Alternative Methods of Adjusting forHeteroscedasticity

in Wheat Growth Data

~

Economics,Statistics, andCooperatives Service

U. S. Departmentof Agriculture

Washington, D. C.20250

ALTERNATIVE METHODS OF ADJUSTING FORHETEROSCEDASTICITY

IN WHEAT GROWTH DATA

By

Greg A. Larsen

Mathematical StatisticianResearch and Development BranchStatistical Research Division

Economicst Statistics and Cooperatives ServiceUnited States Department of Agriculture

Washingtont D. C. February t 1978

Table of Contents

Index of Diagrams, Tables and FiguresAcknowledgementsIntroductionBasic Logistic Growth ModelThe Log TransformationWeighted Least SquaresAn Application To ForecastingConclusionsFigures

i i i

Pagev

vii1

4

5

14

21

24

25

Diagram 1

Table 1

Table 2

Table 3

Table 4

Table 5

Figure

Figure 2Figure 3

Figure 4Figure 5

Index of Diagrams, Tables and Figures

Illustrates the effect of five different log trans-formations on several values of the dependentvariable y.Summary of regression, parameter estimation andcorrelation for the unadjusted model and four logmodels.Estimated standard deviation and number of obser-vations in seven intervals of the independent timevariab 1e t.Summary of regression, parameter estimation andcorrelation for the unadjusted model and the YHATadjusted modelEstimated standard deviation and number of obser-vations in seven intervals of the independent timevariable t after a questionable sample had beenremoved.Summary of regression, parameter estimation andcorrelation for the unadjusted model and threeadjusted models with four weeks of data and thecomplete data set.Plot of the untransformed data and the estimatedvalues of the dependent variable y.Residual plot corresponding to Figure 1Plot of the transformed data and the estimatedvalues of the transformed dependent variable.The log transformation in (6) on page 7 was used.Residual plot corresponding to Figure 3.Plot of the transformed data and the estimatedvalues of the transformed dependent variable.The log transformation in (7) on page 10 was used.

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Page6

9

15

18

19

22

25

2627

28

29

PageFigure 6 Residual plot corresponding to Figure 5. 30Figure 7 Plot of the transformed data and the estimated 31

values of the transformed dependent variable.The log transformation in (8) on page 12 wasused.

Figure 8 Residual plot corresponding to Figure 7. 32Figure 9 Plot of the transformed data and the estimated 33

values of the transformed dependent variable.The double log transformation in (9) on page 13was used.

Figure 10 Residual plot corresponding to Figure 9. 34Figure 11 Plot of the dependent variable at versus the 35

Aindependent variable y and the estimated valuesof the dependent variable.

Figure 12 Plot of the transformed data and the estimated 3uvalues of the transformed dependent variable.Weighted least squares was used with (12) onpage 16 as an estimate of Ilki in (10) on page 14.

Figure 13 Residual plot corresponding to Figure 12. 37Figure 14 Same as Figure 11 except a questionable sample 38

was deleted from the data set.Figure 15 Plot of the transformed data and the estimated 39

values of the transformed dependent variable.Weighted least squares was used with (14) onpage 20 as an estimate of ~ in (10) on page 14.

1

Figure 16 Residual plot corresponding to Figure 15. 40

vi

Acknowledgments

I would like to thank the staff of the Kansas State Statistical Officefor supervising the data collection phase of the 1977 Within-Year WheatGrowth Study. A group of approximately 30 enumerators were employed andthe office staff did a good job in coordinating their efforts. The key-punching, also done in the state office, was nearly perfect.I would also like to recognize the fine job that Jack Nealon did in settingup the sampling plan and preparing all the manuals, forms and suppliesnecessary for the data collection.I am especially appreciative to Mary Jiggetts for her patience and carein typing a difficult manuscript.

vii

IntroductionThe purpose of this report is to suggest some techniques which areuseful in adjusting for heteroscedastic disturbances common to growthdata. While all the results pertain specifically to dried wheat headweights, it is hoped that the concepts involved are general enoughto be of use with other types of growth data or in non-growth situations.Two basic approaches will be discussed; log transformations and weight-ed least squares. Several different log transformations will bedeveloped most of which are of the form ln (Ay + B) where y is thedependent variable and A and B are derived constants based on variouscriteria. The last transformation discussed has four constants to bespecified and the log is taken twice. The second approach is to weight thediagonal elements of the variance-covariance matrix of the dependent vari-able by some appropriate function which creates a homoscedastic disturbanceterm. A linear function is used which relates the expected value of thedependent variable to the population standard deviation.The data used in this report was collected during April to July 1977in 68 winter wheat fields distributed throughout Kansas. These fieldswere a subset of the wheat objective yield sample in which fields wereselected with probability proportional to acreage. This was done sothat field level yield observations could be weighted together equallyto form a state average. Within each field there were two randomlyand independently located plots. Each plot consisted of one row andwas approximately five feet in length. In the case of braodcast wheat,the plots were six inches wide. Stalk counts were made in the fivefoot plot. Every tenth plant was tagged until a total of 30 wasreached. The 30 tagged plants could lie within the five foot sectionor go beyond it depending on the plant density. Stalk counts averagedless than 300 plants and generally the sample extended past the fivefoot area.Weekly visits were made by trained enumerators to observe when thetagged plants had heads fully emerged and when flowering occurred.As a general rule, once 80% of the tagged plants in a particular fieldhad flowered, clipping began. A random sample of four heads per plotper weekly visit was clipped, each head was placed in an air tightplastic tube and mailed to the stat2 lab. In the lab, wet and dryweights were determined for each h2ad. The heads were dried for 46hours at 150 degrees Fahrenheit. Head emergence and floweringcontinued to be observed until two weeks after clipping began or untilall tagged plants had flowered. Heads continued to be clipped on aweekly basis until harvest.

Time since full head emergence and time since flowering were calculatedfor each clipped head. Since the random sample of heads to be clippedeach week was pre-determined, some heads were clipped before head emer-gence and flowering was observed. It was determined in previous work *that time since flowering is the preferred variable. This has been veri-fied and the time variable used in this report is time since flowering.Heads for which flowering was not observed were excluded from the data set.The sample design that has been outlined is a two level nested design.Plots are nested within fields and heads are nested within plots.Consequently, individual head weights are not strictly independent andaggregation*should take place until independent points are obtained.In the past, this has meant averaging head weights for all plantsin a field on a particular weekly visit. Since the success of thegrowth model depends upon the relationship between time since flower-ing and head weight, there is concern that this method of aggregationmight adversely affect the time-growth relationship. The reason forthis is that even though a set of heads might have the same clippingdate, they do not necessarily have the same number of days sinceflowering. Aggregation averages over both time and weight. If timesince flowering is fairly consistent for a particular visit, thereis no problem. However, in the data to be used in this report therewas as much as a two week difference in time values for a particularvisit. Therefore a method of aggregating observations with similartime values was sought. Since weekly visits were made, the floweringdate is actually an average between the date of the visit when flower-ing was first observed and the date of the previous visit. This putsa measurement error of approximately 3.5 days on either side of theflowering date. (Visits have been made every two or three days inearlier research but analysis indicated that weekly visits would besufficiently accurate. *) With this in mind, the data was dividedinto time intervals in each of which the time values were assumed tobe essentially the same. Head weights in each time interval wereaveraged together within a field without regard to plots. So long asthe two plots within a field have the same number of observations,each plot will receive equal weight as the sampling plan intended.However, plots with very few observations would tend to receive lessweight. Averaging within time intervals should more fully preservethe time-growth relationship than would aggregation by visit.

*Nealon, Jack 1976 The Development of Within-Year Forecasting Modelsfor Winter Wheat. Research and Development Branch, StatisticalResearch Division, ESCS, USDA.

2

Since the sampling plan was not conceived with time interval aggre-gation in mind, one consequence is an increased variability in thenumber of observations per mean. Means are comprised of from oneto thirteen observations. To reduce the effect of means with fewobservations, each mean was expanded by the number of observationsgoing into it. In other words. a mean comprised of one observationwould be included once while a mean comprised of eight observationswould be included eight times. This gives the allusion of moredata points than really exist causing a reduction in the parameterstandard error estimates. The reduction is relative, however, anddoes not affect comparisons as long as the expanded data set isused consistently. Another consequence of expanding the data setis that fields are weighted unequally because of varying numbersof observations in each field. This violates the intention of thesampling plan but the purpose of this report is solely to investi-gate methods of adjusting for heteroscedasticity and not to estimatea state yield. It is suggested that the expanded data set is suitablefor this purpose.It should be pointed out that even when we do not expand the aggre-gated means, there is still a problem with the way in which thefields were sampled. This is because the number of visits to clipheads varied from three to seven depending upon when harvest occurred.Therefore. if no expansion is done, fields that had more visits wouldbe more heavily favored. A possible solution to this would be toweight each aggregated mean by the inverse of the number of timeintervals in the corresponding field. In this way, fields with threevisits would receive the same weight as fields with seven. Ideally,weld like to develop a sampling plan that would clip heads a certainnumber of days after flowering rather than clipping a random samplecomprised of various stages of development on a fixed date. Tocontrol the number of heads being clipped in such a plan would notbe easy and this problem remains for future study.

3

The basic growth model which will be referred to throughout this reportis as fo11ows:*

( 1 ) ay. = ---- + E:. wh ere i = 1,2, ...,n1 1 t. 1+ Bp 1

Least squares theory will be used to estimate parameters a, Band p. Thisrequires the following assumptions about the nature of the model.

( a)

( b)

( c)

E (E:.) = a for all i1

2 2V ar (E:.) = E (E:. ) = (J fo r a 11 i

1 1

Co V (E: i'E:j ) = E (E: iE:) = 0 for a11 i f j

Stated another way, assumptions (b) and (c) imply that the variance-co-variance matrix of the dependent variable y is

( 2)2

I (J =n o 1

o . 2(J

nxnHeteroscedasticity is defined to occur when the diagonal elements of theV-C matrix are no longer equal. In matrix notation, the V-C matrix canbe expressed as

Kl 0K22 2

( 3) Vn (J = (J

0 K n

*A good discussion on this model appears in an earlier report. House,Carol C. 1977 A Within-Year Growth Model Approach to Forecasting CornYields. Research and Development Branch, Statistical Research Division,ESCS, USDA. pp 2-3.

4

In the original notation, assumption (b) above would be replaced by(4)

In words, heteroscedasticity means that the variance of the dependentvariable does not remain constant over the range of the independentvariable. In the case of the growth model, the variance of the y'sincreases with time.*The effect that heteroscedasticity has when standard least squares isapplied is often difficult to assess. In generalized linear regression,parameter estimates tend to become unreliable depending upon the degreeof the heteroscedasticity. While they remain unbiased, their standarderrors are underestimated. Also, the estimate of 02 will no longer beunbiased.** Several different methods of adjusting for heteroscedasticitywill now be discussed.The Log TransformationThe basic growth model was fit to the data using the NLIN procedure inSAS.*** A plot of the estimated values of y over1ayed on the data is shownin Figure 1. A plot of the residuals vs. time is shown in Figure 2. Thepresence of heteroscedasticity can be seen in the latter figure. Thecone-shaped plot shows that the deviation of the data from the fittedequation increases with time. A residual plot showing homoscedasticity(and no other model deficiencies) would appear to show a random pattern.The correlation between the absolute value of the residuals and time is.28 using Pearson's R (denoted by R ) and .31 using Spearman's Rho (ap

comparable nonparametric test denoted by R ). Both are significantlysdifferent from zero. See Table 1 on page 9.To adjust for the non-homogeneity of variance, we would like to decreasethe variance as t becomes large so that the absolute value of the residualswill become uncorre1ated with time. It is possible to do this with a logtransformation. The natural logarithm will be used in this discussion butthe logarithm to the base 10 could also be used. To visualize how thisworks refer to Diagram 1 on the next page. Taking the 1n (y) has the effect

*If data is collected well after maturity is reached, the variance willbecome stable for large values of time. While this did not appear tohappen with the wheat data used in this report, it was noted in someearlier corn research (House).**Go1dberger, Arthur S. 1964 Econometric Theory. John Wiley & Sons, Inc.New York, London, Sidney. pp 238-241.***Barr, Anthony J., Goodnight, James H., et. a1. A User's Guide To SAS76. SAS Institute Inc. Raleigh, N. C. pp 193-199.

5

3.01.t1."2. •••1..11..0L,I•• ",."••1-10I.

I.t.r"

O'l .~.l..

0

-.eI-.r--.'-1.0-1.1--1.4-I.~-,.t~l.O-1..1 -1.:~~- %.ef

A,: 3.,QIf 52-I A.,,=/.1octo1o AI.: ,.07offoTo8. :a ,. 0 ~.,. DO" z., z. A 3c ,. 112." ,0

81:'I.00 'z.,~83 Ie .q3!'~1~

of spreading values of y less than 1 farther apart and collapsing valuesof y greater than 1. In growth data, since smaller values of yare lessvariable than larger values, the ln (y) will have more consistent vari-ability over the range of t.Since y represents weight, it would be good to keep the transformedvariable positive also. Values of yare known to be in the range .1 to2.0 so simply using ln (lOy) would produce strictly positive values.Notice that ln (lOy) = ln (y) + ln (lO) = ln (y) + 2.303 so that multi-plying y by a constant doesn't change the spreading and collapsing effectbut only moves the transformed variable up or down the scale. Refer toDi agram 1.The discussion so far suggests that we multiply both sides of (l) by 10and then take the log of both sides. This gives the following model:

( 5) 1n (1 Oy) = 1n lOa + 10E:1+8p t

To get least squares parameter estimates, the model must be expressed withan additive error term. We therefore redefine the model with additiveerror E:' which is different from the E: in (5).

(6) 1n (1 Oy) = 1n lOal+apt

+ E: I

If we are willing to make this change in models, there will be no violationof assumptions in applying least squares theory to estimate the parametersin (6). We are primarily interested in forecasting the average value ofy at the end of the growing season so a (the asymptote) is the parameterwhich we are most interested in estimating. Notice that the least squaresestimate of a (and, for that matter 8 and p) obtained from (6) will be interms of the untransformed dependent variable and therefore no antilogwill be required. This would not be the case if, for example, we had onlytaken the log of the left-hand-side of (1). A plot of the estimated valuesof the transformed dependent variable overlayed on the transformed data isshown in Figure 3. A plot of the residuals versus time is shown in Figure4. The residual plot shows that the heteroscedasticity has reversed it-self, that is, we have over adjusted. Table 1 shows the correlationcoefficients which support this conclusion. R is significantly differentpfrom zero at - .0645 while R is not significantly different from zerosbut is also in the negative direction. The mean square errors (MSE's) of

7

the unadjusted and log models are not directly comparable because the datasets are different. This is particularly true with the log transformationpresently being considered because, as Diagram 1 shows, the transformeddependent variable increases for all but the very smallest values of y.A uniform increase in y (such as adding a constant) would not change the~~E but here the expansion of y values less than 1 is more than the con-traction of y values greater than 1 causing the log model to have a largerMSE.Table 1 also shows the parameter estimates with the corresponding relativestandard error in terms of a percent. Al though the current log model overadjusted, there was a lessening of the affect of non-homogeneous varianceas evidenced by the decrease in the significance of the correlations. Itcan be observed that the relative standard error of ex increased. Thisseems to be consistent with the generalized linear regression situationstated earlier in which the standard error of a parameter estimate tendsto be understated when influenced by heteroscedasticity. Obviously, lineartheory mayor may not hold in a non-linear setting so we only seek to point

A A

out the comparison. The errors for Sand p decrease apparently bringingthe consistency to an end. These same directional changes in the relativestandard errors occu~ in all the adjusted models. The fact that the trans-formation ln (lOy) over adjusted leads us to want to find a transformationwhich does not C3use the small values of y to become more variable than thelarge values. As noted earlier, multiplying y by a constant prior to tak-ing the log does not change the relative relationship between the points.Therefore, we now want to consider transformations of the fonn ln (Ay+B)where A and B are constants to be specified. This will cause changes inthe relative relationship between the points when BfO.Several approaches can be used to adjust for heteroscedasticity. Smallvalues of y could be spread out until they have variability comparable tolarge values of y. Or, conversely, the variability of the small valuescould be held relatively stable and the large values collapsed until acomparable variance was attained. Thirdly, as in the case of (5), therecould be both an expanding and contracting of the y values. Obviously,it would not be desirable to increase the total variation so, a trans-formation maintaining the same approximate range as the untransformeddata would help to keep the total variation from increasing. Therefore,A and B will be determined with this in mind.The y values are known to lie within the interval of 0 to 2 grams. Thefollowing two constraints will keep the transformation of 0 equal to 0and the transformation of 2 equal to 2.

o = 1n (OA 1 + B 1)

2 = ln (2A1 + B1)

8

TABLE 1

MODELS

Unadj 1n (1 Oy) 1n (A1y+B 1) 1n (A2y+B 2) 1n (A31n (A2y+B2) + B 3)

RegressionMSE .0337 .0726 .0309 .0188 .0114

R2 .942 .980 .978 .971 .984Param Estis

a 1 .0360 1 .0307 1 .0292 1.0293 1 .0280

A A

aA/a 1.5 2.0 1.8 1.7 1.91.0 a

S 4.2556 4.2159 4.1674 4.1714 4. 1501

°S/S 5.3 2.8 3.5 3.8 3.1

p .8886 .8947 .8929 .8919 .8946A A

oA/p .5 .4 .4 .5 .4p

Corre 1at ionRp .2780 -.0645 .0770 .1297 -.0162

Prob > I R I .0001 .0041 .0006 .0001 .4712P

Rs .3093 -.0172 .1119 .1626 .0240

Pro b > I R I .0001 .4452 .0001 .0001 .2850s

Exponentiating both sides of each equation gives:e 0 = (OA 1 + B 1,

e2= (2A1 + B1)

Solving the first equation for B1 gives B1=1.Substituting into the second equation and solving for Al gives:

e2 = (2A 1 + 1)A1= e2-1

-2-

The transformation is then ln (3.194528 y + 1). The model now becomes

(7)

where Al = 3.194528

+ E~

The plot of the estimated values of the transformed dependent variableoverlayed on the transformed data is shown in Figure 5. The correspondingresidual plot is shown in Figure 6. The residuals show a slight positivecorrelation with time and Table 1 indicates that the correlation is signifi-cantly different from zero and positive. Both Rand R are, however, sub-p sstantially smaller than in the untransformed data.The MSE in (7) is in the same neighborhood as the MSE in the unadjustedmodel which says that the total variation has not been increased. It doesnot necessarily indicate that the model is superior since the two modelsaren't directly comparable. The R2 value from (7) indicates that more ofthe variability in the transformed data is being accounted for than in theuntransformed data. All the R2 values indicate that good relationships exist,however.

10

While (6) over adjusts, (7) does not adjust enough to cause the variancesof y to become constant over the range of time. This suggests that an-other transformation be developed. The primary function of the growthmodel is to forecast what the value of a will be at the end of the seasonbased on data collected early in the growing season. With this applicationin mind. it would be best not to increase the variance for small values ofy which, in the early part of the growing season, are the main input intothe model. Also, since a is the primary parameter to be estimated, atransformation which causes less distortion around the true asymptote wouldbe preferable.Figure 1 shows that the smallest group of y values (i .e. O<t<5) is approx-imately bounded by .1 and .8. So, we want a transformation that preservesthis same interval thus keeping the variation relatively the same. Basedon previous models, the true value of a is thought to be approximatelyequal to 1. This gives rise to the following two constraints.

1n (.8A 2+B 2) - 1n (.1A 2+B 2 ) ::; • 71n (A2+B2) ::; 1

The first constraint can be expressed as

.8A2+B21n ::; .7.1A 2+B 2

.8A2+B 2 7

.1A 2+B 2= e

7·8A2+B2 ::;e (.lA2+B2)

7 7A2(·8- .1e· ) ::;B 2 (e· - 1)

.598625A2 ::;1.013753B2

A2 ::; 1 • 693469B 2

The second constraint can be expressed as

11

After substituting and solving, the two constraints give the followingvalues for A2 and B2•

A2 = 1.709070B2 = 1.009212

The model now becomes

(8 )

where A2 = 1.709070B2 = 1.009212

The estimated values of the transformed dependent variable overlayed on thetransformed data and the residuals are shown in Figures 7 and 8, respec.tively.The residuals again show a positive correlation with time. The Rand Rp svalues in Table 1 are both significantly different from zero. A comparisonof Figures 1 and 7 reveals that the range of the group of smallest y values(i .e. 0<t<5) is the same and the asymptote is in the neighborhood of 1.0 forboth. While the transformation did keep the variance for small values of ystable, it did not reduce the variation as y increased enough to create ahomogeneous situation.If the present criteria used to select A and B are deemed satisfactory, thenany further attempt to adjust for heteroscedasticity by obtaining other Aand B values would lead to a compromise in the selection criteria. Ther(are no doubt many legitimate ways to select A and B but a feasible alterna-tive to a compromise would be a double log transformation. This can be doneby applying the same criteria to the transformed data in (8) as were appliedto the original data. The two previous constraints will be the same exceptthe transformed points will be substituted.

y . 1

ln (A2y + B2) .165615

12

.8.865615

1.0

1.0

The constraints now becomeln (.865615A3 + B3) - ln (.165615A3 + B3) = .71n ( A 3+B 3 ) = 1

Proceeding as before, the solution isA 3 = 1. 78261 0B3 = .935672

*The model now becomes

where A2 = 1.0709070A3 = 1.782610B 2 = 1.009212B3 = .935672

Figures 9 and 10 show the two plots associated with this model. The residualplot appears to indicate a very slight negative correlation. In Table 1, Rpis negative and Rs is positive but neither is significantly different fromzero. The estimate of the population variance (MSE) is only one-third thatof the unadjusted model. This is reasonable since by holding the variancestable for the smallest values of y, the transformation lessens the variancefor all larger values of y.

*It has been pointed out that (9) has a good deal more potential than isbeing utilized here. We essentially applied the same pair of constraintstwice to obtain values for A2, B2, A3 and B3' Conceivably, since thereare four unknowns, four different constraints could be used. The problem,of course, would be to find four constraints that are solvable.

13

The relative standard errors in (9) increased for a and decreased for Band p which, as discussed earlier, is consistent in all the adjusted models.Compared to the unadjusted model, a from (9) is only .8% less. While het-eroscedasticity tends to cause unreliable parameter estimates, depending onthe degree, the model in (9) seems to show that the unadjusted estimate ofa is quite stable.Weighted Least SquaresAn alternative to the log transftlrmation which sometimes may be effectivein adjusting for heteroscedasticity is to perform a weighted least squaresregression. Recall the assumption concerning the diagonal elements of thevariance-covariance matrix when heteroscedasticity is present.

2E (E. 2) :;; k. a fo r a 11 i

1 1

If we divide (1) by ~ we obtain1

( 10)y.

1

£1

:;; +E·

1

£1

wh ere i :;;1,2, ...•n

The disturbance term is now homoscedastic.

Equivalent sums of squares and parameter estimates are obtained if the di-agonal elements of the V-C matrix of the dependent variable are multiplied by11k .. This fact provides two alternative ways of performing a weighted

1

regression. The NLIN procedure in SAS can be applied to (10) or the weightstatement can be used with the NUN applied to (1). If the weight statemcl)tis used, the value of the weight would be 11k .. Although these two alter-

1

natives give identical sums of squares and parameter estimates, there is auseful difference between the two. The first alternative provides a residualplot in terms of y/r-r- while the other is in terms of the original y values.Therefore, since we need to see the residual plot and correlations to checkfor heteroscedasticity, the first alternative will be employed. The secondalternative could be used if the variance-covariance matrix was obtained andthen a test for differences among diagonal elements would test the hypothesis

14

of homogenious variance. At present, the option of outputting the V-Cmatrix in NLIN is not available.In theory, a weighted regression will provide homoscedastic disturbanceso the problem becomes one of finding a suitable function for k. A re-lationship needs to be found which describes the behavior of at which isthe true standard deviation of y at a point in time. Because of the wayin which the data was collected, each data point has a possible error of~ 3.5 days on the value of time. Therefore, it would exceed the precisionof the data to break it down into one or two day intervals. Since mostobservations were centered around a particular day of the week, the datain Figure 1 already appears grouped. The data readily divides into theseven time intervals listed in Table 2. It is then reasonable to estimateat in each interval by calculating the standard deviation of the y valuesin each group. Denote this estimate by at'

TAB LE 2

Time Interva 1 a No. of(days) (gr£ms ) obs.o <t..s.5 .1115 2105<t<14 .1487 389

14<t<20 .1852 41620<t<28 .1847 42228<t<33 .2210 32433<t<41 .2197 18041<t .2593 42

15

In some applications, the disturbance variance, af' is proportional tothe square of the expected value of y or some linear function of it.*This would make E(y)2 a candidate for k. Since E(y) is, of course, un-

A

known, the y values obtained from a non-linear regression are the "best"estimates. Because of the reasons already stated, we want to apply theNUN procedure to (10) and therefore a relationship for II< is sought.

A A

Figure 1 shows that as y increases so does at' A linear regression ofA A A

at on y weighted by the number of observations in each value of at wasA

run to see if a reasonably good relationship exists. A y value wascalculated for the median of each time interval using the parameterestimates from the unadjusted model. The regression was highly signifi-cant with R2 = .908. Figure 11 shows the estimated values of the dependentvariable and the seven data points. The estimated equation ** is

(11 ) at = . 078111 + .145391 YSince E(E2) = ai, the disturbance variance, it can be seen from (4) that~ = at/a. The MSE from the unadjusted non-linear regression althoughbiased, is an E';;timateof a2• Therefore, dividing (11) by .183683 (seeTable 1) gives a rel ationship for Ik. ***

( 1 2)

*Goldberger, p 245.

A A A

at/a = .425251 + .791535y

**It has been suggested by House, pp 12-13 that a step function be used toestimate at' The steps would then be the at values in Table 2. The timeintervals in the wheat data are fairly wide for reasons already mentionedand there is some concern that a step function of this nature would notadequately describe the unknown continuous function of at'***Dividing by a has no affect on the parameter or standard error estimatesso the choice of a only affects the MSE in the adjusted non-linear regression.

16

There are some obvious deficiencies in using (12) as an estimate of ~.Since y is estimated from a heteroscedastic model t while unbiasedt it isinfluenced by the reliability of the parameter estimates. a is no longerunbiased. SOt we have a situation in which the more the heteroscedasticitYtthe less our ability to estimate at/a. This might suggest an iterative

A A

p~ocedure where y and a are revised based on the previous weighted re-gression.To evaluate how effectively (12) adjusts for heteroscedasticitYt NLIN wasrun on (10) using (12) as an estimate of 11<. Figure 12 shows the estimatedvalues of the transformed dependent variable overlayed on the transformeddata. Figure 13 shows the corresponding residual plot. The residuals stillindicate a positive correlation with time although not nearly as severe asin Figure 2. Table 3 gives the correlations which have been reduced but arestill highly significant. It is interesting to note that the MSE is identi-cal to that obtained in the unadjusted model to four decimal-Elaces. Thisresult is mostly coincidence sincet in theorYt dividing by I k should leavethe true population variance a2• Howevert a2 from the unadjusted model isbiased and therefore the estimate of the population variance in the presentmodel should be somewhat less. ApparentlYt the error in using (12) as anestimate of 11< offset the expected reduction in the bias of ~2. It canalso be seen in Table 3 that the relative standard errors of the parameterestimates responded as they did in the log transformations.At this pointt the deficiencies cited earlier in using (12) as an estimateof ~ prompts us to try an iterative approach in an attempt to improve theestimates of a and y used in the linear regression equation. Since computercosts increase quickly with successive non-linear regressionst at first onlyone iteration was used to assess the effect. The change in the estimate ofpopulation variance and the parameter estimates was very slight after oneiteration. Thust there was also very little difference in the y values.

A A A

The stability of a and y leads one to suspect the validity of the at values.A A A

From (12)t it can be seen that at/a = 1 when y is approximately .73. ThisA A

means that y values are increased when y<.73 and decreased when y>.73.Thereforet if (12) was the proper estimate for /1(, the variance of y valuesto the left of y = .73 would be increased while the lariance to the rightwould be decreased so that the variance over the enL;re range would beconstant. Since (12) didn't adjust enought this in( :cates that the properexpression for 11< should have a steeper slope.

17

TAS LE 3

MO DELSA

Unadj y/(aI+bIy) Unadj* y I (a2+b 2Y)*Regression

MSE .0337 .0337 .0315 .0309R2 .9420 .9365 .9453 .9417

Param Est'sa 1 .0360 1.0479 1 .0338 1 .0510

A AGAia 1.5 1.7 1.5 1 .7

a

S 4.2556 4.1228 4.5903 4.38780::>

A Acr Sl 8 5.3 3.8 5.3 3.0

p .8886 .8925 .8858 .8915A AaA/p .5 .4 .5 .4

pCorrelation

Rp .2780 .0840 .3349 .0482Prob > IR I .0001 .0002 .0001 .0333

PR .3093 .1375 .3409 .0754s

Prob >IR I .0001 .0001 .0001 .0009sa 1 = .425251 a2 = .189735bI = .791535 b2 = 1 .106668

*No te : Model run on a data set which excluded one of the original samp1 es.

The plot of the data in Figure 1 shows several relatively high obser-vations in the first three groups which don't seem to correspond to thecurve being fitted. A check was made on all the observations whichappear to be considerably disjoint from the main body of the data. Theobservations denoted by "F" in the first group, "G" in group two and "G"in the third group all turned out to be from the same sample. The restof the observations which were checked all came from different samples.A close examination of the sample in question revealed no discernibleerrors in recording or keypunching. However, the field observationsrelating to the time variable were highly questionable. Since this sam-

~p1e exerts considerable influence on at' particularly in the first twogroups of data, it was deleted.A non-linear regression using the unadjusted model was run on the resultantdata set. Table 3 shows that the MSE and estimate of a decreased somewhatwhich was to be expected. The correlation coefficients show a slight in-crease in the positive direction.In order to run a weighted regression on the altered data set, estimates forat must be recalculated. Table 4 shows the standard deviation of the yvalues in each time interval. A linear regress ion

TABLE 4

Time Interval a No. 0 f(days) (gr£ms) obs.o <t<5 .0702 2045<t<14 .1296 382

14<t<20 .1751 40920<t<28 .1864 41428<t<33 .2226 31933<t<41 .2197 18041<t .2593 42

19

~of at on y weighted by the number of observations associated with eachvalue of at was run. The regression was again highly significant with animproved R2 value of .948. Figure 14 shows the data along with the esti-mated regression line. The estimated equation is now

(13 )A A

at = .033664 + .196353y

Looking back at (11), it can be seen that the slope of (13) has increased.Taken at face value, (13) should adjust for more of the heteroscedasticity.However, the change in data sets also increased the heteroscedasticity aswas evidenced by the correlation coefficients in Table 3. To get an esti-mate for ~, (13) needs to be divided by the square root of the populationvariance. The estimate of this is ~ producing the following relationship.

(14 )A A A

a /a = .189735 + 1.106668ytA A A

As in (12), at/a = when y is approximately .73. So, an increase in slopehas occurred while the "fulcrum" remained the same. At this point, NUN wasrun on (lO)using (14) as an estimate of~. Figure 15 shows the estimatednon-linear equation overlayed on the transformed data. Figure 16 shows thecorresponding residual plot which, visually, does not seem to depict anycorrelation with time. However, the correlation coefficients in Table 3indicate a significant positive relationship. While not entirely successfulit can be seen that the weighted regression using y is adjusting for nearlyall the heteroscedasticity.Table 3 also shows that the MSE of .0309 is slightly less than the corres-ponding unadjusted MSE of .0315. While this probably evidences a reductionin the bias of a2, there is most likely an offsetting affect, as discussedearlier, because of the error in estimating the components used in obtaining(14). The estimate of a. is approximately 1.7% higher than in the comparableunadjusted model.

20

So far, the weighted regression has utilized a relationship between thevariance and the estimated values of y obtained from an unweighted re-gression. There also exists a strong relationship between the varianceand the time variable. A linear regression of at on t was performed anda weighted regression was run using the resultant linear equation as anestimate of~. Results were similar to those obtained by using (12)in that a large share of the heteroscedasticity was adjusted for butcorrelations remained significantly different from zero. An additionalproblem which occurred when weighting by time was that the residualsretained a very strong correlation with y. In the unadjusted model,heteroscedasticity is evidenced not only by the relationship betweenresiduals and time but also by the correlation between residuals and theestimated y values. The residual plots using time were very similar to

~ A

those using y in all the log models and the weighted regression with yA

and, for that reason, the residuals versus y were not shown in the Figures.Since the weighted regression using the time variable retained a strongcorrelation between the residuals and y, it will not be presented in thisreport.An Application To ForecastingThe main purpose of the growth model is to provi de a forecast of dry headweight at maturity. The earlier a reliable forecast can be made, thebetter. Some of the previously discussed methods of adjusting for hetero-scedasticity are more suited to a forecasting mode than others. Among thelog transformations, (7) is the preferred model to use. It's only require-ment is knowledge of the range of the data. An interval of 0 to 2 gramswas used to find values for A1 and B1 but a simple computer program can bewritten to solve for A1 and B1 using the minimum and maximum dry headweight so that it isn't necessary to examine the data prior to using thelog adjustment. If (7) doesn't adjust for all the heteroscedasticity, adouble log transformation can be applied using the same logic that was em-ployed to obtain (9). (8) would not be particularly useful in a forecast-ing mode since it requires some idea of the value of u. The weighted leastsquares regression approach is fully applicable to forecasting and we willrefer to this as the YHAT adjustment.

21

The data set used in this example is the one used earlier in which aquestionable sample was deleted. This data set and one containing onlythe first four weeks of data will be used to demonstrate how the dryhead weight is forecasted. The fourth week of data corresponds to thefirst week of June, 1977. Harvest was not complete until early Julyand actual harvest data was available somewhat later so, with fourweeks of data, we are forecasting at least one month ahead. The unad-justed model and the YHAT, log and double log adjustments were run withfour weeks and all the data. The results are summarized in Table 5.

~The a'S are consistently higher (about 9%) in all cases with four weeksof data. While this could result from a change in growing conditionsduring the latter part of the season, most likely the higher a'S are dueto a shortage of data points where the growth curve begins to go asymp-totic. This is the penalty of estimating a with most of the data con-centrated on the lower half of the curve. The relative standard errorsdo indicate that the parameter estimates are fairly stable. Rememberthat an expanded data set was used throughout this report which inflatesthe degrees of freedom and reduces the magnitude of the relative standarderror. Therefore, the relative standard errors in Table 5 imply morestability than really exists. When we are mainly interested in forecasting,the data set would not be expanded.As was pointed out earlier, adjustment for heteroscedasticity increasesthe relative standard error of a. This makes the relative standard errora questionable criteria for selection of the preferred model in a partic-ular situation. The relative standard error can be used as an indicationof stability as the growing season progresses and more data becomes avail-able but, beyond t~at, its importance should not be overemphasized.The correlation coefficients then become the primary criteria to use inestablishing a model preference. With four weeks of data, the double logadjustment is the only one that causes a nonsignificant correlation betweenthe absolute value of the residuals and time. When all the data is in-cluded, the double log overadjusts and the YHAT adjustment becomes thepreferred model. The YHAT adjustment has a significant correlation at the5% level but not at the 1% level.

22

TABLE 5Unadj YHAT Log Double Log

4 wee ks All 4 weeks All 4 weeks All 4 weeks All

Regress ionMSE .0212 .0315 .0215 .0309 .0194 .0301 .0160 .0213R2 .9411 .9453 .9374 .9417 .9625 .9714 .9772 .9868

Param Estisa 1 .1339 1.0338 1.1413 1.0510 1.1220 1.0271 1.1191 1.0287

oh/a 4.3 1.5 5.2 1.7 5.0 1.7 5.9 2.1a

N S 4.5117 4.5903 4.5141 4.3878 4.4937 4.4008 4.5384 4.3736wh h

°a/S 4.6 5.3 5.0 3.0 4.8 3.3 5.9 2.6

p .9049 .8858 .9058 .8915 .9063 .8911 .9075 .8951h hOh/p .6 .5 .6 .4 .6 .4 .6 .4

pCarrel ation

Rp • {I1 70 .3349 .1250 .0428 .2372 .1239 .0198 - .1195

Prob >IR I .0001 .0001 .0001 .0333 .0001 .0001 .5238 .0001P

Rs .4307 .3409 .1745 .0754 .2578 .1337 .0623 -.0806

P rob > IR I .0001 .0001 .0001 .0009 .0001 .0001 .0448 .0004s

The R2 values are high in all cases and the MSE's aren't directly com-parable for reasons previously mentioned. On the basis of the corre-lation significances, the double log model is preferred when fittingfour weeks of data while the YHAT adjustment provides the best resultswith the complete data set.ConclusionsIn the application of the two methods to the growth data used in thisreport, one comparison needs to be made. The estimate of a from (9) was1.0280 while the weighted regression on the comparable data set (i .e.(12)) provided a = 1.0479, a difference of nearly 2%. The log estimatewas less than the unadjusted a and the weighted estimate was greater.However, it should also be recognized that the a from (9) is well withina 95% confidence interval on the a from (12) and vice versa. This makesit safe to say that the two adjusted a'S are not significantly differentat the 5% level. The 2% difference in a'S with the complete data set issimilar when only four weeks of data is used.The log transformation and weighted regression have demonstrated the abilityto adjust for at least the largest share of the heteroscedasticity. Itshould be stressed that the success of these two methods depends to a largeextent on the particular set of data being analyzed. However, the generalmethods employed should be of help in analyzing other data sets.

24

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o 8 12 20 2~ 28 32 36 40 44 48 52 56 60____ • __ • -. u • __ • ~ ._. - -- --._-- ---------.----

Time Since Flowering (days)

----- -------------S-T .-r 1ST Ie. L ---.--.A N A L Y S I S S Y S T E M

___________ u __ PLOT OF V*TPLOT OF YHAT*T

LEGEND: #.. " lOBS, I) "'_2 QBSLHC, _SYMBOL USED IS •

-I --1 .32 +

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PCOT-OF-S*X -, ---LE'G-END: A c 1 OPoS,-e-;-Z"OBS, Ele;_____________________ P_L_O_i_O_F~S_H_A._T,_._X S_Y_M_B_OL~QJ~_. _

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to

S T -. TIS 1 I C A L A " A L Y ~ I S $ Y S 1 E '"

PLOT CF S.x LEGE~DI A • lOBS, Il • 2 085, ETC.PLOT OF SHAT.X SYMBOL USED IS •

I0.27 +

IIII

0.24 +I •I ••I A AI •0.21 •II •,I A

o.n +I AI ..", • ...•.

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I &::0.15 •

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I0.12 + •II,I0.09 + •

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0.88 0.96 1.04

S T A T 1 S T ~ C A L II 1'/ A L Y S I S S Y S T E M

PLOi OF Y·T LEGENDr " c 1 08S. 8 c 2 ObS, ETC.PLOT OF YHAT·T SYM80L USED IS •

1.4•••IIII F It It

1.21 +,I Itt It AI: C

1.12 + H HI D DC DG

0 f G I D 8 H"'1 I 1 AE..•. I F GF C 8 ItII)0- 0.96 • FE H Ff.I flC It:c I y F IHD D NIH E H eII) I AB C E H 8 F AA H A H .."QI

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QI , BEE • F U aL I I E KEt C3 I " Ghl1lH DZI 8 FIII- , •• K W L L t t F

t ZE IX E IN tEe E E0.1tI • 0 RE G ,jf t GELG 8 '" H S G E D I

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E J0.32 • I C

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0.16 +I,II

0.00 +

-------+ -----.----+----+ ----+ -- ---+ -- --+ ---- +--- -+----+----'+-----. ----...---- +---..-----.---_.o 8 lZ 16 20 2" 28 32 36

Time Since Flowering (days)40 •••• ••a S2

(S6 60

ST. TIS T 1 C A L A N A L Y 5 I S ~ y ~ T E M

PLCT OF RES.T LEGE"'DI .. ., 1 08 S. B "' 2 C'llS • ETC.0.6 +

t Ft •t 0 Co

0.5 + 1H

1•0.4 + G 0 C

I Ct • II 8 DG

0.3 + I 0 HI E F E BI C FE GF AI c 1 y H Fe 8

0.2 + f I 0 H 8AI H H M 8 0, Nf;.H C At CF E IN EH F • H E H 8;;0 t f G E IV j A Hell ....,

In 0.1 + J TH E P IJI 8 ...•.81<e

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C'I

to IE G 110 E IG C fS I 0 8 8~••• H Z BUF EF E GolQl

3 -c..l + EE RH E I lOLL X F 8Int TG 111 1;/1 0 p 0 FA 8I X 01 1 e EEC Ct 0 L 6 F

-0.2 + CO 6 1 H E L

• DE JE 6 C C CI 1 F S U EE E E FI HK- JF e e

-0.3 + 8'"

o F 10 E 6 0A C 8E 0

E F 0

-0.4 + JtII

-0.5 +

•tt

-0.6 •------+----+--.--. --- +----+ - --+ ----+----+-----.----+-----. ----+---+----+--+---_.o • 8 12 It> 20 24 28 32 3b

Ti me Sin ce Flowe ri n 9 (Days)40 ••• 52 5t> 60