Slope, confidence interval

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

Example 14 For the best-fit line found in Example 13, express the result in terms of confidence intervals for the slope and intercept. We will choose 95% for the confidence interval. 

These standard deviations can be used to establish confidence intervals for the true slope and the true y-intercept... 

Calculate the 95% confidence intervals for the slope and y-intercept determined in Example 5.10. 

Fig. 13. The standard addition method where MB is the confidence interval for the slope of the line = k, and represents 95% confidence interval (14).

The test for the significance of a slope b is formally the same as a t-test (Section 1.5.2) if the confidence interval CI( ) includes zero, b cannot significantly differ from zero, thus ( = 0. If a horizontal line can be fitted between the plotted CL, the same interpretation applies, cf. Figures 2.6a-c. Note that si, corresponds to fx ean). that is, the standard deviation of a mean. In the above example the confidence interval clearly does not include zero this remains so even if a higher confidence level with t(f = 3, p = 0.001) = 12.92 is used. 

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Conclusions the residual standard deviation is somewhat improved by the weighting scheme note that the coefficient of determination gives no clue as to the improvements discussed in the following. In this specific case, weighting improves the relative confidence interval associated with the slope b. However, because the smallest absolute standard deviations. v(v) are found near the origin, the center of mass Xmean/ymean moves toward the origin and the estimated limits of detection resp. quantitation, LOD resp. 

In principle, there is also a test for the intercept. But since the expected value for the intercept depends on the slope, it gets a bit hairy. It also makes the confidence interval so large that the test is nigh on useless - few statisticians recommend it. 

Only 4% of the fracture reduction is explained by the changes in bone density for the treated group. Moreover, the curves for the placebo- and raloxifene-treated cases are quite different (Fig. 8.2) (Lufkin et al. 2001, with different slopes and no overlap between them (including the 95% confidence intervals). In other words, the bone intrinsic properties, well beyond the changes observed in BMD, account for the vast majority (96%) of the antifracture efficacy of the drug. Similar variations in bone density are accompanied by very different risks of fracture during the 3-year observation period in the two groups. 

This model was fitted to the data of all three temperature levels, 375, 400, and 425°C, simultaneously using nonlinear least squares. The parameters were required to be exponentially dependent upon temperature. Part of the results of this analysis (K6) are reported in Fig. 6. Note the positive temperature coefficient of this nitric oxide adsorption constant, indicating an endothermic adsorption. Such behavior appears physically unrealistic if NO is not dissociated and if the confidence interval on this slope is relatively small. Ayen and Peters rejected this model also. 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

This link applies also to the p-value from the unpaired t-test and the confidence interval for p, the mean difference between the treatments, and in addition extends to adjusted analyses including ANOVA and ANCOVA and similarly for regression. For example, if the test for the slope b of the regression line gives a significant p-value (at the 5 per cent level) then the 95 per cent confidence interval for the slope will not contain zero and vice versa. 

A 95% confidence interval on slope or intercept is obtained by multiplying these standard errors by a t value at the desired probability and n — 2 (n - 1) degrees of freedom ... 

What does optimization mean in an analytical chemical laboratory The analyst can optimize responses such as the result of analysis of a standard against its certified value, precision, detection limit, throughput of the analysis, consumption of reagents, time spent by personnel, and overall cost. The factors that influence these potential responses are not always easy to define, and all these factors might not be amenable to the statistical methods described here. However, for precision, the sensitivity of the calibration relation, for example (slope of the calibration curve), would be an obvious candidate, as would the number of replicate measurements needed to achieve a target confidence interval. More examples of factors that have been optimized are given later in this chapter. 

It is usual to test the slope of the regression line to ensure that it is significant using the F ratio, but for most analytical purposes this is an academic exercise. Of more importance is whether the intercept is statistically indistinguishable from zero. In this instance, the 95% confidence interval for the intercept is from — 0.0263 to +0.0266 which indicates that this is the case. 

The in vitro release comparison should be carried out as a two-stage test. At the first stage, two mns of the (six cells) in vitro apparatus should be carried out, yielding six slopes (estimated in vitro release rates) for the prechange lot (R) and six slopes for the postchange lot (T). A 90% confidence interval (to be described below) for the ratio of the median in vitro release rate (in the population) for the postchange lot over the median in vitro release rate (in the population) for the... 

The first step in the computation of the confidence interval is to form the 36 (=6X6) individual T/R ratios. This is illustrated in the following table, where the prechange lot slopes (R) are listed across the top of the table, the postchange lot slopes (T) are listed down the left margin of the table, and the individual T/R ratios are the entries in the body of the table ... 

In the third step, the 8th saA29th ordered individual ratios are the lower and upper limits, respectively, of the 90% confidence interval for the ratio of the median in vitro release rate (slope) for T over the median in vitro release rate for R. In the example, this confidence interval is 1.0343 to 1.2863, or in percentage terms, 103.43% to 128.63%. 

The statistical test described above is based on a standard confidence interval procedure related to the Wilcoxon Rank Sum/Mann-Whimey rank test, apphed to the log slopes. References to this confidence interval procedure include ... 

Then, in the same way that one talks about a certain (percentage) confidence interval for a given series of measurements (see pp 54-57), one can also talk about the (percentage) confidence intervals for the slope and y intercept of a best-fit line. They are related to the standard error of estimate jv/x and, for the desired level of confidence, the t value that corresponds to one less than the number of data pairs, as follows. For the slope,... 

Find the 95% confidence interval of the slope and intercept of the best-fit equation obtained in the problem on p 75 involving thermocouple voltage versus temperature. 

Note Problems 3 through 7 and 10 through 14 are, m part, given at two levels of sophistication and expectation The best" answers involve doing the part marked with an asterisk ( ) instead of the immediately preceding part The parts with an involve the use of the method of least squares, correlation coefficients, and confidence intervals for the slopes and intercepts, the preceding parts do not... 

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