Factorial ANOVA
Introduction
We want to check how the age and different levels of music affects the musical tastes or liking to the music. For this we have two categories of age that is less than 40 years and more than 40 years and three categories of different music that are Fugazi, Abba and Barf Grooks. Basically we want to check whether these variables affect the liking of music and if yes then how they are going to affect the liking of music.
Three Hypotheses can be tested simultaneously with two way analysis of variance.
Statistical Assumption of the test:
1. The data should be collected by random sampling.
2. The sample should be independent from each other.
3. The data should be continuous or we can say that the data should be either interval level or ratio level for t test.
Using the data set you have selected, select independent and dependent variables.
Here we have
Independent Variables: Age, Music
Dependent Variable: Liking
Develop the null and the alternative hypotheses.
First hypothesis:
Null Hypothesis (Ho1): All the age groups have equal music liking on the average or we can say that there is no significant difference in liking music due to age factor.
Alternative Hypothesis (Ha1): All the age groups are not equal music liking on the average or we can say that there is a significant difference in liking music due to age factor.
Mathematically;
Null Hypothesis (Ho1): µ1 = µ2
Alternative Hypothesis (Ha1): µ1 ≠ µ2
Where µ1 = Mean of the liking music for first category of age of less than 40 years.
µ2 = Mean of the liking music for second category of age of more than 40 years.
Second hypothesis:
Null Hypothesis (Ho2): All the music groups have equal music liking on the average or we can say that there is no significant difference in liking music due to music factor.
Alternative Hypothesis (Ha2): All the music groups are not equal music liking on the average or we can say that there is a significant difference in liking music due to music factor.
Mathematically;
Null Hypothesis (Ho2): µ1 = µ2 = µ3
Alternative Hypothesis (Ha2): µi ≠ µj; for some i not equal to j
Where µ1 = Mean of the liking music for first category of music that is Fugazi
µ2 = Mean of the liking music for second category of music that is Abba.
µ3 = Mean of the liking music for third category of music that is Barf Grooks.
Third hypothesis:
Null Hypothesis (Ho3): There is no significant interaction between two main factors that is age and music.
Alternative Hypothesis (Ha3): There is a significant interaction between two main factors that is age and music.
.Level of Significance = .05
The below syntax is used to calculate the Two Way ANOVA test for the above mentioned data.
UNIANOVA liking BY music age
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/POSTHOC=music(BONFERRONI)
/PLOT=PROFILE(music age)
/EMMEANS=TABLES(music)
/EMMEANS=TABLES(age)
/EMMEANS=TABLES(music*age)
/PRINT=OPOWER ETASQ HOMOGENEITY DESCRIPTIVE
/CRITERIA=ALPHA(.05)
/DESIGN=music age music*age.
SPSS Output is given below:
Univariate Analysis of Variance
|
Between-Subjects Factors |
|
|
|
|
|
Value Label |
N |
|
|
Music |
1.00 |
Fugazi |
30 |
|
|
2.00 |
Abba |
30 |
|
|
3.00 |
Barf Grooks |
30 |
|
Age Group |
1.00 |
40+ |
45 |
|
|
2.00 |
0-40 |
45 |
|
Descriptive Statistics |
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Dependent Variable:Liking Rating |
|
|
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|
Music |
Age Group |
Mean |
Std. Deviation |
N |
|
|
Fugazi |
dimension2 |
40+ |
-75.8667 |
14.37193 |
15 |
|
|
|
0-40 |
66.2000 |
19.90406 |
15 |
|
|
|
Total |
-4.8333 |
74.23406 |
30 |
|
Abba |
dimension2 |
40+ |
59.9333 |
19.98380 |
15 |
|
|
|
0-40 |
64.1333 |
16.99524 |
15 |
|
|
|
Total |
62.0333 |
18.35189 |
30 |
|
Barf Grooks |
dimension2 |
40+ |
74.2667 |
22.29499 |
15 |
|
|
|
0-40 |
-71.4667 |
23.17901 |
15 |
|
|
|
Total |
1.4000 |
77.40783 |
30 |
|
Total |
dimension2 |
40+ |
19.4444 |
70.93164 |
45 |
|
|
|
0-40 |
19.6222 |
68.06257 |
45 |
|
|
|
Total |
19.5333 |
69.12035 |
90 |
|
Levene's Test of Equality of Error Variancesa |
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|
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|
Dependent Variable:Liking Rating |
|
|
|
|
F |
df1 |
df2 |
Sig. |
|
1.189 |
5 |
84 |
.322 |
|
Tests the null hypothesis that the error variance of the dependent variable is equal across groups. |
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a. Design: Intercept + music + age + music * age |
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Estimated Marginal Means
|
1. Music |
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Dependent Variable:Liking Rating |
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|
Music |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
Lower Bound |
Upper Bound |
|
Fugazi |
-4.833 |
3.594 |
-11.981 |
2.314 |
|
Abba |
62.033 |
3.594 |
54.886 |
69.181 |
|
Barf Grooks |
1.400 |
3.594 |
-5.747 |
8.547 |
|
2. Age Group |
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|
|
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|
|
Dependent Variable:Liking Rating |
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|
Age Group |
Mean |
Std. Error |
95% Confidence Interval |
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|
|
|
|
|
Lower Bound |
Upper Bound |
|
|
dimension1 |
40+ |
19.444 |
2.935 |
13.609 |
25.280 |
|
|
0-40 |
19.622 |
2.935 |
13.786 |
25.458 |
|
3. Music * Age Group |
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Dependent Variable:Liking Rating |
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|
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|
|
Music |
Age Group |
Mean |
Std. Error |
95% Confidence Interval |
|
|
|
|
|
|
|
Lower Bound |
Upper Bound |
|
|
Fugazi |
dimension2 |
40+ |
-75.867 |
5.083 |
-85.975 |
-65.759 |
|
|
|
0-40 |
66.200 |
5.083 |
56.092 |
76.308 |
|
Abba |
dimension2 |
40+ |
59.933 |
5.083 |
49.825 |
70.041 |
|
|
|
0-40 |
64.133 |
5.083 |
54.025 |
74.241 |
|
Barf Grooks |
dimension2 |
40+ |
74.267 |
5.083 |
64.159 |
84.375 |
|
|
|
0-40 |
-71.467 |
5.083 |
-81.575 |
-61.359 |
Post Hoc Tests
Music
|
Multiple Comparisons |
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Liking Rating Bonferroni |
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|
(I) Music |
(J) Music |
Mean Difference (I-J) |
Std. Error |
Sig. |
95% Confidence Interval |
|
|
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|
|
|
Lower Bound |
Upper Bound |
|
Fugazi |
Abba |
-66.8667* |
5.08292 |
.000 |
-79.2836 |
-54.4498 |
|
|
Barf Grooks |
-6.2333 |
5.08292 |
.671 |
-18.6502 |
6.1836 |
|
Abba |
Fugazi |
66.8667* |
5.08292 |
.000 |
54.4498 |
79.2836 |
|
|
Barf Grooks |
60.6333* |
5.08292 |
.000 |
48.2164 |
73.0502 |
|
Barf Grooks |
Fugazi |
6.2333 |
5.08292 |
.671 |
-6.1836 |
18.6502 |
|
|
Abba |
-60.6333* |
5.08292 |
.000 |
-73.0502 |
-48.2164 |
|
Based on observed means. The error term is Mean Square(Error) = 387.541. |
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*. The mean difference is significant at the .05 level. |
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Profile Plots
From the above analysis and results we can conclude that the assumptions of the models are satisfied because the Levene's Test of Equality of Error Variances has P-value 0.322 which is more than the level of significance, so we can say that the population variances are equal. Also the data is normally distributed and sample is randomly drawn.
So we can conclude that
1) All the age groups have equal music liking on the average or we can say that there is no significant difference in liking music due to age factor because P-value corresponding to this variable is 0.966 which is very high.
2) All the music groups are not equal music liking on the average or we can say that there is a significant difference in liking music due to music factor because the P-value corresponding to this is 0.000.
3) There is a significant interaction between two main factors that is age and music because the P-value corresponding to this is 0.000.
Partial η2 (Partial eta-squared): Partial eta-squared describes the "proportion of total variation attributable to the factor, partialling out (excluding) other factors from the total nonerror variation"
Here the value of R-Square is .923 which means that 92.3% of the variation is explained by these factors. So it means we have obtained the power more than 80% for this analysis.
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