10 MonteCarloGrowth

10 MonteCarloGrowth - EXST7025 - Biological Population...

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Unformatted text preview: EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 1 Monte Carlo Modeling Objective - Usually used to examine the effectiveness of some analytical tool. Statisticians, in particular, often use this to examine the effectiveness of estimators. Approach - A database is stochastically generated with some known characteristics. The mean, the variance, and other properties are established by the investigator in accordance with his objectives. The data is then analyzed with the technique to be examined. Since the properties of the data set are known, the performance of the analytical tool can be evaluated. Some examples of applications. In statistics a frequently used application is the test of hypothesis. A value of "alpha" is set to establish the rate of type I error. The rate of type II error is usually not known. Suppose we develop a new post-hoc test, or wish to evaluate the performance of existing post-hoc tests such as the LSD, or Tukey's or Scheffe's adjustment. We could first generate data sets to do ANOVA with a given number of treatments, say 5. The assumptions would be met, the data would be normally distributed, homogeneous and independent. Many data sets would be generated. The idea is to generate so many datasets that the characteristics would be evident without probabilistic analysis. First, the data generated would have a true null hypothesis, all means equal. It is then analyzed and the probability of rejecting the Ho would be evaluated with alpha=0.05. Approximately 5% of the cases should be rejected for each post-hoc test. The data can then be modified to determine the performance of the tests under various conditions. Unequal means, unequal variance, non-normal distributions, etc. Cases with unequal means should be rejected. The frequency with which these cases are rejected provides an estimate of power under the varying conditions. Any new estimator or modification in an estimator is a candidate for this type of investigation. Other examples Evaluate the effectiveness of transformations under various conditions (e.g. departures from normality). Evaluate estimates of q under differing fishery conditions (fishing sites at random versus systematic site selection versus guided site selection). The program below evaluates the performance of nonlinear estimates for growth in comparison to some linearized estimates. The program below generates 1000 datasets with 500 points each. Each point has a uniformly distributed, randomly-generated age between 0 and 12. Each data set has 100 points that follow a von Bertalanffy growth equation. The variance about the equation is additive and has a mean of 0 and variance of 500. Note that the seed is known. If ranuni() or rannor() is used, SAS will generate a seed from the computer clock date/time value. This makes the exact data generation nearly impossible to reproduce. After the initial generation, SAS continuously generates new seeds and new random numbers. EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 2 dm'log;clear;output;clear'; options ps=512 ls=100 nocenter nodate nonumber; Title1 'Monte Carlo study of Growth model fit'; Title2 '1000 data sets generated, each with 500 observations'; data raw; do rep = 1 to 1000; do obs = 1 to 500; seed = 75611; age = ranuni(seed)*12; size = 300*(1-exp(-0.3*age))+rannor(seed)*50; IntAge = int(age); output; end; end; run; proc sort data=raw; by rep obs; run; options ps=55 ls=99; data raw2; set raw; if rep le 3; run; PROC PLOT DATA=RAW2; BY REP; PLOT SIZE * AGE; RUN; Title3 'Sample plots of the generated data sets'; PROC PLOT DATA=RAW2; BY REP; PLOT SIZE * IntAGE; RUN; options ps=512 ls=111; Monte Carlo study of Growth model fit 1000 data sets generated, each with 500 Observations rep=1 Plot of size*age. Legend: A = 1 obs, B = 2 obs, etc. size | | 400 + | A A A | A A A B A AA | A A A A A B A A BAAA AAA | A A A C A A A A AAAA AA A A A B B | A A A A BAAAA A AAA A AB BA AABB 300 + A AABAA B ABAAB B AA AB AB C B AB A ABAA B AABAA | A AA A A B A A BCA BA AB A C ABBCA BAA | A AAAB BCA A AACAAAAA AAB C BAA AAA AA AA AAA | BA B A A A AA BB C A A AA D A A AA BB A B A | A C BBA B A AA ABA B A AA A AA AA A AB | A AA A A B B B A ACCAA A BA A B A AA A 200 + A C A D A AA AB AAAA BA B A A A A A | A A A AABB A A AAAAA A A A AAA A | A A A AAB BA A A B A | AAAAG A ABB AB A A A A A A A | AAAA BA B B A | A B B AA A 100 + AA BBAA E A AA A | A ABAB A A AA A | AA AB AAA A A A | A BACB A A A | AACA A A A | A A A A 0 + A | AB BAA | A A | A A | | -100 + | ---+------------+------------+------------+------------+------------+------------+-0 2 4 6 8 10 12 age EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 3 rep=2 Plot of size*age. Legend: A = 1 obs, B = 2 obs, etc. size | | 400 + A | A A | A A A AA AA A AB A | AA AC AA A | A A B A A A A A AA A | B A AAA B BA A B CAABA A B ABAA 300 + A AA ABA A ABA A A AABABAC AAAB AA AB | A A AAAA BA AABBA B A B CABA AACA AABAA B | B A A AB AA B CAA A AAAAAA A AC AAAAAAA AAA A | A A AAA A B A AB A A ADAAA A B CAAAAB AA BA A A | AA A B CAAACAACAA BAB A BC A A A B D A A A | A BAA BAAA BB B A AB AAAABA A BA A B A A A 200 + BAA BAABA BAAAAB B A A B BAA AA A A A A A A A | AA C AABABC ABA A AAA A A AA A A | A AA A ABC AAAA AA A AA A B A AA A A | A CA B AA AAA A A | AA AACA A A CA A | BAA BBAB AAAB A A 100 + A A B A A | A A AB AA B AA | A A A A | ACBAA C A | A AA A | A A BAA 0 + AA | AAAA | A A | B A | | A -100 + ---+------------+------------+------------+------------+------------+------------+-0 2 4 6 8 10 12 age Monte Carlo study of Growth model fit 1000 data sets generated, each with 500 observations Sample plots of the generated data sets rep=1 Plot of size*IntAge. Legend: A = 1 obs, B = 2 obs, etc. size | | 400 + | A A A | A B B A B | A B C B A E C | A B E B D D C B | A A B F C C F E 300 + G E G D H D G F | C B B A G D C F J | A E G G F G D B E | C C B F D B F B E D | A H B A B F C B C D | C B F G D D C B A 200 + D A E E D E A A B A | A C F C D A A B B | B A E C C A | K F D B A B A | F C B A | E C 100 + B H E B | A F C B | E D B | I B A | G B | C A 0 + A | G | B | B | | -100 + | ---+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-0 1 2 3 4 5 6 7 8 9 10 11 IntAge EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 4 rep=2 Plot of size*IntAge. Legend: A = 1 obs, B = 2 obs, etc. size | | 400 + A | B | A A A B B D A | B B B B A | A C B B A C | B D E C G D E 300 + A E B E A J F E | A C E G E G G G | C D G C F B C G D | A D D D G E I B C B | C C J F G E C D C | A B F G A F E D C B 200 + G H F B F B C B B | F J C D A A B B | C G E B A D A C A | A F C B A A | G C D A | D G E B 100 + A C B | B E B B | A B A | H C A | C A | D B 0 + B | D | B | B A | | A -100 + ---+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-0 1 2 3 4 5 6 7 8 9 10 11 IntAge PROC NLIN DATA=raw OUTEST=PARMEST1 maxiter=500 noprint; by rep; Title2 'Nonlinear fits to the data sets - decimal age'; PARAMETERS LINF=300 K=0.3 T0=0; MODEL SIZE = LINF*(1-EXP(-K*(AGE-T0))); RUN; DATA PARMEST1; SET PARMEST1; KEEP LINF T0 K SSE; IF _TYPE_ NE 'FINAL' THEN DELETE; SSE=_SSE_; RUN; proc sort data=PARMEST1; by linf t0 k; run; proc univariate DATA=PARMEST1 normal plot; var linf t0 k; Title3 'Summarized data for the nonlinear fits'; run; EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 5 Monte Carlo study of Growth model fit Nonlinear fits to the data sets - decimal age Summarized data for the nonlinear fits The UNIVARIATE Procedure Variable: LINF Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 300.184439 6.34992376 0.16089344 90150978.5 2.11534075 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean 300.1844 Std Deviation Median 300.1323 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 1494.926 Sign M 500 Signed Rank S 250250 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling 6.34992 40.32153 42.39651 8.49712 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 --Statistic--W 0.996519 D 0.023214 W-Sq 0.094327 A-Sq 0.7396 -----p Value-----Pr < W 0.0258 Pr > D >0.1500 Pr > W-Sq 0.1370 Pr > A-Sq 0.0549 Quantiles (Definition 5) Quantile Estimate 100% Max 323.245 99% 316.407 95% 311.084 90% 307.951 75% Q3 304.278 50% Median 300.132 25% Q1 295.781 10% 292.170 5% 290.209 1% 285.850 0% Min 280.848 Extreme Observations ------Lowest----Value Obs 280.848 1 281.416 2 282.066 3 282.226 4 282.405 5 1000 300184.439 40.3215317 0.37116673 40281.2102 0.20080222 -----Highest----Value Obs 318.298 996 320.389 997 320.685 998 321.649 999 323.245 1000 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Histogram Boxplot 323+* .* .* .*** .**** .***** .********** 309+*********** .********************* .********************************** .************************************** .******************************************** .******************************************** .*********************************** 295+************************************ .******************** .*************** .******** .***** .** .** 281+* ----+----+----+----+----+----+----+----+---* may represent up to 3 counts Geaghan Page 6 1 3 1 7 11 15 29 31 63 102 113 130 131 104 107 59 45 22 15 4 5 2 0 0 0 0 | | | | | +-----+ | | *--+--* | | | | +-----+ | | | | | 0 0 Normal Probability Plot 323+ * | * | * | *** | ***++ | ***+ | **** 309+ *** | **** | ***** | **** | **** | **** | *** 295+ ***** | **** | ***** | *** | ***** |+** |* 281+* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 7 Monte Carlo study of Growth model fit Nonlinear fits to the data sets - decimal age Summarized data for the nonlinear fits The UNIVARIATE Procedure Variable: T0 Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 -0.0010626 0.10696607 -0.3869588 11.4314283 -10066.891 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean -0.00106 Std Deviation Median 0.00612 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t -0.31413 Sign M 30 Signed Rank S 5963 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling 0.10697 0.01144 0.79884 0.14086 -----p Value-----Pr > |t| 0.7535 Pr >= |M| 0.0620 Pr >= |S| 0.5142 --Statistic--W 0.991258 D 0.050536 W-Sq 0.281151 A-Sq 1.675404 -----p Value-----Pr < W <0.0001 Pr > D <0.0100 Pr > W-Sq <0.0050 Pr > A-Sq <0.0050 Quantiles (Definition 5) Quantile Estimate 100% Max 0.31075322 99% 0.22043663 95% 0.16243904 90% 0.12970125 75% Q3 0.07140889 50% Median 0.00611816 25% Q1 -0.06945273 10% -0.13673465 5% -0.18460372 1% -0.29090965 0% Min -0.48809026 Extreme Observations ------Lowest-----Value Obs -0.488090 633 -0.348317 865 -0.335127 997 -0.334481 733 -0.331098 644 1000 -1.0625532 0.01144174 0.50448758 11.4302993 0.00338256 ------Highest----Value Obs 0.253595 295 0.261864 90 0.266142 19 0.292442 199 0.310753 305 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Histogram Boxplot 0.325+* .* .**** .********* .********************** .****************************** .******************************************* .******************************** -0.075+*************************** .******************** .******** .***** .*** .** . . -0.475+* ----+----+----+----+----+----+----+----+--* may represent up to 5 counts Geaghan Page 8 1 4 16 42 108 148 211 157 135 98 39 21 12 7 0 0 | | | +-----+ *-----* | + | +-----+ | | | 0 0 1 Normal Probability Plot 0.325+ * | +* | +****** | ****** | ****** | ***** | ******* | *****+ -0.075+ ***** | ****** | +**** | +++**** |+**** |** | | -0.475+* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 0 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 9 Monte Carlo study of Growth model fit Nonlinear fits to the data sets - decimal age Summarized data for the nonlinear fits The UNIVARIATE Procedure Variable: K Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 0.3013297 0.02417052 0.26165061 91.3832172 8.02128637 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean 0.301330 Std Deviation Median 0.301048 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 394.2357 Sign M 500 Signed Rank S 250250 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling 0.02417 0.0005842 0.16138 0.03248 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 --Statistic--W 0.99508 D 0.024572 W-Sq 0.125195 A-Sq 1.025799 -----p Value-----Pr < W 0.0025 Pr > D 0.1472 Pr > W-Sq 0.0517 Pr > A-Sq 0.0108 Quantiles (Definition 5) Quantile Estimate 100% Max 0.390298 99% 0.363897 95% 0.344357 90% 0.331944 75% Q3 0.316649 50% Median 0.301048 25% Q1 0.284165 10% 0.270991 5% 0.264620 1% 0.249356 0% Min 0.228920 Extreme Observations ------Lowest-----Value Obs 0.228920 999 0.231714 1000 0.235704 992 0.239771 995 0.240738 997 1000 301.329699 0.00058421 0.18192614 0.58362973 0.00076434 ------Highest----Value Obs 0.374532 11 0.374895 19 0.376330 5 0.379449 2 0.390298 9 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Histogram Boxplot 0.395+* . .* .** .***** .******** .************* .************************ .********************************** .******************************************** .************************************** .*********************************** .************************** .**************** .****** .** .* 0.225+* ----+----+----+----+----+----+----+----+---* may represent up to 4 counts Geaghan Page 10 1 4 8 19 31 51 93 134 174 151 140 102 61 21 6 3 1 0 0 0 | | | | +-----+ *--+--* | | +-----+ | | | | 0 0 Normal Probability Plot 0.395+ * | | * | **** | ****++ | ****++ | **** | ***** | ***** | ***** | ***** | ****** | ***** | ******* | *****+ |***+ |* 0.225+* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 proc sort data=raw; by rep obs; run; PROC NLIN DATA=raw OUTEST=PARMEST2 maxiter=500 noprint; by rep; Title2 'Nonlinear fits to the data sets - integer age'; PARAMETERS LINF=300 K=0.3 T0=0; MODEL SIZE = LINF*(1-EXP(-K*(IntAGE-T0))); RUN; DATA PARMEST2; SET PARMEST2; KEEP LINF T0 K SSE; IF _TYPE_ NE 'FINAL' THEN DELETE; SSE=_SSE_; RUN; EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models proc sort data=PARMEST2; by Geaghan Page 11 linf t0 k; run; proc univariate DATA=PARMEST2; var linf t0 k; Title3 'Summarized data for the nonlinear fits'; run; Monte Carlo study of Growth model fit Nonlinear fits to the data sets - integer age Summarized data for the nonlinear fits The UNIVARIATE Procedure Variable: LINF Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 300.15825 6.53482255 0.17991704 90137636.4 2.17712575 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean 300.1583 Std Deviation Median 300.1131 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 1452.501 Sign M 500 Signed Rank S 250250 6.53482 42.70391 43.73255 8.56956 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 Quantiles (Definition 5) Quantile Estimate 100% Max 323.590 99% 316.471 95% 311.548 90% 308.484 75% Q3 304.341 Extreme Observations ------Lowest----Value Obs 279.857 1 281.352 2 281.380 3 282.866 4 283.126 5 1000 300158.25 42.7039058 0.30034306 42661.2019 0.20664923 50% Median 25% Q1 10% 5% 1% 0% Min -----Highest----Value Obs 318.332 996 320.546 997 321.541 998 322.505 999 323.590 1000 300.113 295.772 291.825 289.837 284.566 279.857 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 12 Monte Carlo study of Growth model fit Nonlinear fits to the data sets - integer age Summarized data for the nonlinear fits The UNIVARIATE Procedure Variable: T0 Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 -0.4878157 0.1194947 -0.3906396 252.228889 -24.495869 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean -0.48782 Std Deviation Median -0.47891 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t -129.094 Sign M -500 Signed Rank S -250250 0.11949 0.01428 0.78474 0.15613 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 Quantiles (Definition 5) Quantile Estimate 100% Max -0.182535 99% -0.232745 95% -0.303470 90% -0.343102 75% Q3 -0.405964 Extreme Observations ------Lowest-----Value Obs -0.967279 567 -0.921458 999 -0.881968 985 -0.859813 882 -0.853720 988 1000 -487.81573 0.01427898 0.28786232 14.264705 0.00377875 50% Median 25% Q1 10% 5% 1% 0% Min ------Highest----Value Obs -0.216947 345 -0.213541 77 -0.211591 172 -0.205040 140 -0.182535 14 Monte Carlo study of Growth model fit Nonlinear fits to the data sets - integer age Summarized data for the nonlinear fits The UNIVARIATE Procedure Variable: K Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 0.30177374 0.02576957 0.27276049 91.7307984 8.53936874 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean 1000 301.773742 0.00066407 0.16136085 0.6634068 0.00081491 -0.478909 -0.562090 -0.644934 -0.696213 -0.803165 -0.967279 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Basic Statistical Measures Location Variability Mean 0.301774 Std Deviation Median 0.300655 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 370.3175 Sign M 500 Signed Rank S 250250 Geaghan Page 13 0.02577 0.0006641 0.17034 0.03580 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 Quantiles (Definition 5) Quantile Estimate 100% Max 0.398232 99% 0.365108 95% 0.348289 90% 0.335119 75% Q3 0.319080 50% Median 0.300655 25% Q1 0.283283 10% 0.269512 5% 0.264109 1% 0.243177 0% Min 0.227894 Extreme Observations ------Lowest-----Value Obs 0.227894 998 0.230174 985 0.231524 999 0.232350 1000 0.235149 994 ------Highest----Value Obs 0.373791 1 0.381268 13 0.382009 3 0.384751 14 0.398232 8 proc sort data=raw; by rep intage; run; proc means data=raw noprint; by rep intage; var size; output out=means mean=sizemean stderr=se var=sizevar n=sizen; run; data means2; set means; if rep le 3; run; PROC print DATA=means2; Title3 'Sample print of the means for the generated data sets'; RUN; EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 14 Monte Carlo study of Growth model fit Nonlinear fits to the data sets - decimal age Sample print of the means for the generated data sets Obs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 rep 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 Int Age 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 _TYPE_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 _FREQ_ 39 52 32 46 40 43 42 44 45 31 41 45 33 35 48 51 41 41 49 44 39 35 46 38 40 43 41 42 40 44 45 39 39 43 42 42 sizemean 26.678 116.582 155.210 185.482 239.490 241.219 267.474 273.265 277.523 285.941 289.758 286.982 30.218 101.162 162.076 186.258 224.817 253.082 248.346 262.792 262.378 292.295 281.306 284.894 30.231 116.188 157.969 200.333 229.487 243.815 255.492 265.930 275.018 299.758 295.373 297.590 se 6.38382 6.20435 9.41839 8.81655 8.40938 7.29181 8.71038 7.54723 7.23921 9.80106 6.99613 6.68442 9.18995 8.87548 7.31008 6.47938 5.90497 8.38522 7.05622 7.35098 8.66140 6.69268 7.60925 7.93943 7.07662 8.18911 7.59212 7.27777 6.12457 7.31496 8.12963 9.58636 7.33798 7.08208 7.05406 7.81377 PROC SORT DATA=means; BY rep DESCENDING IntAGE; DATA means; SET means; BY rep DESCENDING IntAGE; Title2 'Linear fits to the data sets'; LT = sizemean; IntAGE2 = IntAGE*IntAGE; IntAGE3 = IntAGE*IntAGE*IntAGE; if lt lt 300 then LADJ = LOG((300-LT)/300); else ladj = .; LLT= LOG(LT); LTP1 = LAG1(LT); DLT = LAG1(LT) - LT; if first.rep then LTP1 = .; if first.rep then DLT = .; else do; end; RUN; PROC SORT DATA=means; BY rep IntAGE; data means2; set means; if rep le 3; run; sizevar 1589.37 2001.69 2838.60 3575.66 2828.71 2286.33 3186.57 2506.27 2358.28 2977.88 2006.78 2010.66 2787.02 2757.09 2564.99 2141.10 1429.62 2882.79 2439.72 2377.62 2925.77 1567.72 2663.43 2395.32 2003.14 2883.65 2363.25 2224.57 1500.41 2354.38 2974.09 3584.03 2099.99 2156.70 2089.91 2564.31 sizen 39 52 32 46 40 43 42 44 45 31 41 45 33 35 48 51 41 41 49 44 39 35 46 38 40 43 41 42 40 44 45 39 39 43 42 42 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 15 PROC print DATA=means2; Title3 'Sample print of the means for the generated data sets'; RUN; options ps=52 ls=111; PROC plot DATA=means2; BY rep; plot dlt * IntAge; run; Title3 'Sample print of the means for the generated data sets'; RUN; options ps=512 ls=111; PROC SORT DATA=means; BY rep IntAGE; RUN; PROC REG OUTEST=PARMEST3 DATA=means noprint; by rep; TITLE2 'STEP 1 - Gulland'; MODEL DLT = LT; RUN; DATA PARMEST3; LENGTH MODEL $ 20; SET PARMEST3; MODEL = 'Gulland'; K=ABS(LT); Linf =INTERCEPT/K; RUN; proc univariate DATA=PARMEST3 normal plot; var linf k; Title3 'Summarized data for the Linear fits (Step 1)'; run; Monte Carlo study of Growth model fit STEP 1 - Gulland Summarized data for the Linear fits (Step 1) The UNIVARIATE Procedure Variable: Linf Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 297.972677 7.68498781 0.39872484 88846716.1 2.57909144 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean 297.9727 Std Deviation Median 297.6173 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 1226.121 Sign M 500 Signed Rank S 250250 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling 1000 297972.677 59.0590376 0.49920486 58999.9785 0.24302065 7.68499 59.05904 52.58983 10.24881 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 --Statistic--W 0.990085 D 0.035887 W-Sq 0.245679 A-Sq 1.528039 -----p Value-----Pr < W <0.0001 Pr > D <0.0100 Pr > W-Sq <0.0050 Pr > A-Sq <0.0050 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 16 Quantiles (Definition 5) Quantile Estimate 100% Max 330.860 99% 317.877 95% 310.731 90% 308.070 75% Q3 302.903 50% Median 297.617 25% Q1 292.654 10% 289.077 5% 286.189 1% 281.684 0% Min 278.270 Extreme Observations ------Lowest----Value Obs 278.270 915 278.803 271 279.677 57 279.820 929 280.834 158 -----Highest----Value Obs 320.845 141 325.251 500 325.535 715 330.838 162 330.860 730 Histogram Boxplot 332.5+* .* .* .** .******* .******************** .*********************************** .***************************************** .**************************************** .****************** .****** 277.5+* ----+----+----+----+----+----+----+----+* may represent up to 6 counts 2 2 3 12 40 115 210 241 235 103 33 4 0 0 0 0 | | +-----+ *--+--* +-----+ | | | Normal Probability Plot 332.5+ * | * | * | ****+ | ******+ | ******** | ******* | ******* | ********* | ******** |********+ 277.5+*+ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 17 Monte Carlo study of Growth model fit STEP 1 - Gulland Summarized data for the Linear fits (Step 1) The UNIVARIATE Procedure Variable: K Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 0.26663653 0.02418746 0.11914002 71.6794876 9.07132311 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean 0.266637 Std Deviation Median 0.266239 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 348.6016 Sign M 500 Signed Rank S 250250 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling 0.02419 0.0005850 0.15974 0.03181 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 --Statistic--W 0.998359 D 0.01869 W-Sq 0.057277 A-Sq 0.386954 -----p Value-----Pr < W 0.4660 Pr > D >0.1500 Pr > W-Sq >0.2500 Pr > A-Sq >0.2500 \Quantiles (Definition 5) Quantile Estimate 100% Max 0.345705 99% 0.325082 95% 0.307757 90% 0.297884 75% Q3 0.282134 50% Median 0.266239 25% Q1 0.250326 10% 0.235961 5% 0.226901 1% 0.212324 0% Min 0.185966 Extreme Observations ------Lowest-----Value Obs 0.185966 162 0.189746 141 0.201435 715 0.201745 302 0.204536 538 1000 266.63653 0.00058503 0.08696882 0.58444825 0.00076487 ------Highest----Value Obs 0.331524 20 0.337548 508 0.338947 929 0.344296 158 0.345705 660 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 18 Histogram Boxplot 0.345+* .* .*** .****** .*********** .******************** .******************************* .************************************** 0.265+******************************************* .*************************************** .************************** .******************** .*********** .*** .** . 0.185+* ----+----+----+----+----+----+----+----+--* may represent up to 4 counts 2 4 12 24 43 77 122 150 169 154 104 78 44 10 5 0 0 | | | | +-----+ | | *--+--* +-----+ | | | | 0 2 0 Normal Probability Plot 0.345+ * | * | ***** | *****+ | **** | ***** | ***** | ***** 0.265+ ****** | ***** | ***** | ***** | ******* | ****+ |*+ | 0.185+* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 Monte Carlo study of Growth model fit Step 2 - Regression on Adjusted Lengths Summarized data for the Linear fits (Step 2) The UNIVARIATE Procedure Variable: T0 Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 0.47377081 0.45673733 -0.4213222 432.859158 96.4047004 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean 1000 473.770809 0.20860899 0.45748795 208.400378 0.0144433 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 19 Basic Statistical Measures Location Variability Mean 0.473771 Std Deviation Median 0.504275 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 32.80211 Sign M 356 Signed Rank S 211554 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling 0.45674 0.20861 2.96201 0.57778 -----p Value-----Pr > |t| <.0001 Pr >= |M| <.0001 Pr >= |S| <.0001 --Statistic--W 0.98927 D 0.039681 W-Sq 0.363439 A-Sq 2.274145 -----p Value-----Pr < W <0.0001 Pr > D <0.0100 Pr > W-Sq <0.0050 Pr > A-Sq <0.0050 Quantiles (Definition 5) Quantile Estimate 100% Max 1.666847 99% 1.446516 95% 1.175148 90% 1.035446 75% Q3 0.784467 50% Median 0.504275 25% Q1 0.206683 10% -0.150744 5% -0.319093 1% -0.750166 0% Min -1.295166 Extreme Observations ------Lowest----Value Obs -1.29517 40 -1.27027 392 -1.11124 772 -1.02051 366 -1.00815 718 -----Highest----Value Obs 1.53945 590 1.55807 499 1.65570 487 1.65978 542 1.66685 428 Histogram Boxplot 1.7+* .*** .******** 1.1+****************** .******************************** .******************************************* 0.5+********************************************* .****************************************** .************************** -0.1+************** .*************** .*** -0.7+*** .* .* -1.3+* ----+----+----+----+----+----+----+----+----+ 3 9 29 72 126 170 179 167 101 56 58 11 11 2 4 2 0 | | | | +-----+ *--+--* +-----+ | | | | 0 0 0 0 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 20 * may represent up to 4 counts Normal Probability Plot 1.7+ * | ++*** | +****** 1.1+ ****** | ****** | ****** 0.5+ ***** | ****** | *****+ -0.1+ +*** | ****** | ++*** -0.7++**** |** |* -1.3+* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 PROC REG DATA=means OUTEST=PARMEST4 noprint; by rep; TITLE2 'Step 2 - Regression on Adjusted Lengths'; MODEL LADJ = IntAGE; RUN; DATA PARMEST4; LENGTH MODEL $ 20; SET PARMEST4; MODEL = 'Step 2'; K = -IntAGE; T0 = intercept / intage; RUN; proc univariate DATA=PARMEST4 normal plot; var t0 k; Title3 'Summarized data for the Linear fits (Step 2)'; run; Monte Carlo study of Growth model fit Step 2 - Regression on Adjusted Lengths Summarized data for the Linear fits (Step 2) The UNIVARIATE Procedure Variable: K Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation 1000 0.31003059 0.04418653 1.3446707 98.0694662 14.2523123 Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean Basic Statistical Measures Location Variability Mean 0.310031 Std Deviation Median 0.301998 Variance Mode . Range Interquartile Range Tests for Location: Mu0=0 Test -StatisticStudent's t t 221.8782 1000 310.030594 0.00195245 3.2309141 1.95049686 0.0013973 0.04419 0.00195 0.33518 0.04934 -----p Value-----Pr > |t| <.0001 EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Sign Signed Rank M S 500 250250 Tests for Normality Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling Pr >= |M| Pr >= |S| --Statistic--W 0.920735 D 0.091641 W-Sq 2.825113 A-Sq 16.59205 Geaghan Page 21 <.0001 <.0001 -----p Value-----Pr < W <0.0001 Pr > D <0.0100 Pr > W-Sq <0.0050 Pr > A-Sq <0.0050 Quantiles (Definition 5) Quantile Estimate 100% Max 0.559453 99% 0.452643 95% 0.390749 90% 0.368748 75% Q3 0.329608 50% Median 0.301998 25% Q1 0.280273 10% 0.263346 5% 0.253289 1% 0.237831 0% Min 0.224270 Extreme Observations ------Lowest-----Value Obs 0.224270 428 0.230688 50 0.231042 487 0.232323 542 0.233308 417 ------Highest----Value Obs 0.489338 718 0.498211 785 0.510645 772 0.541237 392 0.559453 40 Histogram Boxplot 0.55+* . .* .** . .* .*** .** 0.39+******** .********* .**************** .************************** .************************************** .*********************************************** .******************************** .*************** 0.23+*** ----+----+----+----+----+----+----+----+----+-* may represent up to 5 counts 2 * 1 6 * * 3 14 10 40 45 78 130 190 235 157 75 14 0 0 0 | | | +-----+ *--+--* +-----+ | | | EXST7025 - Biological Population Statistics Monte Carlo example using Growth Models Geaghan Page 22 Normal Probability Plot 0.55+ * | | * | ** | | * | **** + | ** ++++ 0.39+ *****++ | ***+ | +***** | +***** | ++***** | ******** | ******* | **********+ 0.23+**** +++++ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 Summary: a comparison of parameter estimate results for the fitted models. Model L inf To K Known values 300.0 0.000 0.300 Nlin (age) 300.184439 -0.0010626 ׁ 0.3013297 ׁ Nlin (IntAge) 300.15825 ׁ -0.4878157 ׁ 0.30177374 Gulland 297.972677 Known Linf 0.26663653 0.47377081 0.31003059 ...
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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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