One-factor variance model looks like
Where Xjj- the value of the variable under study obtained on g-level factor (r = 1, 2,..., T) su-th serial number (j- 1,2,..., P);/y - effect due to the influence of the i-th level of the factor; e^. - random component, or disturbance caused by the influence of uncontrollable factors, i.e. variation of a variable within an individual level.
Under factor level refers to some measure or condition of it, for example, the amount of fertilizer applied, the type of metal melting or the batch number of parts, etc.
Basic premises of analysis of variance.
1. Mathematical expectation of disturbance ? (/ - is equal to zero for any i, those.
- 2. The disturbances are mutually independent.
- 3. The dispersion of the disturbance (or variable Xy) is constant for any ij> those.
4. The disturbance e# (or the variable Xy) has a normal distribution law N( 0; a 2).
The influence of factor levels can be like fixed, or systematic(model I), and random(model II).
Let, for example, it is necessary to find out whether there are significant differences between batches of products according to some quality indicator, i.e. check the influence on the quality of one factor - a batch of products. If we include all batches of raw materials in the study, then the influence of the level of such a factor is systematic (model I), and the conclusions obtained are applicable only to those individual batches that were involved in the study; if we include only a randomly selected part of the parties, then the influence of the factor is random (model II). In multifactor complexes, a mixed model III is possible, in which some factors have random levels, while others have fixed levels.
Let's consider this task in more detail. Let there be T batches of products. Selected from each batch accordingly p L, p 2 ,p t products (for simplicity we assume that u = n 2 =... = p t = p). We present the values of the quality indicator of these products in the form of an observation matrix
It is necessary to check the significance of the influence of product batches on their quality.
If we assume that the elements of the rows of the observation matrix are numerical values (realizations) random variables X t , X 2 ,..., X t, expressing the quality of products and having a normal distribution law with mathematical expectations, respectively a v a 2, ..., a t and identical variances a 2, then this task comes down to testing the null hypothesis # 0: a v = a 2l = ... = A t, carried out in analysis of variance.
Let us denote averaging over some index with an asterisk (or dot) instead of an index, then average quality of products of the ith batch, or group average for the ith level of the factor, takes the form
A overall average -
Let us consider the sum of squared deviations of observations from the general average x„:
or Q = Q, + Q 2+ ?>з Last term
since the sum of deviations of the values of a variable from its average, i.e. ? 1.r y - x) is equal to zero. ) =x
The first term can be written in the form
As a result, we obtain the following identity:
etc. _
Where Q = Y, X [ x ij _ x„, I 2 - general, or full, sum of squared deviations; 7=1
Q, - n^ / St. Petersburg. 2011. - 256 p.
Mathematical statistics for psychologists Ermolaev O.Yu [Text] / Moscow_2009 -336s
Lecture 7. Analytical statistics [Electronic resource].
Probability theory and mathematical statistics[Text] / Gmurman V.E. 2010 -479s
