Analysis of variance is a widespread statistical test among analysts, and aims primarily to verify whether there is a significant difference between the means and whether the factors exert influence on any dependent variable.
The proposed factors may be of qualitative or quantitative origin, but the dependent variable must necessarily be continuous.
Since this is a very widespread test and many good statistical software and spreadsheets have the available resource, there will be no deepening of this technique in this chapter, and specialized literature is recommended.
The main application of ANOVA (analysis of variance) is the comparison of averages from different groups, also called treatments, such as historical averages of satisfaction issues, companies operating simultaneously with different incomes, among many other applications.
There are two methods for calculating variance: within groups (MQG) and mean variance (MQR).
In an Anova, these two components of variance are calculated. If the variance calculated using the mean (MQR) is greater than the calculated (MQG) using the data belonging to each individual group, this may indicate that there is a significant difference between the groups.
There are two types of problems to be solved through Anova: at fixed levels or at random levels. Randomness determined the issue of the problem.
In the vast majority of cases these are fixed levels, after all the second type of problem (random) will only arise when a study involving a random choice of factors occurs (in 10 production batches, only 5 out of 15 production machines are chosen). a total of 20, for example).
Variance Analysis Table or ANOVA Table
Source of Variation | SQ | GDL | MQ | F test |
Between Groups | SQG | K - 1 | MQG | MQG / MQR |
Within Groups | SQR | N-K | MQR | |
Total | SQT | N-1 |
- SQT = SQG + SQR (measures the overall variation of all observations).
- SQT is the sum of the total squares, broken down into:
- SQG sum of squared groups (treatments), associated exclusively with a group effect
- SQR sum of the squares of the residuals, due exclusively to the random error, measured within the groups.
- MQG = Square Mean of Groups
- MQR = Square mean residue (between groups)
- SQG and MQG: measure the total variation between the means
- SQR and MQR: measure the variation of observations in each group.
f = MQG
MQR
N - 1 = (K - 1) + (N - K)
SQT = SQG + SQR
MQG = SQG (K - 1)
The null hypothesis will always be rejected when f calculated is greater than the tabulated value. Similarly, if MQG is greater than MQR, the null hypothesis is rejected.
Painting
Source of variation SQ (sum of squares) GDL (g.l) MQ (mean square) Test F |
Between Groups |
Within the groups |
Total |
If the f-test indicates significant differences between the averages and the levels are fixed, then it is interesting to identify which means differ from each other.
Calculate the standard deviation of the means;
Sx = _{ } , where nc is the sum of the number of each variable (group) divided by the number of variables.
Calculate the decision limit (ld)
3 x Sx
Sort the averages in ascending or descending order and compare them two by two. The difference will be significant if greater than Ld.
If the f-test indicates significant differences between the means and the levels are random, then it is interesting to identify the estimate of the variation components.
_{ }
_{ }
_{ }
The value found above will indicate the total variability between groups, indicating whether it is considered significant or not.
Example (fixed levels):
One researcher conducted a study to see which job generated the most employee satisfaction. For this, for a month, 10 employees were interviewed. At the end of a month employees answered a questionnaire generating a score for employee welfare.
Posts | |||
Employees | 1 | 2 | 3 |
1 | 7 | 5 | 8 |
2 | 8 | 6 | 9 |
3 | 7 | 7 | 8 |
4 | 8 | 6 | 9 |
5 | 9 | 5 | 8 |
6 | 7 | 6 | 8 |
7 | 8 | 7 | 9 |
8 | 6 | 5 | 10 |
9 | 7 | 6 | 8 |
10 | 6 | 6 | 9 |
Resume
Group | Score | Sum | Average | Variance |
1 | 10 | 73 | 7,3 | 0,9 |
2 | 10 | 59 | 5,9 | 0,544444 |
3 | 10 | 86 | 8,6 | 0,488889 |
THE NEW
Variation Source | SQ | gl | MQ | F | P-value | F critical |
Between groups | 36,46667 | 2 | 18,23 | 28,29 | 2.37E-07 | 3,35 |
Within the groups | 17,4 | 27 | 0,64 | |||
Total | 53,86667 | 29 |
As f calculated is larger than the tabulated one, the null hypothesis in favor of the 5% risk alternative hypothesis is rejected.
There are significant differences between the groups. MQG is much higher than MQR, indicating a strong variance between the groups.
1. Calculate the standard deviation of the means;
_{ }
2. Calculate the decision limit (Ld)
3 x Sx
3. Sort the averages in ascending or descending order and compare them two by two.
5,9
7,3
8,6
x1 - x2 = - 1.4
x1 - x3 = - 2.7
x2 - x3 = - 1.3
The three differences are smaller than Ld, so it can be concluded that the means differ from each other. Next: Simple Regression (RLS)