ANOVA table calculation

The error term can be approximated in different ways. A first possibility is that, analogous to the above, it is estimated from the multiple-factor interactions (two-, three-factor interactions, etc.) for (fractional) factorial designs [29]. In the example of Table 3.19 the sums of squares of the interactions AB, AC, BC and ABC are summed giving a MS error with 4 degrees of freedom. From this iSmain effects. The ANOVA table and equation (20) give of course the same results. 

From these simple calculations the ANOVA table can be constructed together with the HORRAT ratio. An important difference from the conventional ANOVA table is that it allows for the sample variation to be accounted for. The general ANOVA table and calculated values for S samples are shown in Tables 28 and 29 but the example uses only two. However, if more than two samples are desired/available the procedure can be extended. 

Calculate the sums of squares and mean squares indicated in the ANOVA table. 

Most statistical packages produce ANOVA tables if required, and it is not always necessary to determine these errors manually, although it is important to appreciate the principles behind such calculations. However, for simple examples a manual calculation is often quite and a good alternative to the interpretation of the output of complex statistical packages. 

To complete the hypothesis test, we compare the value of F calculated above with the critical value from the table at a significance level of a. We reject //o if F exceeds the critical value. It is common practice to summarize the results of ANOVA in an ANOVA table, as follows ... 

Construct the ANOVA table Having calculated the total sums of squares from all sources of variation, along with their degrees of freedom, we can now start to construct the ANOVA table. The only other calculations required are the mean squares for among-samples and within-samples (divide each sums of squares by its associated df) and the test statistic, F (divide among-samples mean square by within-samples mean square). All of this information is shown in the partial ANOVA table presented as Table 11.3. Determine if the test statistic is in the rejection region As always, we need to determine if the test statistic F falls in the rejection region. So far, we have not determined the... 

All observations, x, are assigned ranks, r.., and therefore the usual sums of squares can be calculated for the rank scores, q.. For brevity, the expressions for each are provided in Table 11.8, a general one-way ANOVA table, on the basis of ranks. 

Method of moments estimates (also known as ANOVA estimates) can be calculated directly from the raw data as long as the design is balanced. The reader is referred to Searle et al. (11) for a thorough but rather technical presentation of variance components analysis. The equations that follow show the ANOVA estimates for the validation example. First, a two-factor with interaction ANOVA table is computed (Table 7). Then the observed mean squares are equated to the expected mean squares and solved for the variance components (Table 8 and the equations that follow). 

Table 1.4 Commonly used table layout for the analysis of variance (ANOVA) and calculation of the F-value statistic...

The sums of squares are an important component of the ANOVA table that we shall construct. Another important component is known as the degrees of freedom (DF). The total degrees of freedom (DFT) is given by n - 1, because if we know y and n - 1 other of the y values, we can calculate the remaining y value. Likewise, there are m - 1 degrees of freedom for the model (DFM) because y is determined from m parameters (i.e., the model coefficients) and we must subtract 1 DF for y. Finally, there are n - m degrees of freedom in the residual (DFR) because the last term comprise n values of y and m values to determine y. We are now ready to build our ANOVA (Table 3.3). 

Even a casual examination of the data suggests that the calibration plot should be a curve, but it is instructive nonetheless to calculate the least-squares straight line through the points using the method described in Section 5.4. This line turns out to have the equation y = 2.991x -i-1.555. The ANOVA table for the data has the following form ... 

The calculation of the ANOVA table gives the following results ... 

These sums of squares are shown in the analysis of variance (ANOVA) table (Table 6.2). The mean squares are obtained by division of the sums of squares by the appropriate degrees of freedom. One degree of freedom is lost with each parameter calculated from a set of data so the total sum of squares has n — 1 degrees of freedom (where n is the number of data points) due to calculation of the mean. The residual sum of squares has —2 degrees of freedom due to calculation of the mean and the slope of the line. The explained sum of squares has one degree of freedom corresponding to the slope of the regression line. 

According to the calculation in the ANOVA table, Fobs,LoF = 0.886, which is smaller than F(vlof, vpe, a) = 2.1. Thus, the lack of fit is not significant and there is no reason to doubt this model. We will therefore examine whether or not the regression model is significant. 

This value should be compared with the calculated tots = bp /se bp), which are shown in the ANOVA table (Table 7.3). For both parameters Ubs > 2.02, thus both parameters in the model are significant. 

Describe the ANOVA table and how to calculate the different values in the table. 

Let s use a simple example to develop the rationale behind a one-way ANOVA calculation. The data in Table 14.7 show the results obtained by several analysts in determining the purity of a single pharmaceutical preparation of sulfanilamide. Each column in this table lists the results obtained by an individual analyst. For convenience, entries in the table are represented by the symbol where i identifies the analyst and j indicates the replicate number thus 3 5 is the fifth replicate for the third analyst (and is equal to 94.24%). The variability in the results shown in Table 14.7 arises from two sources indeterminate errors associated with the analytical procedure that are experienced equally by all analysts, and systematic or determinate errors introduced by the analysts. 

For the data given in Table 1.15, an ANOVA calculation yields the results shown in Table 1.16. 

When we subject this data to the same ANOVA calculations as the errorless data, we arrive at the following results (Table 10-8) ... 

All of the foregoing calculations are simply to trim the data set by removing outliers in accordance with the set criteria. All that remains is to calculate the mean value for the sample, the repeatability and reproducibility of the trialled method and the HORRAT ratio. These values can be calculated using a one way ANOVA method similar to that described in Table 29 and given in Table 33 and the final values in Table 34. 

Table 2.1 summarizes the single factor ANOVA calculations. The test for the equality of means is a one-tailed variance ratio test, where the groups MS is placed in the numerator so as to inquire whether it is significantly larger than the error MS ... 

This calculated F value is then compared to the critical value, Fa(1) VlV2, where Vj = regression DF = 1, and v2 = residual DF = n — 2. The residual mean square is often written as sfx, a representation denoting that it is the variance of Y after taking into account the dependence of Y on X. The square root of this quantity — that is, Syj—is called the standard error of estimate (occasionally termed the standard error of the regression). The ANOVA calculations are summarized in Table 2.2. 

TABLE 2.2. ANOVA Calculations of Simple Linear Regression... 

The table below shows the summary approach for the ANOVA calculations. 

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