater than 2, these outlying scores were retained for the ANCOVAs.

            The p value for Bartlett’s Box statistic for partiallized scores was .129, and for observed scores the p value was .112. These non-significant p values indicate that there is adequate homogeneity for the ANCOVA.

            Linearity tests for Chanting Frequency and pretest-followup sattva scores produced a p value for a linear explanation of .000, and a p value for a nonlinear explanation of .894. For age charted against pretest-followup sattva scores the linearity p value was .050, and the nonlinear explanation produced a p value of .215. These results indicate that a linear explanation is a better fit for the data than a nonlinear explanation.

            When observed difference scores were regressed on Age, the slope for the maha mantra group was .156, for the alternate mantra group the slope was .056, and for the control group the slope was .147. These similar slopes indicate a linear relationship and absence of interaction, and thus the linearity assumption for the ANCOVA was adequately satisfied.

ANCOVA Using Partiallized Pretest-Followup Difference Scores for Sattva. 

            Hypothesis 8 in the Methodology Chapter stated that the maha mantra group will show significantly decreased sattva, at the .05 level, from pretest to followup compared with the alternate group and the control group, though this decrease was hypothesized to be less than the decrease from pretest to posttest.Table 23 shows the results of a hierarchical ANCOVA, using observed difference scores, assessing the effects of group status on sattva from pretest to followup, with gender, age and chanting frequency as covariates. Effects of gender and age were calculated first, then chanting frequency was assessed, and then the effects of group status were analyzed.

Table 23

Results of ANCOVA Using Partiallized Pretest-Followup Sattva Difference Scores

     Pearson R for this ANCOVA was .42 (Multiple R2= .18). Although the p value for the F test for group status was significant at the .05 level (p = .010), the overall explained variance did not have a significant p value (p = .051). Apparently, the extra degrees of freedom supplied by the covariates, as well as their added standard error, diminished the explanatory ability of the model, which otherwise accounts for 18% of the variance in pretest-followup sattva scores. None of the covariates had a significant p value for the F test.

ANCOVA Using Observed Pretest-Followup Difference Scores for Sattva.

            Using a hierarchical ANCOVA with observed difference scores, assessing the effects of group status on sattva from pretest to followup, with gender, age and chanting frequency as covariates, the p value for the F statistic for the variance explained for the entire model was .058, and for group status the p value was .504, neither of which are significant at the .05 level. Eta2 for group status was .131, and for chanting frequency eta2 was .102. Multiple R2 for the model was .145. These results show that the model as a whole did not explain variance in the dependent variable at a significant level.

Statistical Analysis of the Rajas Variable

Pretest-Posttest Analysis of Rajas

Tests of Assumptions, Interaction and Outliers for Pretest-Posttest Rajas Scores.

            Residual analysis for partiallized difference pretest-posttest rajas scores resulted in 3 standardized residual values greater than 2. Since this represents only 4.84% of the values, the three outlying scores were retained for the ANCOVA. For observed scores there were 4 residuals, whose values were -2.00, 2.37, 2.52, and 2.11. Since none of these outliers had an absolute value greater than 2.52, they were retained for the ANCOVA.

            Bartlett’s Box statistic, F (2, 7601), produced a p value of .118 for partiallized values, and .176 for observed values, indicating that the distributions were adequately homoscedastistic for the ANCOVA.

            A test assessing the appropriateness of a linear model for pretest-posttest rajas scores charted against chanting frequency values produced a p value of .002 for a linear model and .169 for a nonlinear model. For rajas scores charted against age values, a linear model produced a p value of .404, and a nonlinear model had a p value of .415. These results indicate that a linear model was the best fit for the data.

            Observed pretest-posttest difference scores were regressed on the age variable to assess similarity of slopes. For the maha group the slope was -.207, for the Alternate Group the slope was .390, and for the control group the slope was .0568. Sheskin (1997) suggests that a difference in slopes between groups of more than .5 may be considered large. Therefore F tests were performed to determine whether the relationship between age and pretest-posttest rajas scores is best described as linear or quadratic. The p value (.709) for the quadratic F test was larger than the p value for the linear F test, and the quadratic model explained only 4% more variance than the linear model. A quadratic approximation therefore does not significantly add to the explanatory ability of the model, and therefore a linear model was utilized in the statistical analysis of rajas.

ANCOVA for Pretest-Posttest Rajas Scores Using Partiallized Differences.

            Hypothesis 4 in the Methodology Chapter stated that the maha mantra group will show significantly decreased rajas, at the .05 level, from pretest to posttest compared with the Alternate Group and the control group. Using a hierarchical ANCOVA with partiallized difference scores, evaluating the effects of group status on rajas from pretest to posttest, with gender, age, and chanting frequency as covariates, the p value for the F statistic for the variance explained for the entire model was .061, and the p value for the effects of group status was .103.

ANCOVA for Pretest-Posttest Rajas Scores Using Observed Differences.

            A hierarchical ANCOVA using observed difference scores for pretest-posttest rajas values, with gender, age, and chanting frequency as covariates, resulted in a p value for the F statistic for variance explained for the entire model of .053, and a p value for group status of .180. The p value for covariates as a whole was .032, though none of the covariates explained more than 3% of the variance. R2 for the model was .089.

Pretest-Followup Analysis of Rajas

Tests of Assumptions, Interaction and Outliers for Pretest-Followup Rajas Scores.

            For partiallized scores there were 3 outliers, with values of 3.06, -2.47, and -2.39. Since the largest score wwas of opposite sign to the other two outliers, and 3 of 62 scores represented less than 5% of the total scores, these outliers were retained for the ANCOVA. For partiallized scores the Bartlett’s Box p value was .083, indicating that the distribution was adequately homoscedastistic for the ANCOVA.

            For observed scores there were also three outliers, with the highest value being of opposite sign to the other two values, and therefore the outliers were retained for the ANCOVA. The p value for Bartlett’s Box statistic for observed pretest-followup rajas scores was .065, and thus the homoscedasticity assumption was satisfied.

            Linearity tests showed that a linear explanation for the relationship between pretest-followup and Age had a p value of .514 for a linear approximation, and a p value of .536 for a nonlinear explanation. For chanting frequency scores charted against the independent variable, a linear explanation produced a p value of .419, and a nonlinear explanation had a p value of .849. When observed difference scores for each group for pretest-followup were regressed against age, the maha mantra group had a slope of .275, the alternate mantra group had a slope of -.017, and the control group had a slope of -.008. All these statistics provide evidence for the appropriateness of a linear model without interaction between age and group status.

ANCOVA for Pretest-Followup Rajas Scores Using Partiallized Differences.

            Hypothesis 9 in the Methodology Chapter stated that the maha mantra group will show significantly decreased rajas, at the .05 level, from pretest to followup compared with the alternate group and the control group, though this decrease was hypothesized to be less than the decrease from pretest to posttest. Effects of group status on pretest-followup rajas scores were assessed using partiallized difference scores, with gender, age and chanting frequency as covariates. The p value for the F statistic for the variance explained by the entire model was .067, and the p value for group status was .058. The covariates as a whole had a p value of .120, though chanting frequency had a significant p value of .048. Multiple R2 for the model was .12.

ANCOVA for Pretest-Followup Rajas Scores Using Observed Differences.

            Effects of group status on pretest-followup rajas scores were evaluated using observed difference scores, with gender, age and chanting frequency as covariates. This ANCOVA resulted in a p value for group status of .051, though the p value for overall explained variance was .116, indicating that the additional degrees of freedom and error from the covariates decreased the effectiveness of the model to explain the difference scores. Using observed difference scores, chanting frequency had a significant p value (.048) using partiallized scores, and a non-significant p value (.289) using observed scores. Also, with the observed scores ANCOVA the Multiple R2 for the model was less than with ANCOVA using partiallized scores.

Statistical Analysis of the Tamas Variable

Pretest-Posttest Analysis of Tamas

Tests of Assumptions, Interaction and Outliers for Pretest-Posttest Tamas Scores.

            There were four standardized residuals with absolute values greater than 2 for partiallized pretest-posttest tamas scores. Since none of these scores had an absolute value greater than 2.37, these outlying scores were retained for the ANCOVA. For observed difference standardized residuals for pretest-posttest tamas scores there were 3 outliers, none of which had an absolute value greater than 2.52, and one of which had a value of -2.00. Therefore these values were retained for the ANCOVA.

            For partiallized scores, Bartlett’s Box statistic, F (2, 7601), was .294, and for observed scores Bartlett’s Box statistic, F (2, 7601), was .160. Both of these scores provide evidence that the homogeneity assumption was adequately satisfied for ANCOVA.

            A linearity test of the relationship between pretest-posttest tamas scores and chanting frequency produced a p value of .002 for a linear explanation, and a p value of .236 for a nonlinear explanation. For the linearity test of the relationship between pretest-posttest tamas scores and age, the p value for a linear explanation was .243, and for a nonlinear explanation the p value was .611. These statistics provided evidence for a linear model.

            Observed difference scores for each group for pretest-posttest tamas were regressed on the age variable. For the alternate mantra group the slope was -.294, for the maha mantra group the slope was .187, and for the control group the slope was -.062. These similar slopes across groups indicate a linear relationship.

ANCOVA Using Partiallized Pretest-Posttest Tamas Difference Scores

            Hypothesis 5 in the Methodology Chapter stated that the maha mantra group will show significantly decreased tamas, at the .05 level, from pretest to posttest compared with the alternate group and the control group. An ANCOVA was performed assessing partiallized pretest-posttest difference scores for tamas, with gender, age and chanting frequency as covariates. In a hierarchical analysis, effects of gender and age were calculated first, then chanting frequency was evaluated, and then the effects of group status were analyzed. Table 24 shows the results of this analysis.

Table 24

Results of ANCOVA Using Partiallized Pretest-Posttest Tamas Difference Scores

     Pearson R for this ANCOVA was .46 (Multiple R2= .22). Three t tests were performed to identify significant comparisons. A significant p value was found for the maha and control comparison (p= .000; CI for difference= [3.69, 10.00]).

     The results of the F test of the ANCOVA show that group status, controlling for the covariates, had a statistically significant effect on pretest-posttest tamas scores. Significance levels of the t tests demonstrated that the maha-control comparison was the only significant comparison, meaning that the difference between the maha mantra group and the control group was statistically significant, though the difference between the maha mantra group and the alternate mantra group was not significant, nor was the difference between the alternate mantra group and the control group. Group status accounted for 21% of the variance, and chanting frequency, which also had a significant p value (.004) for its F test, explained 7% of the variance. Gender and age did not have a statistically significant effect on the dependent variable.

ANCOVA Using Observed Difference Pretest-Posttest Tamas Scores.

            An ANCOVA was performed assessing observed pretest-posttest difference scores for tamas, with gender, age and chanting frequency as covariates. In a hierarchical analysis, effects of gender and age were calculated first, then chanting frequency was evaluated, and then the effects of group status were analyzed. Table 25 shows the results of this analysis.

Table 25

Results of ANCOVA Using Observed Pretest-Posttest Tamas Difference Scores

     Pearson R for this ANCOVA was .46 (Multiple R2= .21). Three t tests were performed to identify significant comparisons. A significant p value was found for the maha and control comparison (p= .000; CI for difference= [3.79, 10.70]). Using observed difference scores for the ANCOVA, results are essentially the same as using partiallized scores for pretest-posttest tamas values.

Pretest-Followup Analysis of the Tamas Variable

Tests of Assumptions, Interaction and Outliers for Pretest-Followup Tamas Scores.

            Standardized residuals for both partiallized and observed difference scores for pretest-followup tamas values both had 3 outliers, representing 4.84% of all scores. Since this was less than the percentage of standardized residuals that would be expected by chance to have values greater than 2, these outlying scores were retained for the ANCOVAs.  

            Bartlett’s Box statistic, F (2, 7601), for partiallized scores had a p value of .135, and for observed scores the p value was .371, indicating that the distributions were adequately homoscedastistic for ANCOVA.

            Linearity tests between pretest-followup tamas scores and chanting frequency produced a p value for a linear explanation of .078, and for a nonlinear model the p value was .394. For linearity tests between the dependent variable and age, a linear explanation produced a p value of .597, and a nonlinear explanation produced a p value of .659. These values indicate that a linear model was a better fit for the data, and that the linear assumption was sufficiently satisfied.

            Difference scores for each group for pretest-followup tamas values were regressed on the age variable. For the maha mantra group the slope was -.213, for the alternate mantra group the slope was -.282, and for the control group the slope was -.021. These similar slopes across groups indicate a linear relationship and lack of interaction between age and group status.

ANCOVA Using Partiallized Pretest-Followup Tamas Difference Scores.

            Hypothesis 10 in the Methodology Chapter stated that the maha mantra group will show significantly decreased tamas, at the .05 level, from pretest to followup compared with the Alternate Group and the control group, though this decrease was hypothesized to be less than the decrease from pretest to posttest. Using a hierarchical ANCOVA with partiallized difference scores, assessing the effects of group status on tamas from pretest to followup, with gender, age and chanting frequency as covariates, the p value for the F test for variance explained by the complete model was .054, and the p value for variance explained by group status was .116. For chanting frequency F had a significance level of .023, and the other covariates had non-significant p values at the .05 level. For covariates as a whole the p value was .137. Multiple R2 for the model was .175. Partial Eta2, or effect size, for chanting frequency, was 9%. These results indicate that neither group status nor the covariates as a whole had a significant effect on pretest-followup tamas scores using partiallized differences, though the effect of chanting frequency was significant.

ANCOVA Using Observed Pretest-Followup Tamas Difference Scores.

            An ANCOVA using observed scores for pretest-followup tamas values resulted in an explained variance for the model with a p value of .044. Results of this ANCOVA are shown in Table 26.

Table 26

Results of ANCOVA Using Observed Pretest-Followup Tamas Difference Scores

     Pearson R for this ANCOVA was .44 (Multiple R2= .19). Three t tests were performed to identify significant comparisons. A significant p value was found for the maha and control comparison (p= .017; CI for difference= [.07, 6.30]).

            ANCOVA with observed difference scores for pretest-followup tamas values produced a p value for explained variance of the complete model of .044, which is significant at the .05 level, as opposed to the p value using partiallized scores (.054). Also, the p value for group status (.035) was statistically significant, compared with the nonsignificant p value (.116) using partiallized scores. Though chanting frequency had a statistically significant p value using partiallized scores, with observed differences the p value for chanting frequency was nonsignificant. Group status explained 11.3% of the variance using observed scores, and the only comparison that is significantly different for observed scores is Maha-Control.

Data Analytic Summary of Independent Variables

Group Status

            The main hypotheses of this experiment were that subjects in the maha mantra group will decrease their stress, depression, rajas and tamas more than subjects in the alternate mantra group and control group from pretest to posttest, and that the maha mantra group will increase sattva from pretest to posttest more than the other two groups. Statistical analyses, summarized in Table 27, reveal that for four of the five variables the hypotheses are valid at the .05 significance level. Specifically, the maha mantra group showed statistically significant greater differences from pretest to posttest for the variables of stress, depression, tamas and sattva. For stress, depression, and sattva, the maha mantra group showed significantly greater change than both the other groups, and for tamas the maha mantra group changed significantly more than the control group, though not significantly more than the alternate mantra group. In no instance where the effects of group status on the dependent variable were statistically significant did the alternate and control groups significantly differ. For pretest-posttest ANCOVA that produced significant results for group status, effect sizes for group status on the dependent variable ranged from .18 to .33, correlating to 18% to 33% of the variance (see Tables 15, 16, 17, 18, 21, 22, 24, and 25) .

            Secondary hypotheses for this study included predictions that pretest-followup scores for the dependent variables would be effected by group status in the same way as pretest-posttest scores, though it was expected that there would be some reduction in the effect. For depression the effects of group status from pretest to followup were significant at the .05 level, though only for the maha-control comparison. The effects of group status for sattva from pretest to followup were also significant, though the overall explained variance of the model for pretest-followup was not significant at the .05 level. Tamas scores from pretest to followup showed significant changes for the group status variable, with the t test for the maha-control comparison being the only significant t test of the three comparisons. For ANCOVAs resulting in statistically significant results for group status, effect sizes for group status ranged from .11 to .20, corresponding with 11% to 20% of the variance (see Tables 19, 20, 23, and 26). These effect sizes were smaller than for pretest-posttest analyses, as predicted by the hypotheses.

Table 27

Summary of Group Status Effects on the Dependent Variables

            Hypothesis #a                          Sig. or Nonsig. p Valueb Group Effect Sizec, d

            Primary Hypotheses

                        1 (Stress pre-post)                Significant                        .33

                        2 (Depression pre-post)        Significant                         .21

                        3 (Sattva pre-post)                Significant                        .23

                        4 (Rajas pre-post)                Nonsignificant             

                        5 (Tamas pre-post)               Significant                         .21

            Secondary Hypotheses

                        6 (Stress pre-followup)            Nonsignificant

                        7 (Depression pre-followup)     Significant                       .12      

                        8 (Sattva pre-followup)             Significante, f                  .15      

                        9 (Rajas pre-followup)              Nonsignificant             

                        10 (Tamas pre-followup)           Nonsignificantg

aHypothesis # refers to the numbers of the hypotheses given at the end of the methodology chapter (chapter 3).     

bSignificance is determined at a .05 level for partiallized difference values. Unless otherwise noted, observed difference values had the same result with regards to significance or non-significance.

cEffect sizes for non-significant p values are not shown.            

dEffect size values for partiallized difference scores are given.

eThe overall explained variance of the model had a non-significant p value.

fThe observed difference value had a non-significant p value.

gThe observed difference value had a significant p value and an effect size of .11.

Chanting Frequency

            It was hypothesized that chanting frequency for the maha mantra group would correlate positively with sattva, and negatively with stress, depression, rajas and tamas. For the alternate group, chanting frequency was hypothesized to have no correlation with the dependent variables. As shown in Table 11, chanting frequency did correlate significantly with several of the alternate group dependent variables, and with none of the maha mantra group dependent variables.

            For ANCOVAs where group status and explained variance of the complete model had significant F statistics, chanting frequency had statistically significant p values only for pretest-posttest and pretest-followup depression scores, and for pretest-posttest tamas scores. Effect sizes for chanting frequency in these computations ranged from .07 to .17, corresponding with 7% to 17% (see Tables 17, 18, 19, 20, 24, and 25).

Age

            Age was predicted to have no effect on the dependent variables. For ANCOVAs that resulted in significant effects of the group status variable, age had a significant p value only for pretest-posttest stress and depression scores. For these ANCOVAs, the effect sizes of age on the dependent variables ranged from .01 to .06, corresponding with 1% to 6% (see Tables 15, 16, 17, and 18).

Gender

            Gender was hypothesized to have no effect on the dependent variables. As described in the correlations of covariates section, gender did have five statistically significant correlations with the control group, which contained 10 females and 9 males. These correlations indicate that for the control group males had a greater predominance of tamas and females had a greater predominance of sattva. In none of the ANCOVAs for which group status had a significant effect on the dependent variables did gender have an F test with a significant p value.

General Comparison of Partiallized and Observed Difference Scores

            In assessment of the main hypotheses of this study (hypotheses 1-5, as listed in chapter 3), there were not substantial differences between the results derived from partiallized differences and those obtained from observed differences. In all five cases the two methods produced the same results, with regards to significance or non-significance of group status and overall explained variance. For the secondary hypotheses of this study (hypotheses 6-10), the two methods produced the same results, with regards to significance or non-significance, for pretest-followup analysis of stress, depression, and rajas, though the results differed in the analysis of sattva and tamas. Specifically, partiallized difference scores resulted in a significant pretest-followup sattva p value, while observed difference scores resulted in a non-significant p value, and for pretest-followup tamas analyses, partiallized scores resulted in a non-significant p value, and observed scores resulted in a significant p value. For the ten hypotheses, therefore, eight resulted in the same basic result with the two methods. Further, the two instances culminating in different results did not show a pattern of difference, indicating that the methods did not systematically differ in their end results.

            For the four dependent variables for which group status had a significant effect for pretest-posttest, partiallized scores resulted in a larger effect size for group status in three out of four cases, with depression being the only dependent variable for which observed scores resulted in a larger effect size for group status. Also, for two ANCOVAs, pretest-followup depression and pretest-followup rajas, chanting frequency had a significant effect with partiallized scores, though not for observed scores. Further, there is a slight trend in the data for the effect sizes for the covariates to be higher using partiallized scores than using observed scores. Overall, these statistics suggest that partiallized differences resulted in a slightly more favorable analysis of the data, with regard to confirmation of the hypotheses.

            As described earlier in this section, Cohen and Cohen (1983) point out that a partiallized difference score will tend to be less correlated with pretest scores than a non-partiallized difference score, and therefore they assert that partiallized differences are a more objective measure. Nunnally and Bernstein (1994), however, claim that partiallizing scores tends to produce spurious results and to erroneously treat the pretest score as if it were an error-free true score. They conclude, therefore, that observed scores are the best measure of change. With relation to the results of this study, pretest scores between dependent variables tended to vary greatly (see Tables 7-10). Therefore, the partiallized method, which standardizes pretest scores, may be the better choice for data analysis. To clarify, it is typically found that subjects with a relatively low pretest score will have larger gains at posttest than subjects with a relatively high pretest score. This is a manifestation of the statistical phenomenon of regression to the mean (Cohen & Cohen, 1983). In instances where pretest scores vary greatly, partiallizing difference scores adjusts for the regression to the mean, and therefore in such cases this advantage may outweigh the potential disadvantages of partiallizing difference scores.