MEASURE OF SLOPE ROTATABILITY FOR SECOND ORDER RESPONSE SURFACE DESIGNS UNDER INTRA-CLASS CORRELATED STRUCTURE OF ERRORS USING CENTRAL COMPOSITE DESIGNS

In the design of experiments for estimating the slope of the response surface, slope rotatability is a desirable property. In this paper, measure of slope rotatability for second order response surface designs using central composite designs under intra-class correlated structure of errors is suggested and illustrated with examples.


INTRODUCTION
Response surface methodology is a collection of mathematical and statistical techniques useful for analyzing problems where several independent variables influence a dependent variable. The independent variables are often called the input or explanatory variables and the dependent variable is often the response variable. An important step in development of response surface designs was the introduction of rotatable designs by Box and Hunter (1957). Das and Narasimham (1962) constructed rotatable designs using balanced incomplete block designs (BIBD). The study of rotatable designs mainly emphasized on the estimation of absolute response. Estimation of response at two different points in the factor space will often be of great importance. If differences at two points close together, estimation of local slope (rate of change) of the response is of interest. Hader and Park (1978) extended the notion of rotatability to cover the slope for the case of second order models. In view of slope rotatability of response surface methodology, a good estimation of derivatives of the response function is more important than Many authors have studied rotatable designs and slope rotatable designs assuming errors to be uncorrelated and homoscedastic. However, it is not uncommon to come across practical situations when the errors are correlated, violating the usual assumptions. Panda and Das (1994) introduced robust first order rotatable designs. Das (1997Das ( , 1999Das ( , 2003a introduced and studied robust second order rotatable designs. Das (2003b) introduced slope rotatability with correlated errors and gave conditions for the different variance-covariance error structures. Das and Park (2006) studied robust slope-rotatable designs over all directions. Das and Park (2007) Rajyalakshmi (2014) studied some contributions to second order response surface designs under different correlated structure of errors. Rajyalakshmi and Victorbabu (2014a, 14b, 15) studied SOSRD under intra-class structure of errors using CCD, symmetrical unequal block arrangements with two unequal block sizes and BIBD respectively.

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The second order surface model () ij are the parameters of the model and y  is the observed response at the th  design point.

Conditions for slope-rotatability for second order response surface designs with correlated errors (cf. Das (2003b, 2014), Das and Park (2009))
Following Das (2003bDas ( , 2014, Das and Park (2009), the necessary and sufficient conditions for slope-rotatability for second order model with correlated errors are as follows.
The estimated response at x i is given by For the second order model as in (2.2), we have The variance of estimated first order derivative with respect to each independent variable x i as in (2.4) will be a function of The following are the equivalent conditions of (1) to (5) in (2.5) for slope rotatability in second order correlated errors model (2.1)  Intra-class structure is the simplest variance-covariance structure which arises when errors of any two observations have the same correlation and each has the same variance. It is also known as uniform correlation structure. This can happen easily in a situation when all the observations studied are from the same batch or from the same run in a furnace.
Let  is the correlation between errors of any two observations, each having the same variance 2 .  Then intra-class variance covariance structure of errors given by the class: First observe that, Das (1997Das ( , 2003b and 2014))

Conditions of slope rotatability for second order response surface designs under intra-class correlated structure of errors (cf. Das (2003b, 2014))
Following (2.6), the necessary and sufficient conditions for the second order slope rotatability under the intra-class structure after some simplifications turn out to be I 0; for any odd and 4. 1 11 The parameters of the second order slope rotatable design parameters under intra-class structure are as follows 1 Note that (I), (II) and (III) as in (2.14) are second order slope rotatable conditions when errors are uncorrelated and homoscedastic.
Using (2.13), the expression The non-singularity condition (2.8) for the intra-class structure leads to where 3 .
c  = Using (2.13), the condition (5)* in (2.9) rises to Note: Values of SOSRD under intra-class correlated structure of errors using CCD can be obtained by solving the above equation.

Measure of second order slope rotatability for correlated structure of errors (cf. Das and Park (2009))
Following Das and Park (2009) is the proposed measure of slope rotatability for second order response surface designs for any general correlated error structure. Further, it is simplified to Note that 0 ( ) 1, Measure of slope rotatability of second order response surface designs under intra-class correlated structure of errors using CCD can be obtained by By substituting (2.13) and (4.1) in () ii Vb of (2.7) we get above G value.