Standard error of an effect

Using a variance (s ) from R replicate measurements at nominal level [31,39]. The standard error of an effect determined from unreplicated runs is given by equation (12). The number of degrees of freedom for tcriticalequal to R-1. 

The calculated r-value (Eq. 2.15), based on the effect of factor X, Ex, and on the standard error of an effect, (5E) is compared with a (tabulated) critical r-value, tcrmcai- The tcmcd depends on the number of degrees of freedom (d. ) associated with the estimation of SE) and is usually determined at a significance level a = 0.05. All effects with a r-value larger than or equal to t uicai are considered significant. 

Recalling that each value in this example is really an average of two independent observations, we can again apply Eq. (2.15) and write n = 7 /2, where cr is the variance of an individual observation. Using our estimate = 6.5 instead of we obtain an estimate, with 4 degrees of freedom, of the standard error of an effect in our experiment ... 

Another way of obtaining the standard error of an effect is by using Eq. (3.2). Since an effect is a contrast between two averages, that is,... 

The standard error of an effect is the square root of this value, approximately 1.14%. The standard error of the grand average is half this value, 0.57%, because the coefficients of the linear combination in this case are all equal to 1/8, instead of +1/4. Table 3.5 contains the values calculated for aU the effects and their standard errors. 

The square root of this value, s = 0.54, is our estimate of the standard error of an effect. 

In the usual way, we calculate from the duplicate responses a pooled variance of 1.75 x 10 , which corresponds to a standard error of 1.323 x 10 in the response. For this design, the standard error of an effect is half the standard error of the response. Multiplying this value by the t distribution point for 8 degrees of freedom, we find that the limits of the 95% confidence interval for the value of an effect are 1.525 x 10 . This means that only the main effects of factors 1 (time) and 3 (catalyst) and... 

The pooled variance calculated from the 16 duplicate runs is 0.9584. The variance of any effect will be one-eighth of this value, which is 0.1198. The square root of this last value is the standard error of an effect. Multiplying it by tie, we arrive at the limiting value for the statistical significance of the absolute value of an effect, 0.734 (95% confidence level). 

From the three experiments at the center point, we obtain an estimate of 0.40 for the standard error of the response, which in this case is the same as the standard error of an effect. Therefore, the limiting value for statistical significance of the absolute value of an effect (at the 95% confidence level) will be given by... 

G = —1.60, A = —2.11 and GA = 0.52. The standard error of an effect is 0.22, which means that the GA interaction is not significant at the 95% level. The main effects show that the curing time is lowered by 1.6 min when the finer granulation is used (150-200 mesh) and also decreases hy 2.11 min when the residual water increases by 7.5%. 

Since an effect is a difference of means (equation (2)) the standard error of the effect is calculated according to the equation for the standard error on a difference of means ... 

In the previous methods the standard error for an effect was estimated from the variance of the experiments using equation (12). In the methods described in this section, the variance for an effect is estimated with the help of calculated effects that are considered to be negligible. 

Table 3.17 presents the results from an analysis of the standard errors of each effect. Dividing the coefficient value by the standard error for each effect gives the t ratio. The critical t ratio is given by f i, = VF ritlP/ h DPR) ratio. Here we see that 022 exceeds while flu does not. This method is not foolproof since fly and 022 are mutually biased. However, for central composite designs the degree of bias is often slight. 

Figure 8. Effect of maitotoxin (MTX) on the time course of an increase in Ca uptake of cultured rat cardiac myocytes. Cbntrol ( ), and 10 g/mL MTX (o). Vertical lines indicate the standard error of mean (n=3). (Reproduced with permission from Ref. 20. Copyright 1987 Elsevier)...

Figure 4.10 Effect of reduced glutathione (GSH) (0-1.0 mM) on Na/K ATPase inhibition associated with the addition of oxidized glutathione (GSSG) (1 me). Experiments were performed using an isolated bovine ventricular Na/K ATPase preparation. Na/K ATPase activity was quantified by the ouabain-sensitive hydrolysis of ATP to yield inorganic phosphate. The data are presented as means standard errors of the means (n = 6).

Figure 5. Inhibitory effect of NO on Fe -induced lipid peroxidation. Shown is the decreased generation of an oxidative marker (thiobarbituric acid reactive substances, TBARS) as a result of 0.9 iM NO. HL-60 cells (5 x loVral) were placed in an O2 monitor and at the designated time points, butylated hydroxytoluene was added and samples were quick frozen for determination of TBARS. The values represent the mean and standard error of 3-5 independent determinations. Also shown for comparison is the residual concentration of O2 after exposure to the the same conditions. This shows a decrease in utilization of O2 in the presence of NO. We conclude that NO reduces TBARS, and the percent inhibition is similar to the poeent inhibition of O2 consumption. (Modified from our data in Kelley, E.E., Wagner, B.A., Buettner, G.R., and Bums, C.P., 1999, Arch. Biochem. Biophys. 370 97-104).

As I have shown, the response given by the model equation (3.5) has an error term that includes the lack of fit of the model and dispersion due to the measurement (repeatability). For the three-factor example discussed above, there are four estimates of each effect, and in general the number of estimates are equal to half the number of runs. The variance of these estimated effects gives some indication of how well the model and the measurement bear up when experiments are actually done, if this value can be compared with an expected variance due to measurement alone. There are two ways to estimate measurement repeatability. First, if there are repeated measurements, then the standard deviation of these replicates (s) is an estimate of the repeatability. For N/2 estimates of the factor effect, the standard deviation of the effect is... 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

Table 1 summarises the most important results from the investigation of metal doping. In this table the results of MAP treatment are combined with effects of firing temperature and doping. As can be seen in Table 1, y-alumina membranes with pore radii as low as 2.0 nm (Kelvin radius) may be obtained after firing at 600°C. Note that an instrumental standard error of 0.5 nm (90% reliability) is common in permporometry. This technique should therefore only be used for comparison purposes and to obtain a qualitative impression of the pore-size and pore-size distribution of the material under investigation. 

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

For PLS solution basis sets, bulk spectra were generated as described above. Standard error of calibration values (SECV) were determined from prediction residual sum of squares (PRESS) analyses of various permutations of the amide I, II, and III bands (always including amide I) from both Ge and ZnSe spectra. After determination of the effects of different types of normalization on the results, these bands were individually normalized to an area of 100 absorbance units before PLS 1 training. 

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