Wedderburn (1974) analyzes data on the incidence of leaf blotch (Rhynchosporium secalis) on barley.
The data represent the percentage of leaf area affected in a twoway layout with 10 barley varieties at nine sites. The following DATA step converts these data to proportions, as analyzed in McCullagh and Nelder (1989, Ch. 9.2.4). The purpose of the analysis is to make comparisons among the varieties, adjusted for site effects.
data blotch; array p{9} pct1pct9; input variety pct1pct9; do site = 1 to 9; prop = p{site}/100; output; end; drop pct1pct9; datalines; 1 0.05 0.00 1.25 2.50 5.50 1.00 5.00 5.00 17.50 2 0.00 0.05 1.25 0.50 1.00 5.00 0.10 10.00 25.00 3 0.00 0.05 2.50 0.01 6.00 5.00 5.00 5.00 42.50 4 0.10 0.30 16.60 3.00 1.10 5.00 5.00 5.00 50.00 5 0.25 0.75 2.50 2.50 2.50 5.00 50.00 25.00 37.50 6 0.05 0.30 2.50 0.01 8.00 5.00 10.00 75.00 95.00 7 0.50 3.00 0.00 25.00 16.50 10.00 50.00 50.00 62.50 8 1.30 7.50 20.00 55.00 29.50 5.00 25.00 75.00 95.00 9 1.50 1.00 37.50 5.00 20.00 50.00 50.00 75.00 95.00 10 1.50 12.70 26.25 40.00 43.50 75.00 75.00 75.00 95.00 ;
Little is known about the distribution of the leaf area proportions. The outcomes are not binomial proportions, because they
do not represent the ratio of a count over a total number of Bernoulli trials. However, because the mean proportion for variety j on site i must lie in the interval , you can commence the analysis with a model that treats Prop
as a “pseudobinomial” variable:








Here, is the linear predictor for variety j on site i, denotes the ith site effect, and denotes the jth barley variety effect. The logit of the expected leaf area proportions is linearly related to these effects. The variance function of the model is that of a binomial(n,) variable, and is an overdispersion parameter. The moniker “pseudobinomial” derives not from the pseudolikelihood methods used to estimate the parameters in the model, but from treating the response variable as if it had first and second moment properties akin to a binomial random variable.
The model is fit in the GLIMMIX procedure with the following statements:
proc glimmix data=blotch; class site variety; model prop = site variety / link=logit dist=binomial; random _residual_; lsmeans variety / diff=control('1'); run;
The MODEL statement specifies the distribution as binomial and the logit link. Because the variance function of the binomial distribution is , you use the statement
random _residual_;
to specify the scale parameter . The LSMEANS statement requests estimates of the least squares means for the barley variety. The DIFF=CONTROL(’1’) option requests tests of least squares means differences against the first variety.
The “Model Information” table in Output 41.4.1 describes the model and methods used in fitting the statistical model. It is assumed here that the data are binomial proportions.
Output 41.4.1: Model Information in Pseudobinomial Analysis
Model Information  

Data Set  WORK.BLOTCH 
Response Variable  prop 
Response Distribution  Binomial 
Link Function  Logit 
Variance Function  Default 
Variance Matrix  Diagonal 
Estimation Technique  Maximum Likelihood 
Degrees of Freedom Method  Residual 
The “Class Level Information” table in Output 41.4.2 lists the number of levels of the Site
and Variety
effects and their values. All 90 observations read from the data are used in the analysis.
Output 41.4.2: Class Levels and Number of Observations
Class Level Information  

Class  Levels  Values 
site  9  1 2 3 4 5 6 7 8 9 
variety  10  1 2 3 4 5 6 7 8 9 10 
Number of Observations Read  90 

Number of Observations Used  90 
In Output 41.4.3, the “Dimensions” table shows that the model does not contain Gside random effects. There is a single covariance parameter, which corresponds to . The “Optimization Information” table shows that the optimization comprises 18 parameters (Output 41.4.3). These correspond to the 18 nonsingular columns of the matrix.
Output 41.4.3: Model Fit in Pseudobinomial Analysis
Dimensions  

Covariance Parameters  1 
Columns in X  20 
Columns in Z  0 
Subjects (Blocks in V)  1 
Max Obs per Subject  90 
Optimization Information  

Optimization Technique  NewtonRaphson 
Parameters in Optimization  18 
Lower Boundaries  0 
Upper Boundaries  0 
Fixed Effects  Not Profiled 
Fit Statistics  

2 Log Likelihood  57.15 
AIC (smaller is better)  93.15 
AICC (smaller is better)  102.79 
BIC (smaller is better)  138.15 
CAIC (smaller is better)  156.15 
HQIC (smaller is better)  111.30 
Pearson ChiSquare  6.39 
Pearson ChiSquare / DF  0.09 
There are significant site and variety effects in this model based on the approximate Type III F tests (Output 41.4.4).
Output 41.4.4: Tests of Site and Variety Effects in Pseudobinomial Analysis
Type III Tests of Fixed Effects  

Effect  Num DF  Den DF  F Value  Pr > F 
site  8  72  18.25  <.0001 
variety  9  72  13.85  <.0001 
Output 41.4.5 displays the Variety
least squares means for this analysis. These are obtained by averaging

across the sites. In other words, LSmeans are computed on the linked scale where the model effects are additive. Note that the least squares means are ordered by variety. The estimate of the expected proportion of infected leaf area for the first variety is

and that for the last variety is

Output 41.4.5: Variety Least Squares Means in Pseudobinomial Analysis
variety Least Squares Means  

variety  Estimate  Standard Error  DF  t Value  Pr > t 
1  4.3800  0.5643  72  7.76  <.0001 
2  4.2300  0.5383  72  7.86  <.0001 
3  3.6906  0.4623  72  7.98  <.0001 
4  3.3319  0.4239  72  7.86  <.0001 
5  2.7653  0.3768  72  7.34  <.0001 
6  2.0089  0.3320  72  6.05  <.0001 
7  1.8095  0.3228  72  5.61  <.0001 
8  1.0380  0.2960  72  3.51  0.0008 
9  0.8800  0.2921  72  3.01  0.0036 
10  0.1270  0.2808  72  0.45  0.6523 
Because of the ordering of the least squares means, the differences against the first variety are also ordered from smallest to largest (Output 41.4.6).
Output 41.4.6: Variety Differences against the First Variety
Differences of variety Least Squares Means  

variety  _variety  Estimate  Standard Error  DF  t Value  Pr > t 
2  1  0.1501  0.7237  72  0.21  0.8363 
3  1  0.6895  0.6724  72  1.03  0.3086 
4  1  1.0482  0.6494  72  1.61  0.1109 
5  1  1.6147  0.6257  72  2.58  0.0119 
6  1  2.3712  0.6090  72  3.89  0.0002 
7  1  2.5705  0.6065  72  4.24  <.0001 
8  1  3.3420  0.6015  72  5.56  <.0001 
9  1  3.5000  0.6013  72  5.82  <.0001 
10  1  4.2530  0.6042  72  7.04  <.0001 
This analysis depends on your choice for the variance function that was implied by the binomial distribution. You can diagnose the distributional assumption by examining various graphical diagnostics measures. The following statements request a panel display of the Pearsontype residuals:
ods graphics on; ods select PearsonPanel; proc glimmix data=blotch plots=pearsonpanel; class site variety; model prop = site variety / link=logit dist=binomial; random _residual_; run; ods graphics off;
Output 41.4.7 clearly indicates that the chosen variance function is not appropriate for these data. As approaches zero or one, the variability in the residuals is less than that implied by the binomial variance function.
Output 41.4.7: Panel of PearsonType Residuals in Pseudobinomial Analysis
To remedy this situation, McCullagh and Nelder (1989) consider instead the variance function

Imagine two varieties with and . Under the binomial variance function, the variance of the proportion for variety k is 2.77 times larger than that for variety i. Under the revised model this ratio increases to .
The analysis of the revised model is obtained with the next set of GLIMMIX statements. Because you need to model a variance function that does not correspond to any of the builtin distributions, you need to supply a function with an assignment to the automatic variable _VARIANCE_. The GLIMMIX procedure then considers the distribution of the data as unknown. The corresponding estimation technique is quasilikelihood. Because this model does not include an extra scale parameter, you can drop the RANDOM _RESIDUAL_ statement from the analysis.
ods graphics on; ods select ModelInfo FitStatistics LSMeans Diffs PearsonPanel; proc glimmix data=blotch plots=pearsonpanel; class site variety; _variance_ = _mu_**2 * (1_mu_)**2; model prop = site variety / link=logit; lsmeans variety / diff=control('1'); run; ods graphics off;
The “Model Information” table in Output 41.4.8 now displays the distribution as “Unknown,” because of the assignment made in the GLIMMIX statements to _VARIANCE_. The table also shows the expression evaluated as the variance function.
Output 41.4.8: Model Information in Quasilikelihood Analysis
Model Information  

Data Set  WORK.BLOTCH 
Response Variable  prop 
Response Distribution  Unknown 
Link Function  Logit 
Variance Function  _mu_**2 * (1_mu_)**2 
Variance Matrix  Diagonal 
Estimation Technique  QuasiLikelihood 
Degrees of Freedom Method  Residual 
The fit statistics of the model are now expressed in terms of the log quasilikelihood. It is computed as

Twice the negative of this sum equals –85.74, which is displayed in the “Fit Statistics” table (Output 41.4.9).
The scaled Pearson statistic is now 0.99. Inclusion of an extra scale parameter would have little or no effect on the results.
Output 41.4.9: Fit Statistics in Quasilikelihood Analysis
Fit Statistics  

2 Log QuasiLikelihood  85.74 
QuasiAIC (smaller is better)  49.74 
QuasiAICC (smaller is better)  40.11 
QuasiBIC (smaller is better)  4.75 
QuasiCAIC (smaller is better)  13.25 
QuasiHQIC (smaller is better)  31.60 
Pearson ChiSquare  71.17 
Pearson ChiSquare / DF  0.99 
The panel of Pearsontype residuals now shows a much more adequate distribution for the residuals and a reduction in the number of outlying residuals (Output 41.4.10).
Output 41.4.10: Panel of PearsonType Residuals (Quasilikelihood)
The least squares means are no longer ordered in size by variety (Output 41.4.11). For example, . Under the revised model, the second variety has a greater percentage of its leaf area covered by blotch, compared to the first variety. Varieties 5 and 6 and varieties 8 and 9 show similar reversal in ranking.
Output 41.4.11: Variety Least Squares Means in Quasilikelihood Analysis
variety Least Squares Means  

variety  Estimate  Standard Error  DF  t Value  Pr > t 
1  4.0453  0.3333  72  12.14  <.0001 
2  4.5126  0.3333  72  13.54  <.0001 
3  3.9664  0.3333  72  11.90  <.0001 
4  3.0912  0.3333  72  9.27  <.0001 
5  2.6927  0.3333  72  8.08  <.0001 
6  2.7167  0.3333  72  8.15  <.0001 
7  1.7052  0.3333  72  5.12  <.0001 
8  0.7827  0.3333  72  2.35  0.0216 
9  0.9098  0.3333  72  2.73  0.0080 
10  0.1580  0.3333  72  0.47  0.6369 
Interestingly, the standard errors are constant among the LSmeans (Output 41.4.11) and among the LSmeans differences (Output 41.4.12). This is due to the fact that for the logit link

which cancels with the square root of the variance function in the estimating equations. The analysis is thus orthogonal.
Output 41.4.12: Variety Differences in Quasilikelihood Analysis
Differences of variety Least Squares Means  

variety  _variety  Estimate  Standard Error  DF  t Value  Pr > t 
2  1  0.4673  0.4714  72  0.99  0.3249 
3  1  0.07885  0.4714  72  0.17  0.8676 
4  1  0.9541  0.4714  72  2.02  0.0467 
5  1  1.3526  0.4714  72  2.87  0.0054 
6  1  1.3286  0.4714  72  2.82  0.0062 
7  1  2.3401  0.4714  72  4.96  <.0001 
8  1  3.2626  0.4714  72  6.92  <.0001 
9  1  3.1355  0.4714  72  6.65  <.0001 
10  1  3.8873  0.4714  72  8.25  <.0001 