DETERMINATION OF RESPONSE SURFACE FUNCTION USING REPEATED MEASURES

Transcription

1 DETERMINATION OF RESPONSE SURFACE FUNCTION USING REPEATED MEASURES Denise Moyse. Abbott Laboratories David Henry. Abbott Laboratories Hong-Jiong Shi. Abbott Laboratories Robert S. Boger. Abbott Laboratories Clement Maurath. Abbott Laboratories INTRODUCTION AND OBJECTIVE In the clinical trial setting, there are frequently studies which measure effect of study drug over a range of dose levels. In addition, sequential measurements are obtained on subjects prior to, and at multiple times following study drug administration at each of these dose levels. The measurement times may often be unequally spaced. A threedimensional response surface provides a graphical presentation of the results by simultaneously presenting the relationship among drug effect, measurement time, and dose level. The objective of this paper is to show how a threedimensional response surface function was determined using repeated measures. This function was then used to graphically present results from a clinical trial. FULL RESPONSE SURFACE FUNCTION Where E(y) is the expected change from baseline; D is the patient's dose level; T is the time of measurement relative to dosing; b o is the intercept estimate; b 1 and b 2 are the linear effect estimates; b 11 and b 22 are the curvature effect estimates; b 12 is the interaction effect estimate. DETERMINATION OF APPROPRIATE RESPONSE SURFACE FUNCTION Since measurements were made at only four postdosing evaluation times. it was not considered appropriate to use the full quadratic response surface function unless convincing evidence of either nonlinearity or interaction existed. The first step to determine an appropriate response surface function was to look for evidence of interaction by examining significance of the interaction of time with the quadratic dose term, and time with the linear dose term. Although the interaction of time with quadratic dose was not included in the full response surface function, the test for this interaction provided additional confidence in detennining the function. Next, evidence of quadratic effects in time and dose were examined if there was no evidence of interaction. When a covariate is present in a repeated measures model, the appropriate test for quadratic effects in time is obtained at the overall dose mean; however, the SAS* procedure performs the test at the intercept. In order to test for a Quadratic trend in time, an orthogonal contrast was constructed using the coefficients provided in the SAS repeated measures output, and was then examined in an analysis of variance. The final, appropriate response surface function included, at a minimum, linear terms for both time (T) and dose (D). The SAS code used to perform the analyses is shown later, in the example. A flow chart summarizing the algorithm used to determine the appropriate response surface function is shown in Figure 1. EXAMPLE A clinical trial was conducted to examine the effect of study drug on plasma norepinephrine levels. Study drug was administered at several ascending dose levels and measurements were obtained at unequally spaced times following study drug administration. The objective was to find the appropriate response surface function which would describe the change from baseline observed in norepinephrine levels across the ascending dose levels and postdosing evaluations times. N=15 subjects. Study drug dose levels (number of subjects receiving the dose): (n=l); (n=l); (n=l); (n=l); (n=2); (n=l); (n=l); (n=3); (n=2); (n=2). A logarithmic transformation of the dose (loglo(do5e)) was used in 1he analyses. Measurements were made on each subject prior to dosing (baseline), and at four, unequally spaced times following dosing. Time of measurement was recoded so that T took on the following values: Time of Measurement Relative to Dosing +20mins +45 mins +90 mins mins T (per 20 mins) Change from baseline to each of the postdosing evaluation times was calculated for each subject, and these changes from baseline data were used in the analyses

2 The following SAS code was used in the analyses to determine the appropriate response surface: DATA A; SET A; LOGDOSE=LOG10(DOSE); determine 10910(dose level); LOGDOSEQ=LOGDOSE 2; quadratic 1091O(dose level); TITLE 'DETERMINATION OF APPROPRIATE RESPONSE SURFACE FUNCTION'; TITLE2 ''; PROC GLM DATA=A; MODEL Y20 Y45 Y90 Y180 = LOGDOSE LOGDOSEQ I NOUNI; REPEATED TIME 4 ( ) POLYNOMIAL I NOU SUMMARY SHORT PRINTE; TITLE3 'INCLUDING LINEAR AND QUADRATIC TERMS FOR LOGDOSE'; Step 2 : Test for Significance of quadratic dose effect (0 2 ). Result: p = ; not significant. Drop 0 2 from response surface function. Step 3 : Test for significance of quadratic time effect (T2) using an orthogonal contrast. Result: p = ; not significant. Drop T2 from response surface function. Step 4 : Final response surface function: E(y) = b o + b 1 T + b 2 D Figure 2 shows this response surface (shaded plane) and the actual data points (open and solid circles) used in the analyses. SUMMARY PROC GLM DATA=A; MODEL Y20 Y45 Y90 Y180 = LOGOOSEI NOUNI; REPEATED TIME 4 ( ) POLYNOMIAL I NOU SUMMARY SHORT PRINTE PRINTM; TITLE3 'INCLUDING ONLY LINEAR TERM FOR LOGDOSE'; This paper has described an approach to determine an appropriate response surface function using repeated measures in SAS. This was developed to summarize results from a clinical trial so that the relationship among drug effect, evaluation time, and dose Jevel could be simultaneously presented using a three-dimensional response surface. DATA QTIME; SET A; QUA=0.539rY Y rY90 + O.3509*Y180;* quadratic orthogonal contrast. Coefficients obtained from repeated measures model including linear term for logdose;.. the overall average log 1 o(dose level) was ; PROC GLM DATA=QTIME; MODEL QUA = LOGDOSE; ESTIMATE 'EFFECT AT AVG DOSE' INTERCEPT 1 LOGDOSE ; TITLE3 'QUADRATIC TREND IN TIME USING AN ORTHOGONAL CONTRAST'; OUTPUT FROM SAS REPEATED MEASURES ANALYSES AND ANALYSIS OF VARIANCE Selected pages from the SAS output of the repeated measures analyses and analysis of variance are shown in Appendix 1. The entire output was too long to include here, but the appropriate test statistic results have been highlighted to show how the appropriate response surface was determined. Bold face type was added to the output to key the steps: ACKNOWLEDGEMENTS The authors wish to thank Mr. David Jaskela for his assistance. SAS is a registered trademark or trademark of SAS Institute Inc. the USA and other countries. Author contact: Denise Moyse Abbott Laboratories, 0436 AP9Ai2 One Abbott Park Road Abbott Park, IL (708) Step 18 : Test for significance of 0 2 * T interaction. Result: p = ; not significant. Step 1b : Test for significance of 0 * T interaction. Result: p = 0.745; not significant. Drop interaction term from response surface function. 1215

3 Figure 1 Flow Chart Summarizing Analyses Used to Determine an Appropriate Response Surface Function Figure 2 Norepinephrine (pg/ml) d Response Surface " O L 200 I\) (j),... T Q) C.Qi CO E 101) 0 U- Q) Ol 50 c III s::. 0 ;i",. ". ' 'n.) R.I. "'a 10 InfuSion Start

4 Appendix 1 DETERMINATION OF APPROPRIATE RESPONSE SURFACE FUNCTION INCLUDING LINEAR AND QUADRATIC TERMS FOR LOG DOSE General Linear Models Procedure Number of observations In data set = 15 General linear Models Procedure Repeated Measures Analysis of Variance Repeated Measures Level Information Dependent Variable Y20 Y45 Y90 Y180 Level of TIME S=1 M=O.5 N-4 Statistic Value F Num DF Den DF Tests of Hypotheses for Between Subjects Effects Source DF Type III SS hii,g; :: ::i':i:'l:l:;:;i!! m.t:j.;'!iiiii$q;"",:! iif: :::!I:I.I.i!:i!ii!!lj.!!;! :!$i"f Error DETERMINATION OF APPROPRIATE RESPONSE SURFACE FUNCTION INCLUDING ONLY LINEAR TERM FOR LOGDOSE General linear Models Procedure Number of observations In data set = 15 General linear Models Procedure Repeated Measures Analysis of Variance Repeated Measures Level Information Dependent Variable Y20 Y45 Y90 Y180 Level of TIME

5 DETERMINATION OF APPROPRIATE RESPONSE SURFACE FUNCTION INCLUDING ONLY LINEAR TERM FOR LOGDOSE (Cont.) TIME.N represents the nth degree polynomial contrast fortime M1MI \#l!l'j 1ilt.f911,:*il!1 1: Y20 Y45 Y90 Y180 Coefficients for orthogonal contrast to assess quadratic time effect Manova Test CrHerla and Exact F Statistics for thehpti.!;i,jjrnjm.j'!lml:effljil H = Type III SS&CP Matrix for TIME*LOGDOSE E = Error S,,&CP Matrix S=l M=O.5 N=4.5 Statistic Value F Num DF Den DF Analysis of Variance of Contrast Variables TIME.N represents the nth degree polynomial contrast for TIME Contrast Variable: TIME.l Source DF Type III SS MEAN LOGDOSE Error I Contrast Variable: TIME.21 Source DF Type III SS F Value IMEAN LOG DOSE Error Contrast Variable: TIME.3 Source DF Type III SS MEAN LOG DOSE Error

6 DETERMINATION OF APPROPRIATE RESPONSE SURFACE FUNCTION QUADRATIC TREND IN TIME USING AN ORTHOGONAL CONTRAST General Linear Models Procedure Number of observations in data set = 15 General Linear Models Procedure Dependent Variable: QUA Source DF Sum of Squares Model Error Corrected Total R-Square C.V. RootMSE QUA Mean Source DF Type I SS F Value LOG DOSE Source DF Type III SS LOG DOSE Parameter T for HO: Parameter=O Pr> ITI Std Error of Parameter T for HO: Parameter=O Pr> ITI Std Error of INTERCEPT