07 Lecture 02 Growth

07 Lecture 02 Growth - EXST025 - Biological Population...

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Unformatted text preview: EXST025 - Biological Population Statistics Growth Models Page 1 GROWTH CURVES There are many candidate models for growth curves. Linear models will often work adequately for short segments of the live span. Exponential models often fit well for initial growth stages. If only early stage growth is available an asympotic model, such as the the vonBertalanffy, is inadvisable since the asymtote will not be well estimated and may be misleading. Power models and polynomials have also been used to describe segments of growth. These models do not have a very good interpretation for their parameter estimates in terms of growth. Models like the linear, exponential and power models should not be extended much beyond the range of observed data. Most organisms will diminish in growth rate as they age and become asymptotic. vonBERTALANFFY'S GROWTH EQUATION - also asymptotic, but not hyperbolic comparable to the logistic but generally applied to individuals The NEGATIVE EXPONENTIAL CURVE (generic) Size 100 90 80 70 60 50 40 30 von Bertalanffy Growth Models 20 10 0 0 1 2 3 4 5 Age 6 The vonBertalanffy growth model is a member of a family of growth curves (with the Gompertz, Logistic and Richard's). It also has interpretatble parameters and can be extended beyond the range of observed data (since it approaches an asymptote). Page 1 EXST025 - Biological Population Statistics Growth Models Page 2 Parameter interpretation The dependent variable is a measure of size, usuall? length at age (lt ) or s9metimes weight at age (wt )ß and the ind/pendent variable is a measure of age or time (t). a) Each model will have an intercept, expressed as either a value on the Y axis or the X axis. Ricker points out that this is something of a "theoretical value", not necessarily to be take literally. In fisheries these are usually represented as L0 or t0 , but I will use a more generic "2 b) Each of these models will have an asymptote which is interpreted as the mean size of the oldest age group. In fisheries these are usually represented as L, but I will use a more generic "! c) Each model will have a growth rate parameter. This will be represented as "1 in my models, but is usually a "K" in fisheries models. d) Richards model also has a "shape parameter". The commonest form of the equation is l> = "! 1 e"" (t "# ) ‘ but we can also write l> = "! "! e"" (t "# ) Note that "! fixes the asymptote, and "# fixes the “intercept". In fishery applications the intercept is usually called L! or t0 , depending on if it is expressed as a Y axis intercept expressed as an initial length ÐL! Ñ or an X axis intercept expressed as an initial age Ðt0 Ñ with a theoretical length of zero. The value that determines how fast the line goes from "# to "! is the parameter "" . SO "" IS THE “GROWTH RATE". It is an instantaneous rate, a proportionally increasing rate. Page 2 EXST025 - Biological Population Statistics Growth Models Page 3 Two fishes that have the same intercept and same asymptote could have different growth rates. This would result in two different growth curves, both approaching the same asymptote, but at different rates. Size 0 0 1 2 3 4 5 Age 6 It is possible that two fishes have the same growth rate ("" ), but different asymptotes ("! ). Then the growth rates APPEAR different from a linear perspective, but are the same in terms of proportional growth. Size Age Note that both lines get to their respective asymptotes at the same time Page 3 EXST025 - Biological Population Statistics Growth Models Page 4 There are several modifications of the model which aid in understanding it. l> = "! "! e"" (t "# ) if we subtract "! (L_ ) from both sides and change sides, "! l> = "! e"" (t "# ) we get an EXPONENTIAL DECAY CURVE, WITH INTERCEPT "! B0-lt Age if we further modify by dividing through by "! (L_ ), we get "! l> "! œ e"" (t "# ) EXPONENTIAL DECAY CURVE, WITH INTERCEPT 1 1 (B0-lt)/B0 0 Age Page 4 EXST025 - Biological Population Statistics Growth Models Page 5 One additional useful version is expressed as having an intercept instead of t! l> = "! "! e"" (t "# ) leaving off the "# Ðt! Ñ forces the line through the origin l> = "! "! e"" t Size Age This is an exponential decay curve but goes from zero up to an asymptote instead of down to zero. This is the negative exponential curve. another way of getting a nonzero intercept is to simply ADD A CONSTANT to the initial value of the exponential part of the equation eg. l> = "! (C + "! )e"" t this should give exactly the same values of "! and "" as the previous version In fitting, we do not need the two constants, so C and "! can be combined into a new constant l> = "0 "3 e"" t Remember this version for later. It is called the monomolecular equation and is used in various biological applications outside fisheries. Page 5 EXST025 - Biological Population Statistics Growth Models Page 6 We can redefine the parameters biologically as l> = Lw! L_ eKt where Lw! is not really the “intercept", but is the only term containing the intercept (L! L_ ) Recall from an early discussion of the EXPONENTIAL DECAY CURVE, that it was asymptotic on 0.0. In order to get decay to an asymptote other than 0.0, we had to add a constant. Yi = "# "! e"" Xi which is shaped like The negative form is the generic form of the negative exponential Yi = "# "! e"" Xi This is the second form of the vonBertalanffy growth curve presented above. Note that it has an intercept without the exponentiated "# (to). Page 6 EXST025 - Biological Population Statistics Growth Models Page 7 Now suppose that for whatever reason you noticed that using the negative exponential form on the LOG OF Yi gave a better fit. What is this model like? (using the "biological" notation) log(l> ) = L! + L_ L_ eKt log(l> ) = L! + L_ c1 eKt d but we are not estimating parameters L0 anymore, but rather the LOG of L0 log(l> ) = log(L! ) + L_ c1 eKt d take the antilog l> = L! eL_ c1 e w Kt d for this model, when t = 0, then eKt = 1, so c1 eKt d = 0, and lt = L! when t p _, then eKt p 0, so c1 eKt d p 1, and lt = L! eL_ w w w As a result, L_ = L! eL_ , not Lw_ , though it depends on the value of L_ Page 7 EXST025 - Biological Population Statistics Growth Models Page 8 THIS IS Another important GROWTH model - the GOMPERTZ as with the vonBertalanffy, there is a version derived from the vonBertalanffy with an intercept (instead of t! ), and there is also a version with t! . K(tt! ) l> = L_ ee where K is still negative (as for all the models above). Size Age Page 8 EXST025 - Biological Population Statistics Growth Models Page 9 Shape of the vonBertalanffy on lengths and weights on lengths l> = L_ 1 - e-K(t-t! ) ‘ Size 0 Age $ on weights w> = W_ 1 - e-K(t-t! ) ‘ Size Age The Gompertz is an asymmetric, sigmoid curve, and is shaped more like the vBert on weight. The Gompertz curve is also often more associated (and expressed) as a curve on lengths than weights, but can be used for either. K(tt ) w> = W_ ee ! Page 9 EXST025 - Biological Population Statistics In terms of WEIGHT, recall Growth Models Page 10 W = "! L"" Starting with the vonBartalanffy on lengths, l> = L_ 1 - e-K(t-t! ) ‘ $ $ cube both sides , l$ = L_ 1 - e-K(t-t! ) ‘‘ œ L$ ˆ1 - e-K(t-t! ) ‰ > _ $ cube both sides and multiply by "0 , "0 l$ = "0 L$ ˆ1 - e-K(t-t! ) ‰ > _ Wt = W_ 1 - e-K(t-t! ) ‘ $ where we could use a variable power term "p" (instead of 3), The Beverton - Holt Model yield per recruit model, to be discussed soon, uses this equation. Generalized growth model -- [by Richards (1959) in Gulland (1969)] "7 Wt"7 œ W_ 1 " e-K(t-t! ) ‘ Wt œ W_ 1 " e-K(t-t! ) ‘ "7 " which generates various other growth curves by varying the value of “m" if m = 0 and " = 1, this is the von Bertalanffy growth curve for lengths if m = 2 and " = 1, this is the von Bertalanffy growth curve on weights 3 if m = 2, then we have a form of the logistic for biomass W_ Wt œ 1 Be-Kt where B = Wmax W! W! if m = 0 and t=0 then we get a “monomolecular" equation Wt = W! c1-" e-Kt d which is like a vBert with intercept and as m p 1 we get the Gompertz growth curve log(Wt ) œ log(W_ )c1 - " e-Kt d w -Kt Wt œ W! eW_ (1 - "e ) NOTE that the K is related in every equation (though its value is not the same in the double exponential). Page 10 EXST025 - Biological Population Statistics Growth Models Page 11 The equation above (Wt œ W_ 1 " e-K(t-t! ) ‘ "7 ) can be rewritten as " Lt œ L_ 1 e-K(t-t! ) ‘ p This is called the "Generalized vonBertalanffy". another Generalized growth curve (Schnute 1981) Yt = c! + " e#t d $ a) von Bertalanffy model when ! = -" = Y_ , and # = -K, and t! = 0 lengths when $ = 1 weights when $ = 3 Yt = W_ c1 + eKt d 3 b) Richards model when ! = Y_ , " = Y_ p , t! = 0 and $ = -p c) Gompertz growth curve $ Ä 0 d) logistic growth curve (same as Richards when p = 1) Lt œ L_ 1 Be-Kt where B = Lmax L! L! e) and it will also fit a linear growth Page 11 Likelihood Ratio Test The likelihood ratio test is , where -2log () will follow a Chi square distribution ˆ L ˆ L 0 e with degrees of freedom equal to the number of restricted parameters. n n 1 1 2 i 1 The normal density function is L = 2 e 2 2 ˆ Define the restricted likelihood L 0 yi n 2 Then ˆ ˆ ˆ L n y 2 0 i i 1 2 0 ˆ2 1 1 e 2 2 0 2 2 n 2 1 1 ˆ Define the observed likelihood L ˆ2 e 2 ˆ L 0 2 2 n y 2 i ˆ i 1 2 2 ˆ n 1 1 2 n 2 2 e 2 0 2 n 1 1 2 n 2 2 e 2 2 n 2 2 0 2 This is the likelihood ratio, where H0: = 0 versus H1: = a. If we can assume the variation about the nonlinear model is normally distributed, then the non-linear fit is the maximum likelihood solution and the variance is the unrestricted maximum-likelihood estimator of the parameter. The ratio of the variance estimates, under the null and alternate hypothesis, is then , and -2ln() should follow a chi square test statistic, where the degrees of freedom are the d.f. difference under the null and alternate hypothesis. n 2 n 2 2 2 2 ˆ ˆ ˆ ˆ n 2 Where 2 , then -2 ln() = 2 ln 2 2 ln 2 n ln 2 . This should 0 ˆ ˆ ˆ ˆ 2 0 0 0 follow a Chi square distribution. For the Vermillion Lake Ciscos, the tests of Richard’s model against the 3 parameter models are; Ciscos data set from Ricker Model von Bertalanffy Gompertz Logistic Richards C Total df SS 7 1282 7 652 7 412 6 408 10 1022381 MS 128.2043 65.2310 41.1652 40.7784 R² 99.875 99.936 99.960 99.960 Likelihood Ratio Test Ratio -ln(ratio)*n 0.318 11.45 0.625 4.70 0.991 0.09 df 1 1 1 P>Chi2 0.00071 0.03020 0.75865 Note that for the likelihood estimations the degrees of freedom used in calculating the Mean Square is the TOTAL number of observations (n). Chi square Values. df = 1 df = 2 =0.05 3.8415 5.9915 =0.01 6.63490 9.21034 EXST025 - Biological Population Statistics Growth Models Page 13 Fitting the models Originally designed by von Bertalanffy (1938) to describe physiological theory, an OLD theory (archaic) describe difference in effect of anabolism and catabolism i.e. dw dt = hS - KV states that the change in weight over time (= growth rate) is a function of a) hS = anabolism, positive effect a function of cell surface area b) KV = catabolism, negative effect a function of weight or volume Note: V is a function of LENGTH$ -- Volume of sphere = 1 $ 6D S is a function of LENGTH# -- Surface area = 1D# for the length curve ˜l ˜t = h' - K'l = ˜l$ 1 ˜t l# since, ˜W ˜t = hS - kV ˜l$ ˜t = h'l# - K'l$ ˜l$ 1 ˜t l# œ h' - K'l integration yields the von Bertalanffy growth equation Page 13 EXST025 - Biological Population Statistics Growth Models Page 14 The same result can be derived empirically by observing that integation of the formula ˜l ˜t œ K(L_ - l) Change in length will describe the curve given earlier. This formula describes the change in length as a linear, decreasing function of length (with slope K) . Size (t) Since the theory based on anabolism and catabolism is somewhat archaic, an empirical derivation may be more satisfactory. Given the empirical derivation above (or a version of the old theoretical), we can integrate, and obtain the vonBertalanffy Growth curve. ˜l ˜t œ K(L_ - l) ˜l L_ - l =K ˜t -log (L_ - l) = K ˜ t + c (= a constant) or (L_ - l) = e-K˜t * c or l = L_ - c eK˜t and the constant equals L_ as ˜ t p _ Page 14 EXST025 - Biological Population Statistics Growth Models Page 15 then where ˜ t is defined as present time minus initial time (t - t! ), and the length (at time t) is described by, l> = L_ - L_ eK(t t! ) l> = L_ 1 eK(t t! ) ‘ Which is the form of the vonBertalanffy growth equation where; l> = length at time t L_ = the asymptote or maximum length t! = time 0, actually provides a THEORETICAL intercept K = a growth constant, called the Brody - Bertalanffy growth constant classic case - described in GULLAND (1969) situation; a) actual age (time to t) is not known b) age in some intervals is known (interval length = T, where age is usually from scales, so the interval is T = 1 year between annuli) The modern approach to fitting growth models requires a nonlinear fitting approach. Historically, linearized versions were used. These linearized versions are still useful for quick estimates and to obtain initial values for nonlinear fits. Page 15 EXST025 - Biological Population Statistics Growth Models Page 16 The traditional approach to linearizing and fitting the model requires solving the equation in two steps STEP 1: l> = L_ 1 - e-K(t-t! ) ‘ lt+T = L_ 1 - e-K(t+T-t! ) ‘ lt+T - l> = L_ 1 - e-K(t+T-t! ) ‘ - L_ 1-e-K(t-t! ) ‘ lt+T - l> = L_ - L_ e-K(t+T-t! ) - L_ + L_ e-K(t-t! ) Ê L_ cancel lt+T - l> = L_ e-K(t-t! ) (1 - e-Kt ) and since l> = L_ [1 - e-K(t-t! ) ] l> - L_ = -L_ e-K(t-t! ) lt+T - l> = (L_ - l> )(1 - e-Kt ) lt+T - l> = L_ - L_ e-K - l> - l> e-K Ê l> cancel then lt+T = L_ (1 - e-Kt ) + l> e-Kt this is linear slope = b" = e-KT Note: 0 Ÿ b" Ÿ 1 (see Walford plot) Intercept = L_ (1 - e-KT ) = b! = L_ (1 - b" ) so, KT = -log(b" ) and if T = 1 , then -log(b" ) = K L_ = b! 1-b" Page 16 EXST025 - Biological Population Statistics Growth Models Page 17 Another linear version (related to the vonBertalanffy) is Ford's growth model (empirically derived) Instead of the change in size per unit time being a linear, decreasing function of length, Ford described a situation where the change in size at time t+1 was a constant proportion of the change in size at time t lt+2 lt+1 lt+1 lt eg. = k This “k" is not the same as the Brody K, so it is usually designated wtih a lower case k This proportional reduction results from the relationship lt+1 = L_ (1 k) klt where lt+1 = L_ (1 k) klt lt+2 = L_ (1 k) klt+1 then, with some algebra lt+2 lt+1 = k clt+1 lt d Gulland presented a modification for the graph of this relationship as well lt+1 lt = L_ (1 k) (k-1) lt The Ford “k" and Brody “K" are related such that ek = K Page 17 EXST025 - Biological Population Statistics Growth Models Page 18 WALFORD PLOT - from Ricker 350 300 Size (t+1) 250 200 150 Cisco Growth 100 50 0 100 150 200 250 300 350 Size (t) Relationship between lt+1 and lt can be either Fords or the other linearization L_ occurs where lt+T = l> where the fitted line and diagonal intersect note that the slope of the diagonal = 1 so 0 Ÿ b Ÿ 1 for fitted line usually in fisheries T = 1, so we have a slight simplification of the previous “general" equation the typical fish type of growth -- SUSTAINED DIMINISHING GROWTH and will usually be fit well by this curve L_ is an average maximum size, the average size at any particular age should not pass L_ (but this could happen with small samples). With our analysis up to this point (STEP 1), we only have an estimate of L_ we can also get an estimate of [-log(b" ) = k], but this will be re estimated in STEP 2 we do not yet have an estimate of t Page 18 EXST025 - Biological Population Statistics Growth Models Page 19 alternative expression of the WALFORD PLOT - from Gulland Gulland suggested the following modification of this common plot - instead of fitting the size at time t+1 on the size at time t, - fit the change in size (lt+" lt ) on the size at time t (lt ) THEN Change in length 45 Cisco Growth 40 35 30 25 20 15 10 5 0 -5 100 150 200 250 300 350 Size (t) L_ now occurs where ˜ l> reaches the axis Page 19 EXST025 - Biological Population Statistics Growth Models Page 20 from GULLAND STEP 2 USING L_ estimated from STEP 1 l> = L_ [1 - e-K(t-t! ) ] l> = L_ - L_ e-K(t-t! ) l> - L_ = L_ e-K(t-t! ) this can be linearized with logs (Ricker) (L_ - l> ) L_ note switch in order of numerator = e-K(t-t! ) log’ L__ l> “ = -K(t-t! ) L log’ L__ l> “ = (Kt! ) + Kt L This is linear where we regress log’ L__ l> “ on t L intercept = b! = Kt slope = b" = -K t! = b! b" BEVERTON'S METHOD - in GULLAND and in RICKER; from l> - L_ = -L_ e-K(t-t! ) log(L_ - l> ) = (log(L_ ) + Kt ) - Kt where; the intercept b! = (log(L_ ) + Kt ) the slope b" = - Kt BEVERTON USES “EYE - FITTING" to get L_ other simple estimates can be used (mean of older age groups) There are many methods of fitting the von BERTALANFFY This is one of the “simpler" Page 20 EXST025 - Biological Population Statistics Growth Models Page 21 Problem - The K value from STEP 1 probably won't match the K from STEP 2 a) we can develop other solutions (see Ricker) where the value of L is estimated in different steps and L is derived in several iterations to fit both b) we can apply a non - linear technique to solve the whole equation in one step There is much literature and even computer programs available for use in fitting the von BERTALANFFY growth model. Example using data from Ricker (page 226; Vermillion lake Ciscos). SEE SAS HANDOUT STEP 1: Ricker uses the equation lt+1 = b! b" lt he eliminates some ages for various reasons. His results are lt+1 = 93 0.70 lt Ford coefficient then K = -ln(0.70) = 0.37 Brody coefficient L_ = 315 STEP2: to get t! he then plots (using various values of L_ ) until he gets a straight line. He decides on the values L_ = 309 b! = 5.84 then b! = ln(L_ ) kt! b! ln(L_ ) K œ t! 5.84 5.74 0.41 œ t! I could not duplicate Ricker's results exactly. He apparently leaves off age 2 fish because the point is “off", and age 10 and 11 because of the small sample size. All of this is addressed by weighting. We will examine this in a SAS handout and consider 1) Weighted versus unWeighted 2) Linear versus non-linear Page 21 EXST025 - Biological Population Statistics Growth Models Page 22 Alternative method by BEVERTON EXAMPLE: data from Ricker (page 226) start with regression on Walford type plot lt+1 = b + bl> "RICKER" OBTAINS lt+1 = 93 + 0.70 lt (he has eliminated some ages for various reasons) then K = -ln(0.70) = 0.37 L = 315 according to Ricker using various values of L_ , Ricker plots until the line looks straight He decides on L_ = 309, b = 5.84, since b = ln(L) + Kt b - ln(L) K =t (5.84 - 5.74) / 0.41 = 0.24 I could not duplicate Rickers' results exactly, He apparently uses all groups for the first part, but leaves off age 2 (data point “off", due to possible selection) and ages 10 and 11 (small n) for the second part Other differences in part 2 from my calculations probably due to rounding error Page 22 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 1 dm'log;clear;output;clear'; 2 /**************************************************************/ 3 /*** vonBertalanffy Growth Model - EXST7025 Example ***/ 4 /*** ON Vermillion Lake Ciscoe data - Ricker (pg 226 ff) ***/ 5 /**************************************************************/ 6 OPTIONS NOCENTER PS=512 LS=111 NODATE NONUMBER nolabel; 7 ODS HTML style=minimal body='C:\Geaghan\Current\EXST7025\Spring2008\SAS\CiscoGrowth.html' ; NOTE: Writing HTML Body file: C:\Geaghan\Current\EXST7025\Spring2008\SAS\CiscoGrowth.html 8 9 DATA ONE; INFILE CARDS MISSOVER; 10 TITLE1 'Growth Curves fitted to Vermillion Lake Cisco data'; 11 INPUT AGE N WT LT; LAGE=LOG(AGE); 12 CARDS; NOTE: The data set WORK.ONE has 10 observations and 5 variables. NOTE: DATA statement used (Total process time): real time 0.18 seconds cpu time 0.03 seconds 23 ; 24 PROC SORT DATA=ONE; BY DESCENDING AGE; NOTE: There were 10 observations read from the data set WORK.ONE. NOTE: The data set WORK.ONE has 10 observations and 5 variables. NOTE: PROCEDURE SORT used (Total process time): real time 0.10 seconds cpu time 0.03 seconds 25 PROC PRINT DATA=ONE; TITLE2 'Raw data list'; RUN; NOTE: There were 10 observations read from the data set WORK.ONE. NOTE: The PROCEDURE PRINT printed page 1. NOTE: PROCEDURE PRINT used (Total process time): real time 0.39 seconds cpu time 0.01 seconds 26 Growth Curves fitted to Vermillion Lake Cisco data Raw data list Obs AGE N WT LT LAGE 1 2 3 4 5 6 7 8 9 10 11 10 9 8 7 6 5 4 3 2 2 6 20 54 81 67 52 136 14 101 539 539 525 505 477 462 383 298 193 99 306 299 302 294 289 280 265 241 210 172 2.39790 2.30259 2.19722 2.07944 1.94591 1.79176 1.60944 1.38629 1.09861 0.69315 27 DATA ONE; SET ONE; IF AGE EQ . THEN DELETE; 28 AGE2 = AGE*AGE; AGE3 = AGE*AGE*AGE; 29 LLT= LOG(LT); LTP1=LAG1(LT); DLT = LAG1(LT) - LT; RUN; NOTE: Missing values were generated as a result of performing an operation on missing values. Each place is given by: (Number of times) at (Line):(Column). 1 at 29:54 NOTE: There were 10 observations read from the data set WORK.ONE. NOTE: The data set WORK.ONE has 10 observations and 10 variables. NOTE: DATA statement used (Total process time): real time 0.04 seconds cpu time 0.00 seconds Page 23 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 30 31 PROC SORT DATA=ONE; BY AGE LT; RUN; NOTE: There were 10 observations read from the data set WORK.ONE. NOTE: The data set WORK.ONE has 10 observations and 10 variables. NOTE: PROCEDURE SORT used (Total process time): real time 0.00 seconds cpu time 0.00 seconds 32 33 PROC REG OUTEST=PARMEST DATA=ONE; WEIGHT N; 34 TITLE2 'STEP 1 - LINEAR MODELS'; 35 GULLAND:MODEL DLT = LT; 36 WALFORD:MODEL LTP1 = LT; 37 LINEAR:MODEL LT=AGE; 38 QUADRATC:MODEL LT=AGE AGE2; 39 CUBIC:MODEL LT=AGE AGE2 AGE3; 40 POWER:MODEL LLT=LAGE; 41 EXPONENT:MODEL LLT=AGE; RUN; 42 NOTE: The data set WORK.PARMEST has 7 observations and 13 variables. NOTE: The PROCEDURE REG printed pages 2-8. NOTE: PROCEDURE REG used (Total process time): real time 0.50 seconds cpu time 0.15 seconds Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: GULLAND Dependent Variable: DLT Number of Observations Read Number of Observations Used Number of Observations with Missing Values 10 9 1 Weight: N Source Model Error Corrected Total Root MSE Dependent Mean Coeff Var DF 1 7 8 Analysis of Variance Sum of Mean Squares Square 78547 78547 3583.87451 511.98207 82131 22.62702 18.33898 123.38209 R-Square Adj R-Sq F Value 153.42 Pr > F <.0001 0.9564 0.9501 Parameter Estimates Variable Intercept LT DF 1 1 Parameter Estimate 87.84469 -0.27803 Standard Error 5.69681 0.02245 t Value 15.42 -12.39 Pr > |t| <.0001 <.0001 Page 24 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: WALFORD Dependent Variable: LTP1 Number of Observations Read Number of Observations Used Number of Observations with Missing Values Weight: N Source Model Error Corrected Total Root MSE Dependent Mean Coeff Var DF 1 7 8 10 9 1 Analysis of Variance Sum of Mean Squares Square 529650 529650 3583.87451 511.98207 533234 22.62702 268.33333 8.43243 R-Square Adj R-Sq F Value 1034.51 Pr > F <.0001 0.9933 0.9923 Parameter Estimates Variable Intercept LT DF 1 1 Parameter Estimate 87.84469 0.72197 Standard Error 5.69681 0.02245 t Value 15.42 32.16 Pr > |t| <.0001 <.0001 Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: LINEAR Dependent Variable: LT Number of Observations Read Number of Observations Used Number of Observations with Missing Values Weight: N Source Model Error Corrected Total Root MSE Dependent Mean Coeff Var DF 1 7 8 10 9 1 Analysis of Variance Sum of Squares 905641 110490 1016131 125.63538 249.99435 50.25529 R-Square Adj R-Sq Mean Square 905641 15784 F Value 57.38 Pr > F 0.0001 0.8913 0.8757 Parameter Estimates Variable Intercept AGE DF 1 1 Parameter Estimate 152.54883 19.24268 Standard Error 13.97222 2.54038 t Value 10.92 7.57 Pr > |t| <.0001 0.0001 Page 25 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: QUADRATC Dependent Variable: LT Number of Observations Read Number of Observations Used Number of Observations with Missing Values Weight: N Source Model Error Corrected Total Root MSE Dependent Mean Coeff Var DF 2 6 8 10 9 1 Analysis of Variance Sum of Mean Squares Square 1012831 506415 3300.22765 550.03794 1016131 23.45289 249.99435 9.38137 R-Square Adj R-Sq F Value 920.69 Pr > F <.0001 0.9968 0.9957 Parameter Estimates Variable Intercept AGE AGE2 DF 1 1 1 Parameter Estimate 83.20398 51.25906 -3.06729 Standard Error 5.61058 2.34198 0.21972 t Value 14.83 21.89 -13.96 Pr > |t| <.0001 <.0001 <.0001 Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: CUBIC Dependent Variable: LT Number of Observations Read Number of Observations Used Number of Observations with Missing Values Weight: N Source Model Error Corrected Total Root MSE Dependent Mean Coeff Var DF 3 5 8 10 9 1 Analysis of Variance Sum of Mean Squares Square 1015593 338531 537.97358 107.59472 1016131 10.37279 249.99435 4.14921 R-Square Adj R-Sq F Value 3146.35 Pr > F <.0001 0.9995 0.9992 Parameter Estimates Variable Intercept AGE AGE2 AGE3 DF 1 1 1 1 Parameter Estimate 59.52897 68.53766 -6.64157 0.22010 Standard Error 5.29059 3.56398 0.71209 0.04344 t Value 11.25 19.23 -9.33 5.07 Pr > |t| <.0001 <.0001 0.0002 0.0039 Page 26 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: POWER Dependent Variable: LLT Number of Observations Read Number of Observations Used Number of Observations with Missing Values 10 9 1 Weight: N Analysis of Variance Sum of Mean Squares Square 19.02713 19.02713 0.61086 0.08727 19.63799 Source Model Error Corrected Total DF 1 7 8 Root MSE Dependent Mean Coeff Var 0.29541 5.50409 5.36705 R-Square Adj R-Sq F Value 218.04 Pr > F <.0001 0.9689 0.9645 Parameter Estimates Variable Intercept LAGE DF 1 1 Parameter Estimate 4.90944 0.39209 Standard Error 0.04226 0.02655 t Value 116.17 14.77 Pr > |t| <.0001 <.0001 Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS The REG Procedure Model: EXPONENT Dependent Variable: LLT Number of Observations Read Number of Observations Used Number of Observations with Missing Values 10 9 1 Weight: N Source Model Error Corrected Total Root MSE Dependent Mean Coeff Var DF 1 7 8 Analysis of Variance Sum of Mean Squares Square 16.61683 16.61683 3.02116 0.43159 19.63799 0.65696 5.50409 11.93582 R-Square Adj R-Sq F Value 38.50 Pr > F 0.0004 0.8462 0.8242 Parameter Estimates Variable Intercept AGE DF 1 1 Parameter Estimate 5.08668 0.08243 Standard Error 0.07306 0.01328 t Value 69.62 6.20 Pr > |t| <.0001 0.0004 Page 27 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 43 proc print data=parmest; 44 var _MODEL_ _TYPE_ _DEPVAR_ _RMSE_ Intercept LT AGE AGE2 AGE3 LAGE; 45 run; NOTE: There were 7 observations read from the data set WORK.PARMEST. NOTE: The PROCEDURE PRINT printed page 9. NOTE: PROCEDURE PRINT used (Total process time): real time 0.03 seconds cpu time 0.00 seconds Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS Obs _MODEL_ _TYPE_ _DEPVAR_ _RMSE_ Intercept 1 2 3 4 5 6 7 GULLAND WALFORD LINEAR QUADRATC CUBIC POWER EXPONENT PARMS PARMS PARMS PARMS PARMS PARMS PARMS DLT LTP1 LT LT LT LLT LLT 22.627 22.627 125.635 23.453 10.373 0.295 0.657 87.845 87.845 152.549 83.204 59.529 4.909 5.087 LT AGE AGE2 -0.27803 0.72197 -1.00000 -1.00000 -1.00000 . . . . 19.2427 51.2591 68.5377 . 0.0824 . . . -3.06729 -6.64157 . . AGE3 LAGE . . . . . . . . 0.22010 . . 0.39209 . . DATA SUMMARY; LENGTH MODEL $ 20; SET PARMEST; KEEP MODEL G Lo L other1 other2; 47 48 MODEL = _MODEL_; 49 IF MODEL EQ 'GULLAND' THEN DO; G=ABS(LT); L =INTERCEPt/G; END; 50 IF MODEL EQ 'WALFORD' THEN DO; G=1-LT; L =INTERCEPt/G; END; 51 IF MODEL EQ 'LINEAR' THEN DO; G=age; Lo=INTERCEPt; END; 52 IF MODEL EQ 'QUADRATC' THEN DO; Lo=INTERCEPt; other1=age2; END; 53 IF MODEL EQ 'CUBIC' THEN DO; Lo=INTERCEPt; 54 other1=age2; other2=age3; END; 55 IF MODEL EQ 'POWER' THEN DO; G=intercept; Lo=0; other1=lage; END; 56 IF MODEL EQ 'EXPONENT' THEN DO; G=AGE; Lo=INTERCEPt; END; RUN; NOTE: There were 7 observations read from the data set WORK.PARMEST. NOTE: The data set WORK.SUMMARY has 7 observations and 6 variables. NOTE: DATA statement used (Total process time): real time 0.01 seconds cpu time 0.00 seconds 57 proc print data=summary; run; NOTE: There were 7 observations read from the data set WORK.SUMMARY. NOTE: The PROCEDURE PRINT printed page 10. NOTE: PROCEDURE PRINT used (Total process time): real time 0.03 seconds cpu time 0.00 seconds Growth Curves fitted to Vermillion Lake Cisco data STEP 1 - LINEAR MODELS Obs MODEL G L Lo 1 2 3 4 5 6 7 GULLAND WALFORD LINEAR QUADRATC CUBIC POWER EXPONENT 0.2780 0.2780 19.2427 . . 4.9094 0.0824 315.955 315.955 . . . . . . . 152.549 83.204 59.529 0.000 5.087 other1 other2 . . . -3.06729 -6.64157 0.39209 . . . . . 0.22010 . . Page 28 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 350 300 Size (t+1) 250 200 150 100 Cisco Growth 50 0 100 150 200 250 300 350 Size (t) 45 Cisco Growth 40 35 Size (t+1) 30 25 20 15 10 5 0 -5 100 150 200 250 300 350 Size (t) Page 29 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 59 DATA PARMEST; SET SUMMARY; IF MODEL EQ 'GULLAND'; KEEP L; RUN; NOTE: There were 7 observations read from the data set WORK.SUMMARY. NOTE: The data set WORK.PARMEST has 1 observations and 1 variables. NOTE: DATA statement used (Total process time): real time 0.00 seconds cpu time 0.00 seconds 61 DATA ONEB; MERGE ONE PARMEST; 62 IF L EQ . THEN L=OLDL; 63 OLDL=L; LADJ = LOG((L-LT)/L); 64 RETAIN OLDL; 65 KEEP AGE N L LT LLT LADJ; RUN; NOTE: There were 10 observations read from the data set WORK.ONE. NOTE: There were 1 observations read from the data set WORK.PARMEST. NOTE: The data set WORK.ONEB has 10 observations and 6 variables. NOTE: DATA statement used (Total process time): real time 0.00 seconds cpu time 0.00 seconds 66 67 PROC PRINT DATA=ONEB; TITLE2 'Processed data list'; RUN; NOTE: There were 10 observations read from the data set WORK.ONEB. NOTE: The PROCEDURE PRINT printed page 11. NOTE: PROCEDURE PRINT used (Total process time): real time 0.04 seconds cpu time 0.00 seconds Growth Curves fitted Processed data list Obs AGE N 1 2 101 2 3 14 3 4 136 4 5 52 5 6 67 6 7 81 7 8 54 8 9 20 9 10 6 10 11 2 to Vermillion Lake Cisco data LT 172 210 241 265 280 289 294 302 299 306 LLT 5.14749 5.34711 5.48480 5.57973 5.63479 5.66643 5.68358 5.71043 5.70044 5.72359 L 315.955 315.955 315.955 315.955 315.955 315.955 315.955 315.955 315.955 315.955 LADJ -0.78610 -1.09259 -1.43871 -1.82466 -2.17333 -2.46143 -2.66660 -3.11976 -2.92504 -3.45752 69 PROC REG DATA=ONEB OUTEST=PARMEST; WEIGHT N; 70 TITLE2 'Step 2 - Regression on Adjusted Lengths'; 71 STEP2:MODEL LADJ = AGE; RUN; NOTE: The data set WORK.PARMEST has 1 observations and 7 variables. NOTE: The PROCEDURE REG printed page 12. NOTE: PROCEDURE REG used (Total process time): real time 0.09 seconds cpu time 0.04 seconds Growth Curves fitted to Vermillion Lake Cisco data Step 2 - Regression on Adjusted Lengths The REG Procedure Model: STEP2 Dependent Variable: LADJ Number of Observations Read Number of Observations Used Weight: N 10 10 Page 30 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Analysis of Variance Source Model Error Corrected Total DF 1 8 9 Root MSE Dependent Mean Coeff Var 0.56133 -1.80316 -31.13006 Sum of Squares 259.34471 2.52069 261.86539 R-Square Adj R-Sq Mean Square 259.34471 0.31509 F Value 823.09 Pr > F <.0001 0.9904 0.9892 Parameter Estimates Variable Intercept AGE 73 74 75 76 77 78 NOTE: NOTE: NOTE: NOTE: DF 1 1 Parameter Estimate -0.17017 -0.32106 Standard Error 0.06189 0.01119 t Value -2.75 -28.69 Pr > |t| 0.0251 <.0001 DATA SUMMARY; MERGE SUMMARY PARMEST; KEEP MODEL L Lo G XINT YINT other1 other2; IF MODEL EQ 'GULLAND' THEN DO; XINT=-INTERCEPT/AGE; YINT=L*(1-EXP(G*XINT)); END; RUN; There were 7 observations read from the data set WORK.SUMMARY. There were 1 observations read from the data set WORK.PARMEST. The data set WORK.SUMMARY has 7 observations and 8 variables. DATA statement used (Total process time): real time 0.00 seconds cpu time 0.00 seconds 79 80 PROC PRINT DATA=SUMMARY; 81 var MODEL L Lo G XINT YINT other1 other2; 82 TITLE2 'Summary data set after linear models'; RUN; NOTE: There were 7 observations read from the data set WORK.SUMMARY. NOTE: The PROCEDURE PRINT printed page 13. NOTE: PROCEDURE PRINT used (Total process time): real time 0.06 seconds cpu time 0.00 seconds Growth Curves fitted to Vermillion Lake Cisco data Summary data set after linear models Obs MODEL L Lo G 1 2 3 4 5 6 7 GULLAND WALFORD LINEAR QUADRATC CUBIC POWER EXPONENT 315.955 315.955 . . . . . . . 152.549 83.204 59.529 0.000 5.087 0.2780 0.2780 19.2427 . . 4.9094 0.0824 XINT YINT other1 other2 -0.53004 . . . . . . 43.2932 . . . . . . . . . -3.06729 -6.64157 0.39209 . . . . . 0.22010 . . Page 31 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Simple growth models 350 Vermillion Lake Cisco example from Ricker Power 300 Cubic Quadratic Length (mm) 250 200 Linear 150 100 50 0 0 1 2 3 4 5 6 7 AGE (years) 8 9 10 11 12 Page 32 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 The default numerical search method is Gauss-Newton which uses a Taylor series expansion to approximate the nonlinear model with linear terms and then applies OLS to estimate the parameters. One alternative is the method of steepest decent which seeks to minimize the least squares criterion (Q) by iteratively determining the direction of change for the regression coefficients. The Marquardt algorithm uses elements of both of these approaches. Large-sample theory (Asymptotic theorem) - When the error terms are NID(0, 2) and the sample size is reasonably large, the sampling distribution of the estimates, g, is approximately normal. The expected value of the mean vector is approximately E(g) = and the variance-covariance matrix is approximated by s2(g) = MSE(D′D)-1. D is the matrix of partial derivatives evaluated at the last least squares application. How large is large enough? No clear answer, but look for cases where (1) iteration to a solution is quick, (2) boot-strapping can be used to examine the distribution of the estimates for normality. 84 85 86 87 88 NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E-12; _WEIGHT_=N; TITLE2 'TRADITIONAL vBert'; PARAMETERS LINF=315 K=0.278 T0=-0.53; MODEL LT = LINF*(1-EXP(-K*(AGE-T0))); RUN; DER.LINF not initialized or missing. It will be computed automatically. DER.K not initialized or missing. It will be computed automatically. DER.T0 not initialized or missing. It will be computed automatically. PROC NLIN grid search time was 0: 0: 0. Convergence criterion met. The data set WORK.PARMEST has 13 observations and 8 variables. The PROCEDURE NLIN printed page 14. PROCEDURE NLIN used (Total process time): real time 0.46 seconds cpu time 0.07 seconds 89 90 PROC NLIN DATA=ONE CONVERGE=10E-12; _WEIGHT_=N; 91 TITLE2 'vBert with intercept'; 92 PARAMETERS LINF=315 K=0.278 L0=43; 93 MODEL LT = LINF-(LINF-L0)*EXP(-K*AGE); 94 RUN; NOTE: DER.LINF not initialized or missing. It will be computed automatically. NOTE: DER.K not initialized or missing. It will be computed automatically. NOTE: DER.L0 not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The PROCEDURE NLIN printed page 15. NOTE: PROCEDURE NLIN used (Total process time): real time 0.09 seconds cpu time 0.03 seconds 95 Growth Curves fitted to Vermillion Lake Cisco data TRADITIONAL vBert The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF K T0 Weighted SS Page 33 EXST7025 ‐ Biological Population Statistics Growth Models 0 315.0 0.2780 1 310.8 0.3418 2 314.0 0.3431 3 314.1 0.3431 4 314.1 0.3431 NOTE: Convergence criterion met. Estimation Summary Method Iterations R PPC(T0) RPC(T0) Object Objective Observations Read Observations Used Observations Missing Source Model Error Corrected Total Parameter LINF K T0 Estimate 314.1 0.3431 -0.3021 -0.5300 -0.2833 -0.3021 -0.3021 -0.3021 Geaghan Page 1 104464 7205.5 1282.1 1282.0 1282.0 Gauss-Newton 4 1.372E-6 1.071E-6 0.000031 1.483E-9 1282.043 10 10 0 DF 2 7 9 Sum of Squares 1021099 1282.0 1022381 Approx Std Error 2.8780 0.0161 0.0945 Mean Square 510549 183.1 F Value 2787.62 Approx Pr > F <.0001 Approximate 95% Confidence Limits 307.3 320.9 0.3050 0.3811 -0.5256 -0.0785 Approximate Correlation Matrix LINF K LINF 1.0000000 -0.9328561 K -0.9328561 1.0000000 T0 -0.7932938 0.9426013 T0 -0.7932938 0.9426013 1.0000000 growth Curves fitted to Vermillion Lake Cisco data vBert with intercept The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF K 0 315.0 0.2780 1 310.8 0.3418 2 314.0 0.3431 3 314.1 0.3431 4 314.1 0.3431 NOTE: Convergence criterion met. Estimation Summary Method Iterations R PPC(L0) L0 43.0000 33.1309 30.8978 30.9187 30.9175 Weighted SS 105136 4516.9 1282.1 1282.0 1282.0 Gauss-Newton 4 2.046E-6 1.321E-6 Page 34 EXST7025 ‐ Biological Population Statistics Growth Models RPC(L0) Object Objective Observations Read Observations Used Observations Missing Source Model Error Corrected Total Parameter LINF K L0 LINF K L0 96 97 98 99 100 NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: Estimate 314.1 0.3431 30.9175 Geaghan Page 1 0.000038 3.299E-9 1282.043 10 10 0 DF 2 7 9 Sum of Squares 1021099 1282.0 1022381 Approx Std Error 2.8780 0.0161 8.1185 Mean Square 510549 183.1 F Value 2787.62 Approx Pr > F <.0001 Approximate 95% Confidence Limits 307.3 320.9 0.3050 0.3811 11.7202 50.1148 Approximate Correlation Matrix LINF K L0 1.0000000 -0.9328561 0.7742829 -0.9328561 1.0000000 -0.9294754 0.7742829 -0.9294754 1.0000000 PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E-12; _WEIGHT_=N; TITLE2 'GOMPERTZ WITH t0'; PARAMETERS LINF=315 K=0.278 T0=0; MODEL LT = LINF * EXP(-EXP(-K*(AGE-T0))); RUN; DER.LINF not initialized or missing. It will be computed automatically. DER.K not initialized or missing. It will be computed automatically. DER.T0 not initialized or missing. It will be computed automatically. PROC NLIN grid search time was 0: 0: 0. Convergence criterion met. The data set WORK.PARMEST has 13 observations and 8 variables. The PROCEDURE NLIN printed page 16. PROCEDURE NLIN used (Total process time): real time 0.06 seconds cpu time 0.03 seconds 101 102 PROC NLIN DATA=ONE CONVERGE=10E-12; _WEIGHT_=N; 103 TITLE2 'GOMPERTZ WITH INTERCEPT'; 104 *Note: LINF=log(315/43) = 2; 105 PARAMETERS LINF=2 K=0.278 L0=43; 106 MODEL LT = L0*EXP(LINF*(1-EXP(-K*AGE))); 107 RUN; NOTE: DER.LINF not initialized or missing. It will be computed automatically. NOTE: DER.K not initialized or missing. It will be computed automatically. NOTE: DER.L0 not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The PROCEDURE NLIN printed page 17. NOTE: PROCEDURE NLIN used (Total process time): real time 0.06 seconds cpu time 0.03 seconds Page 35 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data GOMPERTZ WITH t0 The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF K 0 315.0 0.2780 1 302.9 0.3658 2 305.7 0.4355 3 308.3 0.4398 4 308.3 0.4397 5 308.3 0.4397 NOTE: Convergence criterion met. T0 0 0.4904 0.7887 0.7820 0.7820 0.7820 Weighted SS 105920 67103.1 4867.0 652.3 652.3 652.3 Estimation Summary Method Iterations Subiterations Average Subiterations R PPC(T0) RPC(T0) Object Objective Observations Read Observations Used Observations Missing Gauss-Newton 5 1 0.2 1.204E-6 9.413E-8 5.196E-6 3.914E-9 652.31 10 10 0 NOTE: An intercept was not specified for this model. Source Model Error Uncorrected Total Parameter LINF K T0 Estimate 308.3 0.4397 0.7820 DF 3 7 10 Sum of Squares 34388751 652.3 34389403 Approx Std Error 1.6216 0.0120 0.0353 Approximate Correlation Matrix LINF K LINF 1.0000000 -0.8880047 K -0.8880047 1.0000000 T0 -0.4105970 0.7240778 Mean Square 11462917 93.1871 F Value 123010 Approx Pr > F <.0001 Approximate 95% Confidence Limits 304.4 312.1 0.4112 0.4682 0.6986 0.8654 T0 -0.4105970 0.7240778 1.0000000 Growth Curves fitted to Vermillion Lake Cisco data GOMPERTZ WITH INTERCEPT The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Page 36 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Iterative Phase Iter LINF K 0 2.0000 0.2780 1 1.4536 0.3158 2 1.3096 0.4184 3 1.4137 0.4418 4 1.4103 0.4397 5 1.4104 0.4397 6 1.4104 0.4397 NOTE: Convergence criterion met. Estimation Summary Method Iterations Subiterations Average Subiterations R PPC(L0) RPC(L0) Object Objective Observations Read Observations Used Observations Missing L0 43.0000 64.4771 78.3799 74.9981 75.2424 75.2330 75.2332 Weighted SS 2220749 1539384 125912 705.7 652.3 652.3 652.3 Gauss-Newton 6 1 0.166667 4.972E-7 4.264E-8 2.217E-6 6.68E-10 652.31 10 10 0 NOTE: An intercept was not specified for this model. Source Model Error Uncorrected Total Parameter LINF K L0 Estimate 1.4104 0.4397 75.2332 DF 3 7 10 Sum of Squares 34388751 652.3 34389403 Approx Std Error 0.0328 0.0120 2.7359 Approximate Correlation Matrix LINF K LINF 1.0000000 0.8879183 K 0.8879183 1.0000000 L0 -0.9937163 -0.9294274 Mean Square 11462917 93.1871 F Value 123010 Approx Pr > F <.0001 Approximate 95% Confidence Limits 1.3328 1.4880 0.4112 0.4682 68.7638 81.7025 L0 -0.9937163 -0.9294274 1.0000000 109 PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E-12; _WEIGHT_=N; 110 TITLE2 'Logistic Growth Curve with t0'; 111 PARAMETERS LINF=315 K=0.278 T0=0; 112 MODEL LT = LINF/(1+EXP(-K*(AGE-T0))); 113 RUN; NOTE: DER.LINF not initialized or missing. It will be computed automatically. NOTE: DER.K not initialized or missing. It will be computed automatically. NOTE: DER.T0 not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The data set WORK.PARMEST has 11 observations and 8 variables. NOTE: The PROCEDURE NLIN printed page 18. NOTE: PROCEDURE NLIN used (Total process time): real time 0.04 seconds Page 37 EXST7025 ‐ Biological Population Statistics Growth Models cpu time 114 115 116 117 118 119 NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: NOTE: Geaghan Page 1 0.03 seconds PROC NLIN DATA=ONE CONVERGE=10E-12; _WEIGHT_=N; TITLE2 'Logistic Growth Curve with L0'; PARAMETERS LINF=315 K=0.278 L0=43; MODEL LT = LINF/(1 + ((LINF-L0)/L0)*EXP(-K*AGE)); RUN; DER.LINF not initialized or missing. It will be computed automatically. DER.K not initialized or missing. It will be computed automatically. DER.L0 not initialized or missing. It will be computed automatically. PROC NLIN grid search time was 0: 0: 0. Convergence criterion met. The PROCEDURE NLIN printed page 19. PROCEDURE NLIN used (Total process time): real time 0.06 seconds cpu time 0.03 seconds Growth Curves fitted to Vermillion Lake Cisco data Logistic Growth Curve with t0 The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF K 0 315.0 0.2780 1 292.4 0.4342 2 300.5 0.5342 3 304.1 0.5413 4 304.2 0.5413 NOTE: Convergence criterion met. Estimation Summary Method Iterations Subiterations Average Subiterations R PPC(K) RPC(T0) Object Objective Observations Read Observations Used Observations Missing T0 0 1.0974 1.5332 1.5145 1.5151 Weighted SS 129343 112175 8979.3 411.7 411.7 Gauss-Newton 4 1 0.25 8.692E-6 4.19E-7 0.000417 0.000195 411.6518 10 10 0 NOTE: An intercept was not specified for this model. Source Model Error Uncorrected Total Parameter LINF K T0 Estimate 304.2 0.5413 1.5151 DF 3 7 10 Sum of Squares 34388991 411.7 34389403 Approx Std Error 1.0754 0.0102 0.0215 Mean Square 11462997 58.8074 F Value 194924 Approx Pr > F <.0001 Approximate 95% Confidence Limits 301.6 306.7 0.5171 0.5655 1.4642 1.5659 Page 38 EXST7025 ‐ Biological Population Statistics Growth Models Approximate Correlation Matrix LINF K LINF 1.0000000 -0.8329084 K -0.8329084 1.0000000 T0 0.1268773 0.3165143 Geaghan Page 1 T0 0.1268773 0.3165143 1.0000000 Growth Curves fitted to Vermillion Lake Cisco data Logistic Growth Curve with L0 The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF L0 0 315.0 0.2780 1 193.1 0.2554 2 274.9 0.9380 3 285.5 0.4910 4 302.2 0.5393 5 304.2 0.5413 6 304.2 0.5413 NOTE: Convergence criterion met. Estimation Summary Method Iterations Subiterations Average Subiterations R PPC(L0) RPC(L0) Object Objective Observations Read Observations Used Observations Missing Weighted SS 43.0000 122.9 74.2112 107.9 93.2580 92.9939 92.9954 8308095 4428546 166056 99645.8 1619.7 411.7 411.7 K Gauss-Newton 6 1 0.166667 3.304E-7 1.511E-8 0.000016 2.523E-7 411.6518 10 10 0 NOTE: An intercept was not specified for this model. Source Model Error Uncorrected Total Parameter LINF K L0 Estimate 304.2 0.5413 92.9954 DF 3 7 10 Sum of Squares 34388991 411.7 34389403 Approx Std Error 1.0754 0.0102 1.6233 1.0000000 -0.8329083 0.6571073 -0.8329083 1.0000000 -0.9315397 F Value 194924 Approx Pr > F <.0001 Approximate 95% Confidence Limits 301.6 306.7 0.5171 0.5655 89.1570 96.8338 Approximate Correlation Matrix LINF K LINF K L0 Mean Square 11462997 58.8074 L0 0.6571073 -0.9315397 1.0000000 Page 39 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 121 PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E-12; _WEIGHT_=N; 122 TITLE2 'Generalized von Bertalanffy - LONG'; 123 PARAMETERS LINF=315 K=0.278 L0=0 P=1.001; 124 MODEL LT = LINF * (1 - L0*EXP(-K*AGE))**P; 125 RUN; NOTE: DER.LINF not initialized or missing. It will be computed automatically. NOTE: DER.K not initialized or missing. It will be computed automatically. NOTE: DER.L0 not initialized or missing. It will be computed automatically. NOTE: DER.P not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The data set WORK.PARMEST has 106 observations and 9 variables. NOTE: The PROCEDURE NLIN printed page 20. NOTE: PROCEDURE NLIN used (Total process time): real time 0.10 seconds cpu time 0.06 seconds 126 127 PROC NLIN DATA=ONE OUTEST=PARMEST CONVERGE=10E-12; _WEIGHT_=N; 128 TITLE2 'Generalized von Bertalanffy - SHORT'; 129 PARAMETERS LINF=315 K=0.278 L0=0 P=1.001; 130 MODEL LT=LINF*(1-L0*EXP(-K*AGE))**P; 131 RUN; NOTE: DER.LINF not initialized or missing. It will be computed automatically. NOTE: DER.K not initialized or missing. It will be computed automatically. NOTE: DER.L0 not initialized or missing. It will be computed automatically. NOTE: DER.P not initialized or missing. It will be computed automatically. NOTE: PROC NLIN grid search time was 0: 0: 0. WARNING: PROC NLIN failed to converge. NOTE: The data set WORK.PARMEST has 106 observations and 9 variables. NOTE: The PROCEDURE NLIN printed page 21. NOTE: PROCEDURE NLIN used (Total process time): real time 0.09 seconds cpu time 0.06 seconds I have fitted the generalized von Bertalanffy with both a +1 and -1 as the transition is numerically difficult and not always successful. Note that the denominator of the Logistic is “1-L0*EXP(-K*AGE)”, so the usually positive value of L0 will be fitted as a negative if the Logistic model is successful. Page 40 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data Generalized von Bertalanffy - start P=1 The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF K L0 P 0 315.0 0.2780 0 1.0010 1 327.4 0.2780 0.8483 1.0010 2 318.3 0.3326 0.6656 1.4124 3 318.0 0.3352 0.6511 1.4583 . . . 498 311.9 0.4073 0.0790 16.0529 499 311.9 0.4073 0.0789 16.0922 500 311.9 0.4073 0.0787 16.1168 WARNING: Maximum number of iterations exceeded. Weighted SS 3260158 11070.7 3742.0 3734.3 3113.5 3113.5 3113.4 WARNING: PROC NLIN failed to converge. Estimation Summary (Not Converged) Method Iterations Subiterations Average Subiterations R PPC(P) RPC(P) Object Objective Observations Read Observations Used Observations Missing Gauss-Newton 500 1167 2.334 0.927989 18.8549 18.8275 0.000053 3113.37 10 10 0 NOTE: An intercept was not specified for this model. Source Model Error Uncorrected Total Parameter LINF K L0 P Estimate 311.9 0.4073 0.0787 16.1168 DF 4 6 10 Sum of Squares 34386290 3113.4 34389403 Approx Std Error 10.0297 0.1627 2.1171 445.5 Mean Square 8596572 518.9 1.0000000 -0.9583757 0.9051448 -0.9056719 -0.9583757 1.0000000 -0.9845134 0.9848119 Approx Pr > F <.0001 Approximate 95% Confidence Limits 287.4 336.4 0.00916 0.8055 -5.1017 5.2592 -1074.0 1106.2 Approximate Correlation Matrix LINF K L0 LINF K L0 P F Value 16567.1 0.9051448 -0.9845134 1.0000000 -0.9999981 P -0.9056719 0.9848119 -0.9999981 1.0000000 Page 41 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Growth Curves fitted to Vermillion Lake Cisco data Generalized von Bertalanffy - start P=-1 The NLIN Procedure Dependent Variable LT Method: Gauss-Newton Iterative Phase Iter LINF L0 0 315.0 0.2780 1 327.4 0.2780 2 326.4 0.2820 3 325.3 0.2874 4 324.1 0.2942 . . . 14 303.7 0.5581 15 303.7 0.5566 16 303.7 0.5566 17 303.7 0.5566 NOTE: Convergence criterion met. Estimation Summary Method Iterations Subiterations Average Subiterations R PPC(L0) RPC(L0) Object Objective Observations Read Observations Used Observations Missing P Weighted SS 0 -0.8483 -0.7466 -0.6668 -0.6077 -1.0010 -1.0010 -1.1161 -1.2384 -1.3613 3260158 282236 281809 276811 266320 -2.9016 -2.7965 -2.7989 -2.7987 -0.8422 -0.8716 -0.8720 -0.8721 575.3 410.0 407.8 407.8 K Gauss-Newton 17 21 1.235294 1.721E-6 2.907E-6 0.000088 2.411E-9 407.7838 10 10 0 NOTE: An intercept was not specified for this model. Source Model Error Uncorrected Total Parameter LINF K L0 P Estimate 303.7 0.5566 -2.7987 -0.8721 DF 4 6 10 Sum of Squares 34388995 407.8 34389403 Approx Std Error 2.2942 0.0639 2.3081 0.4592 Mean Square 8597249 67.9640 1.0000000 -0.9276161 0.8772081 -0.8702797 -0.9276161 1.0000000 -0.9887871 0.9847621 Approx Pr > F <.0001 Approximate 95% Confidence Limits 298.1 309.3 0.4002 0.7130 -8.4464 2.8490 -1.9956 0.2514 Approximate Correlation Matrix LINF K L0 LINF K L0 P F Value 126497 0.8772081 -0.9887871 1.0000000 -0.9995605 P -0.8702797 0.9847621 -0.9995605 1.0000000 Page 42 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Three and four parameter growth models 350 Vermillion Lake Cisco example from Ricker 300 Generalized von Bertalanffy Length (mm) 250 200 150 100 50 von Bertalanffy 0 0 1 2 3 4 5 6 7 AGE (years) 8 9 10 11 12 Page 43 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Three parameter growth models 350 Vermillion Lake Cisco example from Ricker 300 Length (mm) 250 200 150 Logistic 100 50 von Bertalanffy 0 0 1 2 3 4 5 6 7 AGE (years) 8 9 10 11 12 Page 44 EXST7025 ‐ Biological Population Statistics Growth Models Geaghan Page 1 Using Richards model as a full model, the other models can be statistically tested as reduced models The preferred method of testing between the full and reduced models is the Likelihood ratio test (Cerrato 1990). for example, where Richards model is the Full model, and the von Bertalanffy is the Reduced model we jointly test H0: p = 1 and H0: β = 1 both models are fitted, and for large samples (Black drum n=2665) ⎛ 2 n⎞ 2 2 L ( Ω0 ) ⎛ σ ⎞ ⎜ ⎛ σ ⎞ ⎟ = n ln ⎛ σ ⎞ λ= = ⎜ 2 ⎟ and −2 ln ( λ ) = −2 ln ⎜ 2 ⎟ ⎜ 2⎟ ⎜ σ0 ⎟ L (Ω) ⎝ σ 0 ⎠ ⎝ σ0 ⎠ ⎜⎝ ⎠ ⎟ ⎝ ⎠ 2 n 2 where σ2 is estimated by MSE and where the degrees of freedom for maximum likelihood is n Ciscoes data set from Ricker (pg 226 ff) Likelihood Ratio Test Model df SS MS* R² Ratio -ln(ratio)*n df P>Chi2 von Bertalanffy 7 1282 128.2043 99.8746022 0.318 11.45 1 0.00071 Gompertz 7 652 65.2310 99.936197 0.625 4.70 1 0.03020 Logistic 7 412 41.1652 99.959736 0.991 0.09 1 0.75865 Richards 6 408 40.7784 99.9601143 C Total 10 1022381 *Calculated with n instead of d.f. α = 0.05 α = 0.01 ChiSq 1 df = 3.84146 6.63490 ChiSq 2 df = 5.99147 9.21034 Page 45 ...
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This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.

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