Generalized Linear Mixed Models for Categorical and Ordinal Responses

Abstract

According to Agresti (2013), a multinomial distribution is a generalization of a binomial distribution in cases with more than two possible ordered (ordinal) or unordered (nominal) outcomes. Given a response with more than two possible outcomes and independent trials with probabilities of similar category for each trial, the distribution of counts across categories follows a multinomial distribution. Quinn and Keough (2002) believe that several methods exist for multinomial data analysis. The most common form of categorical data analysis in biological sciences, which results in frequency counts, is creating cross-tabulations or contingency tables and chi-squared tests to examine associations between two or more categorical variables. However, such an approach is ill suited for a study aimed at estimating the response when there is a change in the explanatory variable(s), as contingency tables are used to analyze the association between variables without considering a predictor or response variable. In this analysis, the results are valid as long as less than 20% of the cells have an expected count less than five and none are less than one (Logan 2010). Fisher’s exact test extends the chi-squared test in studies involving small sample sizes.

8.1 Introduction

According to Agresti (2013), a multinomial distribution is a generalization of a binomial distribution in cases with more than two possible ordered (ordinal) or unordered (nominal) outcomes. Given a response with more than two possible outcomes and independent trials with probabilities of similar category for each trial, the distribution of counts across categories follows a multinomial distribution. Quinn and Keough (2002) believe that several methods exist for multinomial data analysis. The most common form of categorical data analysis in biological sciences, which results in frequency counts, is creating cross-tabulations or contingency tables and chi-squared tests to examine associations between two or more categorical variables. However, such an approach is ill suited for a study aimed at estimating the response when there is a change in the explanatory variable(s), as contingency tables are used to analyze the association between variables without considering a predictor or response variable. In this analysis, the results are valid as long as less than 20% of the cells have an expected count less than five and none are less than one (Logan 2010). Fisher’s exact test extends the chi-squared test in studies involving small sample sizes.

There are several methods for modeling multinomial data; traditional methods of multinomial data analysis include frequency analysis (counts), which uses the chi-squared test and the log-linear model for contingency tables. This chapter focuses on describing multinomial logit and probit models in detail.

8.2 Concepts and Definitions

For the multinomial distribution each observation drawn from a total of N observations belongs to exactly one of the mutually and exclusive c = 1, ⋯, C categories and each category has a probability π_c (c = 1, ⋯, C) of belonging to the category c. A multinomial distribution refers to the probability that exactly one randomly sampled observation from the population belongs to category y₁, that is, it belongs to category 1, y₂ observations belong to category 2, and so forth up to category C,where $\sum \limits_{c=1}^C{y}_c=N$ and $\sum \limits_{c=1}^C{\pi}_c=1$. The density function of this distribution is equal to

$$ f\left({y}_1,{y}_2,\dots, {y}_C\right)=\frac{N!}{y_1!{y}_2!\dots {y}_C!}{\pi}_1^{y_1}{\pi}_2^{y_2}\dots {\pi}_C^{y_c} $$

Multinomial models are applied in data analysis where the categorical response variable has more than two possible outcomes while the independent variables can be continuous, categorical, or both (Hosmer and Lemeshow 2000). The categorical response variable can be either ordinal (ordered) or nominal (unordered). Ordinal response variables are single values that represent a rank order on some dimension, but there are not enough values to be treated as a continuous variable. Nominal (unordered) response variables are those whose values provide a rank but do not provide an indication of order. Models for multinomial data are constructed in a similar way as for binomial data. The link functions used in these types of models are similar to the logit and probit functions used for binomial data. Cumulative logit and cumulative probit models define the link function such that when properly fitted to the data, they allow for parsimonious modeling of ordinal or multinomial data. Generalized logit and probit models do not require ordered categories and are therefore suitable for multinomial nominal data.

In terms of generalized linear models (GLMs) and generalized linear mixed models (GLMMs), a multinomial distribution with C categories requires C − 1 link functions to fully specify a model that relates the response probabilities (π₁, π₂, …, π_C) to the linear predictor. The commonly used models are the cumulative logit model, also known as the proportional odds model proposed by McCullagh (1980), and the cumulative probit model, also known as the threshold model. Throughout this chapter, we will use either of these two link functions interchangeably.

The link functions for a cumulative logit model with C categories are

$$ {\displaystyle \begin{array}{c}{\boldsymbol{\eta}}_1=\log \left(\frac{\pi_1}{1-{\pi}_1}\right)={\eta}_1+\boldsymbol{X}\boldsymbol{\beta } +\boldsymbol{Zb}\\ {}{\boldsymbol{\eta}}_2=\log \left(\frac{\pi_1+{\pi}_2}{1-\left({\pi}_1+{\pi}_2\right)}\right)={\eta}_2+\boldsymbol{X}\boldsymbol{\beta } +\boldsymbol{Zb}\\ {}\vdots \\ {}{\boldsymbol{\eta}}_{\boldsymbol{C}-1}=\log \left(\frac{\pi_1+{\pi}_2+\cdots +{\pi}_{C-1}}{1-\left({\pi}_1+{\pi}_2+\cdots +{\pi}_{C-1}\right)}\right)={\eta}_{C-1}+\boldsymbol{X}\boldsymbol{\beta } +\boldsymbol{Zb}\end{array}} $$

where X and Z are the design matrices, whereas β and b are the vectors of fixed and random effects parameters, respectively. The inverse links of each of the functions are as follows:

$$ {\displaystyle \begin{array}{c}{\pi}_1=\frac{1}{1+{e}^{-{\boldsymbol{\eta}}_1}}=h\left({\boldsymbol{\eta}}_1\right)\\ {}{\pi}_1+{\pi}_2=\frac{1}{1+{e}^{-{\boldsymbol{\eta}}_2}}=h\left({\boldsymbol{\eta}}_2\right)\\ {}\vdots \\ {}{\pi}_1+{\pi}_2+\cdots +{\pi}_{C-1}=\frac{1}{1+{e}^{-{\boldsymbol{\eta}}_{\boldsymbol{c}-1}}}=h\left({\boldsymbol{\eta}}_{\boldsymbol{C}-1}\right).\end{array}} $$

Once h(η₁), h(η₂), ... h(η_c − 1) have been estimated, we can then estimate the probabilities ${\hat{\pi}}_1$, ${\hat{\pi}}_2$, ..., ${\hat{\pi}}_c$.

8.3 Cumulative Logit Models (Proportional Odds Models)

Multinomial logit models are used to model the relationships between a polytomous response variable and a set of predictor variables. These polytomous response models can be classified – as mentioned above – into two different types, depending on whether the response variable has an ordered or an unordered structure.

In a proportional odds model, the covariates (linear predictor η) have the same effect on the probabilities that the response variable has in any category when considering different values of the covariates, thus shifting the response distribution to the right (or left) without changing the shape of the distribution. In a proportional odds model, the cumulative logits model the effect of the covariates on the response probabilities below or equal to the category cutoff.

A multinomial logit model assumes independence of categories, which implies that the probabilities of choosing a category c relative to a category c^′ are independent of the category characteristics of c and c^′ for c ≠ c^′. The assumption requires that if a new category is available, then the prior probabilities are precisely adjusted to preserve the original probabilities between all pairs of outcomes. The proportional odds model employs a strict assumption that the odds ratio does not depend on the category, and, therefore, we need to test the proportional odds assumption, which is also called the “parallel regression assumption.”

8.3.1 Complete Randomize Design (CRD) with a Multinomial Response: Ordinal

Data are obtained from an experiment related to red core disease in strawberries, which is caused by the fungus Phytophthora fragariae. In this example, 12 strawberry populations were evaluated in a completely randomized experiment with 4 replications (Table 8.1). Plots generally consisted of 10 plants; in some cases, only 9 plants were observed. At the end of the experiment, each plant was assigned to one of three ordered categories representing fungal damage (1 = no damage, 2 = moderate damage, and 3 = severe damage).

Table 8.1 Evaluation of red core disease in strawberry plants

A total of 12 populations were obtained by crossing 3 genotypes of male parents with 4 genotypes of female parents. The variation between and within plots is considered minimal, whereas the genetic and nongenetic effects are more significant, as plants from the same cross are not genetically identical.

The model that fits these data for the cumulative probabilities is a GLMM, which exhibit a classification effect on the treatment variable (population resulting from crossing genotypes). Thus, the GLMM for multinomial ordered outcomes with C categories requires C − 1 link function equations to fully specify the model that relates the response probabilities (π₁, π₂, …, π_C) to the linear predictor η_ij (Stroup 2013). The C − 1 multinomial logit equations are tested against each of the remaining categories 1, 2, . . , C − 1.

The components of the GLMM with an ordinal multinomial response are as follows:

• Distributions: y_1ij, y_2ij, y_3ij|r_j~Multinomial(N_ij, π_1ij, π_2ij, π_3ij), where y_1ij, y_2ij, and y_3ij are the observed frequencies of responses (damage level) in each category C (1 = no damage, 2 = moderate damage, and 3 = severe damage) and r_j is the random effect due to repetition, assuming ${r}_j\sim N\left(0,{\sigma}_{\mathrm{r}}^2\right)$.

• Linear predictor: η_cij = η_c + τ_i + r_j, where η_cij is the cth link (c = 1, 2, 3) that relates the mean and the linear predictor for the treatment i (i = 1, 2, …, 12) and the jth block (j = 1, 2, 3, 4); η_c is the intercept for the cth link; τ_i is the fixed effect due to the ith treatment (cross); and r_j is the random effect due to the jth repetition $\left({r}_j\sim N\left(0,{\sigma}_{\mathrm{r}}^2\right)\right)$. The link functions for each category are as follows:

  $$ \log \left(\frac{\pi_{1 ij}}{1-{\pi}_{1 ij}}\right)={\eta}_{1 ij} $$
  $$ \log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}\right)}\right)={\eta}_{2 ij} $$

The following GLIMMIX program fits a cumulative logit model with an ordinal multinomial response in a CRD.

proc glimmix data=FRESA; class rep trt cat; model cat(order=data)= trt/dist=Multinomial link=clogit solution oddsratio; random intercept/subject=rep solution ; estimate ’c=1, t=1’ intercept 1 0 trt 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, ’c=2, t=1’ intercept 0 1 trt 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, ’c=1, t=2’ intercept 1 0 trt 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, ’c=2, t=2’ intercept 0 1 trt 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, ’c=1, t=3’ intercept 1 0 trt 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0, ’c=2, t=3’ intercept 0 1 trt 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0, ’c=1, t=4’ intercept 1 0 trt 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0, ’c=2, t=4’ intercept 0 1 trt 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0, ’c=1, t=5’ intercept 1 0 trt 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0, ’c=2, t=5’ intercept 0 1 trt 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0, ’c=1, t=6’ intercept 1 0 trt 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0, ’c=2, t=6’ intercept 0 1 trt 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0, ’c=1, t=7’ intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0, ’c=2, t=7’ intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0, ’c=1, t=8’ intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0, ’c=2, t=8’ intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0, ’c=1, t=9’ intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0, ’c=2, t=9’ intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0, ’c=1, t=10’ intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0, ’c=2, t=10’ intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0, ’c=1, t=11’ intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0, ’c=2, t=11’ intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0, ’c=1, t=12’ intercept 1 0 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1, ’c=2, t=12’ intercept 0 1 trt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1/ilink; freq freq; run;

Although most of the GLIMMIX commands have already been described in previous examples, it is important to emphasize that the data should be structured in a logical way as follows: one line for repetition, treatment, lesion category, and the frequency or number of observations (Y), which, in this case, is referenced by the variables rep, trt (trt = cross), cat (category), and freq, respectively. Part of the data arrangement can be seen in Table 8.2, whereas the rest of the dataset can be found in the Appendix (Data: CRD with multinomial response: ordinal).

Table 8.2 Data ordered

In the program commands of this example, “order = data” indicates that the order in which the categories are arranged in the dataset is under an order (ordinal) category. Consider that the observations in each line always have order categories such as no injury (Without), moderate injury (Moderate), and severe injury (Severe). If there is no congruent order in the arrangement of the dataset to be analyzed, then GLIMMIX will reorder the categories in an alphabetical or numerical order depending on the initial coding of the data. The “estimate” command specifies the estimable functions that form the boundaries between the categories for each of the populations (trt). Finally, the “freq command” instructs GLIMMIX to use “freq” as the number of observations (frequency) under the corresponding categorization. In this way, the first estimate ^“c = 1, t = 1^” defines the predictor η₁ + τ₁, that is, the boundary between the “Without” and “Moderate” categories for treatment 1 with its corresponding logit $\log \left(\frac{\pi_{11}}{1-{\pi}_{11}}\right)$, whereas the second estimate ^“c = 2, t = 1^” defines the boundary between the categories of “Moderate” and “Severe” damage with the logit $\mathit{\log}\left(\frac{\pi_{11}+{\pi}_{21}}{1-\left({\pi}_{11}+{\pi}_{21}\right)}\right),$ which estimates the probability of observing a plant from population1 (M1H1 = trt) “Without” damage and “Moderate” damage when exposed to the fungus (Phytophthora fragariae). Part of the output is presented in Table 8.3.

Table 8.3 Results of the multinomial analysis of variance for injury level in strawberry plants

The estimated variance component (part (a)) due to plants is ${\hat{\sigma}}_{\mathrm{r}}^2=0.1453$, whereas the hypothesis tests for type III effects (part (b)) (“Type III tests of fixed effects”) indicate that the crosses have different significant tolerance levels to fungal attacks (Pr > F = P = 0.0032). The results of the fixed effects solution, obtained by specifying the “solution” option in the model, are shown in Table 8.4.

Table 8.4 Fixed effects solution for injury categories

From the fixed effects solution, we can estimate the linear predictors for the two categories of each treatment, which are in terms of the model scale. For example, for treatment 1, the first category of injury ${\hat{\eta}}_{11}={\hat{\eta}}_1+{\hat{\tau}}_1=-0.4571+\left(-1.1456\right)=-1.6027,$ where ${\hat{\eta}}_1$ defines the boundary between the categories “Without” damage and “Moderate” damage and ${\hat{\eta}}_2$ defines the boundary between the categories “Moderate” damage and “Severe” damage, and the linear predictor is ${\hat{\eta}}_{11}={\hat{\eta}}_2+{\hat{\tau}}_1=1.0631+\left(-1.1456\right)=-0.0825$. Note that for the proportional odds, the τ_i values are not category-specific; treatment effects move the boundaries as a group.

The odds ratio (Table 8.5) is the result of taking ${e}^{{\hat{\tau}}_i}$ for crosses 1–12. Since odds ratios are not specific to a particular category, this value is the same for all three categories and hence the name odds ratio.

Table 8.5 Estimated odds ratio

In Table 8.6, we show the maximum likelihood estimates of the linear predictors ${\hat{\eta}}_{ci}={\hat{\eta}}_C+{\hat{\tau}}_i$ in the “Estimate” column, in terms of the model scale, as well as the means on the data scale for each of the categories of the treatments tested (“Mean”).

Table 8.6 Estimates on the model scale (Estimate) and on the data scale (Mean) for the damage categories in strawberry plants

Thus, for c = 1, t = 1 (response category “Without” damage and treatment 1), the estimator is ${\hat{\eta}}_{11}=-1.6027$ and for c = 2, t = 1 (“Moderate” damage and treatment 1), the linear predictor is ${\hat{\eta}}_{21}=-0.0825$. Taking the inverse of the link function yields the probability of ${\hat{\pi}}_{11}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1+{e}^{1.6027}$}\right.=0.1676$. This is the estimated probability for which the cross (treatment) M1H1 has a response score of “Without damage.” This inverse value is presented under the “Mean” column (Table 8.6).

Now, for c = 2, t = 1, the inverse of the link yields the following probability: ${\hat{\pi}}_{11}+{\hat{\pi}}_{21}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1+{e}^{0.0825}$}\right.=0.4794$ (cumulative probability). From this value, we deduce the probability of observing a “Moderate” damage and a “Severe” damage in the plant of the cross M1H1. For “Moderate” damage, $\mathrm{the}\ \mathrm{probability}\ \mathrm{is}\ {\hat{\pi}}_{21}=0.4794-{\hat{\pi}}_{11}=0.4794-0.1676=0.3118$, and, for “Severe” damage, $\mathrm{it}\ \mathrm{is}\ {\hat{\pi}}_{31}=1-{\hat{\pi}}_{11}+{\hat{\pi}}_{21}=1-0.4794=0.5206$. Similarly, the rest of the probabilities in the different crosses are estimated.

8.3.2 Randomized Complete Block Design (RCBD) with a Multinomial Response: Ordinal

In recent years, poultry production has become conscious of animal welfare, which is associated with bird mortality, behavior, and health, among others (Stanley 1981; Martrenchar et al. 2002). One of the diseases related to animal welfare is footpad dermatitis, and, among many repercussions, it affects a bird’s ability to walk (Bilgili et al. 2009). Pododermatitis is known as contact dermatitis or footpad dermatitis and is characterized by inflammation and necrotic lesions from the plantar surface to deep within the footpads of chicken. Deep ulcers may result in abscesses and in the thickening of the underlying tissues and structures (Greene et al. 1985).

Chicken feet have great economic importance because they are in high demand in the foreign market, mainly in Southeast Asia and China; however, due to diseases or alterations such as pododermatitis, there are significant economic losses since diseased feet are not suitable for human consumption and this, subsequently, reflects in market prices (Taira et al. 2014). Due to the economic importance of this product, Garcia et al. (2010) have focused on studying the factors that cause this disease and on finding strategies to reduce leg and carcass lesions in poultry. Important factors in broiler fattening are the type of litter, litter height, nutrition and feeding programs, and bird health, among others.

The objective of this study was to evaluate the effect of litter density and organic minerals (Availa Zn and Availa Mn), with an extract of Yucca schidigera (Micro-Aid) as a supplement to a traditional fattening program, on the development of footpad dermatitis in broilers. The genetic material used in this experiment was mainly male Ross line chickens. The traditional broiler fattening program by the poultry farm consists of three phases: a starter diet (1–18 days), a grower diet (19–35 days), and a finisher diet (36–50 days), applied for a period of 50 days, where rice husk is used as bedding material at a density of 1 kg m⁻². In this research, a foot health program was implemented in addition to the traditional fattening program, which included the addition of 125 ppm of Micro-Aid (Yucca schidigera extract), 40 ppm of Availa Zn, and 40 ppm of Availa Mn to the fattening diet.

Based on the above information, four treatments were evaluated at two poultry farms, as described below:

• Treatment 1 involved the application of the company’s traditional fattening program (Trt1).

• Treatment 2 was the company’s traditional fattening program plus an increase in litter density from 1 to 2 kg m⁻² (Trt2).

• Treatment 3 was the traditional fattening program plus the implementation of the foot health program during the fattening period until completion (Trt3).

• Treatment 4 consisted of the traditional fattening program plus the implementation of the foot health program and an increase in litter density from 1 to 2 kg m⁻² (Trt4). The following table lists the treatments studied (Table 8.7):

Table 8.7 Treatment design

The response variable evaluated was the degree of foot lesion (pododermatitis) at the end of the fattening period (50 days). The response variable was evaluated on 1250 chickens per treatment. The degree of a footpad lesion was determined according to a visual guide for lesions in chickens based on the method of De Jong and Guémené (2012). This method entails defining three grades: grade 0 is attributed to legs with no lesions, grade one is if lesions exist in some areas of the footpad (<50%), and grade two is if the leg has extensive lesions in areas of the footpad (50–100%). Table 8.8 shows the dataset indicating the block, treatment, level of lesion, and the number of birds observed with a given lesion (frequency).

Table 8.8 Pododermatitis in broilers

The GLMM for multinomial ordered results with C categories requires C − 1 link function equations instead of one to fully specify a model that relates the response probabilities (π₁, π₂, …, π_C) to the linear predictor η_ij (Stroup 2013). The C − 1 multinomial logit equations are tested against each of the categories 1, 2, …, C − 1.

The link functions for the cumulative logit model to describe the response variable with C categories are as follows:

$$ {\displaystyle \begin{array}{c}{\eta}_{(1) ij}=\log \left(\frac{\pi_{1 ij}}{1-{\pi}_{1 ij}}\right)={\eta}_1+{\tau}_i+{b}_j\\ {}{\eta}_{(2) ij}=\log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}\right)}\right)={\eta}_2+{\tau}_i+{b}_j\\ {}\vdots \\ {}{\eta}_{\left(C-1\right) ij}=\log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}+\cdots +{\pi}_{\left(C-1\right) ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}+\cdots +{\pi}_{\left(C-1\right) ij}\right)}\right)={\eta}_{C-1}+{\tau}_i+{b}_j\end{array}} $$

The components of the GLMM with an ordinal multinomial response variable are as follows:

• Distributions: y_oij, y_1ij, y_2ij|b_j ~ Multinomial(N_ij, π_0ij, π_1ij, π_2ij), where y_oij, y_1ij, and y_2ij are the observed frequencies of the responses (paw injury) in each category (none, mild, and severe) and b_j is the random effect due to block assuming ${b}_j\sim N\left(0,{\sigma}_{\mathrm{b}}^2\right)$.

• Linear predictor: η_(c)ij = η_c + τ_i + b_j, where η_(c)ij is cth link (c = 0, 1) for processing i and block j, η_c is the intercept for the cth link, τ_i is the fixed effect due to the ith treatment, and b_j is the random effect due to the jth block $\left({b}_j\sim N\left(0,{\sigma}_{\mathrm{b}}^2\right)\right)$. The link functions for each category are as follows:

  $$ \log \left(\frac{\pi_{0 ij}}{1-{\pi}_{0 ij}}\right)={\eta}_{(0) ij} $$

$$ \log \left(\frac{\pi_{0 ij}+{\pi}_{1 ij}}{1-\left({\pi}_{0 ij}+{\pi}_{1 ij}\right)}\right)={\eta}_{(1) ij} $$

The following GLIMMIX commands fit a cumulative logit model with an ordinal multinomial response.

proc glimmix data=multinomial_ord; class block trt; model categoria (order=data)= trt/dist=Multinomial link=clogit solution oddsratio(DIFF=LAST LABEL); random intercept/subject=block; estimate ’c=0, t=1’ intercept 1 0 trt 1 0, ’c=1, t=1’ intercept 0 1 trt 1 0, ’c=0, t=2’ intercept 1 0 trt 0 1 0, ’c=1, t=2’ intercept 0 1 trt 0 1 0, ’c=0, t=3’ intercept 1 0 trt 0 0 0 1 0, ’c=1, t=3’ intercept 0 1 trt 0 0 0 1 0, ’c=0, t=4’ intercept 0 1 trt 0 0 0 0 1, ’c=1, t=4’ intercept 1 0 trt 0 0 0 0 1/ilink; freq y; run;

The data should have one column for block, treatment, lesion category, and frequency or number of observations (Y), which, in this case, is referenced by the variables block, trt, category, and frequency, respectively.

Most of the options in the above syntax have already been explained previously; the “order = data” option specifies that the order in which the categories appear in the dataset will be treated as ordinal categories from the lowest to the highest for the analysis. If this option is not used with the response variable in the model specification, “proc GLIMMIX” will rearrange its categories in an alphabetical or numerical order, but this will depend on whether the categories are entered as a number or a name. The “freq y^” option orders GLIMMIX to use y as the number of observations in the corresponding category. The “estimate” command specifies the estimable functions that form the boundaries between categories of each of the four treatments. For example, the first estimate “c = 0, t = 1” defines η₀ + τ₁, that is, the boundary between the categories “Without” (no lesion) and “Moderate” (slight lesion) for treatment 1. This first estimate corresponds to logit $\log \left(\frac{\pi_{01}}{1-{\pi}_{01}}\right)$, which is the probability that a chicken that received treatment 1 will respond to a degree of lesion classified under category 0 (no lesion). The second estimation “c = 1, t = 1” defines η₁ + τ₁, that is, the boundary between the categories “Moderate” (slight lesion) and “Severe” (severe lesion) for treatment 1 and corresponds to logit $\log \left(\frac{\pi_{11}}{1-{\pi}_{11}}\right)$, and so on. By taking the inverse of these links values, we can obtain the estimated probabilities of π₀₁ and π₁₁. Part of the Statistical Analysis Software (SAS) glimmix output is presented below:

The results of the analysis of variance in part (a) of Table 8.9 indicate that the degree of lesion in the chicken footpad (pododermatitis) in the treatments tested were significantly different (P < 0.0001). Therefore, the hypothesis of proportional odds of treatments is rejected (H₀ : τ_i = 0 for all i, that is, oddsratio = 1).

Table 8.9 Results of the analysis of variance in the multinomial cumulative logit model

In part (b) of Table 8.9, we can see that the estimated intercepts ${\hat{\eta}}_1=0.6144\kern0.5em$and ${\hat{\eta}}_2=3.8787$ define the boundary between the categories “Without” lesion and “Moderate” lesion and the boundary between the categories “Moderate” lesion and “Severe” lesion, respectively. The estimated effect of the treatments $\left({\hat{\tau}}_i\right)$ shows that the boundaries move either upward or downward when a certain treatment is applied. In this sense, all estimated treatment coefficients have a negative effect with respect to treatment 4. This means that chickens under treatments 1–3 have a low probability of developing a moderate lesion and a higher probability of developing a severe lesion than when treatment 4 is applied.

To calculate the probability that a chicken will not develop footpad dermatitis (c = 0) when receiving treatment 1, that is, “c = 0, Trt = 1,” we first estimate the linear predictor ${\hat{\eta}}_{01}={\hat{\eta}}_0+{\hat{\tau}}_1=0.6144+\left(-1.5034\right)=-0.889$, and, taking the inverse, we obtain ${\hat{\pi}}_{01}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1+{e}^{-\left(-0.889\right)}$}\right.=0.29$. This value is the estimated probability that a chicken will not develop footpad dermatitis when receiving treatment 1. However, now, for “c = 1, Trt = 1,” ${\hat{\eta}}_{11}={\hat{\eta}}_1+{\hat{\tau}}_1=3.8787+\left(-1.5034\right)=2.3753,$whose inverse value is 0.915. This value is an estimate of the probability ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}$. From this value, we obtain the probability that a chicken will develop a moderate lesion and a severe lesion. For a moderate lesion, the probability is ${\hat{\pi}}_{11}=0.915-{\hat{\pi}}_{01}=0.915-0.29=0.624,$and, for a severe lesion, the probability is ${\hat{\pi}}_{21}=1-0.915=0.085$. In a similar way the probabilities for the categories (c = 0, 1, 2) of the rest of the treatments are computed.

The odds ratios tabulated in Table 8.10 are the odds ratios for treatments 1 through 4, i.e., ${e}^{{\hat{\tau}}_i}$ for treatments 1–4. These are the estimated odds ratios of adjacent categories of treatments i (i = 1, 2, 3) relative to treatment 4. Values of τ_i are not category-specific; the odds ratios for “Without” lesion versus “Moderate” lesion and those for “Moderate” lesion versus “Severe” lesion are listed below (hence the name “proportional odds”).

Table 8.10 Estimated odds ratio

From the above odds ratio results, it should be obvious why the F- and P-values in the fixed effects tests are what they are. Adding the “ilink” option to the end of the “estimate” command prompts GLIMMIX to estimate the inverse of the linear predictors $\left({\hat{\eta}}_{ci}\right)$, i.e., the probabilities per category ${\hat{\pi}}_{ci}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1+{e}^{-{\hat{\eta}}_{ci}}$}\right.$ (Table 8.11).

Table 8.11 Estimates on the model scale (Estimate) and on the data scale (Mean) for footpad dermatitis categories in the multinomial cumulative logit model

In the above table, several estimates are shown for ${\hat{\eta}}_c+{\hat{\tau}}_i$. For example, the probability that a chicken will not develop a lesion under treatment 1 can be represented by “c = 0, t = 1,” that is, ${\hat{\eta}}_c+{\hat{\tau}}_1$ = −0.8893. This result matches the one obtained from the fixed effects table “Solutions for fixed effects” previously shown. Taking the inverse of the link yields the probability ${\hat{\pi}}_{01}=1/\left(1+{e}^{0.8893}\right)=0.2914$. This probability is the maximum likelihood estimate that a chicken will have no footpad lesion with treatment 1. The inverse of the link function is under the “Mean” column of Table 8.11. Now, for the category “c = 1, t = 1,” the inverse of the linear predictor is 0.9149, this is the estimate of ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}$. From this value, we can obtain the probability of a chicken showing a “Moderate” lesion when receiving treatment 1, that is, ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}=0.9149$, and, substituting the value of ${\hat{\pi}}_{01},$ we obtain the value ${\hat{\pi}}_{11}=0.9141-0.2914=0.6227$. Finally, for a “Severe” lesion (category^“c = 2, t = 1^”), the probability that a chicken will present a severe lesion is ${\hat{\pi}}_{21}=1-0.9141=0.0859$. Following the same procedure, we can obtain the probabilities for each of the following categories (c = 0, 1, 2) of the rest of the treatments (2–4).

Figure 8.1 shows that under the traditional feeding program with a litter density of 1 kg m⁻² of rice husks (Trt1), there is a high probability that broilers will develop moderate and severe footpad lesions, as shown by ${\hat{\pi}}_{11}=0.624$ and ${\hat{\pi}}_{21}=0.085,\mathrm{respectively}$. When the litter density was increased from 1 to 2 kg m⁻² of rice husks under the traditional broiler program (Trt2), the probability of the risk of developing moderate and severe footpad lesions in broilers decreased significantly to ${\hat{\pi}}_{12}=0.384$ and ${\hat{\pi}}_{22}=0.026$, respectively, compared to Trt1, whereas the probability of not developing a footpad lesion increased to ${\hat{\pi}}_{02}=0.590\ \left(\mathrm{Trt}2\right)$ compared to ${\hat{\pi}}_{01}=0.291\ \left(\mathrm{Trt}1\right)$. Regarding the implementation of the two foot care programs plus the litter density of 2 kg husk m⁻² of rice husks, the probability of chickens of not developing a footpad lesion is ${\hat{\pi}}_{04}=0.649$ (Trt4) compared to ${\hat{\pi}}_{03}=0.396$ in Trt3, whereas the probability of chickens developing moderate and severe lesions decreased from ${\hat{\pi}}_{14}=0.331$ and ${\hat{\pi}}_{24}=0.025$ in Trt4 compared to ${\hat{\pi}}_{13}=0.549$ and ${\hat{\pi}}_{23}=0.055$ in Trt3.

Fig. 8.1

Estimated probabilities for the footpad lesion categories in the treatments tested, using the cumulative logit model

8.4 Cumulative Probit Models

An ordinal cumulative probit model, first considered by Aitchison and Silvey (1957), generalizes a binary probit model to ordinal responses. This model results from the probit modeling of the cumulative probabilities as a linear function of the covariates. The link functions for the cumulative probit model with C categories are listed below:

$$ {\displaystyle \begin{array}{c}{\boldsymbol{\eta}}_1={\Phi}^{-1}\left({\pi}_1\right)={\eta}_1+\boldsymbol{X}\boldsymbol{\beta } +\boldsymbol{Zb}\\ {}{\boldsymbol{\eta}}_2={\Phi}^{-1}\left({\pi}_1+{\pi}_2\right)={\eta}_2+\boldsymbol{X}\boldsymbol{\beta } +\boldsymbol{Zb}\\ {}\vdots \\ {}{\boldsymbol{\eta}}_{\boldsymbol{C}-1}={\Phi}^{-1}\left({\pi}_1+{\pi}_2+\cdots +{\pi}_{C-1}\right)={\eta}_{C-1}+\boldsymbol{X}\boldsymbol{\beta } +\boldsymbol{Zb}\end{array}} $$

where X and Z are the design matrices, β and b are the vectors of fixed and random effects parameters, respectively, and Φ⁻¹() is the inverse function of the standard normal cumulative distribution. The inverse link of each of the link functions is as follows:

$$ {\displaystyle \begin{array}{c}{\pi}_1=\Phi \left({\boldsymbol{\eta}}_1\right)=h\left({\boldsymbol{\eta}}_1\right)\\ {}{\pi}_1+{\pi}_2=\Phi \left({\boldsymbol{\eta}}_2\right)=h\left({\boldsymbol{\eta}}_2\right)\\ {}\vdots \\ {}{\pi}_1+{\pi}_2+\cdots +{\pi}_{C-1}=\Phi \left({\boldsymbol{\eta}}_{\boldsymbol{c}-1}\right)=h\left({\boldsymbol{\eta}}_{\boldsymbol{c}-1}\right).\end{array}} $$

Once h(η₁), h(η₂), ... h(η_c − 1) are estimated, we can estimate ${\hat{\pi}}_1$, ... , ${\hat{\pi}}_C$. The quality of the estimates of the ordinal cumulative probit model are usually very similar to those of an ordinal cumulative logit model for some datasets but not all. Both involve stochastic ordering at different levels of the response variable and are designed to detect the location of changes in the response variable.

Returning to Example 8.3.1, for the cumulative probit model, we change the “LINK = CPROBIT” option in the model’s definition of the above program syntax. The output will contain all the same elements, except the odds ratios. The analysis for the cumulative probit is exactly the same as that one we performed in the cumulative logit model. Part of the output is shown in parts (a)–(c) of Table 8.12.

Table 8.12 Results of the analysis of variance in the multinomial cumulative probit model

The estimated variance component due to blocks is ${\hat{\sigma}}_{\mathrm{block}}^2=0.0092$. The results of the analysis of variance showed that the degrees of lesion in the chickens’ footpad (pododermatitis) in the tested treatments differ significantly (P < 0.0001).

In part (b) of Table 8.12, it is possible to observe that the estimated intercepts ${\hat{\eta}}_1=0.3880$ and ${\hat{\eta}}_2=2.2407$define the boundary between the “Without” lesion and “Moderate” lesion categories and the boundary between the “Moderate” lesion and “Severe” lesion categories, respectively. The estimated effect of the treatments $\left({\hat{\tau}}_i\right)$ moves the boundaries either upward or downward, when a certain treatment is applied. In this sense, all estimated treatment coefficients have a negative effect with respect to treatment 4. This means that chickens under treatments 1–3 have a low probability of developing a footpad lesion and a higher probability of developing a severe lesion with respect to treatment 4.

From “Type III tests of fixed effects” (Table 8.12, part (b)), the probabilities for each of the categories can be obtained. For the probability that a chicken will not develop a footpad lesion (c = 0) under treatment 1, i.e., ^“c = 0, Trt = 1, ^” the estimated linear predictor is obtained as ${\hat{\eta}}_{01}={\hat{\eta}}_0+{\hat{\tau}}_1=0.3880+\left(-0.9278\right)=-0.5398$ and, taking the inverse, gives ${\hat{\pi}}_{01}=\Phi \left(-0.5398\right)=0.2946$, that is, the estimated probability that a chicken will not develop a footpad lesion when receiving treatment 1. For ^“c = 1, Trt = 1, ^” ${\hat{\eta}}_{11}={\hat{\eta}}_1+{\hat{\tau}}_1=2.2407+\left(-0.9278\right)=1.3129$, whose inverse value is 0.9054. This value is an estimator of ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}$. From this value, we can obtain the probability that a chicken will develop a moderate lesion and a severe lesion. For a moderate lesion, ${\hat{\pi}}_{11}=0.9054-{\hat{\pi}}_{01}=0.9054-0.2946=0.6108$, and, for a severe lesion, ${\hat{\pi}}_{21}=1-0.9054=0.0946$. Similarly, we can obtain the probabilities of the categories for the other treatments (c = 0, 1, 2) for the rest of the treatments.

Similar to the previous example, adding the “ILINK” option to the end of the “ESTIMATE” command prompts GLIMMIX to estimate the values of the linear predictors $\left({\hat{\eta}}_{ci}\right)$ and the inverse of the linear predictors, which are the probabilities per category $\left({\hat{\pi}}_{ci}=\Phi \left({\hat{\eta}}_{ci}\right)\right)$. Table 8.13 shows the estimates of the linear predictors as well as their inverse values (probabilities in this case).

Table 8.13 Estimates on the model scale (Estimate) and on the data scale (Mean) for footpad lesion categories in the multinomial cumulative probit model

From the above table, we show the estimates of ${\hat{\eta}}_c+{\hat{\tau}}_i$. For example, the estimated linear predictor that a chicken will not develop a footpad lesion under treatment 1, i.e., ^“c = 0, t = 1, ^” is calculated as ${\hat{\eta}}_c+{\hat{\tau}}_1=-0.5398$. This result matches the values obtained from the fixed effects table (“Solutions for fixed effects”) previously shown. Taking the inverse of the link function, ${\hat{\pi}}_{01}=\Phi (0.5398)=0.2947$. This is the probability that a chicken will not develop a footpad lesion when receiving treatment 1. This probability is under the “Mean” column.

Now, for the category ^“c = 1, t = 1, ^” the inverse of the link function is a probability of 0.9054, which results from the inverse value of the linear predictor ${\hat{\eta}}_1+{\hat{\tau}}_1=1.3129$. This value is the estimate in terms of probability ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}$. From this value, we can obtain the probability that a chicken presents a “Moderate” lesion when receiving treatment 1, that is, ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}=0.9054$, and, using the value of ${\hat{\pi}}_{01}$, we obtain the values ${\hat{\pi}}_{11}=0.9054-0.2947=0.6107$ and ${\hat{\pi}}_{21}=1-0.9054=0.0946$. Following the same procedure, we can obtain the rest of the probabilities for each one of the categories (c = 0, 1, 2) and for the rest of the treatments (2–4).

8.5 Effect of Judges’ Experience on Canned Bean Quality Ratings

Canning quality is one of the most essential traits required in all new dry bean (Phaseolus vulgaris L.) varieties, and the selection for this trait is a critical part of bean breeding programs. Advanced lines that are candidates for release as varieties must be evaluated for canning quality for at least 3 years from samples grown at different locations. Quality is evaluated by a panel of judges with varying levels of experience in evaluating breeding lines for visual quality traits. A total of 264 bean breeding lines from 4 commercial classes were retained according to the procedures described by Walters et al. (1997). These included 62 white (navy), 65 black, 55 kidney, and 82 pinto bean lines plus control or “check” lines. The visual appearance of the processed beans was determined subjectively by a panel of 13 judges on a 7-point hedonic scale (1 = very undesirable, ..., 4 = neither desirable nor undesirable,..., 7 = very desirable). Beans were presented to the panel of judges in random order at the same time. Before evaluating the samples, all judges were shown examples of samples rated as satisfactory.

There is concern that certain judges, due to lack of experience, may not be able to correctly score the canned samples. From attribute-based product evaluations, inferences about the effects of experience can be drawn from the psychology literature (Wallsten and Budescu 1981). Prior to the bean canning quality rating experiment, it was postulated that not only do less experienced judges have a more severe rating than do more experienced judges but also that experience should have little or no effect on white beans, for which the canning procedure was developed. Judges are stratified for the purpose of analysis by experience (less than 5 years, greater than 5 years). Counts by canning quality, judge experience, and bean breeding lines are listed in the following table (Table 8.14).

Table 8.14 Frequency of ratings of different types of beans as a function of the bean-rating experience

The link functions for the cumulative logit model for describing a variable with C categories are as follows:

$$ {\displaystyle \begin{array}{c}{\eta}_{(1) ij}=\log \left(\frac{\pi_{1 ij}}{1-{\pi}_{1 ij}}\right)={\eta}_1+{\alpha}_i+{\beta}_j+{\left(\alpha \beta \right)}_{ij}\\ {}{\eta}_{(2) ij}=\log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}\right)}\right)={\eta}_2+{\alpha}_i+{\beta}_j+{\left(\alpha \beta \right)}_{ij}\\ {}\vdots \\ {}{\eta}_{C-1}=\log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}+\cdots +{\pi}_{\left(C-1\right) ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}+\cdots +{\pi}_{\left(C-1\right) ij}\right)}\right)={\eta}_{C-1}+{\alpha}_i+{\beta}_j+{\left(\alpha \beta \right)}_{ij}\end{array}} $$

The components of the GLMM with an ordinal multinomial response are as follows:

• Distributions: y_1ij, y_2ij, y_3ij, y_4ij, y_5ij, y_6ij,y_7ij~Multinomial(N_ij, π_1ij, π_2ij, π_3ij, π_4ij, π_5ij, π_6ij, π_7ij), where y_1ij, y_2ij, y_3ij, y_4ij, y_5ij, y_6ij, and y_7ij are the observed frequencies of the responses in each category c of the hedonic scale (1 = very undesirable, ..., 4 = neither desirable nor undesirable, ..., 7 = very desirable).

• Linear predictor: η_(c)ij = η_c + α_i + β_j + (αβ)_ij, where η_(c)ij is the cth link (c = 1, 2,...,6) for bean type i and judge’s experience j; η_c is the intercept for the cth link; α_i is the fixed effect due to the bean type for ith bean class; β_j is the fixed effect due to the jth experience of the judge; and (αβ)_ij is the fixed effect due to the interaction between bean class and judge experience. The link functions for each category are as follows:

$$ \log \left(\frac{\pi_{1 ij}}{1-{\pi}_{1 ij}}\right)={\eta}_{1 ij} $$

$$ \log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}\right)}\right)={\eta}_{2 ij} $$

$$ \log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}\right)}\right)={\eta}_{3 ij} $$

$$ \log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}+{\pi}_{4 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}+{\pi}_{4 ij}\right)}\right)={\eta}_{4 ij} $$

$$ \log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}+{\pi}_{4 ij}+{\pi}_{5 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}+{\pi}_{4 ij}+{\pi}_{5 ij}\right)}\right)={\eta}_{5 ij} $$

$$ \log \left(\frac{\pi_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}+{\pi}_{4 ij}+{\pi}_{5 ij}+{\pi}_{6 ij}}{1-\left({\pi}_{1 ij}+{\pi}_{2 ij}+{\pi}_{3 ij}+{\pi}_{4 ij}+{\pi}_{5 ij}+{\pi}_{6 ij}\right)}\right)={\eta}_{6 ij} $$

The following GLIMMIX commands fit a cumulative logit model with an ordinal multinomial response.

proc glimmix data=beans ; class Exper; model cal(order=data)= Exper|Class/dist=Multinomial link=clogit solution oddsratio; Contrast ’Effect of Experience on Black bean’ exper 1 -1 class*exper 1 -1 0 0 0 0 0 0 0 0 0 0; Contrast ’Effect of Experience on Kidney Bean’ exper 1 -1 class*exper 0 0 1 -1 0 0 0 0 0 0 0 0; Contrast ’Effect of Experience on Navies bean’ exper 1 -1 class*exper 0 0 0 0 0 0 0 1 -1 0 0 0; Contrast ’Effect of Experience on Pinto beans’ exper 1 -1 class*exper 0 0 0 0 0 0 0 0 0 0 0 1 -1; estimate ’Black, < 5 year, Rating = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 1 0 0 0 0 0 0 exper 1 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, < 5 year, Rating <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 1 0 0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, < 5 year, Rating <= 3’ Intercept 0 0 0 1 0 0 0 0 0 class 1 0 0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, < 5 year, Rating <= 4’ Intercept 0 0 0 0 1 0 0 0 class 1 0 0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, < 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 1 0 0 class 1 0 0 0 0 0 0 exper 1 0 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating <= 6’ Intercept 0 0 0 0 0 0 0 0 1 class 1 0 0 0 0 0 exper 1 0 class*exper 1 0 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 1 0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 1 0 0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating <= 3’ Intercept 0 0 0 1 0 0 0 0 0 class 1 0 0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating <= 4’ Intercept 0 0 0 0 1 0 0 0 class 1 0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 1 0 0 class 1 0 0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0 0/ilink; estimate ’Black, > 5 year, Rating <= 6’ Intercept 0 0 0 0 0 0 0 0 1 class 1 0 0 0 0 0 exper 0 1 class*exper 0 1 0 0 0 0 0 0 0 0 0/ilink; estimate ’Kidney, < 5 year, Rating = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 0 1 0 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink; estimate ’Kidney, < 5 year, Rating <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 0 1 0 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink; estimate ’Kidney, < 5 yr, Rating <= 3’ Intercept 0 0 0 1 0 0 0 0 0 class 0 1 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink; estimate ’Kidney, < 5 year, Rating <= 4’ Intercept 0 0 0 0 0 1 0 0 0 class 0 1 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink; estimate ’Kidney, < 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 1 0 0 class 0 1 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink; estimate ’Kidney, < 5 year, Rating <= 6’ Intercept 0 0 0 0 0 0 0 0 1 class 0 1 0 0 0 0 exper 1 0 0 class*exper 0 0 0 1 0 0 0 0 0 0 0/ilink; estimate ’Kidney, > 5 year, Rating = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 0 1 0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink; estimate ’Kidney, > 5 year, Rating <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 0 1 0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink; estimate ’Kidney, > 5 year, Rating <= 3’ Intercept 0 0 0 1 0 0 0 0 0 class 0 1 0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink; estimate ’Kidney, > 5 year, Rating <= 4’ Intercept 0 0 0 0 0 1 0 0 0 class 0 1 0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink; estimate ’Kidney, > 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 1 0 0 class 0 1 0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink; estimate ’Kidney, > 5 year, Rating <= 6’ Intercept 0 0 0 0 0 0 0 0 1 class 0 1 0 0 0 0 exper 0 1 class*exper 0 0 0 0 1 0 0 0 0 0 0/ilink; estimate ’Navies, < 5 year, Rating = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink; estimate ’Navies, < 5 year, Qualification <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink; estimate ’Navies, < 5 year, Qualification <= 3’ Intercept 0 0 0 0 1 0 0 0 0 0 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink; estimate ’Navies, < 5 year, Rating <= 4’ Intercept 0 0 0 0 0 1 0 0 0 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink; estimate ’Navies, < 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 0 1 0 0 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink; estimate ’Navies, <5 year, Qualification <= 6’ Intercept 0 0 0 0 0 0 0 0 0 1 class 0 0 0 1 0 0 exper 1 0 0 class*exper 0 0 0 0 0 1 0 0 0 0 0/ilink; estimate ’Navies, > 5 year, Qualification = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink; estimate ’Navies, > 5 year, Qualification <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink; estimate ’Navies, > 5 year, Rating <= 3’ Intercept 0 0 0 1 0 0 0 0 0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink; estimate ’Navies, > 5 year, Rating <= 4’ Intercept 0 0 0 0 0 1 0 0 0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink; estimate ’Navies, > 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 0 1 0 0 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink; estimate ’Navies, > 5 year, Rating <= 6’ Intercept 0 0 0 0 0 0 0 0 0 1 class 0 0 0 1 0 0 exper 0 1 class*exper 0 0 0 0 0 0 0 1 0 0 0/ilink; estimate ’Pinto, < 5 year, Qualification = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 0 0 0 0 0 0 1 exper 1 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink; estimate ’Pinto, < 5 year, Qualification <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 0 0 0 0 0 0 1 exper 1 0 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink; estimate ’Pinto, < 5 year, Qualification <= 3’ Intercept 0 0 0 0 1 0 0 0 0 0 class 0 0 0 0 0 1 exper 1 0 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink; estimate ’Pinto, < 5 year, Rating <= 4’ Intercept 0 0 0 0 0 1 0 0 0 class 0 0 0 0 0 1 exper 1 0 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink; estimate ’Pinto, < 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 0 1 0 0 class 0 0 0 0 0 1 exper 1 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink; estimate ’Pinto, < 5 year, Rating <= 6’ Intercept 0 0 0 0 0 0 0 0 0 1 class 0 0 0 0 0 1 exper 1 0 class*exper 0 0 0 0 0 0 0 0 1 0 0/ilink; estimate ’Pinto, > 5 years, Qualification = 1’ Intercept 1 0 0 0 0 0 0 0 0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 1/ilink; estimate ’Pinto, > 5 year, Qualification <= 2’ Intercept 0 1 0 0 0 0 0 0 0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink; estimate ’Pinto, > 5 year, Qualification <= 3’ Intercept 0 0 0 1 0 0 0 0 0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink; estimate ’Pinto, > 5 year, Rating <= 4’ Intercept 0 0 0 0 0 1 0 0 0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink; estimate ’Pinto, > 5 year, Rating <= 5’ Intercept 0 0 0 0 0 0 0 1 0 0 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 0 1/ilink; estimate ’Pinto, > 5 year, Qualification <= 6’ Intercept 0 0 0 0 0 0 0 0 0 1 class 0 0 0 0 0 1 exper 0 1 class*exper 0 0 0 0 0 0 0 0 0 1/ilink; freq y; run;

Part of the results is shown below. The results of the analysis of variance show that the class of bean (Class), experience of the evaluator (Exper), and the interaction between class and experience (Class×Exper) on bean canning scores differ significantly (P = 0.0001). That is, the results of comparing judges with more and less years of experience will depend on the line (variety) of beans (Table 8.15).

Table 8.15 Fixed effects hypothesis testing in the multinomial cumulative logit model

The contrasts address this interaction (Table 8.16). Hypothesis testing is as follows: π_{class of bean, < 5 years of experience} = π_{class of bean, > 5 years of experience.}

Table 8.16 Hypothesis testing in quality assessment

The results show that judges with more than 5 years of experience differ from those with less than 5 years of experience in evaluating the quality of canned kidney and pinto beans (Table 8.16). With the “solution” option in the model specification, the fixed parameter estimates table shows the solution of the fixed effects parameters under maximum likelihood. In this table, we can observe the values of the estimated intercepts: ${\hat{\eta}}_1=-4.6421$ defines the boundary between the categories, “1 = highly undesirable” and “2 = moderately undesirable”, whereas ${\hat{\eta}}_2=-2.9316$ defines the boundary between the categories “2 = moderately undesirable” and “3 = slightly undesirable.” The third intercept defines the boundary between the categories “3 = moderately undesirable” and “3 = slightly undesirable,” ${\hat{\eta}}_3=-1.3995$ defines the boundary between the categories “3 = slightly undesirable” and “4 = neither undesirable nor desirable,” and so on.

The estimated effects of bean type $\left({\hat{\alpha}}_i\right),$evaluator $\left({\hat{\beta}}_i\right),$ and their interaction $\left({\hat{\alpha \beta}}_{ij}\right)$ are shown below. From these values, we can estimate the linear predictors for each of the categories. For example, the linear predictor for canned black beans evaluated by an inexperienced judge who assigns the category “1 = very undesirable” is ${\hat{\eta}}_{111}={\hat{\eta}}_1+{\hat{\alpha}}_1+{\hat{\beta}}_1+{\hat{\alpha \beta}}_{11}=-4.6421+1.9670+1.0284-0.8066=-2.4533,$ for category “2 = moderately undesirable,” it is ${\hat{\eta}}_{211}={\hat{\eta}}_2+{\hat{\alpha}}_1+{\hat{\beta}}_1+{\hat{\alpha \beta}}_{11}=-2.9316+1.9670+1.0284-0.8066=-0.7428,$ for category “3 = slightly undesirable,” it is ${\hat{\eta}}_{311}={\hat{\eta}}_3+{\hat{\alpha}}_1+{\hat{\beta}}_1+{\hat{\alpha \beta}}_{11}=-1.3995+1.9670+1.0284-0.8066=0.7893$, and, for category “4 = neither undesirable nor desirable,” it is ${\hat{\eta}}_{411}={\hat{\eta}}_4+{\hat{\alpha}}_1+{\hat{\beta}}_1+{\hat{\alpha \beta}}_{11}=0.004287+1.9670+1.0284-0.8066=2.1931$. This is how the other categories are calculated for each type of bean and assessor (Table 8.17).

Table 8.17 Maximum likelihood estimation of the estimated parameters in the fixed effects solution of canned bean quality ratings in the multinomial cumulative logit model

The results of Table 8.18 were obtained with the “estimate” command in conjunction with the “ilink” option that prompts GLIMMIX to compute the values of the linear predictors, ${\hat{\eta}}_{cij},$ tabulated under the “Estimate” column, and the estimated probabilities ${\hat{\pi}}_{cij}$ for all categories of each treatment are tabulated under the “Mean” column $\left({\hat{\pi}}_{cij}\right)$, except the reference category.

Table 8.18 Estimates on the model scale (Estimate) and on the data scale (Mean) based on judges’ experience in canned bean quality ratings in the multinomial cumulative logit model

From Table 8.18 (“Estimates”), we can obtain the probabilities reported under the “Mean” column in which an inexperienced (<5 years) panelist (judge) would rate canned black beans as category 1 (1 = highly undesirable) with a probability of ${\hat{\pi}}_{111}=0.08$ compared to an experienced panelist (>5 years) who would give a probability of ${\hat{\pi}}_{112}=0.0646$. To calculate the probability that a judge with less than 5 years experience would assign a rating of 2 (2 = moderately undesirable) to canned black beans, we derive this probability from the cumulative probability of 0.3224, which corresponds to ${\hat{\pi}}_{211}+{\hat{\pi}}_{111}$, from which we get ${\hat{\pi}}_{211}=0.3224-{\hat{\pi}}_{111}=0.3224-0.08=0.24$. On the other hand, for a judge with experience (>5 years), the probability of assigning a score of 2 to canned black beans is ${\hat{\pi}}_{212}=0.2760-{\hat{\pi}}_{112}=0.2760-0.06446=0.2115$.

Following the same procedure, the other probabilities for the rest of the categories are obtained. The probabilities calculated for each of the categories are shown in Table 8.19 and can be seen in Fig. 8.2.

Table 8.19 Probabilities calculated for each of the canned bean grades

Fig. 8.2

Estimated probabilities for each category of the acceptability of canned beans, according to the experience of the panelist (judge)

8.6 Generalized Logit Models: Nominal Response Variables

In a model with unordered data, the polytomous response variable does not have an ordered structure. Two classes of models, generalized logit models and conditional logit models, can be used with nominal response data. A generalized logit model consists of a combination of several binary logits estimated simultaneously. A logit model is the simplest and best-known probabilistic choice model. However, there are problems in making use of a multinomial logit model because of its inflexibility. A generalized logit model is essentially more flexible than the traditional multinomial cumulative logit model.

A generalized logit model shows the same flexibility as a probit model but is much more tractable. Like cumulative logit and probit models, a generalized logit model has C – 1 link functions, where C denotes the number of response categories. Moreover, in this class of models, a category is first defined as the reference category. This may be arbitrary or it may make compelling logical sense in the study to designate a particular response category as the reference. In practice and throughout the analysis, the category used as the reference is irrelevant, as long as we are consistent about it. For example, if C is used as the reference category, then the generalized logits are defined as shown below:

$$ {\displaystyle \begin{array}{c}{\boldsymbol{\eta}}_1=\log \left(\frac{\pi_{1 ij}}{\pi_{cij}}\right)={\alpha}_1+\boldsymbol{X}{\boldsymbol{\beta}}_1+\boldsymbol{Z}{\boldsymbol{b}}_1\\ {}{\boldsymbol{\eta}}_2=\log \left(\frac{\pi_{2 ij}}{\pi_{Cij}}\right)={\alpha}_2+\boldsymbol{X}{\boldsymbol{\beta}}_2+\boldsymbol{Z}{\boldsymbol{b}}_2\\ {}\vdots \\ {}{\boldsymbol{\eta}}_{C-1}=\log \left(\frac{\pi_{\left(C-1\right) ij}}{\pi_{Cij}}\right)={\alpha}_{c-1}+\boldsymbol{X}{\boldsymbol{\beta}}_{C-1}+\boldsymbol{Z}{\boldsymbol{b}}_{C-1}\end{array}} $$

Given the different effects in the models, the intercepts (α^´s), β^´s, and b^´s vary across the pairs of response variable categories for each link function. Using algebra, it can be shown that the general form of the inverse of the link functions is given by

$$ {\pi}_c=\frac{e^{\eta_c}}{1+\sum \limits_{c=1}^{C-1}{e}^{\eta_c}},\kern1.5em c=1,2,\dots, C-1 $$

Once π₁, π₁, … . π_C − 1 are estimated, the reference category is estimated as ${\pi}_C=1-\sum \limits_{c=1}^{C-1}{\pi}_c$.

8.6.1 CRDs with a Nominal Multinomial Response

In practice, cumulative models are used for analyzing ordinal data and generalized logit models for nominal data. Returning to Example 8.3.1, we will now implement the analysis of a generalized logit model. This model relaxes the assumptions of proportionality; but it is less parsimonious than the “odds ratio” model since they fit C − 1 binary logit models, where C is the number of categories of the response variable. The linear predictor and distribution are the same as in the previous example.

The following GLIMMIX syntax implements the analysis of the generalized logit model:

proc glimmix data=chickens ; class trt block category; model category(reference=’severe’)= trt/dist=Multinomial link=glogit oddsratio; random intercept/subject=block solution group=category; estimate ’t=1’ intercept 1 trt 1 0, t=2’ intercept 1 trt 0 1 0, ’t=3’ intercept 1 trt 0 0 0 1 0, ’t=4’ intercept 1 trt 0 0 0 0 1/ilink bycat; freq y; run;

Most of the syntax of the program has already been explained. The “reference=” option is new to this program in the command, where the model is defined and is used to designate the reference category. By not specifying the “reference=” option, GLIMMIX by default uses the last category in the dataset. Moreover, the “link = glogit” option prompts GLIMMIX to fit a generalized logit model. The “bycat” option in the “estimate” command is unique to the generalized logit model. Finally, the “ilink” option asks GLIMMIX to estimate all category probabilities for each treatment, except those for the reference category. Part of the output is shown in Table 8.20. The fixed effects test shows that there are highly significant differences (P = 0.0001) on the average percentage of footpad lesion level between treatments.

Table 8.20 Analysis of variance in the generalized multinomial logit model

Unlike the cumulative logit model, in the generalized logit model, the estimates of the fixed effects (treatments), as well as the intercepts, are separated for each link function. For the estimation of linear predictors, we use the estimated values of Table 8.21 (“Solutions for fixed effects”). The estimated intercepts ${\hat{\alpha}}_1=4.8525$ and ${\hat{\alpha}}_2=4.2485$ define the boundary between the categories “Without” lesion and “Moderate” lesion and the boundary between the categories “Moderate” lesion and “Severe” lesion, respectively. For treatment 1, the treatment effects (${\hat{\tau}}_i\Big)$ estimated for the “Without” lesion category is ${\hat{\tau}}_1=-3.8447$ and for the “Moderate” lesion category, it is ${\hat{\tau}}_1=-2.6478$. With these values, the linear predictors for the “Without” lesion and “Moderate” lesion categories under treatment 1 are ${\hat{\eta}}_{01}=4.8525-3.8447=1.0077$ and ${\hat{\eta}}_{11}=4.2485-2.6478=1.6007$, respectively.

Table 8.21 Maximum likelihood estimates on the model scale (Estimate) for footpad lesion level in the multinomial generalized logit model

The estimated probabilities for each of the categories (“Without” lesion and “Moderate” lesion) in each treatment, except for the reference category, are found under the “Mean” column of Table 8.22. The probability that a chick has no footpad lesion when receiving treatment 1 is ${\hat{\pi}}_{01}=0.315$, whereas the value 0.57 corresponds to the cumulative probability ${\hat{\pi}}_{01}+{\hat{\pi}}_{11}$. From this value, we can calculate the probability of observing a moderate lesion, which is ${\hat{\pi}}_{11}=0.57-{\hat{\pi}}_{01}=0.57-0.315=0.255$. From these probabilities, we can estimate the probability of observing a severe footpad lesion under treatment 1 as ${\hat{\pi}}_{21}=1-(0.57)=0.43$. Following the same logic, we can estimate the reference probabilities for the rest of the other treatments.

Table 8.22 Estimates on the model scale (“Estimate”) and on the data scale (“Mean”) for footpad lesion level observed in treatments in the multinomial generalized logit model

Another important result is the odds ratio estimates. These estimates are shown in Table 8.23.

Table 8.23 Estimated odds ratio

These odds ratios compare the odds for the labeled category to those for the reference category for treatments 1–3 relative to treatment 4. These odds ratio values are derived from the estimated probabilities in each of the categories. For example, the probabilities that a chicken does not present a lesion and a moderate lesion are ${\hat{\pi}}_{04}=0.6433$ and ${\hat{\pi}}_{14}=0.3517,$ respectively. From these probabilities, we can estimate the probability of observing a severe lesion as follows: ${\hat{\pi}}_{24}=1-\left(0.6433+0.3517\right)=0.005$. The estimated odds ratio of not observing a lesion (“Without” lesion) between treatments 1 and 4 is

$$ {\hat{\mathrm{Odds}\ \mathrm{ratio}}}_{\mathrm{Trt}1,\mathrm{Trt}4}=\frac{{\hat{\pi}}_{01}}{{\hat{\pi}}_{21}}/\frac{{\hat{\pi}}_{04}}{{\hat{\pi}}_{24}}=\frac{0.315}{0.115}/\frac{0.6433}{0.005}=0.0213 $$

the value provided in the odds ratio estimates table. If we compare the analysis using the cumulative logit link and the generalized logit link, we observe insignificant changes in the estimated category probabilities by treatment as well as in the significance level in the test of treatment effects.

8.6.2 CRD: Cheese Tasting

Consider a study in which you want to know the effects of various additives on the flavor of cheese. Researchers tested 4 cheese additives and obtained 52 response ratings for each additive. Each response was measured on a scale of 9 categories ranging from: I dislike it very much (1) to I like it very much or excellent flavor (9). Data are obtained from the study by McCullagh and Nelder (1989) (Table 8.24).

Table 8.24 Effect of additives on cheese flavor

The components of the GLMM with an ordinal multinomial response are as follows:

• Distributions: y_1i, y_2i, y_3i, y_4i, y_5i, y_6i,y_7i, y_8i, y_9i~Multinomial(N_i, π_1i, π_2i, π_3i, π_4i, π_5i, π_6i, π_7i,π_8i, π_9i), where y_1i, y_2i, y_3i, y_4i, y_5i, y_6i,y_7i, y_8i, and y_9i are the observed frequencies of the responses in each category c of the hedonic scale (1 = very undesirable, ..., 5 = neither desirable nor undesirable, ... , 9 = very desirable).

• Linear predictor: η_(c)i = η_c + α_i, where η_(c)ij is cth link (c = 1, 2, …, 8) for the additive type i, η_c is the intercept for the cth link, and α_i is the fixed effect due to the ith additive. The link functions for each category are as follows:

$$ \log \left(\frac{\pi_{1i}}{1-{\pi}_{1i}}\right)={\eta}_{1i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}}{1-\left({\pi}_{1i}+{\pi}_{2i}\right)}\right)={\eta}_{2i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}+{\pi}_{3i}}{1-\left({\pi}_{1i}+{\pi}_{2i}+{\pi}_{3i}\right)}\right)={\eta}_{3i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}}{1-\left({\pi}_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}\right)}\right)={\eta}_{4i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}}{1-\left({\pi}_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}\right)}\right)={\eta}_{5i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}+{\pi}_{6i}}{1-\left({\pi}_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}+{\pi}_{6i}\right)}\right)={\eta}_{6i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}+{\pi}_{6i}+{\pi}_{7i}}{1-\left({\pi}_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}+{\pi}_{6i}+{\pi}_{7i}\right)}\right)={\eta}_{7i} $$

$$ \log \left(\frac{\pi_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}+{\pi}_{6i}+{\pi}_{7i}+{\pi}_{8i}}{1-\left({\pi}_{1i}+{\pi}_{2i}+{\pi}_{3i}+{\pi}_{4i}+{\pi}_{5i}+{\pi}_{6i}+{\pi}_{7i}+{\pi}_{8i}\right)}\right)={\eta}_{8i} $$

The following GLIMMIX commands fit a cumulative logit model with an ordinal multinomial response.

proc glimmix ; class id additive scale; model scale(order=data)= additive/dist=Multinomial link=clogit solution oddsratio; estimate ’c=1, a=1’ intercept 1 0 0 0 0 0 0 0 additive 1 0 0 0, ’c=2, a=1’ intercept 0 1 0 0 0 0 0 0 additive 1 0 0 0, ’c=3, a=1’ intercept 0 0 1 0 0 0 0 0 additive 1 0 0 0, ’c=4, a=1’ intercept 0 0 0 1 0 0 0 0 additive 1 0 0 0, ’c=5, a=1’ intercept 0 0 0 0 1 0 0 0 additive 1 0 0 0, ’c=6, a=1’ intercept 0 0 0 0 0 1 0 0 additive 1 0 0 0, ’c=7, a=1’ intercept 0 0 0 0 0 0 1 0 additive 1 0 0 0, ’c=8, a=1’ intercept 0 0 0 0 0 0 0 1 additive 1 0 0 0, ’c=1, a=2’ intercept 1 0 0 0 0 0 0 0 additive 0 1 0 0, ’c=2, a=2’ intercept 0 1 0 0 0 0 0 0 additive 0 1 0 0, ’c=3, a=2’ intercept 0 0 1 0 0 0 0 0 additive 0 1 0 0, ’c=4, a=2’ intercept 0 0 0 1 0 0 0 0 additive 0 1 0 0, ’c=5, a=2’ intercept 0 0 0 0 1 0 0 0 additive 0 1 0 0, ’c=6, a=2’ intercept 0 0 0 0 0 1 0 0 additive 0 1 0 0, ’c=7, a=2’ intercept 0 0 0 0 0 0 1 0 additive 0 1 0 0, ’c=8, a=2’ intercept 0 0 0 0 0 0 0 1 additive 0 1 0 0, ’c=1, a=3’ intercept 1 0 0 0 0 0 0 0 additive 0 0 1 0, ’c=2, a=3’ intercept 0 1 0 0 0 0 0 0 additive 0 0 1 0, ’c=3, a=3’ intercept 0 0 1 0 0 0 0 0 additive 0 0 1 0, ’c=4, a=3’ intercept 0 0 0 1 0 0 0 0 additive 0 0 1 0, ’c=5, a=3’ intercept 0 0 0 0 1 0 0 0 additive 0 0 1 0, ’c=6, a=3’ intercept 0 0 0 0 0 1 0 0 additive 0 0 1 0, ’c=7, a=3’ intercept 0 0 0 0 0 0 1 0 additive 0 0 1 0, ’c=8, a=3’ intercept 0 0 0 0 0 0 0 1 additive 0 0 1 0, ’c=1, a=4’ intercept 1 0 0 0 0 0 0 0 additive 0 0 0 1, ’c=2, a=4’ intercept 0 1 0 0 0 0 0 0 additive 0 0 0 1, ’c=3, a=4’ intercept 0 0 1 0 0 0 0 0 additive 0 0 0 1, ’c=4, a=4’ intercept 0 0 0 1 0 0 0 0 additive 0 0 0 1, ’c=5, a=4’ intercept 0 0 0 0 1 0 0 0 additive 0 0 0 1, ’c=6, a=4’ intercept 0 0 0 0 0 1 0 0 additive 0 0 0 1, ’c=7, a=4’ intercept 0 0 0 0 0 0 1 0 additive 0 0 0 1, ’c=8, a=4’ intercept 0 0 0 0 0 0 0 1 additive 0 0 0 1/ilink; freq freq; run;

Part of the results is shown in Table 8.25. The results of the analysis of variance show that the type of additive used in the manufacture of cheese significantly affects the degree of consumer acceptance (P = 0.0001). That is, the type of additive affects the sensory characteristics of the cheese.

Table 8.25 Fixed effects tests in the multinomial cumulative logit model

The contrast of hypothesis are presented in Table 8.26. The hypothesis tests are as follows:

Table 8.26 Contrast of hypothesis in the acceptance of cheese made with four additives

$$ {\pi}_{{\mathrm{additive}}_i}={\pi}_{{\mathrm{additive}}_j};\forall i\ne j $$

The results show that the additives provide different sensory characteristics that are reflected in the evaluation of preference.

With the “solution” option in the model specification, Table 8.27 (fixed parameter estimates) shows the solution of the maximum likelihood estimates for the fixed effects parameters. In this table, we observe the values of the estimated intercepts: ${\hat{\eta}}_1=-7.0802$ defines the boundary between categories “1” and “2,” whereas ${\hat{\eta}}_2=-6.0250$ defines the boundary between categories “2” and “3.” The third intercept, ${\hat{\eta}}_3=-4.9254$, defines the boundary between categories “3” and “4” and so forth. The estimated effects of the additive type $\left({\hat{\alpha}}_i,i=1,2,3,\mathrm{and}\ 4\right)$ are 1.628, 4.9646, 3.3227, and 0, respectively. From these values, linear predictors are estimated for each of the categories.

Table 8.27 Maximum likelihood estimates of the fixed effects in the preference ratings of cheese made with different types of additives in the multinomial cumulative logit model

For example, the estimated linear predictor for a cheese made with additive 1, where the evaluator (consumer) assigns it category “1 = highly undesirable,” is represented as ${\hat{\eta}}_{11}={\hat{\eta}}_1+{\hat{\alpha}}_1=-7.0802+1.6128=-5.4674$; for the category “2 = moderately undesirable,” it is ${\hat{\eta}}_{21}={\hat{\eta}}_2+{\hat{\alpha}}_1=-6.0250+1.6128=-4.4122$; for the category “3 = slightly undesirable,” it is ${\hat{\eta}}_{31}={\hat{\eta}}_3+{\hat{\alpha}}_1=-4.9254+1.6128=-3.3126$; and for the category “4 = neither undesirable nor desirable,” it is ${\hat{\eta}}_{41}={\hat{\eta}}_4+{\hat{\alpha}}_1=-3.8568+1.6128=-2.2440$. These values are shown in the “Estimate” column of Table 8.28; other categories are similarly calculated for each type of additive.

Table 8.28 Estimates on the model scale (Estimate) and on the data scale (Mean) based on judges’ preference ratings of cheese made with different types of additives in the multinomial cumulative logit model

The estimated values in Table 8.27 obtained with the “estimate” command in conjunction with the “ilink” option prompts GLIMMIX to calculate the values of the linear predictors ${\hat{\eta}}_{Ci}$ tabulated in the “Estimate” column and estimated probabilities ${\hat{\pi}}_{Ci}$of all categories of each treatment, tabulated in the “Mean” column $\left({\hat{\pi}}_{cij}\right)$, except for the reference category.

From Table 8.28 (Estimates), we obtain the probabilities for each category that is reported under the “Mean” column. In this case, the probability for ${\hat{\pi}}_{11}=0.004205$. This value is obtained by taking the inverse value of the linear predictor ${\hat{\eta}}_{11}=-5.4674$ $\left({\hat{\pi}}_{11}=1/\left(1+{\exp}^{(5.4674)}\right)=0.004205\right)$. To calculate the probability that a panelist would assign a rating of 2 (2 = moderately undesirable) to cheese made with additive 1, we use the cumulative probability of 0.01198, which corresponds to ${\hat{\pi}}_{21}+{\hat{\pi}}_{11}$. From this value, we obtain ${\hat{\pi}}_{21}=0.01198-{\hat{\pi}}_{11}=0.01198-0.004205=0.007775$ and for the probability of assigning a rating of 3 to cheese made with additive 1, ${\hat{\pi}}_{31}=0.03514-\left({\hat{\pi}}_{21}+{\hat{\pi}}_{11}\right)=0.03514-0.001198=0.033942.$ Following the same procedure, we obtain the other probabilities for the rest of the categories of each of the additives used in the manufacturing of cheese, which are tabulated in Table 8.29 and can be seen in Fig. 8.3.

Table 8.29 Probabilities calculated for each of the ratings by additives used in the manufacture of cheese

Fig. 8.3

Estimated probabilities for the categories of acceptability for the cheese according to the type of additive

Figure 8.3 shows the probability results of each flavor rating for each of the additives (it should be noted that some probability values were suppressed to avoid overwriting). It can be seen that additive 1 primarily receives ratings of 5–7; additive 2 primarily receives ratings of 2–5; additive 3 primarily receives ratings of 4–6; and additive 4 primarily receives ratings of 7–9.

The odds ratio results (Table 8.30) show the preferences more clearly. For example, the odds ratio additive 1 vs. 4 states that the first additive is 5.017 times more likely to receive a lower score than the fourth additive.

Table 8.30 Odds ratio

8.7 Exercises

Exercise 8.7.1

The dataset for this exercise corresponds to the results of 9 judges who rated 2 classes of wine, namely, white wine (WW = 1) and red wine (RW = 2), and, within each wine class, they rated 10 wines on a scale of 1–20 points. The minimum rating for a particular wine was 7, and the maximum rating was 19.5. For didactic purposes, ratings between 7 and 11 were assigned low quality, a rating between 12 and 15 as medium quality, and anything above 15 was considered excellent quality. The data are shown in Table 8.31 of the wine evaluation experiment under columns “Judge” (wine evaluator panelist), “Wine_type” (white wine: 1, red wine: 2), “Quality” (low, medium, and excellent), and the frequency of the observed qualities (“y”).

Table 8.31 Results of the wine evaluation experiment

• (a) Fit the cumulative logit proportional odds model to these data. Perform a complete and appropriate analysis of the data, focusing on:

  • (i) An evaluation of the effects of the combination of treatments

  • (ii) Interpretation of the odds ratios

  • (iii) The expected probability per category for each treatment

Exercise 8.7.2

Data were obtained from a series of experiments conducted to reduce damage to potato tubers due to a potato lifter. The experiments were conducted at the Institute of Agricultural Engineering (IMAG-DLO) in Wageningen, the Netherlands. One source of damage was the type of rod used in the lifter. In the experiment – under consideration – eight types of rods were compared. It is an empirical fact that the degree of damage varies considerably between potato varieties with the type of rope used in the lifting of full potato sacks. Three blocks of observations were obtained for the combinations of varieties and rope types. Most of the combinations involved about 20 potatoes. For some combinations, there are no data due to an insufficient number of large potatoes. Tubers were dropped from a given height. To determine the damage, all tubers were peeled and the degree of blue coloration was classified into one of four classes (class 1: no damage; class 2: slight damage; class 3: moderate damage; and class 4: severe damage). The observations, in the form of counts per class and combination, are shown in Table 8.32 of the tuber experiment whose columns are “Variety” (1, 2, 3, 4, 4, 5, 6), “String” (1, 2, 3, 4, 5, 6, 7, 8), “Block” (1, 2, 3), Type of damage (sd = no damage, dl = light damage, dm = moderate damage, ds = severe damage), and the observed frequency (Y).

Table 8.32 Results of the tuber experiment. V = variety, C = string, B = block, D = damage (sd = no damage, dl = slight damage, dm = moderate damage, ds = severe damage), and Y = observed frequency

• (a) List the components of the multinomial GLMM.

• (b) Fit the cumulative logit proportional odds model to these data. Perform a complete and appropriate analysis of the data, focusing on:

  • (i) An evaluation of the effects of the combination of treatments

  • (ii) Interpretation of the odds ratios

  • (iii) The expected probability per category for each treatment

• (c) Test whether the proportional odds assumption is viable. Cite relevant evidence to support your conclusion regarding the adequacy of the assumption.

• (d) If as a result of b), you consider that an alternative cumulative logit model is better, then revise your analysis in a) accordingly.

Exercise 8.7.3

Fit a generalized multinomial logit model using the dataset from Exercise 8.7.2 of this section, following the instructions:

• (a) List the components of this model.

• (b) Perform a thorough and appropriate analysis of the data, focusing on:

  • (i) An evaluation of the main effects and treatment interaction

  • (ii) Odds ratio interpretation

  • (iii) The expected probability per category for each treatment

• (c) Comment on and discuss your results. Cite relevant evidence to support your conclusion regarding the adequacy of the assumption.

Exercise 8.7.4

In this exercise, the effects of judges’ experience on quality ratings of canned beans are assessed. Canning quality is one of the most essential traits required in all new dry bean (Phaseolus vulgaris L.) varieties, and selection for this trait is a critical part of bean breeding programs. Advanced lines, which are candidates for release as varieties, must be evaluated for canning quality for at least 3 years from samples grown at different locations. Quality is evaluated by a panel of judges with varying levels of experience in evaluating breeding lines for visual quality traits. In all, 264 bean breeding lines from 4 commercial classes were conserved according to the procedures described by Walters et al. (1997).

These included 62 white (navy), 65 black, 55 kidney, and 82 pinto bean lines plus control lines and “checks.” The visual appearance of the processed beans was determined subjectively by a panel of 13 judges on a 7-point hedonic scale (1 = very undesirable, 4 = neither desirable nor undesirable, 7 = very desirable). The beans were presented to the panel of judges in random order at the same time. Prior to evaluating the samples, all judges were shown examples of samples rated as satisfactory (4). There is concern that certain judges, due to lack of experience, may not be able to score canned samples correctly.

From attribute-based product evaluations, inferences about the effects of experience can be drawn from the psychology literature (Wallsten and Budescu (1981). Prior to the bean canning quality rating experiment, it was postulated that not only do less experienced judges have a more severe rating than do more experienced judges but also that experience should have little or no effect on the white beans for which the canning procedure was developed. Judges are stratified for the purpose of analysis by experience (less than 5 years, greater than 5 years).

Counts by canning quality, judge experience, and bean breeding lines are listed in the following table (Table 8.33).

Table 8.33 Bean experiment results

• (a) Fit the generalized logit model to these data. Perform a complete and appropriate analysis of the data, focusing on:

  • (i) An evaluation of the effects of the combination of treatments

  • (ii) Interpretation of the odds ratios

  • (iii) The expected probability per category for each treatment

• (b) Test whether the proportional odds assumption is viable. Cite relevant evidence to support your conclusion regarding the adequacy of the assumption.

Exercise 8.7.5

An experiment was conducted to look at the damage levels (ordinal categories 0–4) of Picea sitchensis shoots in two time periods (10 November and 8 December), at four temperatures (different on each date), and at four ozone levels (Table 8.34).

Table 8.34 Experimental results of Picea sitchensis sprouts

• (a) Fit the cumulative logit proportional odds model to these data. Perform a complete and appropriate analysis of the data, focusing on:

  • (i) An evaluation of the effects of the combination of treatments

  • (ii) Interpretation of the odds ratios

  • (iii) The expected probability per category for each treatment

• (b) Test whether the proportional odds assumption is viable. Cite relevant evidence to support your conclusion regarding the adequacy of the assumption.

Appendix

Data: CRD with a multinomial response: ordinal
Rep	Trt	Cat	Freq	Rep	Trt	Cat	Freq
rep1	M1H1	Without	0	rep1	M2H3	Moderate	3
rep2	M1H1	Without	2	rep2	M2H3	Moderate	1
rep3	M1H1	Without	2	rep3	M2H3	Moderate	3
rep4	M1H1	Without	2	rep4	M2H3	Moderate	2
rep1	M1H2	Without	2	rep1	M2H4	Moderate	4
rep2	M1H2	Without	0	rep2	M2H4	Moderate	2
rep3	M1H2	Without	4	rep3	M2H4	Moderate	2
rep4	M1H2	Without	2	rep4	M2H4	Moderate	5
rep1	M1H3	Without	3	rep1	M2H1	Severe	4
rep2	M1H3	Without	7	rep2	M2H1	Severe	6
rep3	M1H3	Without	1	rep3	M2H1	Severe	7
rep4	M1H3	Without	2	rep4	M2H1	Severe	4
rep1	M1H4	Without	0	rep1	M2H2	Severe	5
rep2	M1H4	Without	5	rep2	M2H2	Severe	2
rep3	M1H4	Without	2	rep3	M2H2	Severe	3
rep4	M1H4	Without	1	rep4	M2H2	Severe	4
rep1	M1H1	Moderate	3	rep1	M2H3	Severe	3
rep2	M1H1	Moderate	2	rep2	M2H3	Severe	4
rep3	M1H1	Moderate	3	rep3	M2H3	Severe	4
rep4	M1H1	Moderate	5	rep4	M2H3	Severe	4
rep1	M1H2	Moderate	3	rep1	M2H4	Severe	5
rep2	M1H2	Moderate	3	rep2	M2H4	Severe	6
rep3	M1H2	Moderate	6	rep3	M2H4	Severe	0
rep4	M1H2	Moderate	3	rep4	M2H4	Severe	3
rep1	M1H3	Moderate	4	rep1	M3H1	Without	0
rep2	M1H3	Moderate	2	rep2	M3H1	Without	3
rep3	M1H3	Moderate	1	rep3	M3H1	Without	2
rep4	M1H3	Moderate	3	rep4	M3H1	Without	0
rep1	M1H4	Moderate	5	rep1	M3H2	Without	5
rep2	M1H4	Moderate	4	rep2	M3H2	Without	3
rep3	M1H4	Moderate	8	rep3	M3H2	Without	3
rep4	M1H4	Moderate	4	rep4	M3H2	Without	2
rep1	M1H1	Severe	6	rep1	M3H3	Without	0
rep2	M1H1	Severe	6	rep2	M3H3	Without	2
rep3	M1H1	Severe	5	rep3	M3H3	Without	1
rep4	M1H1	Severe	3	rep4	M3H3	Without	0
rep1	M1H2	Severe	5	rep1	M3H4	Without	3
rep2	M1H2	Severe	7	rep2	M3H4	Without	5
rep3	M1H2	Severe	0	rep3	M3H4	Without	7
rep4	M1H2	Severe	5	rep4	M3H4	Without	3
rep1	M1H3	Severe	3	rep1	M3H1	Moderate	0
rep2	M1H3	Severe	1	rep2	M3H1	Moderate	5
rep3	M1H3	Severe	7	rep3	M3H1	Moderate	5
rep4	M1H3	Severe	5	rep4	M3H1	Moderate	0
rep1	M1H4	Severe	5	rep1	M3H2	Moderate	3
rep2	M1H4	Severe	1	rep2	M3H2	Moderate	2
rep3	M1H4	Severe	0	rep3	M3H2	Moderate	6
rep4	M1H4	Severe	5	rep4	M3H2	Moderate	1
rep1	M2H1	Without	1	rep1	M3H3	Moderate	3
rep2	M2H1	Without	2	rep2	M3H3	Moderate	5
rep3	M2H1	Without	1	rep3	M3H3	Moderate	3
rep4	M2H1	Without	1	rep4	M3H3	Moderate	3
rep1	M2H2	Without	1	rep1	M3H4	Moderate	0
rep2	M2H2	Without	3	rep2	M3H4	Moderate	2
rep3	M2H2	Without	1	rep3	M3H4	Moderate	3
rep4	M2H2	Without	4	rep4	M3H4	Moderate	4
rep1	M2H3	Without	4	rep1	M3H1	Severe	9
rep2	M2H3	Without	5	rep2	M3H1	Severe	2
rep3	M2H3	Without	3	rep3	M3H1	Severe	3
rep4	M2H3	Without	4	rep4	M3H1	Severe	10
rep1	M2H4	Without	1	rep1	M3H2	Severe	2
rep2	M2H4	Without	1	rep2	M3H2	Severe	5
rep3	M2H4	Without	8	rep3	M3H2	Severe	1
rep4	M2H4	Without	2	rep4	M3H2	Severe	7
rep1	M2H1	Moderate	4	rep1	M3H3	Severe	6
rep2	M2H1	Moderate	2	rep2	M3H3	Severe	3
rep3	M2H1	Moderate	2	rep3	M3H3	Severe	6
rep4	M2H1	Moderate	5	rep4	M3H3	Severe	7
rep1	M2H2	Moderate	4	rep1	M3H4	Severe	7
rep2	M2H2	Moderate	4	rep2	M3H4	Severe	3
rep3	M2H2	Moderate	6	rep3	M3H4	Severe	0
rep4	M2H2	Moderate	2	rep4	M3H4	Severe	3