1. Air temperature Metabolic rate Fatreserves burned Notion of a latent variable [O 2 ]

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

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

4. Latent variable X1X1 X2X2 X3X3 X4X4 Observed (indicator) variables 11 22 33 44 Error variables Latent variable

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

6. Length X 1 X 2 X 3 X 4 11 22 33 44 L=N(0,  )  1 =N(0,  )  2 =N(0,  )  3 =N(0,  )  4 =N(0,  ) X 1 =1L +  1 X 2 =a 2 L +  2 X 3 =a 3 L +  3 X 4 =a 4 L +  4 Cov(  1,  2 )=Cov(  1,  3 )=Cov(  1,  4 )=Cov(  2,  3 )= Cov(  2,  4 )=Cov(  3,  4 )=0 1

9. 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 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.

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

11. 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 Difference between observed and predicted variances & covariances ~log likelihood

12. 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 Maximum likelihood estimate Standard error of the estimateZ- value of a normal distribution testing H 0 : coefficient=0 in population X1=1L X 2 =0.07L X 3 =0.39L X 4 =1L

13. Body size Visual estimate of body weight Total body length Neck circumference Chest circumference  1  2  3  4 Body size is difficult to measure in free-ranging animals

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

15. Left horn: Right horn: - Basal diameter - horn length General size factor Left horn length Left horn basal diameter Right horn basal diameter Right horn length  1  1  1  1

16. Left horn: Right horn: - Basal diameter - horn length General size factor Left horn length Left horn basal diameter Right horn basal diameter Right horn length  1  2  3  4 MLX 2 =759.106, 2 df, p<0.000001 This causal structure is wrong

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

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

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