Statistical Consulting Topics. RCBD with a covariate

Transcription

1 Statistical Consulting Topics RCBD with a covariate Goal: to determine the optimal level of feed additive to maximize the average daily gain of steers. VARIABLES Y = Average Daily Gain of steers for 160 days FACTOR = Diet (4 levels: 0, 10, 20, 30) BLOCK = Barn (8 levels, 4 animals in each barn) COVARIATE = initial weight BLOCK is random, and the other terms are fixed. We will assume a linear relationship between the covariate, or initial weight (iwt), and the response, average daily gain (adg). There were 32 steers altogether, randomly assigned to barns. Diet levels were randomly assigned to animals within each barn. The animals were individually fed over the 160 days. 1

2 We have 8 observations on each level of Diet (one from each barn). Observations within a barn are correlated. In this set-up, we get to compare treatments within a block (or barn) after accounting for the initial weight. adg diet 0 diet 10 diet 20 diet iwt Y ij = α i + β i x ij + b j + ɛ ij (1) where b j iid N(0, σ 2 b ) and ɛ ij iid N(0, σ 2 ɛ ) for i = 1, 2, 3, 4 and j = 1, 2,..., 8 α i intercept of i th diet β i slope of i th diet x ij iwt of the steer on diet i in block j b j random block effect 2

3 The non-common slope model SAS code for model with separately fit lines for diets: proc mixed data=gain; class trt blk; model adg=trt iwt iwt*trt/solution ddfm=satterth; **Subset of output follows** Covariance Parameter Estimates Cov Parm Estimate blk Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt iwt iwt*trt

4 The common slope model SAS code for model with parallel lines for diets: proc mixed data=gain; class trt blk; model adg=trt iwt/solution ddfm=satterth; **Subset of output follows** Covariance Parameter Estimates Cov Parm Estimate blk Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt iwt

5 The model with parallel lines is complex enough to capture the relationship between the variables. Solution for Fixed Effects Standard Effect trt Estimate Error DF t Value Pr> t Intrcpt trt trt tr trt iwt Parameters for the common slope model Thus, the common slope is , and the intercepts are , , , , respectively. Because trt was significant, at least one of the lines is different from the others. 5

6 Though I ve plotted all 4 fitted lines below, some may not be significantly different from each other. adg diet 0 diet 10 diet 20 diet I ts common in an ANCOVA to report and compare the treatment groups at the average value of the covariate (shown above with dotted line). /*Get mean of covariate.*/ proc means data=gain; var iwt; Analysis Variable : iwt iwt N Mean Std Dev Minimum Maximum

7 proc mixed data=gain; class trt blk; model adg=trt iwt/ solution ddfm=satterth; lsmeans trt/adjust=tukey at iwt=389.6; Least Squares Means Standard Effect trt iwt Estimate Error DF t Value Pr > t trt <.0001 trt <.0001 trt <.0001 trt <.0001 Differences of Least Squares Means Standard Effect trt trt iwt Estimate Error DF t Value trt trt trt trt trt trt

8 Differences of Least Squares Means Effect trt _trt Pr > t Adjustment Adj P trt Tukey-Kramer trt Tukey-Kramer trt Tukey-Kramer trt Tukey-Kramer trt Tukey-Kramer trt Tukey-Kramer Diet 0 is statistically significantly different than the others. I should note that the LSMEANS statement would have compared the treatments at the average value of the covariate even without specifically asking for it. proc mixed data=gain; class trt blk; model adg=trt iwt/ solution ddfm=satterth; lsmeans trt/adjust=tukey; 8

9 If you ask for the LSMEANS of trt at iwt=0, you ll get the estimated intercepts: proc mixed data=gain; class trt blk; model adg=trt iwt/ solution ddfm=satterth; lsmeans trt/adjust=tukey at iwt=0; Standard Effect trt iwt Estimate Error DF t Value Pr > t trt trt trt trt

10 Dose-Response Curve Because the levels of the factor of interest actually represents a quantitative value, we can model this with a trend or dose-response curve (rather than doing pairwise comparisons of the four levels). The relationship between the covariate and adg is still linear, but the relationship between diet level and adg can be fit with a polynomial. adg(lsmean at x=389.6) trt 10

11 proc mixed data=gain; class trt blk; model adg=trt iwt/solution ddfm=satterth; estimate linear trt ; estimate quad trt ; estimate cubic trt ; Estimates Standard Label Estimate Error DF t Value Pr > t linear quad cubic The results suggest a quadratic is sufficient for modeling the trend. The quadratic model proc mixed data=gain; class blk; model adg=trt trt*trt iwt/solution ddfm=satterth; 11

12 adg(lsmean at x=389.6) trt But, perhaps a threshold model or piece-wise linear might also work well. We would need more levels of the additive to get at comparing such models. 12