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One-Way Manova Essay


Introduction

In an effort to use aptitude as well as students’ achievement among several demographic variables in determining high school seniors’ choice of career after high school, a survey was conducted among over 500 students.

A one way multivariate analysis of variance was then conducted with “collplan” being the predictor variable whereas student’s career plan in college after high school had nine categorical variables including: agricultural college, no plans yet, liberal arts, none, engineering college, music/arts, teacher college, other and university.

The quantitative outcome variables in this dataset were “abstract”- a test of abstract reasoning and “creative”- a test of creativity. The One-Way MANOVA was conducted in an attempt to answer the following question:

How well do the categorical predictor variable “factors” (levels) predict scores on a.) a measure of abstract reasoning ability, and b.) a measure of creativity? After conducting a One-Way MANOVA on the dataset using SPSS, the results of the analyses were presented and interpreted as described in below.

The author hypothesizes that categorical predictor variables (none, Teacher College, agricultural college, engineering college, liberal arts, music/arts, university, other, and no plans yet) are significant predictors of scores on a measure of abstract thinking ability and scores on a measure of creativity.

This has been explained by the descriptive statistics as well as the MANOVA test, specifically the Wilk’s lambda and the between-subject effects of the variables.

Descriptive Statistics

The GLM statistics for between-subjects factors indicated that there were 178 seniors who did not plan to join any of the listed institutions in this study and this was the highest number of students. This was followed by students who had plans of joining university and these amounted to 88 students. The third largest category of seniors had plans of doing liberal arts and these amounted to 59 students.

These were followed by 57 students who had “other” plans after high school. The number of those who had no plans yet was equal to that of students who wished to join a teachers college (38 students). There were 29 seniors who had plans of joining an engineering college, 11 who wished to do music/arts and finally only 4 students had plans of joining an agricultural college after high school (Table 1).

From the descriptive statistics (Table 2), it is evident that having significant differences between the dependent variable and the independent variables may be somewhat impossible since some categorical variables have very large cell sizes (N) which are many times larger than the smallest cell size.

For instance, the cell size for “none” is 178 whereas “agricultural college”, the smallest cell size has a size of N =4. For the fixed factor “abstract”, the mean abstract thinking for seniors who did not have plans after college (“none”) was 8.94, SD = 2.616 whereas the mean for those who had plans of joining a teacher college was the same as that of students who had plans of joining university i.e. 10.37, SD= 2.509 and 10.37, SD = 2.709 respectively and these were the highest means for the “abstract” category.

The lowest mean was for students who wished to join an agricultural college, mean = 7.25, SD = 2.50 followed by those who did not have plans yet, mean = 8.84, SD = 2.881. The means for students who aspired to join an engineering college, do liberal arts and those who planned to do music/arts were 10.17 SD = 2.156, 9.97 SD = 3.129 and 10.09 SD= 2.914 respectively. Finally, the mean for abstract thinking for seniors who had other plans other than those included in the study was 9.74, SD = 2.482.

Table 3 indicates that the 95% CI for “none” in predicting the abstract reasoning ability of high school students was 8.549 – 9.338 whereas the CI for “teacher college” on predicting the abstract reasoning ability was 95% CI (9.515 – 11.222). The 95 percent CI for “agricultural college” on determining abstract thinking was 4.619 – 9.881 whereas that of “engineering college” was 9.915 – 11.150.

The confidence interval for “liberal arts” in determining abstract thinking was 95% CI (9.281 – 10.651) while the 95 percent CI for “music/arts” was 8.504 – 11.678. The 95% CI for “university” as a predictor of abstract thinking was 9.814 – 10.936 while the confidence interval for “other” was 95% CI (9.040 – 10.434).

Finally the CI for “no plans yet,” as a determinant of abstract thinking was 95% CI (7.988 – 9.696). It is clear that all the categorical variables have their CI ranging from positive lower boundary value to a positive upper boundary value. This implies that the set of data is somewhat normally distributed as earlier confirmed by the Levene’s F statistic.

Table 3 also indicates that the 95% CI for “none” in predicting the creativity level of high school students was 8.084 – -9.163 whereas the CI for “teacher college” on predicting the creativity level was 95% CI (8.516 – 10.852). The 95 percent CI for “agricultural college” on determining creativity was 7.901 – 15.099 whereas that of “engineering college” was 11.146 – 13.820.

The confidence interval for “liberal arts” in determining creativity level of high school seniors was 95% CI (10.673 – 12.547) while the 95 percent CI for “music/arts” was 86.920 – 11.261. The 95% CI for “university” as a predictor of creativity was 10.426 – 11.961 while the confidence interval for “other” was 95% CI (8.625 – 10.532). Lastly, the CI for “no plans yet,” as a determinant of creativity was 95% CI (7.648 – 9.984).

Again, it is clear that all the categorical variables have their CI ranging from positive lower boundary value to a positive upper boundary value. This implies that the set of data is somewhat normally distributed as earlier confirmed by the Levene’s F statistic.

According to Table 1, the mean for creativity test score (“creative”) for students who had plans of joining an engineering college was the highest, 12.48, SD = 3.203 whereas the creativity score for seniors who did not want to do anything after college was the lowest, 8.62, SD = 3.378.

Students who had plans of doing liberal arts after school had a higher mean creativity score, 11.61 SD = 4.115, compared to those who had plans of joining an agricultural college,11.50 SD = 4.796, or joining university, 11.19 SD= 3.977.

The mean creativity score for seniors who had “other plans” after college was relatively low, 9.58 SD =3.822, but this was higher than the mean of those who had “no plans yet”, 8.82 SD = 3.220 or those who planned to do music/arts, 9.09 SD= 4.636. Finally, the mean creativity test score for seniors who had plans of joining a teacher college was 9.68, SD = 3.557.

Box’s M Statistic and Wilk’s Lambda

The Box’s M statistic is useful for determining homogeneity of covariance existing across the various groups of categorical variables. The significance level is usually set at p<.001. In this analyses, the Box’s M = 23.586. The F Test for Box’s M= 23.586, F (24, 3373.80) =.925, p =.568, which is greater than p =.001 (Table 4).

This implies that there existed no significant differences between the covariance matrices and therefore the assumption of homogeneity of covariance across the groups was not violated. This also gives us a green light to use the Wilk’s Lambda test for the analyses.

Since the Box’s M test is non-significant and has proved Wilk’s Lambda as a good test for MANOVA, a MANOVA test was conducted and interpreted using the Wilk’s Lambda test. The significance level was considered at p<.05. Table 5 therefore indicates the Wilk’s Lambda =.851, F (18, 984) = 4.603, p =.001.

The F value for Wilk’s Lambda is significant indicating that significant differences existed among the plans of seniors’ (“collplan”) after completing high school on a linear combination of the abstract test score and creativity test scores (dependent variables).

In addition, the Wilk’s lambda is large i.e. greater than.8 thus indicating that the null hypothesis that the categorical factors can be used to determine the student’s creativity and abstract reasoning ability, is supported.

Levene’s F Test

In a MANOVA test, the Levene’s test is useful in determining whether there are any differences in variances/covariance of every variable across the groups. For the assumption to be maintained that no variance exists across the groups, the Levene’s F should be non-significant, otherwise the assumption is violated (Field, 2009).

The Levene’s F for “abstract” was F(9, 493) =.844, p =.576, indicating that the Levene’s F was not statistically significance (Table 6). It therefore means that there are no significant group differences in variance on the variable “abstract.” Moreover, the F value is small hence doubts that are brought about by large values of F regarding the null hypothesis are excluded (Tabachnick & Fidell, 2001).

On the other hand, the Levene’s F value for the variable “creative” was F(9, 493) = 1.400, p =.185 which also indicates that there are no significant differences in variance on the variable “creative”. Overall, it can be assumed that the dataset is normally distributed since variances differ insignificantly.

Between-Subjects Effects

The Wilk’s Lambda indicated that the MANOVA is significant thus it is appropriate to examine Table 7 which essentially provides the univariate results for the dependent variables (abstract and creative). The test of between-subjects effects indicate that the pairs of means for collplan i.e. abstract and creative are statistically different.

For instance, the Mean Square for abstract was 24.322, F(9, 493) = 3.390, p =.001 whereas the Mean Square for creative was 99.880, F (9, 493) = 7.440, p =.001. The R squared value for abstract was.058 indicating that abstract reasoning equivalent to 5.8 percent of multivariate variance in the model was contributed by the student’s career choice after high school i.e. students’ plans after high school.

On the other hand, the R squared value for creative was.120 indicating that creative thinking contributed to 12 percent of multivariate variance in the model i.e. determining the students’ plans after college. It is therefore evident that creativity level of a student has a highly contributed by the student’s plans after high school compared to the contribution on the student’s abstract reasoning ability by the same.

However it is important to note that both abstract reasoning ability and student’s creativity levels are significantly affected by the student’s career plans after high school. This is confirmed by the fact that F values for both variables are significant at the level of.001.

In other words seniors’ plans after high school were significantly different depending on the student’s abstract reasoning ability (F(9, 493) = 3.390, p=.001) and student’s creativity level (F(9, 493) = 7.440, p =.001).

Summary

The decisions of high school seniors regarding their plans on career choices after completing high school were evaluated based on the student’s abstract reasoning ability and creativity.

Factors such as having no plans of a career choice after school, joining a teacher college, an engineering college, doing liberal arts, music/arts, joining university, any other plans or those who had no plans yet were used to determine the student’s abstract reasoning and the student’s creativity. It is evident that overall, most high school seniors did not have any career choice after completing high school.

It is evident that most high school seniors do not prefer joining an agricultural college after high school as demonstrated by a low number of students (4) preferring to join an agricultural college. The highest number of high school seniors (88) would prefer to join university after high school, followed by those who would like to do liberal arts (59), and those who had other plans (57).

The preference for joining either a teacher’s college or an engineering college was relatively high (38 and 29 students) whereas the preference for doing music/arts was relatively low (11 students only).

Having plans of joining a teacher college and/or joining university translated to a high level of abstract reasoning among high school seniors. However, having plans of joining a teacher college translated to a lower creativity score compared to abstract reasoning ability.

On the other hand, the creativity level increased with having plans of joining university compared to the effect of the same plan on abstract reasoning ability. Having plans of joining an engineering college was associated with a high creativity score which was beyond the abstract reasoning ability resulting from the same plans.

While the lowest creativity score resulted from students not wanting to do anything after high schools, the lowest abstract reasoning ability emanated from planning to join an agricultural college after high school.

Having plans of doing liberal arts translated to a higher creativity score than abstract reasoning score whereas having plans of doing music/arts after high school translated to a higher abstract reasoning ability compared to creativity level. There was only a very small difference in creativity level and abstract reasoning ability as a result of having “no plans yet” after completing high school.

The creativity and abstract reasoning ability of high school senior students is demonstrated as being significantly affected by the student’s choice of career after high school. Creativity and abstract reasoning differs depending on whether the student has any plans of joining a specific career after high school or not.

Overall, there is a higher creativity among high school students as a result of future career choice compared to the abstract reasoning ability emanating from the same.

In essence, up to 5.8 percent of abstract reasoning is as a result of the career choice a student has after high school whereas 12 percent of creativity is as a result of the student’s choice of career after completing high school.

Reference

Field, A. (2009). Discovering statistics using SPSS, Third Edition. San Diego, CA: SAGE Publications Ltd.

Tabachnick, B. G. and Fidell, L. S. (2001). Using multivariate statistics. Boston: Allyn and Bacon.

Appendix

Table 1: Between-Subjects Factors

Between-Subjects Factors
Value Label N
collplan 1 none 178
2 teacher college 38
3 agricultural college 4
4 engineering college 29
5 liberal arts 59
6 music/arts 11
7 university 88
8 other 57
9 no plans yet 38
10 10 1

Table 2: Descriptive Statistics for “Collplan”

Descriptive Statistics
collplan Mean Std. Deviation N
abstract none 8.94 2.616 178
teacher college 10.37 2.509 38
agricultural college 7.25 2.500 4
engineering college 10.17 2.156 29
liberal arts 9.97 3.129 59
music/arts 10.09 2.914 11
university 10.37 2.709 88
other 9.74 2.482 57
no plans yet 8.84 2.881 38
10 11.00 . 1
Total 9.59 2.735 503
creative none 8.62 3.378 178
teacher college 9.68 3.557 38
agricultural college 11.50 4.796 4
engineering college 12.48 3.203 29
liberal arts 11.61 4.115 59
music/arts 9.09 4.636 11
university 11.19 3.977 88
other 9.58 3.822 57
no plans yet 8.82 3.220 38
10 15.00 . 1
Total 9.89 3.870 503

Table 3: Estimated Marginal Means and Related 95% Confidence Intervals for Collplan

collplan
Dependent Variable collplan Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
abstract none 8.944 .201 8.549 9.338
teacher college 10.368 .434 9.515 11.222
agricultural college 7.250 1.339 4.619 9.881
engineering college 10.172 .497 9.195 11.150
liberal arts 9.966 .349 9.281 10.651
music/arts 10.091 .808 8.504 11.678
university 10.375 .286 9.814 10.936
other 9.737 .355 9.040 10.434
no plans yet 8.842 .434 7.988 9.696
10 11.000 2.678 5.737 16.263
creative none 8.624 .275 8.084 9.163
teacher college 9.684 .594 8.516 10.852
agricultural college 11.500 1.832 7.901 15.099
engineering college 12.483 .680 11.146 13.820
liberal arts 11.610 .477 10.673 12.547
music/arts 9.091 1.105 6.920 11.261
university 11.193 .391 10.426 11.961
other 9.579 .485 8.625 10.532
no plans yet 8.816 .594 7.648 9.984
10 15.000 3.664 7.801 22.199

Table 4: Box’s M Test

Box’s Test of Equality of Covariance Matricesa
Box’s M 23.586
F .925
df1 24
df2 3373.800
Sig. .568
Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups.
a. Design: Intercept + collplan

Table 5: Multivariate Tests- Wilk’s Lambda

Multivariate Testsc
Effect Value F Hypothesis df Error df Sig.
Intercept Pillai’s Trace .684 532.157a 2.000 492.000 .000
Wilks’ Lambda .316 532.157a 2.000 492.000 .000
Hotelling’s Trace 2.163 532.157a 2.000 492.000 .000
Roy’s Largest Root 2.163 532.157a 2.000 492.000 .000
collplan Pillai’s Trace .153 4.542 18.000 986.000 .000
Wilks’ Lambda .851 4.603a 18.000 984.000 .000
Hotelling’s Trace .171 4.663 18.000 982.000 .000
Roy’s Largest Root .138 7.570b 9.000 493.000 .000
  • Exact statistic
  • The statistic is an upper bound on F that yields a lower bound on the significance level.
  • Design: Intercept + collplan

Table 6: Levene’s Test

Levene’s Test of Equality of Error Variancesa
F df1 df2 Sig.
abstract .844 9 493 .576
creative 1.400 9 493 .185
Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
a. Design: Intercept + collplan

Table 7: Test of Between-Subjects Effects

Tests of Between-Subjects Effects
Source Dependent Variable Type III Sum of Squares df Mean Square F Sig.
Corrected Model abstract 218.894a 9 24.322 3.390 .000
creative 898.919b 9 99.880 7.440 .000
Intercept abstract 6326.276 1 6326.276 881.844 .000
creative 7822.492 1 7822.492 582.683 .000
collplan abstract 218.894 9 24.322 3.390 .000
creative 898.919 9 99.880 7.440 .000
Error abstract 3536.740 493 7.174
creative 6618.497 493 13.425
Total abstract 50020.000 503
creative 56763.000 503
Corrected Total abstract 3755.634 502
creative 7517.416 502
a. R Squared =.058 (Adjusted R Squared =.041)
b. R Squared =.120 (Adjusted R Squared =.104)
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