Analysis of Variance

Analysis of variance (ANOVA) is a useful tool which helps the user to identify sources of variability from one or more potential sources, sometimes referred to as "treatments" or "factors." This method is widely used in industry to help identify the source of potential problems in the production process, and identify whether variation in measured output values is due to variability between various manufacturing processes, or within them. By varying the factors in a predetermined pattern and analyzing the output, one can use statistical techniques to make an accurate assessment as to the cause of variation in a manufacturing process.

This methodology represents a basic level of the field known as design of experiments (DOE), which can be a powerful tool for statistical analysis. However, DOE methodology is essentially a field unto itself, which by and large exceeds the scope of this text and the Weibull++ software, so we will concern ourselves with the ANOVA in its most basic form.

One-Way ANOVA

The one-way ANOVA is a method of analysis that requires multiple experiments or readings to be taken from a source that can take on two or more different inputs or settings. The one-way ANOVA performs a comparison of the means of a number of replications of experiments performed where a single input factor is varied at different settings or levels. The object of this comparison is to determine the proportion of the variability of the data that is due to the different treatment levels or factors as opposed to variability due to random error. The model deals with specific treatment levels and is involved with testing the null hypothesis $image\h0eqn.gif$ where $image\ui.gif$ represents the level mean. Basically, rejection of the null hypothesis indicates that variation in the output is due to variation between the treatment levels, and not due to random error. If the null hypothesis is rejected, there is a difference in the output of the different levels at a significance $image\alpha.gif$ , and it remains to be determined between which treatment levels the actual differences lie.

Inputs

Required inputs to perform a one-way ANOVA are the number of levels being compared, a, and the number of replications at each level, $image\ni2.gif$ . Typically, the user is asked to provide a value for the risk factor, $image\alpha.gif$ , which represents the Type 1 error probability the user is willing to live with.

Once this information has been obtained, the actual results of the experiments need to be entered. Ideally, an input grid based on the values of a and $image\nj.gif$ would facilitate the input of this information. The experiment result data will be denoted as,

$image\11_62_1.gif$

where,

i = 1, 2, … a is the number of levels being tested.

j = 1, 2, … $image\ni2.gif$ is the number of replicates at each level.

The resulting input grid would appear like the following:

$image\11_62_2.gif$

Note that the number of replications at each level need not be equal, or that $image\n1.gif$ need not be equal to $image\n2.gif$ , and so forth.

The total number of data points for a given data set will be calculated as,

$image\11_62_3.gif$

It should be noted that these N data points should be collected in a random order to insure that the effects of unknown "nuisance variables" will not affect the results of the experiment. A common example of this is a warm-up effect of production machinery, in which the output will vary as the machine accumulates more usage time from initial start-up.

Data Analysis

In ANOVA analysis, the output of each experiment, or observation, is thought to consist of variations of an overall mean value. These variations can have two sources: variation due to the factor or level and variation due to random error. The model used for the data in ANOVA analysis follows the form,

$image\11_63_1.gif$

where,

$image\u2.gif$ = the overall mean,

$image\taoi.gif$ = the level effect, and

$image\epsilonij.gif$ = the random error component.

Since the purpose of this analysis is to determine if there is a significant difference in the effects of the factors or levels, the null hypothesis can also be written as $image\h0eqn2.gif$ .

The sum of the responses over a level is denoted as,

$image\11_63_2.gif$

while the level mean is denoted as,

$image\11_63_3.gif$

The grand sum of all responses is denoted as,

$image\11_63_4.gif$

while the overall mean of the data is,

$image\11_63_5.gif$

The analysis is broken down into "sums of squares" that measure the variability due to the levels and due to the errors. The general form is,

$image\11_63_6.gif$

where,

$image\sst.gif$ = the total sum of squares,

$image\ssl.gif$ = the sum of squares due to the levels, and

$image\sse.gif$ = the sum of squares due to the errors.

The equation for the total sum of squares, which is a measure of the overall variability of the data, is,

(18)

$image\11_63_18.gif$

The equation for the sum of squares for the levels, which measures the variability due to the levels or factors, is,

(19)

$image\11_63_19.gif$

With $image\sst.gif$ and $image\ssl.gif$ known, $image\sse.gif$ can be calculated by subtracting $image\ssl.gif$ from $image\sst.gif$ , or $image\sse.gif$ = $image\sst.gif$ - $image\ssl.gif$ . The $image\sse.gif$ term measures the variability of the data due to random error.

There are degrees of freedom terms associated with each of the sums of squares. The degrees of freedom are given by,

$image\11_63_8.gif$

Mean square values are calculated by dividing the sum of square terms for the level and error by their respective degrees of freedom values. These values represent the variance of the level and error components of the data. Mean squares values for levels and errors are,

$image\11_63_9.gif$

Cochran's theory [25] states that the ratio,

$image\11_63_11.gif$

is F-distributed with degrees of freedom a - 1 and N - a. Therefore, if,

$image\11_63_12.gif$

where,

(20)

$image\11_63_20.gif$

we reject the null hypothesis and conclude that some of the variability of the data is due to differences in the treatment levels.

Output

The general format for output for this type of analysis is an ANOVA table, which contains basic information about the analysis:

$image\11_64_1.gif$

ANOVA Example

An insulation manufacturer is investigating variations in the thickness of one of its products. It is thought that the speed at which the manufacturing machinery is operated has an effect on the thickness of the insulation. An experiment was run where the machine was run at five different speeds, and a number of samples were taken at each speed. It is desired to know if the difference in thicknesses is due to the manufacturing speeds, at a 10% significance level. In order to avoid biasing the data, the samples were manufactured in a random order. The results of the experimentation appear in the following table:

$image\11_65_1.gif$

First, the total number of data points is calculated as,

$image\11_65_2.gif$

Further calculations can be aided by the construction of the following table:

$image\11_65_3.gif$

The total sum of squares is calculated by,

$image\11_65_4.gif$

The sum of squares for the factors or levels is calculated by,

$image\11_65_5.gif$

The sum of squares for the error is then calculated by,

$image\11_65_6.gif$

The mean square values are then calculated for the level and error components,

$image\11_65_7.gif$

Finally, the value of $image\f0.gif$ is calculated by,

$image\11_65_8.gif$

This value must now be compared to a value of $image\fcrit.gif$ . This value is calculated by,

$image\11_65_9.gif$

Since $image\f0.gif$ > $image\fcrit.gif$ , we must reject the null hypothesis and conclude that there is a difference in the level treatments at a significance of $image\alpha.gif$ = 10% (or more).

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