Pooled sample variance

In eq. (33.3) and (33.4) x, and Xj are the sample mean vectors, that describe the location of the centroids in m-dimensional space and S is the pooled sample variance-covariance matrix of the training sets of the two classes. 

If the two variances are unknown but presumed equal, we calculate a pooled sample variance ... 

In the pooled estimate, we calculate the sample variance for each group (each column of data in a one-way analysis). Then we weigh each of these estimates by its degrees of freedom to obtain a pooled sample variance. For any column i, the sample variance is ... 

Further, if m i = m2, the pooled sample variance becomes the average of the two sample variances and the degrees of freedom would be Sp = /2(s + s ) = 33.1, df = mi+M2-2 = 18. In this example, we assume that the population variances are unequal and thus there are 14 degrees of freedom. The r-statistic for a two-tailed confidence interval of 95% with 14 degrees of freedom equals 2.145 and the inequality for the null hypothesis is expressed as ... 

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... 

Comparing two variances. To compare the averages of two independent samples, we combine the two sample variances to form a single pooled estimate. Since the pooled value has more degrees of freedom, the confidence interval becomes narrower and the test more sensitive, that is, it becomes capable of detecting smaller systematic differences. Obviously it only makes sense to combine sample variances if they are estimates of the same population variance. To justify calculating a pooled estimate, we need to test the nuU hypothesis that and s are estimates of identical population variances, = cr - This can be done by performing an F-test, which is based on Eq. (2.23). If the two population variances are equal, Eq. (2.23) becomes... 

Now we only need to compare the ratio of the two sample variances with the tabulated value for the F distribution with the appropriate numbers of degrees of freedom. Using Table A.4 we see that F4 4 = 6.39, at the 95% confidence level. To reject the null hypothesis, the variance ratio will have to be larger than this value. This would also imply that the pooled estimate should not be calculated. Since in our example... 

Substituting the appropriate values and using as the standard deviation the value obtained by pooling the variances of the two samples, we have... 

For example, the gene expression values are 12.79,12.53, and 12.46 for the naive condition and 11.12, 10.77, and 11.38 for the 48-h activated condition from the T-cell immune response data. The sample sizes are nj = 2 = 3. The sample means are 12.60 and 11.09 and the sample variances are 0.0299 and 0.0937, resulting in a pooled variance of (0.2029). The i-statistic is (12.60 - 11.09)/0.2029 = 7.44 and the degree of freedom is ni -I- 2 2 = 3 -I- 3 — 2 = 4. Then we ean find a p-value of 0.003. If using Welch s t-test, the t-statistic is still 7.42 sinee i = n, but we find the p-value of 0.0055 since the degree of freedom is 3.364 rather than 4. We claim that the probe set is differentially expressed under the two eonditions because its p-value is less than a predetermined significance level (e.g., 0.05). In this manner, p-values for the other probe sets ean be calculated and interpreted. In Section 4.4, the overall interpretation for p-values of all of the probe sets is described with adjustments for multiple testing. The Student s t-test and Weleh s t-test are used for samples drawn independently from two eonditions. When samples from the two conditions are paired, a different version ealled the paired t-test is more appropriate than independent t-tests ... 

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... 

The PCA on valve measurements is presented in Fig. 7.6. The loadings on PCI (90% of the variance) are all large and positive, indicating that most shape variation among these specimens is related to size. The second component, PC2 (9%) contrasts an increase in thickness and a decrease in width, relative to valve length (consistent with Peck et al., 1987). Plots of PC2 vs PCI from the pooled sample indicate separation between L. neozelanica and L. uva, but also a broad region of overlap in values between the two, which both vary widely. 

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. 

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... 

Note that the FGLS estimator of the slope is closer to the 1.46885 of sample 3 (the highest of the three OLS estimates). This is to be expected since the third group has the smallest residual variance. The LM test statistic is based on the pooled regression,... 

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