- 171 Views
- Uploaded on

Download Presentation
## WHY ARE YOU USING THAT REGRESSION?

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

FOCUS

- POPULATIONS
- VARIANCE IN RELATIONSHIPS
- OBJECTIVES
- USE OF REGRESSION
- TECHNIQUES are SECONDARY

TWO WORLDS

- SURVEY SAMPLING
- Fixed Populations
- Objective refers to Population
- REGRESSION ANALYSIS
- Relationships between variables
- Objectives refer to individuals or populations

SURVEY SAMPLING

- Fixed Population.
- Specified probability-sampling processes.
- Estimation of population parameters
- unbiased estimators.

SURVEY SAMPLING

“If we are to infer from sample to population, the selection process is an integral part of the inference.” - Stuart (1984, p. 4)

REGRESSIONS IN SURVEY SAMPLING

- AUXILIARY INFORMATION (X)
- known for population.
- Increased precision.
- MODEL-ASSISTED ESTIMATORS COMMON (Särndal et al.,1992)
- MODEL-BASED ESTIMATORS

MODEL-ASSISTED SURVEY SAMPLING

Ratio of Means Estimator:

Asymptotically unbiased,

whether or not y proportional to x.Could be used to estimate individual y’s.

No claim of unbiasedness here.

MODEL-BASED SURVEY SAMPLING

- Assumptions from Regression Analysis.
- True model
- E(e|x) = 0
- Errors are independent.
- Random selection avoids a source of bias.
- Inference from regression theory, not the distribution of samples.
- Theory from Royall (1970).

REGRESSION ANALYSIS

- Least Squares - Legrendre (1805) and Gauss.
- Sir Francis Galton (1877, 1885):

Offspring of seeds “did not tend to resemble their parent seeds in size, but to be always more mediocre [i.e., more average] than they - to be smaller than the parents, if the parents were large … the mean filial regression towards mediocrity was directly proportional to the parental deviation from it.” (quoted from Draper & Smith)

HEIGHT-AGE CURVES

- Site Curves (Curtis)
- Site Index Prediction Functions
- Geometric Mean Regression
- Stochastic Differential Equation
- Height Growth Models
- Percentile Models

Site Curves and SI Prediction Functions

- Curtis et al. (1974)
- Site Curve - Yield table construction
- H = f(A, SI).
- SI Prediction Function - Site Classification
- SI = f(A, H).

SITE CURVES, SI PREDICTION, and GMR

SI = H (index age)

HA = H (age A)

3 Lines:

All at mean (HA, SI)

Slope = SI/HA { , 1, 1/ }

Straight-line assumption valid for bivariate normal.

Stochastic Differential Equation (Garcia, 1979)

- dH/dt = (b/c)H{(a/H)c -1}
- b is plot-specific, (a, c) are global.
- Integrates to Chapman-Richards.
- Add Wiener process error to growth.
- Add measurement errors at intervals.
- Fit with Maximum Likelihood.
- It’s a growth-model; also base-age invariant site curve.

Height Growth Model

- Family of H-A curves.
- From any one age, predict height difference to next or previous age.
- Parameters adjusted to minimize errors in predicted growth. (Bonnor et al, 1995), Flewelling et al (2001).
- Crude, ignores measurement errors, and correlations between periods. Flexible model form.
- It’s a growth model - attempts to model H-A trajectories of plots. Base-age invariant.

Percentile Models

- Concept by Pienaar and Clutter (Clutter et al, 1983).
- Example by Bi (2002).
- Extends to irregular data. (Flewelling, 1982, unpublished).
- Current econometrics theory, rich history.

Percentile Models

- Pienaar and Clutter:

Percentiles as a labeling device: “useful in illustrating the fact that index age is not a fundamental or required concept in the use of site index to express site quality.”

Percentile Models, Example

- Bi et al ( 2002)
- Temporary plots (age and site assumed orthogonal).
- H(t) assumed to have normal distribution.
- Q0.75 and Q0.25, fit as functions of t.
- methodology from Koenker and Bassett (1978)
- Mean H(t) fit with weighted regression.

Percentile Model, Irregular data.

- Sectioned tree data, height every year.
- Younger ages: full data set.
- Older ages: reduced data set.
- Establish tree percentiles at young age.
- Reassign censored percentiles older ages.
- Compute (and model) means and standard deviations from heights and percentiles.

Percentile models, econometrics

- Koenker (2000):
- wonderful discussion of least squares, alternative methods, and statistical history.
- Minimization of summed absolute errors dates from 1760’s.

Height-Age Curves. Questions

- Should height growth models be the same as constant percentile curves?
- Are regressions from one age to another wanted?
- Is there any use for an index age other than as a label?

POPULATIONS

WHICH PROJECTION IS WANTED?

TREE GROWTH MODELS

- DBH
- Mortality fractions.
- What ensures that the variance of projected stand table is correct?
- Need variance models as constraints?
- Different fitting techniques?
- Good luck and occasional checking?

RIGHT INDEPENDENT VARIABLES?

Regional H-DBH

Curves.

Biased by Age or

position in stand.

Alternative:

local curves,

another variable.

Bayesian Regression

- Neglected in Forestry?
- Empirical Bayes used in volume equations (Green and Strawderman, 1985).
- Taper and volume equations by forest district (McTague, Stansfield and Lan, 1992).
- Other opportunities?

Bayesian Opportunity

- Fit y = a0 + a1x1 + a2x2 + a3x3 + …..
- Often by species or other category.
- Coefficients tested and omitted if non-significant.
- Or, selected coefficients fit in common for all species.
- Bayesian regression or other methods better?

OTHER REGRESSION TECHNIQUES

- ML with better error characterization.
- Mixed models.
- Systems: Seemingly unrelated regression, 2SLS, 3SLS ……..
- Generally are more efficient, better estimates of parameter variance, possibly avoid some biases. Necessary?
- Imputation?

SUMMARY

- What does population look like?
- What should be described?
- What techniques allow that?

REFERENCES

- Bi, H., A.D. Kozek and I.S. Ferguson. 2002. Quantile-based site index curves: a brief introductory note. Proc of IUFRO Symposium on Statistics and Technology in Forestry, Sept 8-12, 2002 Blacksburg. [ May be a related 2003 paper in J of Agr, Biological, and Environmental Statistics.]
- Bonnor, G.M., R. J. DeJong, P. Boudewyn and J. Flewelling. 1995. A guide to the STIM growth model. Nat. Res. Canada. Info Rpt X-353.
- Clutter, J.L., J.C. Fortson, L.V. Pienaar, G.H. Brister and R.L. Bailey. 1983. Timber management: a quantitative approach. Krieger Publ., Malamar, FL. 333 p.
- Curtis, R.O., D.J. Demars, F.R. Herman 1974. Which dependent variable is site index - height - age relationships? For. Sci. 20: 74-87
- Draper, N. R. and H. Smith. 1998. Applied Regression Analysis. Wiley. New York. 706 p.
- Flewelling, J. 1982. Dominant height trends for plantations of loblolly pine at the Mississippi/Alabama region of Weyerhaeuser Company. Research Rpt 050-3415/3. Weyerhaeuser Forestry Research, Hot Springs. (unpublished)
- Flewelling, J., R. Collier, B. Gonyea, D. Marshall and E. Turnblom. 2001. Height-age curves for planted stands of Douglas fir, with adjustments for density. SMC Working Paper No. 1, Univ. of WA, Seattle.

REFERENCES

- Garcia, O., 1979. A stochastic Differential Equation Model for height growth of forest stands. Biometrics 39: 1059-1072.
- Green, E. and W.E. Strawderman. 1985. The use of Bayes/Empirical Bayes Estimation in Individual Tree Volume Equation Development. For. Sci. 31: 975-990.
- Koenker, R. 2000. Galton, Edgeworth, Frisch, and prospects for quantile regression in econometrics. J of Econometrics 95: 347-374.
- Koenker, R.W. and G.W. Basset. 1978. Regression Quantiles. Econometrica 50, 43-61.
- McTague, J.P., W.F. Stansfield, Z. Lan. 1992. Southwestern ponderosa pine, Douglas fir and white fir volume and taper functions. Report to USFS. Northern Arizona University.
- Ricker, W.E. Computation and uses of central trend lines. Can. J. Zool. 62:1897-1905
- Royall, R.M. 1970. On finite population sampling theory under certain linear regression models. Biometrika 57: 377-387.
- Särndal, C., B. Swensson, J. Wretman . 1992. Model assisted survey sampling. Springer-Verlag, New York. 694 p.
- Stuart, A. 1984. The ideas of sampling. Macmillan, New York. 91 p.

QUESTIONS?

Download Presentation

Connecting to Server..