Pooled variance estimates

The univariate response data on all standard biomarker data were analysed, ineluding analysis of variance for unbalaneed design, using Genstat v7.1 statistical software (VSN, 2003). In addition, a-priori pairwise t-tests were performed with the mean reference value, using the pooled variance estimate from the ANOVA. The real value data were not transformed. The average values for the KMBA and WOP biomarkers were not based on different flounder eaptured at the sites, but on replicate measurements of pooled liver tissue. The nominal response data of the immunohistochemical biomarkers (elassification of effects) were analysed by means of a Monte... 

The statistical model also requires that the random errors (xi,X2) be independently distributed, with zero mean and equal variance, at all level combinations of the factorial design. This is exactly what we assumed when we combined all observations to obtain a pooled variance estimate. If we wish to perform a or an F-test on our results, we must also assume that the errors are normally distributed (Section 2.6). 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

To cany out a Lagrange multiplier test of the hypothesis of equal variances, we require the separate and common variance estimators based on the restricted slope estimator. This, in turn, is the pooled least squares estimator. For the combined sample, we obtain... 

Since an analysis of the three-factor variance is with no replications, for an estimate of residual variance one may pool variances of the three last interactions, as they cannot be significant theoretically. 

Standard Two-Stage Method. Population characteristics of each parameter are estimated as the empirical mean (arithmetic or geometric) and variance of the individual estimates pharmacokinetic parameters derived from experimental pharmacokinetic studies. ° The advantage of the STS approach is its simplicity, but the validity of its results should not be overemphasized. It has been shown from simulation studies that the STS approach tends to overestimate parameter dispersion. ... 

As before, these sample statistics are estimates of the unknown population parameters, the population means, and the population variances. If the population variances are assumed to be equal, each sample statistic is a different estimate of the same population variance. It is then reasonable to average or "pool" these estimates to obtain the following ... 

To start, the point estimate for the between-group (active minus placebo) difference in mean change from baseline is (-0.86) - (-0.37) = -0.49. To calculate the standard error we first need to obtain an estimate of the pooled variance, which is calculated as follows ... 

If overhtting occurs, then the prediction ability will be much worse than the classihcation ability. To avoid it, it is very important that the sample size is adequate to the problem and to the technique. A general rule is that the number of objects should be more than hve times (at least, no less than three times) the number of parameters to be estimated. LDA works on a pooled variance-covariance matrix this means that the total number of objects should be at least hve times the number of variables. QDA computes a variance-covariance matrix for each category, which makes it a more powerful method than LDA, but this also means that each category should have a number of objects at least hve times higher than the number of variables. This is a good example of how the more complex, and therefore better methods, sometimes cannot be used in a safe way because their requirements do not correspond to the characterishcs of the data set. 

Let 7 be the variance of the pooled treatment estimate, /. The variance of the unbiased (separate) treatment estimate, tf, of Xf, being based on the fraction / of the total patients, is y /f. Hence the mean square errors are... 

Note that the smallest value in Table A.4 is 1.000. This means that in the F-test the numerator is always the larger of the two variances. According to a much used rule of thumb that dispenses with Table A.4, we can always pool variances to obtain a single variance estimate as long as the ratio of the larger and smaller variances does not exceed four. 

Each one of the runs was performed twice, and therefore furnishes a variance estimate with only 1 degree of freedom. To obtain a pooled estimate, with 4 degrees of freedom, we generahze Eq. (2.27) and calculate an average of all the estimates, weighted by their respective numbers of degrees of freedom. Including the variances calculated for the other three runs (8, 2 and 8, respectively), we have... 

The pooled variance of the duplicate runs is 44.91. The standard deviation of a response is the square root of this value, 6.70. The variance of an effect is half that of the pooled variance, 22.45, and therefore its standard error is 4.74. Since the estimate of the pooled variance has 4... 

The process of pooling or averaging the individual variance estimates (each with only a few degrees of freedom) is equivalent to a weighted average calculation. Thus the pooled variance is obtained from the sum of each individual variance estimate S (i) multiplied by the number of values in its replicate set a, divided by the sum of the individual number of allies in each replicate set. The number of degrees of freedom [Pg.24]

We now have two estimates of the individual population variance the first obtained from the variation in treatment means, and the second. S p the pooled variance obtained from the replicates. If there are no real effects of the treatments (all /), = 0). both of these are estimating the underlying variance of the population. An F-lest can be u.sed to decide if the two estimates are equal. Using the hypotheses as given above and using the technically justified assertion that if the variances are not really equal the between-treat-ments variance should be larger than the within-treatment variance, the F-ratio is defined as... 

The data for a water quality sample containing arsenic as tested at three laboratories are as shown in Table 4. Assuming that the three standard deviations are estimates of one and the same population standard deviation, it is quite proper to pool the variances, and take the square root of the pooled variance. Using this procedure, the best estimate of the within-laboratory standard deviation (S ) is obtained as... 

It is unclear, however, how many degrees of freedom are associated with f(a, v) since there are two sets of independent measurements. If the variances sa and sb estimate the same O, then the two standard deviations can be factored out of equation 4.19 and replaced by a pooled standard deviation. Spool, which provides a better estimate for the precision of the analysis. Thus, equation 4.19 becomes... 

A second way to work with the data in Table 14.7 is to treat the results for each analyst separately. Because the repeatability for any analyst is influenced by indeterminate errors, the variance, s, of the data in each column provides an estimate of O rand- A better estimate is obtained by pooling the individual variances. The result, which is called the within-sample variance (s ), is calculated by summing the squares of the differences between the replicates for each sample and that sample s mean, and dividing by the degrees of freedom. 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

One uses ANOVA when comparing differences between three or more means. For two samples, the one-way ANOVA is the equivalent of the two-sample (unpaired) t test. The basic assumptions are (a) within each sample, the values are independent and identically normally distributed (i. e., they have the same mean and variance) (b) samples are independent of each other (c) the different samples are all assumed to come from populations having the same variance, thereby allowing for a pooled estimate of the variance and (d) for a multiple comparisons test of the sample means to be meaningful, the populations are viewed as fixed, meaning that the populations in the experiment include all those of interest. 

Suppose that the data used for the CRF values were mean values from a duplicate experiment. Then it would be possible to obtain an estimate of the error by pooling the data. By taking the mean difference squared of the data pairs a run variance is obtained. A pooled estimate is calculated by summing all eight run variances and taking the mean value. This calculation is shown in Table 9. 

If it is assumed that the pooled run variance is a reasonable estimate for the residual variance. Table 7 can be reworked and the variance ratios (F values) calculated for each of the effects. The results of this rework are shown in Table 10. This approach confirms that the methanol effect is the largest by a very long way. The F value (1,8 df) is 5.32. Whilst this confirms that A is not significant. 

Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal variances, we refer %2 = 401n.58333 - 201n.847071 - 201n.320506 to the chi-squared table. (Under the null hypothesis, the pooled least squares estimator is maximum likelihood.) Thus, %2 = 4.5164, which is roughly equal to the LM statistic and leads once again to rejection of the null hypothesis. 

Finally, we allow for cross sectional correlation of the disturbances. Our initial estimate of b is the pooled least squares estimator, 2/3. The estimates of the two variances are. 84444 and. 32222 as before while the cross sectional covariance estimate is... 

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