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WK6Assgn1RiddickJ.extension.docxRunning Head: FACTORIAL ANALYSIS OF VARIANCE
FACTORIAL ANALYSIS OF VARIANCE
Jamiah Riddick
Walden University
RSCH - 8260F; Advanced Quantitative
Dr. Marker
July 12th, 2021
Factorial Analysis of Variance:
ANOVA (Factorial analysis of variance), being the statistical procedure, allows to investigate or determine the influence of independent variables (two or more) on the dependent variable. In this way, factorial ANOVA is regarded as the powerful test for exploring this interaction as it reduces the potential error variance. Therefore, factorial ANOVA deals with the comparison of different means between the groups which are split into independent variables (called factors). In regard, factorial ANOVA has been applied to see that whether there is any interaction between the gun owners, race, citizenship on the family income, where guns own (yes/no/refused), citizenship (yes/no), and race (white/black/other) are independent variables or factors, and family income is the dependent variable. Age has been excluded as it was not a categorical variable in the data set i.e., instead of age groups general continuous limits about age are mentioned. In this way, the factorial ANOVA is using the three independent variables (with two or more categories) for predicting the change in the dependent variable i.e., family income. Therefore, for factorial ANOVA or univariate variance analysis regarding the underpinned factors, the following assumptions must be held, in case of violations, the results or findings will not be validated. These assumptions are six and they are:
· Assumption 1: In the factorial ANOVA, the dependent variable is always a continuous variable i.e., measured at a continuous level.
· Assumption 2: In the factorial ANOVA, the independent variables are always categorical variables i.e., categorical predictor variables that consist of categories or independent groups.
· Assumption 3: In the factorial ANOVA, observations of each group are always independent of each other.
· Assumption 4: In the factorial ANOVA, there is no significant outliers i.e., the data points do not follow the usual pattern.
· Assumption 5: In the factorial ANOVA, the dependent variable is always normally distributed for the combination of the groups of independent variables.
· Assumption 6: In the factorial ANOVA, each combination of groups of the independent variables has homogenous variances.
The validation of assumptions 4, 5, and 6 can be analyzed only through the SPSS statistics while 1, 2, and 3 are not possible to check through the SPSS. Therefore, these assumptions are checked through the theory and are not violated in the case. Therefore, the factorial ANOVA based on the given hypothesis has been estimated by using the SPSS.
Based on the research question of the study, the hypothesis is:
Hypothesis:
HO: There is no interaction of citizenship, race, and gun ownership on family income.
H1: There is a significant interaction of citizenship, race, and gun ownership on family income.
Therefore, the results of the three categorical variables (factors) on the response variable through testing the hypothesis are given in the upcoming section of the conclusion.
Conclusion:
The results have been arrived at by using the social set database on SPSS. In this regard, the Univariate Analysis of Variance has is presented in appendices. The statistics in the concern are presented in appendix 1 regarding the between-subjects factors whose sample size is slightly different from the sample size before estimating the factorial analysis and the sample size changes because of factorial ANOVA. Based on the given table, the three factors i.e., citizenship, race, and gun ownership are presented along with their groups and their sample size are presented.
The descriptive statistics estimated through the ANOVA analysis is presented in appendix 02 (see appendices). These statistics are presenting the different means as compared to the observed means which are depicting clearly in marginal expected means appendices. Therefore, Levene’s test of equability of error variances is presenting that model is normally distributed and actors are significant (see appendix 03). Therefore, based on the mean approach not no assumption of equal variances was violated. Hence, the results of between-subjects effects in this regard are presented in appendix 04. These are the key results that postulate the statistics regarding the corrected model, intercept, one-way ANOVA, two-way ANOVA, and three-way ANOVA. According to these statistics corrected model has a significant value of 0.008 while the partial Eta squared value is 0.130, additionally, the significant probability value for the intercept is 0.000 and the partial Eta Squared is 0.132. these statistics are depicting that the corrected model and the value of the intercept are highly significant. In the original models 1 and 2 (appendix 3 and 4), constant or intercept was also found highly significant. Moreover, in one-way factorial ANOVA the probability values of citizenship, race, and gun ownership are 0.74, 0.46, and 0.01, these values are representing the insignificant interaction with the family income other than the gun owners whose value is depicting the significant interaction with the family income. Furthermore, in the case of two-way ANOVA, it has found that citizenship and race have an insignificant interaction while there was no interaction found between citizenship and gun owners. Similarly, in two-way ANOVA it has also been found an insignificant interaction between race and gun ownership. In the way, the three-way ANOVA results postulate that there are no interactions found between citizenship, race, and gun ownership on the family income. Therefore, a null hypothesis is expected i.e., there is no interaction found between the factors and dependent variable. Moreover, the estimated marginal means of the given factors in one-way, two-way, and three-way factorial ANOVA are presented in appendix 5. The plot graph in this regard is presented below.
APPENDICES
Appendix 1: Univariate Analysis of Variance
|
Between-Subjects Factors |
|||
|
|
Value Label |
N |
|
|
ARE YOU A CITIZEN OF AMERICA? |
1 |
YES |
143 |
|
|
2 |
NO |
12 |
|
RACE OF RESPONDENT |
1 |
WHITE |
104 |
|
|
2 |
BLACK |
26 |
|
|
3 |
OTHER |
25 |
|
HAVE GUN IN HOME |
1 |
YES |
14 |
|
|
2 |
NO |
139 |
|
|
3 |
REFUSED |
2 |
Appendix 2: Descriptive Statistics
|
|
|||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
|||||
|
ARE YOU A CITIZEN OF AMERICA? |
RACE OF RESPONDENT |
HAVE GUN IN HOME |
Mean |
Std. Deviation |
N |
|
YES |
WHITE |
YES |
99129.22 |
55911.545 |
13 |
|
|
|
NO |
65581.50 |
52313.103 |
84 |
|
|
|
REFUSED |
17920.75 |
21685.904 |
2 |
|
|
|
Total |
69023.91 |
53749.903 |
99 |
|
|
BLACK |
NO |
41082.12 |
39977.686 |
22 |
|
|
|
Total |
41082.12 |
39977.686 |
22 |
|
|
OTHER |
YES |
160742.22 |
. |
1 |
|
|
|
NO |
65236.66 |
57698.459 |
21 |
|
|
|
Total |
69577.82 |
59876.440 |
22 |
|
|
Total |
YES |
103530.15 |
56185.282 |
14 |
|
|
|
NO |
61280.49 |
51804.395 |
127 |
|
|
|
REFUSED |
17920.75 |
21685.904 |
2 |
|
|
|
Total |
64810.39 |
53512.069 |
143 |
|
NO |
WHITE |
NO |
36432.70 |
36626.053 |
5 |
|
|
|
Total |
36432.70 |
36626.053 |
5 |
|
|
BLACK |
NO |
22631.88 |
9614.695 |
4 |
|
|
|
Total |
22631.88 |
9614.695 |
4 |
|
|
OTHER |
NO |
27404.58 |
16941.014 |
3 |
|
|
|
Total |
27404.58 |
16941.014 |
3 |
|
|
Total |
NO |
29575.40 |
24604.682 |
12 |
|
|
|
Total |
29575.40 |
24604.682 |
12 |
|
Total |
WHITE |
YES |
99129.22 |
55911.545 |
13 |
|
|
|
NO |
63943.93 |
51843.111 |
89 |
|
|
|
REFUSED |
17920.75 |
21685.904 |
2 |
|
|
|
Total |
67457.03 |
53385.261 |
104 |
|
|
BLACK |
NO |
38243.62 |
37412.304 |
26 |
|
|
|
Total |
38243.62 |
37412.304 |
26 |
|
|
OTHER |
YES |
160742.22 |
. |
1 |
|
|
|
NO |
60507.65 |
55526.458 |
24 |
|
|
|
Total |
64517.03 |
57936.173 |
25 |
|
|
Total |
YES |
103530.15 |
56185.282 |
14 |
|
|
|
NO |
58543.36 |
50778.440 |
139 |
|
|
|
REFUSED |
17920.75 |
21685.904 |
2 |
|
|
|
Total |
62082.52 |
52658.348 |
155 |
Appendix 3: Levene's Test of Equality of Error Variancesa
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
df1 |
df2 |
Sig. |
|
2.099 |
8 |
146 |
.039 |
|
Tests the null hypothesis that the error variance of the dependent variable is equal across groups. |
|||
|
a. Design: Intercept + citizen + race + owngun + citizen * race + citizen * owngun + race * owngun + citizen * race * owngun |
Appendix 4: Tests of Between-Subjects Effects
|
Tests of Between-Subjects Effects |
||||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
||||||
|
Source |
Type III Sum of Squares |
df |
Mean Square |
F |
Sig. |
Partial Eta Squared |
|
Corrected Model |
55538558582.154a |
8 |
6942319822.769 |
2.728 |
.008 |
.130 |
|
Intercept |
56352155100.249 |
1 |
56352155100.249 |
22.147 |
.000 |
.132 |
|
citizen |
8216122291.620 |
1 |
8216122291.620 |
3.229 |
.074 |
.022 |
|
race |
3898512396.712 |
2 |
1949256198.356 |
.766 |
.467 |
.010 |
|
owngun |
21733302617.516 |
2 |
10866651308.758 |
4.271 |
.016 |
.055 |
|
citizen * race |
568544245.832 |
2 |
284272122.916 |
.112 |
.894 |
.002 |
|
citizen * owngun |
.000 |
0 |
. |
. |
. |
.000 |
|
race * owngun |
3377873954.877 |
1 |
3377873954.877 |
1.328 |
.251 |
.009 |
|
citizen * race * owngun |
.000 |
0 |
. |
. |
. |
.000 |
|
Error |
371488293338.152 |
146 |
2544440365.330 |
|
|
|
|
Total |
1024434003858.308 |
155 |
|
|
|
|
|
Corrected Total |
427026851920.307 |
154 |
|
|
|
|
|
a. R Squared = .130 (Adjusted R Squared = .082) |
Appendix 5: Estimated Marginal Means
|
1. Grand Mean |
|||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
|||
|
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
Lower Bound |
Upper Bound |
|
59573.514a |
8800.002 |
42181.669 |
76965.360 |
|
a. Based on modified population marginal mean. |
|
2. ARE YOU A CITIZEN OF AMERICA? |
||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
||||
|
ARE YOU A CITIZEN OF AMERICA? |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
Lower Bound |
Upper Bound |
|
YES |
74948.745a |
10902.979 |
53400.691 |
96496.800 |
|
NO |
28823.053a |
14881.551 |
-588.035 |
58234.141 |
|
a. Based on modified population marginal mean. |
|
3. RACE OF RESPONDENT |
||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
||||
|
RACE OF RESPONDENT |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
Lower Bound |
Upper Bound |
|
WHITE |
54766.043a |
11200.238 |
32630.501 |
76901.585 |
|
BLACK |
31856.999a |
13709.180 |
4762.921 |
58951.078 |
|
OTHER |
84461.153a |
19758.967 |
45410.605 |
123511.702 |
|
a. Based on modified population marginal mean. |
|
4. HAVE GUN IN HOME |
||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
||||
|
HAVE GUN IN HOME |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
Lower Bound |
Upper Bound |
|
YES |
129935.718a |
26173.300 |
78208.230 |
181663.206 |
|
NO |
43061.574 |
7923.692 |
27401.620 |
58721.528 |
|
REFUSED |
17920.750a |
35668.196 |
-52571.934 |
88413.434 |
|
a. Based on modified population marginal mean. |
|
5. ARE YOU A CITIZEN OF AMERICA? * RACE OF RESPONDENT |
|||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
|||||
|
ARE YOU A CITIZEN OF AMERICA? |
RACE OF RESPONDENT |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
|
Lower Bound |
Upper Bound |
|
YES |
WHITE |
60877.157 |
12902.357 |
35377.640 |
86376.674 |
|
|
BLACK |
41082.124a |
10754.366 |
19827.780 |
62336.467 |
|
|
OTHER |
112989.438a |
25814.745 |
61970.580 |
164008.297 |
|
NO |
WHITE |
36432.700a |
22558.548 |
-8150.788 |
81016.188 |
|
|
BLACK |
22631.875a |
25221.223 |
-27213.980 |
72477.730 |
|
|
OTHER |
27404.583a |
29122.960 |
-30152.452 |
84961.618 |
|
a. Based on modified population marginal mean. |
|
6. ARE YOU A CITIZEN OF AMERICA? * HAVE GUN IN HOME |
|||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
|||||
|
ARE YOU A CITIZEN OF AMERICA? |
HAVE GUN IN HOME |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
|
Lower Bound |
Upper Bound |
|
YES |
YES |
129935.718a |
26173.300 |
78208.230 |
181663.206 |
|
|
NO |
57300.095 |
5447.845 |
46533.270 |
68066.920 |
|
|
REFUSED |
17920.750a |
35668.196 |
-52571.934 |
88413.434 |
|
NO |
YES |
.b |
. |
. |
. |
|
|
NO |
28823.053 |
14881.551 |
-588.035 |
58234.141 |
|
|
REFUSED |
.b |
. |
. |
. |
|
a. Based on modified population marginal mean. |
|||||
|
b. This level combination of factors is not observed, thus the corresponding population marginal mean is not estimable. |
|
7. RACE OF RESPONDENT * HAVE GUN IN HOME |
|||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
|||||
|
RACE OF RESPONDENT |
HAVE GUN IN HOME |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
|
Lower Bound |
Upper Bound |
|
WHITE |
YES |
99129.218a |
13990.217 |
71479.713 |
126778.724 |
|
|
NO |
51007.101 |
11610.114 |
28061.502 |
73952.701 |
|
|
REFUSED |
17920.750a |
35668.196 |
-52571.934 |
88413.434 |
|
BLACK |
YES |
.b |
. |
. |
. |
|
|
NO |
31856.999 |
13709.180 |
4762.921 |
58951.078 |
|
|
REFUSED |
.b |
. |
. |
. |
|
OTHER |
YES |
160742.218a |
50442.446 |
61050.509 |
260433.927 |
|
|
NO |
46320.621 |
15566.877 |
15555.092 |
77086.151 |
|
|
REFUSED |
.b |
. |
. |
. |
|
a. Based on modified population marginal mean. |
|||||
|
b. This level combination of factors is not observed, thus the corresponding population marginal mean is not estimable. |
|
8. ARE YOU A CITIZEN OF AMERICA? * RACE OF RESPONDENT * HAVE GUN IN HOME |
||||||
|
Dependent Variable: FAMILY INCOME IN CONSTANT DOLLARS |
||||||
|
ARE YOU A CITIZEN OF AMERICA? |
RACE OF RESPONDENT |
HAVE GUN IN HOME |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
|
|
Lower Bound |
Upper Bound |
|
YES |
WHITE |
YES |
99129.218 |
13990.217 |
71479.713 |
126778.724 |
|
|
|
NO |
65581.503 |
5503.722 |
54704.246 |
76458.760 |
|
|
|
REFUSED |
17920.750 |
35668.196 |
-52571.934 |
88413.434 |
|
|
BLACK |
YES |
.a |
. |
. |
. |
|
|
|
NO |
41082.124 |
10754.366 |
19827.780 |
62336.467 |
|
|
|
REFUSED |
.a |
. |
. |
. |
|
|
OTHER |
YES |
160742.218 |
50442.446 |
61050.509 |
260433.927 |
|
|
|
NO |
65236.659 |
11007.444 |
43482.145 |
86991.173 |
|
|
|
REFUSED |
.a |
. |
. |
. |
|
NO |
WHITE |
YES |
.a |
. |
. |
. |
|
|
|
NO |
36432.700 |
22558.548 |
-8150.788 |
81016.188 |
|
|
|
REFUSED |
.a |
. |
. |
. |
|
|
BLACK |
YES |
.a |
. |
. |
. |
|
|
|
NO |
22631.875 |
25221.223 |
-27213.980 |
72477.730 |
|
|
|
REFUSED |
.a |
. |
. |
. |
|
|
OTHER |
YES |
.a |
. |
. |
. |
|
|
|
NO |
27404.583 |
29122.960 |
-30152.452 |
84961.618 |
|
|
|
REFUSED |
.a |
. |
. |
. |
|
a. This level combination of factors is not observed, thus the corresponding population marginal mean is not estimable. |
