Diamondback Moth Egg Diamondback Moth Egg Counts on Counts on Braya Braya speciesspecies

Susan TilleyBiology 7932

Can egg numbers be explained by Can egg numbers be explained by species, year, disturbance or plant type?species, year, disturbance or plant type?

► Response Variable: Number of Eggs (E)► Explanatory Variable:

►Species (S) – 2 ►Year (Y) - 3►Disturbance (D) - 2►Plant Type (T) - 4

E= B0 + BSXS + BYXY + BDXD + BTXT + BS*YXS*Y + BS*DXS*D + BS*TXS*T + BY*DXY*D + BY*TXY*T + BD*TXD*T + BS*Y*DXS*Y*D + BS*Y*TXS*Y*T + BS*D*TXS*D*T + BY*D*TXY*D*T + BS*Y*D*TXS*Y*D*T + error

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Normal Probabil ity Plot of the Residuals(response is Eggs)

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Histogram of the Residuals(response is Eggs)

Evaluation of Model Using Evaluation of Model Using ResidualsResiduals

ANOVAANOVASource DF Seq SS Adj SS Adj MS F PS 1 211.259 114.405 114.405 46.60 0.000Y 2 188.106 91.729 45.865 18.68 0.000D 1 89.351 39.490 39.490 16.08 0.000T 3 777.429 518.975 172.992 70.46 0.000S*Y 2 28.577 25.269 12.635 5.15 0.006S*D 1 46.306 33.451 33.451 13.63 0.000S*T 3 136.640 86.249 28.750 11.71 0.000Y*D 2 3.621 3.150 1.575 0.64 0.527Y*T 6 126.922 88.758 14.793 6.03 0.000D*T 3 85.836 52.000 17.333 7.06 0.000S*Y*D 2 31.148 19.914 9.957 4.06 0.017S*Y*T 6 30.202 25.190 4.198 1.71 0.115S*D*T 3 25.409 25.350 8.450 3.44 0.016Y*D*T 6 8.220 6.533 1.089 0.44 0.850S*Y*D*T 6 17.977 17.977 2.996 1.22 0.292Error 2643 6488.822 6488.822 2.455Total 2690 8295.825

Poisson or Binomial?Poisson or Binomial?

► Lots of zeros in count data therefore Poisson is next step

BUT…► 1 egg = damage

THEREFORE► Can use presence/absence of eggs in analysis

Generalized Linear Model: Binomial Generalized Linear Model: Binomial Distribution Distribution

Model Information Data Set WORK.COUNTS Distribution Binomial Link Function Logit Dependent Variable PE

Number of Observations Read 2692

Number of Observations Used 2692

Number of Events 662 Number of Trials 2692

Class Level Information Class Levels Values S 2 1 2 Y 3 1 2 3 D 2 1 2 T 4 1 2 3 4

Response Profile Ordered Total Value PE Frequency 1 1 662 2 2 2030

1 = Presence

2 = Absence

Criteria For Assessing Goodness Of Criteria For Assessing Goodness Of FitFit

Criterion DF Value Value/DF Deviance 2644 2422.0342 0.9160 Scaled Deviance 2644 2422.0342 0.9160 Pearson Chi-Square 2644 2601.0000 0.9837 Scaled Pearson X2 2644 2601.0000 0.983 Log Likelihood -1211.0171

LR Statistics For Type 1 AnalysisLR Statistics For Type 1 Analysis Chi-Source Deviance DF Square Pr > ChiSqIntercept 3003.2023S 2930.6693 1 72.53 <.0001Y 2859.5837 2 71.09 <.0001D 2817.4975 1 42.09 <.0001T 2492.4354 3 325.06 <.0001S*Y 2484.5770 2 7.86 0.0197S*D 2473.0021 1 11.57 0.0007S*T 2471.0216 3 1.98 0.5765Y*D 2462.6702 2 8.35 0.0154Y*T 2457.9981 6 4.67 0.5865D*T 2453.7222 3 4.28 0.2332S*Y*D 2448.1246 2 5.60 0.0609S*Y*T 2443.2222 6 4.90 0.5564S*D*T 2440.3382 3 2.88 0.4099Y*D*T 2435.4702 6 4.87 0.5608S*Y*D*T 2422.0342 6 13.44 0.0366

Generalized Linear Model: Binomial Generalized Linear Model: Binomial DistributionDistribution

Braya longiiBraya longii Braya fernaldiiBraya fernaldii

Model InformationDistribution BinomialLink Function LogitDependent Variable PE Number of Observations Read 1573Number of Observations Used 1573Number of Events 479Number of Trials 1573

Model InformationDistribution BinomialLink Function LogitDependent Variable PE Number of Observations Read 1119Number of Observations Used 1119Number of Events 183Number of Trials 1119

Criteria For Assessing Goodness Of Criteria For Assessing Goodness Of FitFit

Braya longiiCriterion DF Value Value/DFDeviance 1549 1578.9682 1.0193Scaled Deviance 1549 1578.9682 1.0193Pearson Chi-Square 1549 1523.0000 0.9832Scaled Pearson X2 1549 1523.0000 0.9832Log Likelihood -789.4841

Braya fernaldiiCriterion DF Value Value/DFDeviance 1095 843.0660 0.7699Scaled Deviance 1095 843.0660 0.7699Pearson Chi-Square 1095 1078.0000 0.9845Scaled Pearson X2 1095 1078.0000 0.9845Log Likelihood -421.5330

LR Statistics For Type 1 AnalysisLR Statistics For Type 1 Analysis

Chi-Source Deviance DF Square Pr > ChiSqIntercept 1933.6586Y 1882.6239 2 51.03 <.0001D 1842.9210 1 39.70 <.0001T 1600.3273 3 242.59 <.0001Y*D 1598.7296 2 1.60 0.4499Y*T 1592.3175 6 6.41 0.3786D*T 1587.1275 3 5.19 0.1584Y*D*T 1578.9682 6 8.16 0.2267

Braya longii

LR Statistics For Type 1 AnalysisLR Statistics For Type 1 Analysis

Chi-Source Deviance DF Square Pr > ChiSqIntercept 997.0107Y 968.0396 2 28.97 <.0001D 963.7795 1 4.26 0.0390T 870.6944 3 93.09 <.0001Y*D 859.8225 2 10.87 0.0044Y*T 856.2523 6 3.57 0.7346D*T 853.2107 3 3.04 0.3853Y*D*T 843.0660 6 10.14 0.1187

Braya fernaldii

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Generalized Linear Model: Poisson Generalized Linear Model: Poisson Distribution Distribution

Model InformationData Set

WORK.COUNTSDistribution PoissonLink Function LogDependent Variable E

Number of Observations Read 2692Number of Observations Used 2692

Class Level InformationClass Levels ValuesS 2 1 2Y 3 1 2 3D 2 1 2T 4 1 2 3 4

Criteria For Assessing Goodness Of Criteria For Assessing Goodness Of FitFit

Criterion DF Value Value/DFDeviance 2644 3989.7034 1.5090Scaled Deviance 2644 3989.7034 1.5090Pearson Chi-Square 2644 6275.5418 2.3735Scaled Pearson X2 2644 6275.5418 2.3735Log Likelihood -1431.6801

LR Statistics For Type 1 AnalysisLR Statistics For Type 1 Analysis Chi-

Source Deviance DF Square Pr > ChiSqIntercept 6139.6847S 5786.3804 1 353.30 <.0001Y 5483.9630 2 302.42 <.0001D 5358.0698 1 125.89 <.0001T 4157.7239 3 1200.35 <.0001S*Y 4126.7328 2 30.99 <.0001S*D 4103.3052 1 23.43 <.0001S*T 4099.5266 3 3.78 0.2864Y*D 4091.1411 2 8.39 0.0151Y*T 4069.7446 6 21.40 0.0016D*T 4059.2600 3 10.48 0.0149S*Y*D 4035.4374 2 23.82 <.0001S*Y*T 4028.7876 6 6.65 0.3544S*D*T 4022.4713 3 6.32 0.0972Y*D*T 4009.7178 6 12.75 0.0471S*Y*D*T 3989.7034 6 20.01 0.0028

Pearson Pearson Chi-Chi-

SquareSquare

0.98 2.37 1.08 1.93 1.14

BinomiBinomialal

PoissonPoisson Negative Negative BinomialBinomial

Poisson Poisson Eggs OnlyEggs Only

Negative Negative Binomial Eggs Binomial Eggs

OnlyOnly

SS 72.53 353.3 90.91 48.6 27.36

YY 71.09 302.42 81.57 35.73 20.48

DD 42.09 125.89 35.14 7.65 3.99

TT 325.06 1200.35 376.31 135.09 80.1

S*YS*Y 7.86 30.99 13.06 3.00 2.09

S*DS*D 11.57 23.43 5.15 2.09 1.04

S*TS*T 1.98 3.78 0.99 3.08 2.29

Y*DY*D 8.35 8.39 3.51 2.44 1.88

Y*TY*T 4.67 21.4 8.28 5.09 3.32

D*TD*T 4.28 10.48 6.08 3.16 2.09

S*Y*DS*Y*D 5.6 23.82 9.64 7.67 4.79

S*Y*TS*Y*T 4.9 6.65 5.32 3.61 2.76

S*D*TS*D*T 2.88 6.32 4.45 0.71 0.51

Y*D*TY*D*T 4.87 12.75 8.72 10.2 6.86

S*Y*D*TS*Y*D*T 13.44 20.01 16.52 2.91 2.18

Pearson Pearson Chi-Chi-

SquareSquare

0.98 2.37 1.08 1.93 1.14

BinomiBinomialal

PoissonPoisson Negative Negative BinomialBinomial

Poisson Poisson Eggs OnlyEggs Only

Negative Negative Binomial Eggs Binomial Eggs

OnlyOnly

SS 72.53 353.3 90.91 48.6 27.36

YY 71.09 302.42 81.57 35.73 20.48

DD 42.09 125.89 35.14 7.65 3.99

TT 325.06 1200.35 376.31 135.09 80.1

S*YS*Y 7.86 30.99 13.06 3.00 2.09

S*DS*D 11.57 23.43 5.15 2.09 1.04

S*TS*T 1.98 3.78 0.99 3.08 2.29

Y*DY*D 8.35 8.39 3.51 2.44 1.88

Y*TY*T 4.67 21.4 8.28 5.09 3.32

D*TD*T 4.28 10.48 6.08 3.16 2.09

S*Y*DS*Y*D 5.6 23.82 9.64 7.67 4.79

S*Y*TS*Y*T 4.9 6.65 5.32 3.61 2.76

S*D*TS*D*T 2.88 6.32 4.45 0.71 0.51

Y*D*TY*D*T 4.87 12.75 8.72 10.2 6.86

S*Y*D*TS*Y*D*T 13.44 20.01 16.52 2.91 2.18

Pearson Pearson Chi-Chi-

SquareSquare

0.98 2.37 1.08 1.93 1.14

BinomiBinomialal

PoissonPoisson Negative Negative BinomialBinomial

Poisson Poisson Eggs OnlyEggs Only

Negative Negative Binomial Eggs Binomial Eggs

OnlyOnly

SS 72.53 353.3 90.91 48.6 27.36

YY 71.09 302.42 81.57 35.73 20.48

DD 42.09 125.89 35.14 7.65 3.99

TT 325.06 1200.35 376.31 135.09 80.1

S*YS*Y 7.86 30.99 13.06 3.00 2.09

S*DS*D 11.57 23.43 5.15 2.09 1.04

S*TS*T 1.98 3.78 0.99 3.08 2.29

Y*DY*D 8.35 8.39 3.51 2.44 1.88

Y*TY*T 4.67 21.4 8.28 5.09 3.32

D*TD*T 4.28 10.48 6.08 3.16 2.09

S*Y*DS*Y*D 5.6 23.82 9.64 7.67 4.79

S*Y*TS*Y*T 4.9 6.65 5.32 3.61 2.76

S*D*TS*D*T 2.88 6.32 4.45 0.71 0.51

Y*D*TY*D*T 4.87 12.75 8.72 10.2 6.86

S*Y*D*TS*Y*D*T 13.44 20.01 16.52 2.91 2.18

ConclusionsConclusions► Binomial model is better than Poisson model because

Pearson Chi-Square is closer to 1.

► The questions of:

1. why is an organism present or absent? 2. what controls the abundance of an organism that is

present?

are very different and therefore should be analyzed separately.

Presence/Absence = Binomial What controls abundance once present = Poisson,

Negative Binomial, and Poisson with scale factor

Littell, et al. (2002) SAS for Linear Models

http://faculty.ucr.edu/~hanneman/linear_models/index.html

► Contains SAS files used in textbook

The very longThe very long““Analysis of Parameter Analysis of Parameter

EstimatesEstimates””TableTable