Meaning response

We begin by determining the confidence interval for the response at the center of the factorial design. The mean response is 0.335, with a standard deviation of 0.0094. The 90% confidence interval, therefore, is... 

Quantitation is performed by the single-point calibration technique. The calibration standard should be at a level similar to the expected residues and should be injected after every 3 samples throughout the sample batch. The mean response of the calibration standards which bracket the sample should be used for the residue calculation. 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

The covariance matrix COV(k ) is obtained by Equation 3.30. Let us now concentrate on the expected mean response of a particular response variable. The (l-a)100% confidence interval of yl0 (i=l.,w). the i,h element of the response vector y0 at x0 is given below... 

As we mentioned in Chapter 2, the user specified matrix Qj should be equal to the inverse of COV(e,). However, in many occasions we have very little information about the nature of the error in the measurements. In such cases, we have found it very useful to use Q, as a normalization matrix to make the measured responses of the same order of magnitude. If the measurements do not change substantially from data point to data point, we can use a constant Q. The simplest form of Q that we have found adequate is to use a diagonal matrix whose jth element in the diagonal is the inverse of the squared mean response of the j variable,... 

Having determined the uncertainty in the parameter estimates, we can proceed and obtain confidence intervals for the expected mean response. Let us first consider models described by a set of nonlinear algebraic equations, y=f(x,k). The 100(1 -a)% confidence interval of the expected mean response of the variable y at x0 is given by... 

Moerman, Daniel E., The Meaning Response Thinking About Placebos , Pain Practice 6, no. 4 (2006) 233-36... 

Ensure that the responses from samples are close to the mean response, y, of the calibration set. This will decrease the error contribution from the least-squares estimate of the regression line. 

Clinical phase I and II data reveal arzoxifene to be safe, well tolerated, and efficacious. Two multi-institutional phase II trials including 100 women with metastatic or recurrent endometrial cancer have demonstrated significant activity of arzoxifene at 20 mg/d in patients with metastatic or recurrent endometrial cancer. The observed clinical response rates were 25 and 31%, with a mean response duration of 19.3 and 13.9 months, respectively. Progression of the disease was stabilised in a substantial number of women. Toxicity was mild, except for two cases of pulmonary embolism that might have been drug related (Burke et al. 2003). 

The toxicological experiment is repeated for a number of different doses, and normal curves similar to Figure 2-3 are drawn. The standard deviation and mean response are determined from the data for each dose. 

A complete dose-response curve is produced by plotting the cumulative mean response at each dose. Error bars are drawn at cr around the mean. A typical result is shown in Figure 2-6. 

Fig. 8.2 Mean response frequency or duration by (a-c) female, F, and (d-f) male, M, L. catta to conspecific glandular secretions, (a) F sniffing all odorants as a function of her reproductive state (breed > non F 3 =28.57, P =0.013 ). (b) F licking labial odorant as a function of the donors reproductive state (breed > non t =3.00, P= 0.58, n.s.). (c) F frequency and site-specificity of scent marking as a function of odorant type Fs counter marked the unscented dowel in response to scrotal scent, but over-marked scented dowels in response to labial scent (ti = 3.87, P =0.030 ). (d) M response as a function of odorant type (antebrachial was sniffed least = 6.75, P = 0.011 brachial was wrist marked most Fs = 7.16, P = 0.009 ). (e) M...

For any given set of data, the specification of a mean value allows each individual response to be viewed as consisting of two components - a part that is described by the mean response (y,), and a residual part (r,) consisting of the difference or... 

The concept is illustrated in Figure 3.3 for data point seven of the example in Section 3.1. The total response (32.57) can be thought of as consisting of two parts - the contribution from the mean (32.53) and that part contributed by a deviation from the mean (0.04). Similarly, data point five of the same data set can be viewed as being made up of the mean response (32.53) and a deviation from the mean (-0.05), which add up to give the observed response (32.48). The residuals for the whole data set are shown in Figure 3.4. The set of residuals has n-1 degrees of freedom [Youden (1951)]. 

Replication is the independent performance of two or more experiments at the same set of levels of all controlled factors. Replication allows both the calculation of a mean response, y and the estimation of the purely experimental uncertainty, 5, at that set of factor levels. 

Note that b is used with the fixed level of j , = 6 to estimate the mean response for the replicate observations. 

The mean response can be subtracted from each of the individual responses to produce the so-called responses corrected for the mean. This terminology is unfortunate because it wrongly implies that the original data was somehow incorrect responses adjusted for the mean might be a better description, but we will use the traditional terminology here. It will be convenient to define a matrix of responses corrected for the mean, C. 

Some of the variation of the responses about their mean is caused by variation of the factors. The effect of the factors as they appear in the model can be measured by the differences between the predicted responses (y,) and the mean response (y,). For this purpose, it is convenient to define a matrix of factor contributions, F. 

Before discussing the sum of squares due to lack of fit and, later, the sum of squares due to purely experimental uncertainty, it is computationally useful to define a matrix of mean replicate responses, J, which is structured the same as the Y matrix, but contains mean values of response from replicates. For those experiments that were not replicated, the mean response is simply the single value of response. The J matrix is of the form... 

Thus, the parameter estimate bl is the estimated response at pH = 10.00. The overall mean response is... 

The matrix Y expressing the contribution of the overall mean response to each experiment is... 

Although the derivation is beyond the scope of this presentation, it can be shown that the estimated variance of predicting a single new value of response at a given point in factor space, is equal to the purely experimental uncertainty variance, plus the variance of estimating the mean response at that point, 5, 0 that is. 

For a given experimental design (such as that of Equation 11.15), the variance of predicting the mean response at a point in factor space is... 

Figure 11.8 Confidence bands (95% level) for predicting a single new value of response (outer band), a single mean of four new values of response (middle band), and a single estimate of the true mean response (inner band).

For a data array X containing responses of M variables over N different samples, this method involves the calculation of the mean response of each of the M variables over the N samples, and, for each of the M variables, subsequent subtraction of this mean from the original responses. Mean-centering can be represented by the following equation ... 

Another series of trials, all identical to each other (no changes). This time, the results should be tabulated, and a mean and a standard deviation for the blank and each standard should be calculated and the data graphed (mean response values vs. concentration) to create the standard curve (Figure 3.2). In addition, the slope of the line and the y-intercept are determined, as well as the correlation coefficient. If the results look good, one moves on to Step 5, or makes some change to try to improve the results and repeat the above process. 

---