Using PROC MIXED on Animal Growth Curves (Graham F.Healey, Huntingdon Research Centre, UK)

Transcription

1 Using PROC MIXED on Animal Growth Curves (Graham F.Healey, Huntingdon Research Centre, UK) The Motivation Consider the problem of analysing growth curve data from a long-term study in rats. Group mean bodyweight curves look well-behaved (Figure 1), but there may be considerable variation between animals (Figure 2). Often an analysis is performed separately at each timepoint. However, because each timepoint has the same animals, the high correlation between adjacent timepoints can lead to a misleading impression of the significance by the perpetuation of chance significance occurring locally. A comparison of groups over time might therefore be more useful. Also of interest may be the investigation of sources of variability. This paper describes some simple approaches to such a problem using procedures PROC GLM, PROC NUN and PROC MIXED in SAS/STA T~ The intention is to explore possibilities from a practical viewpoint, and statistical theory will be of only secondary interest. The Data Bodyweight was measured every week over two years. There were control and three dose groups initially with 40 rats or "subjects" per group. To reduce problems of computing time and space, only the first 10 animals from each group and weeks 0,4,8,12,20 and 28 timepoints were used. All animals were measured at the same timepoints. Exponential growth curve model Typically, growth follows a damped exponential curve, rising steeply at first and then approaching a plateau. Many functions have been described in the literature. The following differential equation generalises most 9f the simpler ones: When ex == (1-11 d) and B= -c, the general form of the above differential equation yields on integration (for animal i, time j, with an intercept a): Yij = a + b. (1 - exp(-c. tij))d Modelling over groups Because all animals were measured at the same timepoints, a variety of approaches (labelled A1-A7 for reference) were possible. Some simple methods not using mixed models will be described first. Strictly formal, statistical criteria for assessing and comp:u-ing models should rarely be used. Some useful, more informal, criteria used in this study were 1) Plots of residuals versus time, 2) Analysis of the absolute value of residuals and 3) Changes in sums of squares with successive models. Carrying out model development requires the ability to output to file. This can be achieved for datasets (predicted values, residuals, etc) and results (parameter estimates, etc) in PROC GLM (OUTPUT and OUTES1), PROC NLIN (OUTPUT and OUTES1) and PROC MIXED (MAKE). 770

2 Results of analyses not using PROC MIXED At. Split-plot analysis PROC GLM was used. There are two levels of variation: the between-animal (within-group) and the within-animal (around the time curve). The TEST statement is used for the Group effect: TABLE 1: Analysis of variance output Source df ss ms Group F=14.42 *** Animal(Group) E(ms) = a.z+6a. z Week Group*Week F=12.45 *** Error E(ms) = a Z Total The estimates of the variance components were: The split-plot approach is not strictly applicable here, since the "sub-plots", ie the timepoints, are not applied randomly. The nonlinearity is not addressed directly and the variance is not modelled adequately. The significant Group-by-Week interaction could be modelled using a multivariate decomposition. The curves may still be parallel in some nonlinear sense.. A2: Subject fit The above exponential curve was fitted to each of the 40 animals separately using PROC NUN (with BWT/lOO and T/lOO). The parameter estimates were averaged over groups. Because of the correlations between parameters, and the resulting irregular sums of squares surface, the distribution of the parameters over groups will not usually be normal. Non-parametric summary in the form of medians, standard deviations estimated as the inter-quartile range over 1.35, and comparisons across groups using Kruskal-Wallis tests in PROC NPARlWAY (for B: X2(3)-l5.63), was therefore used. The parameter medians were then put into a further PROC NUN run to estimate a between-animal variance. TABLE 2: Median parameter estlmates Group n a b c d significance p>0.5 p= p>0.5 p>0.2 If more than one parameter seems to be varying over groups, plots of pairs of parameters can be used to facilitate examination of the correlations. 771 _' -J:--:_-_-.-~p-' ~ <,. -,."",", r _ ~ :, ~ -~'-.: '~ :..

3 The pooled variances of the parameter estimates were: var(a) = 0.013, var(b) = 1.154, var(c) = 4.080, var(d) = 0.012, o}=0.0123, U/+Uk2= The estimate of the within-subject mean square was obtained by pooling the 40 individual estimates (2 df each). This analysis assumes that each subject has sufficient measurements for the curve to be estimable. Also, it is not clear how to obtain the best average group estimate from the individual estimates. Analysis with a,c and d fixed (1.7, 5.7 and 0.68) gave: var(b) = 0.385, u/=0.0298, u/+uk2= A3. Pooled within groups fit. A single nonlinear fit was applied to pooled individual data, ignoring subjects, using PROC NUN (Figure 3 shows fit for Group 1). TABLE 3: Parameter estimates Group a b c d I pooled se I 0.13 I 0.73 I 2.70 I 0.14 I The standard errors produced by PROC NUN are only first-order. Generally, the more correlated are the parameter estimates, the more unreliable are such standard errors. Successive fitting showed that only parameter B needed to be varied across groups. The other three could be held constant or "common" (ie still estimated, as opposed to "fixed" a prion). Such modelling is achieved easily in PROC NUN by replacing a common parameter, say a, by a set of group-dependent parameters: a = 11 *al + I2*a2 + I3*a3 + I4*a4 The constants, Ii, are indicator variables taking the value 1 when the group = i, and 0 otherwise. TABLE 4: Breakdown of successive model fitting parameters common df error ss ms 4 a,b,c,d a,c,d none ComparIson of group means (reduction In 88): F(3,233) = (40.28/3)/ = TABLE 5' Parameter estimates (common a, cd)', a b1 b2 b3 b4 c d

4 The pooled variances of the parameter estimates were: None common: var(a) = 0.018, var(b) = 0.540, var(c) = 7.303, var(d) = 0.020, U/+Uk2= Common a,c,d: var(a) = 0.004, var(b) = 0.135, var(c) = 1.597, var(d) = 0.005, u e 2 +Uk2= A4: Group mean fit Taking advantage of the balance, group means were calculated at each timepoint. The exponential curves were then fitted once to each group using PROC NUN. There is no theory to say that the functional form of the mean curves will be the same as that for the individuals, although in the present case this appears to be a reasonable assumption. The parameter estimates are in fact identical to those obtained with the previous fit (see analysis A3 above). This analysis of means requires the same timepoints for each subject, and does not model the within-group variation. However, it is simple to apply and provides a reasonable test of groups (reduction in SS): F(3,17) = (4.028/3)/ = Mixed modelling TABLE 6: Breakdown of successive model fitting parameters common df error ss ms 4 a,b,c,d a,c,d none Two important features of mixed modelling are :- 1) Flexibility: the ability to model quite complex covariance structures. Briefly, for both fixed effects and mixed effects modelling, the fixed effects are estimated with the weighted regression equation (where the inverses may be generalized) : ~ = (X'V-IX) -IX'V-Iy The random effects are then estimated by: The variance matrix, V, is given by: V = GZ'V- I (y-x~) v = ZGZ' + R where Z is the between-subject, within-group design matrix, G is the matrix of covariance estimates and R is the within-subject error matrix. The separation into between- and within- components is more clear and allows the modelling of the R matrix free of subject effects. In particular, this separation can be achieved for unbalanced designs. 773

5 2) Parsimony: the ability to replace estimates for each subject by single covariance terms. If there are too few timepoints for each subject, then a separate curve cannot be fitted to each subject. Mixed modelling allows the variance of the between- and within-subject parameters to be estimated without necessarily performing the separate fits. This is particularly useful in areas such as population pharmacokinetics where the sampling may be sparse, or indeed any situation where the design is unbalanced. Nonlinear mixed modelling Useful work on the fitting of nonlinear mixed models is now appearing in the literature (eg Vonesh and Carter 1992, Lindstrom and Bates 1990). One form of nonlinear mixed modelling involves minimizing the weighted regression objective function: y = (y-f(x,li)) 'V- 1 (y-f(x,id) Most nonlinear fitting programs, including PROC NUN, allow the inclusion of weights in the form of a diagonal covariance matrix V. For mixed modelling, the matrix V is not in general diagonal, so special methods must be used. The problem is usually simplified in some way: For example: Ykij = fi (2fJ' 13 k, ~ij' V i) ~ Y = f (X{3 + Zv) Most of the methods described in the literature (see Vonesh and Carter) require special nonlinear fitting routines with various levels of approximation and iteration. They are certainly not yet available in any commercial software. The question then is "Can something reasonably simple be carried out in SAS, without any external programming?".. Modelling the within-subject variability Exploratory analyses showed that the residual variance is likely to be a function of both y and t. Both PROC GLM and PROC NUN do allow weights, although in this study it was decided not to include weights in PROC NUN fits in order to facilitate the comparison between different models. With PROC MIXED, weighting based on within-animal variability can be incorporated by modelling the R matrix, although it may require patience. With bodyweight data, there is usually an "autocorrelation" of as much as 0.5 between the residuals for adjacent days after fitting a bodyweight curve. However, the timepoints in the data considered here were up to 30 days apart and any autocorrelation had been lost. Applying PROCMIXED with an AR(l) term produced an estimate for the correlation not significantly different from zero. Results of analyses using PROC MIXED AS: Random effects model The above split-plot analysis (A1), was repeated using PROC MIXED. One way to obtain exactly the same fixed effects tests and variance estimates is to repeat the GROUP effect in the RANDOM statement. This is equivalent to an ANIMAL(GROUP) term: 774

6 MODEL Y = GROUP WEEK GROUP*WEEK/NOINT; RANDOM GROUP/SUBjECT=ANIMAL; Here, the balanced nature of the data allowed the use of a "SUBJECT =" term in PROC MIXED. This creates block diagonal covariance matrices and much quicker fitting. An attempt was then made to model the increase in variance across time: MODEL Y = GROUP WEEK GROUP*WEEK/NOINT; REPEATED WEEK/SUBjECT=ANIMAL TYPE=UN{l); This led to variance estimates for each week: u/= (0.01, 0.08, 0.11, 0.15, 0.25, 0.40) The mean of these estimates, , shows that the combined between-animal and within-animal components (see analysis Al) have been modelled. The within-group divergence of the data is clearly shown. Attempts to model the within-animal variance itself (using both RANDOM and REPEATED terms) have so far failed because of excessive computing time. The test of Groups was F{3,216) = A6: Residual mixed modelling Vonesh and Carter describe a four stage process: 1. Unweighted least squares nonlinear regression to each group. 2. Mixed modelling of the residuals (y-xb). 3. Estimation of the covariance matrix V. 4. Weighted (V-l) nonlinear regression. Stage 1: The exponential function was fitted to individual data within each group, ignoring the animal term, exactly as in analysis A3 above, with no parameters held common: Stages 2 and 3: The fitting of parameters varying over groups removes most of the group-to-group variation in the data. A check of the residuals confirmed that the variability increased with time, due mainly to the fact that the between-animal variation had not yet been removed. The functional form of the residuals is unknown, but employing a first-order approximation to the exponential function allows estimation of the covariance parameters: y = a.der a + b.der b + (c-c).der c + (d-d).der d In PROC MIXED, the residuals can be modelled as follows: MODEL YRES=/NOINT; RANDOM DERA DERB DERC DERDISUBjECT=ANIMAL; Covariance estimates from Maximum Likelihood and REML fitting were identical, differing from those for the MIVQUEO method: ML/REML: MIVQUEO: u.2 = 0., Ub2 = , u c 2 = 0, ui = , u/ == u.2 = , Ub2 = , u c 2 = 0, ui = 0, u e 2 =

7 Stage 4: With four covariance parameters, the matrix V is not diagonal. 'It is currently difficult to extract it from SAS without programming. Transformation of the problem may be possible (Vonesh and Carter), possibly using a decomposition such as W = (ZGZ' +R)-l = HDH'. Here, with the block diagonal set-up, and the low values for some of the estimates, the matrix may be close to diagonal. Considering it to be diagonal makes the re-fitting of the nonlinear curve easy. Weights were formed from the diagonal terms of V: p=4 v ij = L Op2. dex p lj + o2e p=l PROC NUN was re-executed with these weights; The estimates for b and c were slightly modified (compare with Table 3): TABLE 7: Parameter estimates (none common) Group a b c d pooled se The weighted within-animal variance was (::::: when corrected for total weight). If parameters a,c and d were made common, then the V matrix would be diagonal. Covariance estimates from such a fit were: REML: MIVQUEO: (1b2 = , (1/ = (1b2 = , (1e2 = Re-fitting the nonlinear function produced estimation problems for group 4. Also, the different weighting makes it difficult to compare the various fits. A7: First-order modelling The difficulty involved with automating stages 3 and 4 of the above procedure prompted the search for a simpler solution. Therefore, the nonlinear function was replaced by the first-order approximation in both the fixed effect and random effects parts: y = J x f3 + Jzv where, for example, (expansion around initial values a,b, a, d) 776

8 The simple form for parameters a and b results from their linear inclusion in the function. This procedure is equivalent to the random coefficients model with Jacobians as 'regressors: y = (ao+ai)der.a + (bo+bi ) Der.b + (C;+ci)Der.c + (d;+di)der.d (c;, d; being deviations around c, d) The Jacobians are functions of the parameters and therefore require initial values. If these values are not very good, the fixed effect parameter estimates will also be poor. However, the random part of the model may take up some of the incompletely explained variability (see below). On the other hand, the worse the initial values, the more the residuals were time-dependent. With good initial estimates based on a visual assessment of the overall curve, all four parameters were well estimated. TABLE 8' Parameter estimates (none common) Group A B C D pooled se The REML covariance estimates for various sets of initial values were: Good Poor Very poor [1.7,6.3,5.8,0.7]: [0,10,5,1]: [0,20,1,1] u a 2 = 0., Ub2 = , u c 2 = 0, ui = u/ = , Ub 2 = , u c 2 = 0, ui = u a 2 = , Ub2 = , u/ = 0, ui = The estimates of components were: Good: Poor: Very poor: u e 2 = , Uk2 = , Ue 2 +Uk2 = u e 2 = , Uk 2 = , U/+Uk2 = u/ = , Uk2 = , Ue 2 +Uk2 = As the initial values get poorer, there is a shift in variance allocation from the within-subject to the between-subject level. Comparison of fixed effects is carried out using Wald tests. Summary For the methods Al, A2 and A3, the variances of the parameter estimates can be compared: Analysis var(a} var(b} var(c} var(d} Split-plot Subject fit " " " (B only) Pooled within groups " " " (B only)

9 The estimates for the Subject and Pooled within-groups fits were of the saine order of magnitude. Including fixed or common parameters clearly reduces the approximate standard errors but increases the estimate of the within-subject component. For the mixed modelling methods (AS and A6), the covariance estimates can be compared: Analysis (1e2 +(1k2 (12 e (1k 2 (12 a (1b 2 (12 c (1i Residual REML (0.1623) o o MIVQUEO (0.1623) (0.1342) O. O. REML (B only) (0.1623) REML (re-fit) (0.1623) First-order (good) O O (poor) O The important feature is that the estimates for the Residual REML and First-order methods (with good initial values) were very similar. This suggests that First-order modelling method might be a useful substitute to direct nonlinear fitting. Conclusion I! The simple approaches are probably inadequate for comparing groups if the data are unbalanced. Also, they do not address nonlinearity explicitly. However, as in all modelling, remember Box's dictum "All models are wrong, but some are useful." The simple methods may be adequate for certain situations. The residual mixed modelling method described above is clearly useful, but correct estimation of the covariance parameters is not obvious. The first-order modelling is promising but may require some iteration. It is not yet clear how to achieve this. Many other problems remain: degrees of freedom, Wald tests of fixed effects, the estimation method and, very importandy, execution time. References Lindstrom,M.J. and Bates,D.M.(1990) "Nonlinear mixed effects models for repeated measures data." Biometrics 46: pp Vonesh,E.F. and Carter,R.L.(1992) "Mixed-effects nonlinear regression for unbalanced repeated measures." Biometrics 48: pp SAS/STAT is a registered trademark of SAS Institute Inc., Cary, NC, USA. Mr.Graham F.Healey, Head of Statistics, Huntingdon Research Centre, P.O.Box 2, Huntingdon, Cambs, UK, PE18 6ES ( ) 778

10 .y FIGURE 1 - Plot of mean bodyweight (n= 10) BODYWEIGHT(G.'OOJ r , o WEEK OF STUOY e-e-e CONTROL EI- El-E1 LOW oir-a-6 MEDIUM... HIGH FIGURE 2 - Plot of individual bodyweights BODYWEIGHT(G.,O?& , o WEEK OF STUDY 30 FIGURE 3 - Plot of fitted curve (Group 1) BODYWEIGHT(G.I ooj r ;::----, o WEEK OF STUDY 779