Mesurement models and maximum likelihood

1. [O2] Metabolic Fat reserves Air [O2] temperature rate burned Mesurement models and maximum likelihood Notion of a latent variable

2. [O2] [O2] Metabolic Air Fat reserves temperature rate burned Thermometer gas Change in exchange reading body weight Measurement Measurement Measurement error 1 error 2 error 3 Mesurement models and maximum likelihood Notion of a latent variable

3. Metabolic Air Fat reserves temperature rate burned Thermometer gas Change in exchange reading body weight Measurement Measurement Measurement error 1 error 2 error 3 Thermometer Gas Change in reading = exchange= body weight= air temperature metabolic fat reserves rate burned Mesurement models and maximum likelihood

4. Latent variable 1 2 3 4 Mesurement models and maximum likelihood Latent variable Observed (indicator) variables X1 X2 X3 X4 Error variables

5. True length of strings (latent) Ruler ± 1cm Her hand ± 0.07 hand Ruler ± 1 inch Visual estimation ± 10 cm Mesurement models and maximum likelihood

6. Length X1 X2 X3 X4 1 2 3 4 2=N(0, ) 3=N(0, ) 4=N(0, ) L=N(0,) 1=N(0, ) X1=1L + 1 X2=a2L + 2 X3=a3L + 3 X4=a4L + 4 Mesurement models and maximum likelihood 1 Cov(1, 2)=Cov(1, 3)=Cov(1, 4)=Cov(2, 3)= Cov(2, 4)=Cov(3, 4)=0

7. Mesurement models and maximum likelihood

8. Mesurement models and maximum likelihood Now, analyze this model using EQS...

9. Starting values in the iterations for maximum likelihood With latent variable models, if the starting values are too far from the real ones, one will get “convergence” problems - local minima. Mesurement models and maximum likelihood Structural equations used by EQS: 10 /EQUATIONS 11 V1= + 1F1 + E1; 12 V2= + 1*F1 + E2; 13 V3= + 1*F1 + E3; 14 V4= + 1*F1 + E4; 15 /VARIANCES 16 F1= 100*; 17 E1= 0.01*; 18 E2= 0.1*; 19 E3= 10*; 20 E4= 100*; 21 /COVARIANCES 22 /END

10. likelihood likelihood Global maximum Global maximum Starting value Starting value Local maximum Value of free parameter Better starting value Value of free parameter Mesurement models and maximum likelihood “Convergence problems”

11. Difference between observed and predicted variances & covariances ~log likelihood Mesurement models and maximum likelihood PARAMETER ESTIMATES APPEAR IN ORDER, NO SPECIAL PROBLEMS WERE ENCOUNTERED DURING OPTIMIZATION. ITERATIVE SUMMARY PARAMETER ITERATION ABS CHANGE ALPHA FUNCTION 1 44.628872 0.50000 5.67518 2 20.353161 1.00000 2.87032 3 20.187513 1.00000 0.56798 4 1.278098 1.00000 0.04411 5 0.682669 1.00000 0.01115 6 0.065675 1.00000 0.01114 7 0.006249 1.00000 0.01114 8 0.000533 1.00000 0.01114

12. X1=1L X2=0.07L X3=0.39L X4=1L Maximum likelihood estimate Standard error of the estimate Z- value of a normal distribution testing H0: coefficient=0 in population Mesurement models and maximum likelihood MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS X1 =V1 = 1.000 F1 + 1.000 E1 X2 =V2 = .069*F1 + 1.000 E2 .004 17.105 X3 =V3 = .368*F1 + 1.000 E3 .021 17.529 X4 =V4 = .998*F1 + 1.000 E4 .061 16.315

13. Body size Chest Visual estimate circumference of body weight Total body Neck length circumference e e 1 4 e e 2 3 Mesurement models and maximum likelihood Body size is difficult to measure in free-ranging animals

14. Mesurement models and maximum likelihood Body size is difficult to measure in free-ranging animals MLX2=0.971, 2 df p=0.615 (measurement model fits the data well) Units: Kg Ln(estimated weight)=1Ln(“Body size”)+N(0,0.023) r2=0.893 Ln(total length)=0.370Ln(“Body size”)+N(0.0.023) r2=0.911 Ln(neck circumference)=0.42Ln(“Body size”)+N(0,0.005) r2=0.883 Ln(chest circumference)=0.387Ln(“Body size”)+N(0,0.001) r2=0.982

15. Left horn: Right horn: - Basal diameter - Basal diameter General size - horn length - horn length factor Left horn Right horn length length Left horn Right horn basal diameter basal diameter e 1 e 1 e e 1 1 Mesurement models and maximum likelihood

16. Left horn: Right horn: - Basal diameter - Basal diameter - horn length - horn length General size factor Left horn Right horn length length Left horn Right horn basal diameter basal diameter e 4 e 1 e e 2 3 Mesurement models and maximum likelihood MLX2=759.106, 2 df, p<0.000001 This causal structure is wrong

17. Left horn: Right horn: - Basal diameter - Basal diameter - horn length - horn length Growth factor 2 1 Length growth Diameter growth Left horn length Right horn length Left horn diameter Right horn diameter 3 4 5 6 Mesurement models and maximum likelihood

18. Left horn: Right horn: - Basal diameter - Basal diameter - horn length - horn length Growth factor 2 Length growth 1 Diameter growth Right horn diameter Left horn length Right horn length Left horn diameter 6 3 4 5 Mesurement models and maximum likelihood MLX2=3.948, 1df, p=0.05

19. Mesurement models and maximum likelihood Left horn: Right horn: - Basal diameter - Basal diameter - horn length - horn length Growth factor Is this latent variable really “a growth factor”? 2 Length growth 1 Diameter growth Are these latent variables really growth of diameter and length? Right horn diameter Left horn length Right horn length Left horn diameter 6 3 4 5 The naming fallacy