13. The Cochran-Satterthwaite Approximation for Linear Combinations of Mean Squares

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1 13. The Cochran-Satterthwaite Approximation for Linear Combinations of Mean Squares opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

2 Suppose M 1,..., M k are independent mean squares and that It follows that d i M i E(M i ) χ2 d i i = 1,..., k. [ ] [ ] di M i di M i E = d i, Var E(M i ) E(M i ) for all i = 1,..., k. = 2d i, and M i E(M i) χ 2 d d i i Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

3 Consider the random variable M = a 1 M 1 + a 2 M a k M k, where a 1, a 2,..., a k are known constants in R. Note that M is a linear combination of scaled χ 2 random variables. The Cochran-Satterthwaite approximation works by assuming that M is approximately distributed as a scaled χ 2, just like each of the variables in the linear combination. dm E(M) χ 2 d M E(M) d χ2 d. What choice for d makes the approximation most reasonable? opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

4 If then Var(M) M E(M) d χ2 d, = ( ) 2 E(M) Var ( ) χ 2 d d ( ) 2 E(M) (2d) d = 2 [E(M)]2 d 2M2 d. Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

5 Now note that Var(M) = a 2 1 Var(M 1) + + a 2 k Var(M K) = a 2 1 [ E(M1 ) d 1 = 2 k a 2 i [E(M i)] 2 i=1 d i ] 2 2d1 + + a 2 k [ E(Mk ) d k ] 2 2dk 2 k i=1 a2 i M 2 i /d i. Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

6 Equating these two variance approximations yields 2M 2 d = 2 k a 2 i Mi 2 /d i. i=1 Solving for d yields d = M 2 k i=1 a2 i M2 i /d i = ( k ) 2 i=1 a im i k i=1 a2 i M2 i /d. i This is the Cochran-Satterthwaite formula for the approximate degrees of freedom associated with the linear combination of mean squares defined by M. Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

7 Recall the first example from the last slide set. y = y 111 y 121 y 211 y 212, X = , Z = X 1 = 1, X 2 = X, X 3 = Z y (P 2 P 1 )y + y (P 3 P 2 )y + y (I P 3 )y = y (I P 1 )y opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

8 Expected Mean Squares SOURCE EMS trt 1.5σ 2 u + σ 2 e + (τ 1 τ 2 ) 2 xu(trt) ou(xu, trt) σ 2 u + σ 2 e σ 2 e E(1.5MS xu(trt) 0.5MS ou(xu,trt) ) = 1.5(σu 2 + σe) 2 0.5σe 2 = 1.5σu 2 + σe 2 Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

9 An Approximate F Test The statistic F = MS trt 1.5MS xu(trt) 0.5MS ou(xu,trt) is approximately F distributed with 1 numerator degree of freedom and denominator degrees freedom approximated by the Cochran-Satterthwaite Method: d = (1.5MS xu(trt) 0.5MS ou(xu,trt) ) 2 (1.5) 2 [ MS xu(trt) ] 2 + ( 0.5) 2 [ MS ou(xu,trt) ] 2. Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

10 SAS Code for Example data d; input trt xu y; cards; ; run; opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

11 SAS Code for Example proc mixed method=type1; class trt xu; model y=trt / ddfm=satterthwaite; random xu(trt); run; opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

12 The Mixed Procedure Model Information Data Set WORK.D Dependent Variable y Covariance Structure Variance Components Estimation Method Type 1 Residual Variance Method Factor Fixed Effects SE Method Model-Based Degrees of Freedom Method Satterthwaite Class Level Information Class Levels Values trt xu Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

13 Dimensions Covariance Parameters 2 Columns in X 3 Columns in Z 3 Subjects 1 Max Obs Per Subject 4 Number of Observations Number of Observations Read 4 Number of Observations Used 4 Number of Observations Not Used 0 opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

14 Type 1 Analysis of Variance Sum of Source DF Squares Mean Square trt xu(trt) Residual Source Expected Mean Square Error Term trt Var(Residual) MS(xu(trt)) Var(xu(trt))+Q(trt) MS(Residual) xu(trt) Var(Residual)+Var(xu(trt)) MS(Residual) Residual Var(Residual). opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

15 Degrees of Freedom for Satterthwaite Approximation d = (1.5MS xu(trt) 0.5MS ou(xu,trt) ) 2 (1.5) 2 [ MS xu(trt) ] 2 + ( 0.5) 2 [ MS ou(xu,trt) ] 2 = ( ) 2 (1.5) 2 [2.42] 2 + ( 0.5) 2 [0.18] 2 = Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

16 Type 1 Analysis of Variance Error Source DF F Value Pr > F trt xu(trt) Residual... Covariance Parameter Estimates Cov Parm Estimate xu(trt) Residual Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

17 Concluding Remarks This example was chosen to be small so that we could write out all the data and see how each observation was involved in the analysis. Because of the very low sample size, it would be surprising if the approximate F test worked well for this example. It would be difficult to draw any meaningful conclusions with 4 observations of the response on 3 experimental units. We will see more practically relevant examples where the samples sizes are larger and the approximate F-based inferences may be reasonable. Copyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18

18 Concluding Remarks In more complicated examples, there may be more than one linear combination of mean squares with the desired expectation. In such cases, linear combinations with non-negative coefficients are recommended over those with a mix of positive and negative coefficients. opyright c 2018 Dan Nettleton (Iowa State University) 13. Statistics / 18