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[O 2 ]. Metabolic. Fat. reserves. Air. [O 2 ]. temperature. rate. burned. Mesurement models and maximum likelihood. Notion of a latent variable. [O 2 ]. [O 2 ]. Metabolic. Air. Fat reserves. temperature. rate. burned. Thermometer. gas. Change in. exchange. reading.

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Presentation Transcript
slide1

[O2]

Metabolic

Fat

reserves

Air

[O2]

temperature

rate

burned

Mesurement models and maximum likelihood

Notion of a latent variable

slide2

[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

slide3

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

slide4

Latent

variable

1

2

3

4

Mesurement models and maximum likelihood

Latent variable

Observed

(indicator)

variables

X1

X2

X3

X4

Error variables

slide5

True length

of strings

(latent)

Ruler

± 1cm

Her hand

± 0.07 hand

Ruler

± 1 inch

Visual estimation

± 10 cm

Mesurement models and maximum likelihood

slide6

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

slide8

Mesurement models and maximum likelihood

Now, analyze this model using EQS...

slide9

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

slide10

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”

slide11

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

slide12

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

slide13

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

slide14

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

slide15

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

slide16

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

slide17

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

slide18

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

slide19

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

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