Discrete Data Examples
dm "output;clear;log;clear";
******************************************************;
* DiscreteData.sas -- Examples of generating Poisson *;
* and Negative Binomial data and the analysis of it. *;
* Further examples of data from Ludwig and Reynolds *;
* with goodness-of-fit statistics. *;
******************************************************;
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Title1 "Discrete Data Models";
Title2 "Poisson Data";
Data Poi;
Retain Seed 0 lambda 16 N 100;
Do i=1 To N;
Y=RanPoi(Seed,lambda);
Keep Y;
Output;
End;
Run;
Proc GChart Data=Poi;
* VBar Y / Discrete;
VBar Y;
Run;
Quit;
/*
* GENMOD estimates the mean of the response.
* The extra-dispersion should be zero here
* since variance=mean in the Poisson.
*/
Proc Genmod Data=Poi;
Model Y = / Dist=Poisson Link=Log LRCI MaxIter=50;
Estimate "Population Mean" Intercept 1 / Exp;
Run;
Quit;
Title3 "Consider an Overdispersed Model";
Proc Genmod Data=Poi;
Model Y = / Dist=Poisson Link=Log DScale LRCI MaxIter=50;
Estimate "Population Mean" Intercept 1 / Exp;
Run;
Quit;
Title3 "Fit the Negative Binomial Model";
Proc Genmod Data=NB;
Model Y = / Dist=Negbin Link=Log LRCI MaxIter=50;
Estimate "Population Mean" Intercept 1 / Exp;
Run;
Quit;
Title2 "Negative Binomial Data";
Data NB;
/*
* Generate Negative Binomial NB(r,p) data.
* Use result that negative binomial can result
* from generalizing of Poisson(X/p), where
* X is Gamma(r).
*/
Retain Seed 0 r 8 p 0.5 N 100;
OneMinusP=1-p;
Mu=r*OneMinusP/p;
MuPlusR=Mu+r;
Sigma2=r*OneMinusP/p**2;
k=(Sigma2-Mu)/Mu**2;
Put Mu= Sigma2= MuPlusR= k=;
Do i=1 To N;
X=RanGam(Seed,r);
lambda=X/p;
Y=RanPoi(Seed,lambda);
Keep X Y;
Output;
End;
Run;
Proc GChart Data=NB;
* VBar Y / Discrete;
VBar Y;
Run;
Quit;
/*
* GENMOD estimates the mean of the response, so
* is estimating Mu+r from distribution above.
* Also the dispersion parameter k is from
* V(Y) = mu + k*mu**2
*/
Proc Genmod Data=NB;
Model Y = / Dist=Negbin Link=Log LRCI MaxIter=50;
Estimate "Population Mean" Intercept 1 / Exp;
Run;
Quit;
Title3 "Treat as Overdispersed Poisson Model";
Proc Genmod Data=NB;
Model Y = / Dist=Poisson Link=Log DScale LRCI MaxIter=50;
Estimate "Population Mean" Intercept 1 / Exp;
Run;
Quit;
/*
* Example: Carpenter Bee Larvae in soap-tree yucca plants
* From pages 29-35 in Ludwig and Reynolds 1988.
* The data come in a summarized form. Though it is possible
* to do the analysis in this format, some procedures do
* not handle all aspects of the data properly. Thus,
* take the data back to individual observations.
*/
Title2 "Carpenter Bee Larvae Counts in Soap-tree Yucca";
Data CarpenterBees;
Input Y Frequency;
Do i=1 To Frequency;
Output;
End;
Keep Y;
Datalines;
0 114
1 25
2 15
3 10
4 6
5 5
6 2
7 1
8 1
9 0
10 1
;
/*
* Show original frequency table
*/
Proc Freq Data=CarpenterBees;
Table Y;
Run;
/*
* Examine a histogram of the data
*/
Proc GChart Data=CarpenterBees;
VBar Y / Discrete;
Run;
Proc Univariate Data=CarpenterBees;
Var Y;
Run;
/*
* Fit a Poisson distribution to the data
*/
Title3 "Poisson Model";
Proc Genmod Data=CarpenterBees;
Model Y = / Dist=Poisson Link=Log LRCI;
Estimate "Population Mean" Intercept 1 / Exp;
ODS Output ParameterEstimates=Parms;
Run;
/*
* Compute Expected Probabilities. These
* will be used in a GOF test to follow.
*/
Data Expected;
If _N_=1 Then
Do;
Set Parms;
Lambda=Exp(Estimate); /* First obs is ln(lambda) */
ELambda=Exp(-Lambda);
Retain Lambda ELambda;
End;
Do Y=0 To 10;
Prob=(Lambda**Y)*ELambda/Gamma(Y+1); /* Poisson Probability */
Expected=180*Prob;
Cummulative+Prob;
InvCum=1-Cummulative+Prob;
Output;
End;
Stop;
Keep Y Prob Expected Lambda Cummulative InvCum;
Run;
Title4 "Expected Probabilities";
Proc Print Data=Expected;
Run;
/*
* Can use PROC FREQ to do GOF test, though
* d.f. are not correct. Since some expected
* values will be less than 1, we will group
* the data for Y>=4 into a common group.
*/
Proc Format;
Value YGroup 4-High="4+";
Run;
/*
* Since there will be 5 cells in this table,
* PROC FREQ will compute the d.f. to be 5-1=4.
* However, the probabilities were predicted
* by estimating the parameter Lambda using the
* same data. Thus we need to lose 1 more d.f.
* Thus, d.f.=5-1-1=3.
*/
Title4 "Pearson Chi-square Goodness-of-fit Test";
Title5 "Note: Degrees of Freedom Should Be 3";
Proc Freq Data=CarpenterBees;
Table Y / Chisq NoCum TestP=(38.674 36.740 17.452 5.526 1.607);
Format Y YGroup.;
Run;
/*
* Repeat analysis using the Negative Binomial Model.
*/
Title3 "Negative Binomial Model";
Proc Genmod Data=CarpenterBees;
Model Y = / Dist=NegBin Link=Log LRCI MaxIter=500;
Estimate "Population Mean" Intercept 1 / Exp;
ODS Output ParameterEstimates=Parms;
Run;
Data Expected;
If _N_=1 Then
Do;
i=1;
Set Parms Point=i Nobs=Nobs;
Mu=Exp(Estimate); /* First obs is ln(Mu) */
i=2;
Set Parms Point=i Nobs=Nobs;
k=Estimate; /* Second obs is dispersion parameter */
kinv=1/k;
VarY=Mu+k*Mu**2;
Retain Mu k VarY kinv;
End;
Do Y=0 To 10;
Prob=Gamma(Y+kinv)/(Gamma(Y+1)*Gamma(kinv))*(k*mu)**Y/((1+k*mu)**(Y+kinv)); /* Neg binomial Probability */
Expected=180*Prob;
Cummulative+Prob;
InvCum=1-Cummulative+Prob;
Output;
End;
Stop;
Keep Y Prob Expected Mu k kinv VarY Cummulative InvCum;
Run;
Title4 "Expected Probabilities";
Proc Print Data=Expected;
Run;
Proc Format;
Value YGroup 5-High="5+";
Run;
Title4 "Pearson Chi-square Goodness-of-fit Test";
Title5 "Note: Degrees of Freedom Should Be 3";
Proc Freq Data=CarpenterBees;
Table Y / Chisq NoCum TestP=(62.617 16.530 8.150 4.641 2.820 5.243);
Format Y YGroup.;
Run;
/*
* Example: Mites on Apple Leaves
* Source: Pages 37-38 in Ludwig and Reynolds 1988
*/
Title2 "Mites on Apple Leaves";
Data AppleLeaves;
Input Y Frequency;
Do i=1 To Frequency;
Output;
End;
Keep Y;
Datalines;
0 70
1 38
2 17
3 10
4 9
5 3
6 2
7 1
;
/*
* Show original frequency table
*/
Proc Freq Data=AppleLeaves;
Table Y;
Run;
/*
* Examine a histogram of the data
*/
Proc GChart Data=AppleLeaves;
VBar Y / Discrete;
Run;
Proc Univariate Data=AppleLeaves;
Var Y;
Run;
/*
* Fit a Poisson distribution to the data
*/
Title3 "Poisson Model";
Proc Genmod Data=AppleLeaves;
Model Y = / Dist=Poisson Link=Log LRCI;
Estimate "Population Mean" Intercept 1 / Exp;
ODS Output ParameterEstimates=Parms;
Run;
/*
* Compute Expected Probabilities. These
* will be used in a GOF test to follow.
*/
Data Expected;
If _N_=1 Then
Do;
Set Parms;
Lambda=Exp(Estimate); /* First obs is ln(lambda) */
ELambda=Exp(-Lambda);
Retain Lambda ELambda;
End;
Do Y=0 to 10;
Prob=(Lambda**Y)*ELambda/Gamma(Y+1); /* Poisson Probability */
Expected=150*Prob;
Cummulative+Prob;
InvCum=1-Cummulative+Prob;
Output;
End;
Stop;
Keep Y Prob Expected Lambda Cummulative InvCum;
Run;
Title4 "Expected Probabilities";
Proc Print Data=Expected;
Run;
/*
* Can use PROC FREQ to do GOF test, though
* d.f. are not correct. Since some expected
* values will be less than 1, we will group
* the data for Y>=4 into a common group.
*/
Proc Format;
Value YGroup 4-High="4+";
Run;
/*
* Since there will be 5 cells in this table,
* PROC FREQ will compute the d.f. to be 5-1=4.
* However, the probabilities were predicted
* by estimating the parameter Lambda using the
* same data. Thus we need to lose 1 more d.f.
* Thus, d.f.=5-1-1=3.
*/
Title4 "Pearson Chi-square Goodness-of-fit Test";
Title5 "Note: Degrees of Freedom Should Be 3";
Proc Freq Data=AppleLeaves;
Table Y / Chisq NoCum TestP=(31.769 36.429 20.886 7.983 2.933);
Format Y YGroup.;
Run;
/*
* Repeat analysis using the Negative Binomial Model.
* For these data the numerical algorithm would not
* properly converge, thus I fixed the INTERCEPT
* parameter at LN(MU)=0.1369.
*/
Title3 "Negative Binomial Model";
Proc Genmod Data=AppleLeaves;
Model Y = / Dist=NegBin Link=Log LRCI MaxIter=500;
Estimate "Population Mean" Intercept 1 / Exp;
ODS Output ParameterEstimates=Parms;
Run;
Data Expected;
If _N_=1 Then
Do;
i=1;
Set Parms Point=i Nobs=Nobs;
Mu=Exp(Estimate); /* First obs is ln(Mu) */
i=2;
Set Parms Point=i Nobs=Nobs;
k=Estimate; /* Second obs is dispersion parameter */
kinv=1/k;
VarY=Mu+k*Mu**2;
Retain Mu k VarY kinv;
End;
Do Y=0 to 10;
Prob=Gamma(Y+kinv)/(Gamma(Y+1)*Gamma(kinv))*(k*mu)**Y/((1+k*mu)**(Y+kinv)); /* Neg binomial Probability */
Expected=150*Prob;
Cummulative+Prob;
InvCum=1-Cummulative+Prob;
Output;
End;
Stop;
Keep Y Frequency Prob Expected Mu k kinv VarY Cummulative InvCum;
Run;
Title4 "Expected Probabilities";
Proc Print Data=Expected;
Run;
Proc Format;
Value YGroup 5-High="5+";
Run;
Title4 "Pearson Chi-square Goodness-of-fit Test";
Title5 "Note: Degrees of Freedom Should Be 3";
Proc Freq Data=AppleLeaves;
Table Y / Chisq NoCum TestP=(46.325 25.067 12.401 7.135 3.791 4.281);
Format Y YGroup.;
Run;
| Discrete Data Models |
| Poisson Data |
| Model Information |
| Data Set |
WORK.POI |
| Distribution |
Poisson |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
100 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
99 |
95.3649 |
0.9633 |
| Scaled Deviance |
99 |
95.3649 |
0.9633 |
| Pearson Chi-Square |
99 |
92.9479 |
0.9389 |
| Scaled Pearson X2 |
99 |
92.9479 |
0.9389 |
| Log Likelihood |
|
2657.2623 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
2.7311 |
0.0255 |
2.6807 |
2.7807 |
11449.6 |
<.0001 |
| Scale |
0 |
1.0000 |
0.0000 |
1.0000 |
1.0000 |
|
|
| NOTE: |
The scale parameter was held fixed. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
2.7311 |
0.0255 |
0.05 |
2.6811 |
2.7811 |
11450 |
<.0001 |
| Exp(Population Mean) |
15.3500 |
0.3918 |
0.05 |
14.6010 |
16.1374 |
|
|
| Discrete Data Models |
| Poisson Data |
| Consider an Overdispersed Model |
| Model Information |
| Data Set |
WORK.POI |
| Distribution |
Poisson |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
100 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
99 |
95.3649 |
0.9633 |
| Scaled Deviance |
99 |
99.0000 |
1.0000 |
| Pearson Chi-Square |
99 |
92.9479 |
0.9389 |
| Scaled Pearson X2 |
99 |
96.4908 |
0.9747 |
| Log Likelihood |
|
2758.5502 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
2.7311 |
0.0251 |
2.6816 |
2.7798 |
11886.0 |
<.0001 |
| Scale |
0 |
0.9815 |
0.0000 |
0.9815 |
0.9815 |
|
|
| NOTE: |
The scale parameter was estimated by the square root of DEVIANCE/DOF. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
2.7311 |
0.0251 |
0.05 |
2.6820 |
2.7802 |
11886 |
<.0001 |
| Exp(Population Mean) |
15.3500 |
0.3845 |
0.05 |
14.6145 |
16.1225 |
|
|
| Discrete Data Models |
| Negative Binomial Data |
| Model Information |
| Data Set |
WORK.NB |
| Distribution |
Negative Binomial |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
100 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
99 |
102.7304 |
1.0377 |
| Scaled Deviance |
99 |
102.7304 |
1.0377 |
| Pearson Chi-Square |
99 |
98.2249 |
0.9922 |
| Scaled Pearson X2 |
99 |
98.2249 |
0.9922 |
| Log Likelihood |
|
2648.3986 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
2.7180 |
0.0451 |
2.6292 |
2.8078 |
3627.96 |
<.0001 |
| Dispersion |
1 |
0.1376 |
0.0291 |
0.0895 |
0.2066 |
|
|
| NOTE: |
The negative binomial dispersion parameter was estimated by maximum likelihood. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
2.7180 |
0.0451 |
0.05 |
2.6296 |
2.8064 |
3628.0 |
<.0001 |
| Exp(Population Mean) |
15.1500 |
0.6836 |
0.05 |
13.8676 |
16.5510 |
|
|
| Discrete Data Models |
| Negative Binomial Data |
| Treat as Overdispersed Poisson Model |
| Model Information |
| Data Set |
WORK.NB |
| Distribution |
Poisson |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
100 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
99 |
303.5469 |
3.0661 |
| Scaled Deviance |
99 |
99.0000 |
1.0000 |
| Pearson Chi-Square |
99 |
303.0198 |
3.0608 |
| Scaled Pearson X2 |
99 |
98.8281 |
0.9983 |
| Log Likelihood |
|
848.8781 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
2.7180 |
0.0450 |
2.6285 |
2.8049 |
3650.24 |
<.0001 |
| Scale |
0 |
1.7510 |
0.0000 |
1.7510 |
1.7510 |
|
|
| NOTE: |
The scale parameter was estimated by the square root of DEVIANCE/DOF. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
2.7180 |
0.0450 |
0.05 |
2.6298 |
2.8062 |
3650.2 |
<.0001 |
| Exp(Population Mean) |
15.1500 |
0.6816 |
0.05 |
13.8714 |
16.5465 |
|
|
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Y |
Frequency |
Percent |
Cumulative
Frequency |
Cumulative
Percent |
| 0 |
114 |
63.33 |
114 |
63.33 |
| 1 |
25 |
13.89 |
139 |
77.22 |
| 2 |
15 |
8.33 |
154 |
85.56 |
| 3 |
10 |
5.56 |
164 |
91.11 |
| 4 |
6 |
3.33 |
170 |
94.44 |
| 5 |
5 |
2.78 |
175 |
97.22 |
| 6 |
2 |
1.11 |
177 |
98.33 |
| 7 |
1 |
0.56 |
178 |
98.89 |
| 8 |
1 |
0.56 |
179 |
99.44 |
| 10 |
1 |
0.56 |
180 |
100.00 |
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| The UNIVARIATE Procedure |
| Variable: Y |
| Moments |
| N |
180 |
Sum Weights |
180 |
| Mean |
0.95 |
Sum Observations |
171 |
| Std Deviation |
1.70203624 |
Variance |
2.89692737 |
| Skewness |
2.38180835 |
Kurtosis |
6.56643091 |
| Uncorrected SS |
681 |
Corrected SS |
518.55 |
| Coeff Variation |
179.16171 |
Std Error Mean |
0.12686229 |
| Basic Statistical Measures |
| Location |
Variability |
| Mean |
0.950000 |
Std Deviation |
1.70204 |
| Median |
0.000000 |
Variance |
2.89693 |
| Mode |
0.000000 |
Range |
10.00000 |
| |
|
Interquartile Range |
1.00000 |
| Tests for Location: Mu0=0 |
| Test |
Statistic |
p Value |
| Student's t |
t |
7.488435 |
Pr > |t| |
<.0001 |
| Sign |
M |
33 |
Pr >= |M| |
<.0001 |
| Signed Rank |
S |
1105.5 |
Pr >= |S| |
<.0001 |
| Quantiles (Definition 5) |
| Quantile |
Estimate |
| 100% Max |
10 |
| 99% |
8 |
| 95% |
5 |
| 90% |
3 |
| 75% Q3 |
1 |
| 50% Median |
0 |
| 25% Q1 |
0 |
| 10% |
0 |
| 5% |
0 |
| 1% |
0 |
| 0% Min |
0 |
| Extreme Observations |
| Lowest |
Highest |
| Value |
Obs |
Value |
Obs |
| 0 |
114 |
6 |
176 |
| 0 |
113 |
6 |
177 |
| 0 |
112 |
7 |
178 |
| 0 |
111 |
8 |
179 |
| 0 |
110 |
10 |
180 |
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Poisson Model |
| Model Information |
| Data Set |
WORK.CARPENTERBEES |
| Distribution |
Poisson |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
180 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
179 |
421.6296 |
2.3555 |
| Scaled Deviance |
179 |
421.6296 |
2.3555 |
| Pearson Chi-Square |
179 |
545.8421 |
3.0494 |
| Scaled Pearson X2 |
179 |
545.8421 |
3.0494 |
| Log Likelihood |
|
-179.7712 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
-0.0513 |
0.0765 |
-0.2050 |
0.0949 |
0.45 |
0.5024 |
| Scale |
0 |
1.0000 |
0.0000 |
1.0000 |
1.0000 |
|
|
| NOTE: |
The scale parameter was held fixed. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
-0.0513 |
0.0765 |
0.05 |
-0.2012 |
0.0986 |
0.45 |
0.5024 |
| Exp(Population Mean) |
0.9500 |
0.0726 |
0.05 |
0.8178 |
1.1036 |
|
|
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Poisson Model |
| Expected Probabilities |
| Obs |
Lambda |
Y |
Prob |
Expected |
Cummulative |
InvCum |
| 1 |
0.95000 |
0 |
0.38674 |
69.6134 |
0.38674 |
1.00000 |
| 2 |
0.95000 |
1 |
0.36740 |
66.1327 |
0.75414 |
0.61326 |
| 3 |
0.95000 |
2 |
0.17452 |
31.4130 |
0.92866 |
0.24586 |
| 4 |
0.95000 |
3 |
0.05526 |
9.9475 |
0.98393 |
0.07134 |
| 5 |
0.95000 |
4 |
0.01313 |
2.3625 |
0.99705 |
0.01607 |
| 6 |
0.95000 |
5 |
0.00249 |
0.4489 |
0.99954 |
0.00295 |
| 7 |
0.95000 |
6 |
0.00039 |
0.0711 |
0.99994 |
0.00046 |
| 8 |
0.95000 |
7 |
0.00005 |
0.0096 |
0.99999 |
0.00006 |
| 9 |
0.95000 |
8 |
0.00001 |
0.0011 |
1.00000 |
0.00001 |
| 10 |
0.95000 |
9 |
0.00000 |
0.0001 |
1.00000 |
0.00000 |
| 11 |
0.95000 |
10 |
0.00000 |
0.0000 |
1.00000 |
0.00000 |
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Poisson Model |
| Pearson Chi-square Goodness-of-fit Test |
| Note: Degrees of Freedom Should Be 3 |
| Y |
Frequency |
Percent |
Test
Percent |
| 0 |
114 |
63.33 |
38.67 |
| 1 |
25 |
13.89 |
36.74 |
| 2 |
15 |
8.33 |
17.45 |
| 3 |
10 |
5.56 |
5.53 |
| 4+ |
16 |
8.89 |
1.61 |
Chi-Square Test
for Specified Proportions |
| Chi-Square |
121.8554 |
| DF |
4 |
| Pr > ChiSq |
<.0001 |
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Negative Binomial Model |
| Model Information |
| Data Set |
WORK.CARPENTERBEES |
| Distribution |
Negative Binomial |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
180 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
179 |
143.4229 |
0.8012 |
| Scaled Deviance |
179 |
143.4229 |
0.8012 |
| Pearson Chi-Square |
179 |
151.6760 |
0.8474 |
| Scaled Pearson X2 |
179 |
151.6760 |
0.8474 |
| Log Likelihood |
|
-114.7281 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
-0.0513 |
0.1451 |
-0.3329 |
0.2433 |
0.13 |
0.7237 |
| Dispersion |
1 |
2.7355 |
0.5787 |
1.7886 |
4.1172 |
|
|
| NOTE: |
The negative binomial dispersion parameter was estimated by maximum likelihood. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
-0.0513 |
0.1451 |
0.05 |
-0.3356 |
0.2330 |
0.13 |
0.7237 |
| Exp(Population Mean) |
0.9500 |
0.1378 |
0.05 |
0.7149 |
1.2624 |
|
|
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Negative Binomial Model |
| Expected Probabilities |
| Obs |
Mu |
k |
kinv |
VarY |
Y |
Prob |
Expected |
Cummulative |
InvCum |
| 1 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
0 |
0.62617 |
112.711 |
0.62617 |
1.00000 |
| 2 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
1 |
0.16530 |
29.754 |
0.79147 |
0.37383 |
| 3 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
2 |
0.08150 |
14.670 |
0.87297 |
0.20853 |
| 4 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
3 |
0.04641 |
8.353 |
0.91938 |
0.12703 |
| 5 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
4 |
0.02820 |
5.075 |
0.94757 |
0.08062 |
| 6 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
5 |
0.01778 |
3.200 |
0.96535 |
0.05243 |
| 7 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
6 |
0.01148 |
2.066 |
0.97683 |
0.03465 |
| 8 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
7 |
0.00754 |
1.357 |
0.98437 |
0.02317 |
| 9 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
8 |
0.00501 |
0.902 |
0.98938 |
0.01563 |
| 10 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
9 |
0.00336 |
0.606 |
0.99275 |
0.01062 |
| 11 |
0.95000 |
2.73552 |
0.36556 |
3.41880 |
10 |
0.00228 |
0.410 |
0.99502 |
0.00725 |
| Discrete Data Models |
| Carpenter Bee Larvae Counts in Soap-tree Yucca |
| Negative Binomial Model |
| Pearson Chi-square Goodness-of-fit Test |
| Note: Degrees of Freedom Should Be 3 |
| Y |
Frequency |
Percent |
Test
Percent |
| 0 |
114 |
63.33 |
62.62 |
| 1 |
25 |
13.89 |
16.53 |
| 2 |
15 |
8.33 |
8.15 |
| 3 |
10 |
5.56 |
4.64 |
| 4 |
6 |
3.33 |
2.82 |
| 5+ |
10 |
5.56 |
5.24 |
Chi-Square Test
for Specified Proportions |
| Chi-Square |
1.3079 |
| DF |
5 |
| Pr > ChiSq |
0.9341 |
| Discrete Data Models |
| Mites on Apple Leaves |
| Y |
Frequency |
Percent |
Cumulative
Frequency |
Cumulative
Percent |
| 0 |
70 |
46.67 |
70 |
46.67 |
| 1 |
38 |
25.33 |
108 |
72.00 |
| 2 |
17 |
11.33 |
125 |
83.33 |
| 3 |
10 |
6.67 |
135 |
90.00 |
| 4 |
9 |
6.00 |
144 |
96.00 |
| 5 |
3 |
2.00 |
147 |
98.00 |
| 6 |
2 |
1.33 |
149 |
99.33 |
| 7 |
1 |
0.67 |
150 |
100.00 |
| Discrete Data Models |
| Mites on Apple Leaves |
| The UNIVARIATE Procedure |
| Variable: Y |
| Moments |
| N |
150 |
Sum Weights |
150 |
| Mean |
1.14666667 |
Sum Observations |
172 |
| Std Deviation |
1.50786158 |
Variance |
2.27364653 |
| Skewness |
1.54453863 |
Kurtosis |
2.06063766 |
| Uncorrected SS |
536 |
Corrected SS |
338.773333 |
| Coeff Variation |
131.499556 |
Std Error Mean |
0.12311638 |
| Basic Statistical Measures |
| Location |
Variability |
| Mean |
1.146667 |
Std Deviation |
1.50786 |
| Median |
1.000000 |
Variance |
2.27365 |
| Mode |
0.000000 |
Range |
7.00000 |
| |
|
Interquartile Range |
2.00000 |
| Tests for Location: Mu0=0 |
| Test |
Statistic |
p Value |
| Student's t |
t |
9.313681 |
Pr > |t| |
<.0001 |
| Sign |
M |
40 |
Pr >= |M| |
<.0001 |
| Signed Rank |
S |
1620 |
Pr >= |S| |
<.0001 |
| Quantiles (Definition 5) |
| Quantile |
Estimate |
| 100% Max |
7.0 |
| 99% |
6.0 |
| 95% |
4.0 |
| 90% |
3.5 |
| 75% Q3 |
2.0 |
| 50% Median |
1.0 |
| 25% Q1 |
0.0 |
| 10% |
0.0 |
| 5% |
0.0 |
| 1% |
0.0 |
| 0% Min |
0.0 |
| Extreme Observations |
| Lowest |
Highest |
| Value |
Obs |
Value |
Obs |
| 0 |
70 |
5 |
146 |
| 0 |
69 |
5 |
147 |
| 0 |
68 |
6 |
148 |
| 0 |
67 |
6 |
149 |
| 0 |
66 |
7 |
150 |
| Discrete Data Models |
| Mites on Apple Leaves |
| Poisson Model |
| Model Information |
| Data Set |
WORK.APPLELEAVES |
| Distribution |
Poisson |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
150 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
149 |
284.3125 |
1.9081 |
| Scaled Deviance |
149 |
284.3125 |
1.9081 |
| Pearson Chi-Square |
149 |
295.4418 |
1.9828 |
| Scaled Pearson X2 |
149 |
295.4418 |
1.9828 |
| Log Likelihood |
|
-148.4602 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
0.1369 |
0.0762 |
-0.0164 |
0.2827 |
3.22 |
0.0727 |
| Scale |
0 |
1.0000 |
0.0000 |
1.0000 |
1.0000 |
|
|
| NOTE: |
The scale parameter was held fixed. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
0.1369 |
0.0762 |
0.05 |
-0.0126 |
0.2863 |
3.22 |
0.0727 |
| Exp(Population Mean) |
1.1467 |
0.0874 |
0.05 |
0.9875 |
1.3315 |
|
|
| Discrete Data Models |
| Mites on Apple Leaves |
| Poisson Model |
| Expected Probabilities |
| Obs |
Lambda |
Y |
Prob |
Expected |
Cummulative |
InvCum |
| 1 |
1.14667 |
0 |
0.31769 |
47.6541 |
0.31769 |
1.00000 |
| 2 |
1.14667 |
1 |
0.36429 |
54.6434 |
0.68198 |
0.68231 |
| 3 |
1.14667 |
2 |
0.20886 |
31.3289 |
0.89084 |
0.31802 |
| 4 |
1.14667 |
3 |
0.07983 |
11.9746 |
0.97067 |
0.10916 |
| 5 |
1.14667 |
4 |
0.02288 |
3.4327 |
0.99356 |
0.02933 |
| 6 |
1.14667 |
5 |
0.00525 |
0.7872 |
0.99881 |
0.00644 |
| 7 |
1.14667 |
6 |
0.00100 |
0.1504 |
0.99981 |
0.00119 |
| 8 |
1.14667 |
7 |
0.00016 |
0.0246 |
0.99997 |
0.00019 |
| 9 |
1.14667 |
8 |
0.00002 |
0.0035 |
1.00000 |
0.00003 |
| 10 |
1.14667 |
9 |
0.00000 |
0.0005 |
1.00000 |
0.00000 |
| 11 |
1.14667 |
10 |
0.00000 |
0.0001 |
1.00000 |
0.00000 |
| Discrete Data Models |
| Mites on Apple Leaves |
| Poisson Model |
| Pearson Chi-square Goodness-of-fit Test |
| Note: Degrees of Freedom Should Be 3 |
| Y |
Frequency |
Percent |
Test
Percent |
| 0 |
70 |
46.67 |
31.77 |
| 1 |
38 |
25.33 |
36.43 |
| 2 |
17 |
11.33 |
20.89 |
| 3 |
10 |
6.67 |
7.98 |
| 4+ |
15 |
10.00 |
2.93 |
Chi-Square Test
for Specified Proportions |
| Chi-Square |
47.9694 |
| DF |
4 |
| Pr > ChiSq |
<.0001 |
| Discrete Data Models |
| Mites on Apple Leaves |
| Negative Binomial Model |
| Model Information |
| Data Set |
WORK.APPLELEAVES |
| Distribution |
Negative Binomial |
| Link Function |
Log |
| Dependent Variable |
Y |
| Observations Used |
150 |
| Parameter Information |
| Parameter |
Effect |
| Prm1 |
Intercept |
| Criteria For Assessing Goodness Of Fit |
| Criterion |
DF |
Value |
Value/DF |
| Deviance |
149 |
152.2965 |
1.0221 |
| Scaled Deviance |
149 |
152.2965 |
1.0221 |
| Pearson Chi-Square |
149 |
139.4156 |
0.9357 |
| Scaled Pearson X2 |
149 |
139.4156 |
0.9357 |
| Log Likelihood |
|
-128.0874 |
|
| Analysis Of Parameter Estimates |
| Parameter |
DF |
Estimate |
Standard Error |
Likelihood Ratio 95%
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Intercept |
1 |
0.1369 |
0.1110 |
-0.0823 |
0.3575 |
1.52 |
0.2176 |
| Dispersion |
1 |
0.9760 |
0.2628 |
0.5446 |
1.5989 |
|
|
| NOTE: |
The negative binomial dispersion parameter was estimated by maximum likelihood. |
|
| Contrast Estimate Results |
| Label |
Estimate |
Standard Error |
Alpha |
Confidence Limits |
Chi-Square |
Pr > ChiSq |
| Population Mean |
0.1369 |
0.1110 |
0.05 |
-0.0807 |
0.3544 |
1.52 |
0.2176 |
| Exp(Population Mean) |
1.1467 |
0.1273 |
0.05 |
0.9225 |
1.4253 |
|
|
| Discrete Data Models |
| Mites on Apple Leaves |
| Negative Binomial Model |
| Expected Probabilities |
| Obs |
Mu |
k |
kinv |
VarY |
Y |
Prob |
Expected |
Cummulative |
InvCum |
| 1 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
0 |
0.46325 |
69.4880 |
0.46325 |
1.00000 |
| 2 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
1 |
0.25067 |
37.5999 |
0.71392 |
0.53675 |
| 3 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
2 |
0.13401 |
20.1011 |
0.84793 |
0.28608 |
| 4 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
3 |
0.07135 |
10.7026 |
0.91928 |
0.15207 |
| 5 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
4 |
0.03791 |
5.6869 |
0.95719 |
0.08072 |
| 6 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
5 |
0.02012 |
3.0181 |
0.97731 |
0.04281 |
| 7 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
6 |
0.01067 |
1.6004 |
0.98798 |
0.02269 |
| 8 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
7 |
0.00565 |
0.8482 |
0.99363 |
0.01202 |
| 9 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
8 |
0.00300 |
0.4493 |
0.99663 |
0.00637 |
| 10 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
9 |
0.00159 |
0.2379 |
0.99822 |
0.00337 |
| 11 |
1.14667 |
0.97600 |
1.02459 |
2.42995 |
10 |
0.00084 |
0.1260 |
0.99906 |
0.00178 |
| Discrete Data Models |
| Mites on Apple Leaves |
| Negative Binomial Model |
| Pearson Chi-square Goodness-of-fit Test |
| Note: Degrees of Freedom Should Be 3 |
| Y |
Frequency |
Percent |
Test
Percent |
| 0 |
70 |
46.67 |
46.33 |
| 1 |
38 |
25.33 |
25.07 |
| 2 |
17 |
11.33 |
12.40 |
| 3 |
10 |
6.67 |
7.14 |
| 4 |
9 |
6.00 |
3.79 |
| 5+ |
6 |
4.00 |
4.28 |
Chi-Square Test
for Specified Proportions |
| Chi-Square |
2.1504 |
| DF |
5 |
| Pr > ChiSq |
0.8280 |