Analysis of Variance(ANOVA) Technique.
Highlights;
- ANOVA
- Assumptions of ANOVA
- Post-hoc
ANOVA, also known as analysis of variance technique is test statistics used to when a researcher is interested in finding out the difference that exist between the levels of an independent variable usually with 3 levels and above.
Reasons why t- test is not employed in a question that requires ANOVA is because;
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- t-test will have multiple pairwise comparison
- Because of error( type I error rate will not be maintained)
To determine the number of pairwise comparison that can be carried out, this formula is employed
n = k(k – 1)/2. for example if k =5 the n = 5(5-1)/2 = 10 comparisons.
Type one error rate will not not be maintained. the formula is
= 1 – (1-Alpha)^n
assuming alpha level = 0.05
the 1 – (1-0.05)^10
= 0.40 = alpha
This means our level of significance have changed from 0.05 to 0.40. This is why it is not advisable to carry analysis involving more than two levels with t-test. Instead we use ANOVA.
ANOVA talks about variation (errors). it is grouped into;
- Between group variation
- Within group variation
Within group variation arise as a result of random assignment. Between group variation may come as a result of;
- Error due to random assignment
- Error due to treatment effect like teaching methods applied to a particular group.
Analysis of Variance(ANOVA) Technique
Assumptions of ANOVA
- Random and independent sample assumption
- Normality assumption
- Equality of variance assumption
- The variables must be measure on iinterval/ratio level
Random and independent sample assumption means the sample should be sampled differently such that there are independent of one another.
Normality assumption means the scores or trait or attributes should be normally distributed. i.e every member in the distribution should have an atom of the trait.
Equality of variance means the errors should be normaly distributed such that error in group A = group B = group C.
Vital Parameters
Total sum of square(TSS) = summation(X^2 -T^2/N)
Sum of square between(SSb) = Summation(T^2/nk)- T^2/N)
Sum of square within(SSw) = summationX^2-summation(T^2/nk)
or SSW = SST – SSB
The Degree of freedom
Degree of freedom total(DFT) = N -1
Degree of freedom between(DFB) = K-1
Also, degree of freedom within(DF) = N-K
The Mean sum of squares
Mean sum of square between(MSB) = SSB/DFB = SSB/K-1
Mean sum of square within(MSW) = SSW/DFW = SSW/N-k
F-ratio = MSB/MSW
Example on Analysis of variance
Assuming a researcher is interested in finding out the effect of 3 teaching methods on student’s achievement in mathematics.
| SN | A | B | C | A^2 | B^2 | C^2 |
| 1 | 10 | 2 | 5 | 100 | 4 | 25 |
| 2 | 9 | 3 | 8 | 81 | 9 | 64 |
| 3 | 8 | 5 | 6 | 64 | 25 | 36 |
| 4 | 10 | 6 | 4 | 100 | 36 | 16 |
| 5 | 6 | 2 | 3 | 36 | 4 | 9 |
| Total | 43 | 18 | 26 | 381 | 78 | 150 |
| Mean | 8.6 | 3.6 | 5.2 |
A^2 is gotten by squaring each values of A. so does B and C
H0: There is no significant differences between the mean scores of students exposed to the 3 teaching methods.
n = 5, N = 15
Summation(T) = 42+18+26=87
SummationX^2 = 381+78+150 =609
TSS = summation(X^2 -T^2/N) = 609 – 87^2/15 = 104.4
SSb = Summation(T^2/nk)- T^2/N) = 43^2/5+18^2/5+26^2/5 -87^2/15 = 65.2
SSw = SST – SSB = 104.4 – 65.2 = 39.2
calculating the degree of fredom
DFB = k-1 = 3 -1 =2
DFW = N-K =15 -3 = 12
determining the mean squares
MSB = SSB/DFB = 65.2/2 = 32.6
MSW = SSW/DFW = 39.2/12 = 3.27
F-ratio = MSB/MSW = 32.6/3.27 = 9.98
since F-cL(9.98) is greater than F-critical(3.89), hence the null hypothesis is rejected. which implies that a significant difference exist among the three teaching methods.
Since a significant difference exist, these leads to a concept called Post -Hoc multiple comparison procedure.
Post -Hoc multiple comparison procedure
This is used to find the significant difference among the levels of the independent variable. The analysis is conducted after an omnibus test and f-test. Post hoc is applied when result of an ANOVA analysis shows a significant result.
The computation of an ANOVA statistics is referred to as an omnibus test or f- statistics. The post-hoc can be classsified into’
- Omnibus test;
- Pairwise comparison
- Complex comparison
- Planned/aprori contrast
Pairwise comparison entails running multiple comparison involving two mean at a time.
Types of post-hoc
- tukey
- tukey krammer
- ducan LSD
- bonferronni
- newman keuls etc
Conditions for using the types of post-hoc
The following post hoc types are used when we have equal sample size;
- Tukey HSD
- Duncan LSD
- Newman keuls
The following are used when the sample are not equal;
- Tukey Krammer
- Bonferronni
All these are bade on studentized Q-distribution.
in complex comparison, we have the sheffe test that involve comparing more than two mean at a time.
Planned/aprori contrast
Planned or apropri contrast are specific hypothesis which the researcher formulate before the actual conduct of a research investigation.
This is classified into Orthogonal contrast and trend analysis/
Orthogonal contrast are contrast that are mutually independent of each other. contrast refers to algebraic computation of means and coefficients. Note: contrast refers to hypothesis. orthogonal contrast is used when we have qualitative data.
Trend analysis is used when we have quantitative data. The trend analysis enables us to determine the nature, pattern or sequence of relationship between variables under investigation. it makes use of coefficient of polynomials.
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