Squared deviations
{{#invoke:see also|seealso}} In probability theory and statistics, the definition of variance is either the expected value (when considering a theoretical distribution), or average value (for actual experimental data), of squared deviations from the mean. Computations for analysis of variance involve the partitioning of a sum of squared deviations. An understanding of the complex computations involved is greatly enhanced by a detailed study of the statistical value:
It is well known that for a random variable with mean and variance :
Therefore
From the above, the following are easily derived:
If is a vector of n predictions, and is the vector of the true values, then the SSE of the predictor is:
Sample variance
{{#invoke:main|main}} The sum of squared deviations needed to calculate sample variance (before deciding whether to divide by n or n − 1) is most easily calculated as
From the two derived expectations above the expected value of this sum is
which implies
This effectively proves the use of the divisor n − 1 in the calculation of an unbiased sample estimate of σ^{2}.
Partition — analysis of variance
In the situation where data is available for k different treatment groups having size n_{i} where i varies from 1 to k, then it is assumed that the expected mean of each group is
and the variance of each treatment group is unchanged from the population variance .
Under the Null Hypothesis that the treatments have no effect, then each of the will be zero.
It is now possible to calculate three sums of squares:
- Individual
- Treatments
Under the null hypothesis that the treatments cause no differences and all the are zero, the expectation simplifies to
- Combination
Sums of squared deviations
Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on , only .
- total squared deviations aka total sum of squares
- treatment squared deviations aka explained sum of squares
- residual squared deviations aka residual sum of squares
The constants (n − 1), (k − 1), and (n − k) are normally referred to as the number of degrees of freedom.
Example
In a very simple example, 5 observations arise from two treatments. The first treatment gives three values 1, 2, and 3, and the second treatment gives two values 4, and 6.
Giving
- Total squared deviations = 66 − 51.2 = 14.8 with 4 degrees of freedom.
- Treatment squared deviations = 62 − 51.2 = 10.8 with 1 degree of freedom.
- Residual squared deviations = 66 − 62 = 4 with 3 degrees of freedom.
Two-way analysis of variance
The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.
Extra CO_{2} | Extra humidity | |
---|---|---|
No fertiliser | 7, 2, 1 | 7, 6 |
Nitrate | 11, 6 | 10, 7, 3 |
Phosphate | 5, 3, 4 | 11, 4 |
Five sums of squares are calculated:
Factor | Calculation | Sum | |
---|---|---|---|
Individual | 641 | 15 | |
Fertiliser × Environment | 556.1667 | 6 | |
Fertiliser | 525.4 | 3 | |
Environment | 519.2679 | 2 | |
Composite | 504.6 | 1 |
Finally, the sums of squared deviations required for the analysis of variance can be calculated.
See also
References
- ↑ Mood & Graybill: An introduction to the Theory of Statistics (McGraw Hill)