If treating the PAS as quantitative violates model assumptions and is not best-practice, what is the alternative? The usual answer for ordinal data is to use non-parametric statistics. While these are still a good option in very simple studies, their limitations make them unsuitable as an all-around tool. Most non-parametric tests are based on rank statistics. Common rank-based statistics include the Wilcoxon Signed Rank Test, Mann–Whitney
U Test, and Spearman correlations [
67]. These tests work well for ordinal outcome variables and should be a strong consideration for the PAS when the design and research questions allow. Their disadvantage, though, is a big one: they are tests, not models [
68]. Most research questions require more sophisticated analysis than simple group comparisons. Non-parametric statistics cannot handle interactions, covariates, or more than the simplest repeated measures. Although it is tempting to turn instead to statistical models designed for quantitative data, despite the inability of the PAS to meet their assumptions, there are better alternatives. In the following section, we will discuss descriptive and modeling options for categorical data that test all necessary hypotheses without making untenable assumptions.
Models for Analyzing Nominal Multicategory Data
In this section, we will discuss several different types of logistic regression (binary, multinomial, and ordinal) as approaches that are suitable for analyzing the PAS provided that it is treated as a categorical variable. A major advantage of using logistic regression models (compared to the non-parametric rank statistic approaches mentioned previously) is that repeated measures, covariates, interactions, and quadratic effects can all be easily included.
It is assumed that the reader will already be familiar with ordinary linear regression, in which a continuous and unbounded dependent variable with interval or ratio properties,
Y, is modeled as a function of different levels of a predictor variable,
X
1
, X
2
, X
3, …
X
k. The equation for linear regression is written as follows:
$$Y_{i} = \beta _{0} + \beta _{1} X_{{1i}} + \beta _{2} X_{{2i}} + \ldots + \beta _{k} X_{{ki}} + \varepsilon _{i}.$$
The beta functions in this equation reflect the intercept, β
0, which is the average value of Y when all values of X = 0, and the coefficients β
1
X
1i, β
2
X
2i, etc., which must be added to the intercept to reach the values of Y for each unit of the predictor, X. The function ε
I at the end of the equation is the error term in the model.
Logistic regression is similar to linear regression, except that the dependent variable is no longer a continuous parameter with interval or ratio properties, but rather the natural log-odds of a particular value of a categorical parameter occurring. The odds ratio is calculated as the probability (Pr) of a given categorical value,
P, divided by the inverse probability of that categorical value (1−
P). This can be easily illustrated using the example of a binary version of the PAS comparing “healthy” scores of 1 and 2 to scores of concern (≥3). Here, the odds ratio for obtaining a score of concern would be written as
$${\text{Odds (PAS }} \ge { 3) = }\frac{{\Pr \;({\text{PAS }} \ge { 3})}}{{\Pr \;({\text{PAS < 3}})}} = \frac{\Pr \;(P)}{\Pr \;(1 - P)}.$$
An odds ratio of 1 would reflect an equal probability of obtaining a score from either category. Odds ratios >1 reflect a greater probability of obtaining a score in the category of interest (in this case, PAS ≥ 3), whereas odds ratios <1 reflect a greater probability of obtaining a score in the comparison or reference category (in this case, PAS < 3). For the interested reader, additional information regarding odds ratios, as well as both linear and logistic regression, can be accessed through the free webinar series at
www.analysisfactor.com.
Binary Logistic Regression
If the PAS is reduced to 2 categories (i.e., healthy vs unhealthy), binary logistic regression will usually be the best option for analysis. The equation for binary logistic regression is written as follows:
$${\text{Ln}}\left( {{\text{Odds}}} \right){\mkern 1mu} = \beta _{0} + \beta _{1} X_{1} + \beta _{2} X_{2} + {\mkern 1mu} \ldots {\mkern 1mu} + \beta _{{kXk}} + \varepsilon _{i}.$$
Rather than using ordinary least squares, logistic regression explores the estimated values of all parameters in an iterative fashion until it finds the most likely value for the model (i.e., Maximum Likelihood Estimation). The outcome is expressed in the log-odds of a specific response category, with each β coefficient reflecting the difference in log-odds. The exponent of the log-odds value is an odds ratio, such that the β exponent reflects the amount by which the odds ratio for the dependent variable category needs to be multiplied for each one-unit change in the predictor variable, X.
Multinomial Logistic Regression
Although binary models are common and familiar to most readers, many clinicians and scientists may feel that reduction of the PAS to a two-category variable sacrifices important levels of detail. Multinomial logistic models can be used for situations in which the dependent variable has 3 or more categories; these are essentially a set of binary models [
69,
70], which provide detailed estimates of treatment effect and a separate set of coefficients for each categorical response option in comparison to one category that is treated as the reference category. In principle, this could be achieved by running several parallel binary logistic regression models; however, the multinomial approach provides a more accurate result based on the inclusion of a single error variance term. The process is as follows:
-
The dependent variable is coded into multiple binary 1/0 variables for each outcome category except one (the reference category). There will be M-1 binary outcome variables for M categories. The reference category is assigned a value of 0 for each of these binary variables.
-
The multinomial logistic regression then estimates a separate binary logistic regression model for each of those binary variables. The result is M-1 binary logistic regression models. Each one measures the effect of the independent variables on the log-odds of that outcome category, in comparison to the reference category. Each model has its own intercept and regression coefficients—the predictors can affect each category differently.
-
The multinomial logistic regression equation, with the subscript h indicating each category of the dependent variable, is written as follows:
$${\text{Ln}}\left( {{\text{Odds}}_{h} } \right){\mkern 1mu} = \beta _{{0h}} + \beta _{{1h}} X_{1} + \beta _{{2h}} X_{2} + {\mkern 1mu} \ldots {\mkern 1mu} + \beta _{{kh}} X_{k} + \varepsilon _{i}.$$
The main disadvantage of this approach (i.e., complexity) is also its main advantage (i.e., detail). There are a few consequences of this complexity. The first is that there are many coefficients to interpret, which can make patterns difficult to see. Secondly, when the independent variables are also categorical, every combination of outcome category and independent variable category must be present in the data in order for the model to compute. In the illustrative dataset, there are no cases of PAS = 4 or 6. This issue, which is called zero cell counts, will cause a failure of model convergence if these values occur in one condition but not another (e.g., pre-treatment but not post-treatment). A work-around for the current data set would be to recode the PAS levels that are not missing data as purely nominal or categorical variables: A = 1, B = 2, C = 3, D = 5, E = 7 and F = 8. In a multinomial logistic regression model, the odds of each of these 6 possible outcome categories for the PAS would be compared using a reference category of PAS = 1 (i.e., category A). The effect of each predictor (treatment group and consistency) would be estimated separately for each outcome category.
However, two other instances of the zero cell count problem are likely to occur in swallowing outcomes research and these may pose challenges to the suitability of the multinomial logistic regression approach. First, for the current data set, we stated an expectation that all participants were included in the trial on the basis of demonstrating a worst baseline PAS score of 3 or higher. In using multinomial logistic regression with the example data set, we determined that the absence of PAS scores in the healthy range (i.e., 1 and 2) at baseline led to a failure of model convergence assuming a repeated measures design with both pre- and post-treatment data. A similar dilemma might well have occurred in the case that no observations of PAS = 8 were present for a given combination of Consistency and Group in the post-treatment data: the model would fail to converge. In other words, if a treatment is so good that no participant continues to display silent aspiration at the post-treatment assessment, the model will not be able to calculate the probability of silent aspiration in that treatment condition. The zero cell count can occur in binary logistic regression as well, but it becomes more likely in multinomial models because there are so many individual outcome categories.
Ordinal Logistic Regression
In the introduction, we discussed historical survey data that suggest that some clinicians are uncertain whether the 8 categories of the PAS are correctly ordered as numerated [
6]. Nevertheless, we have proposed that the PAS could be re-ordered or collapsed into categories that are ordered based on a physiological framework, as suggested in the proposed Categorical PAS with possible labels of A, B, C, and D. If one accepts that category A is less severe than category B and so forth, ordinal logistic regression becomes a suitable approach to analysis. In our opinion, this is not only the most appropriate approach to analysis of PAS data, but it will also provide the richest information with respect to differences in airway protection status. Furthermore, this approach has the advantage of reduced complexity for interpretation compared to the multinomial approach described above.
There are a number of versions of ordinal logistic regression. For the purposes of illustration, we will describe the most common approach, the Proportional Odds Model [
69,
71‐
73], which is available in most statistical software packages (e.g., SPSS PLUM, SAS proc logistic, Stata’s ologit). Here, the equation is written as follows:
$${\text{Ln}}\left( {{\text{Odds}}_{j} } \right){\mkern 1mu} = \theta j - {\mkern 1mu} (\beta _{0} + \beta _{1} X_{1} + \beta _{2} X_{2} + {\mkern 1mu} \ldots {\mkern 1mu} + \beta _{{kXk}} + \varepsilon _{i} ).$$
1.
Here, the intercept β
0
has been replaced by a new function, θj, where j represents the number of ordered categories. This intercept represents a threshold value in the model, at which the odds shifts from the dependent variable being in a lower-ordered category into the next higher-ordered category. An important aspect of this approach is the proportional odds assumption, which states that the difference in odds for different categories in the model lies in this threshold value. Consequently each outcome category has its own intercept, but all outcome categories share the same regression coefficient.
Remember that in the multinomial model, the odds of each category are compared to a single reference category. Here, we measure the odds of any lower category in comparison to any higher category.
The calculation of unique intercepts allows for the fact that some outcome categories are simply more frequent than others, while the single regression coefficient measures how the independent variables relate to the odds of being in
any lower category. It says, for example, that the effect of using a thicker bolus consistency is generalized in the sense that it has the same effect on the odds of a PAS score = 7 compared to a PAS score = 8 as it does on the odds of a PAS score = 5 compared to PAS scores of either 7 or 8. This assumption is important to understand but is often violated in real data and needs to be checked. Table
6, below, provides frequency data for the illustrative dataset with PAS scores collapsed according to the scheme suggested above.
Table 6
Frequencies and percentages for categorical PAS scores in the hypothetical dataset by treatment group and consistency
Control | Thin | Count | 19 | 13 | 5 | 3 |
% | 47.5% | 32.5% | 13% | 7% |
Mildly thick | Count | 20 | 11 | 3 | 6 |
% | 50% | 28% | 7% | 15% |
Experimental | Thin | Count | 23 | 9 | 3 | 5 |
% | 57.5% | 22.5% | 7% | 13% |
Mildly thick | Count | 33 | 4 | 2 | 1 |
% | 82.5% | 10% | 5% | 2.5% |
Table
7 provides regression coefficients (intercepts) and odds ratios for the Ordinal PAS categories of A vs. B, C, and D; A or B vs. C and D; and A, B or C vs D, respectively, given factors of group (experimental vs. control) and consistency (mildly thick vs. thin) and their interaction. The threshold coefficients are not usually interpreted individually, but they represent the intercepts, specifically the point (in terms of a logit or log-odds function) where PAS scores can be predicted to shift into higher-ordered categories. The odds ratios for the effects of group and consistency and their interaction tell us how these predictors affect the odds of a swallow having a penetration-aspiration score in a higher-ordered category. Table
8 provides greater detail regarding the interaction, showing that being in the experimental group has a stronger effect on the odds of a seeing a higher-ordered (i.e., less severe) PAS category with the mildly thick liquid compared to the thin liquid.
Table 7
Output for ordinal logistic regression analysis of treatment group differences in categorical PAS scores by consistency
PAS category A vs. categories B, C and D | 0.251 | 0.309 | 0.661 | 0.416 | 1.286 |
PAS categories A or B vs. categories C and D | 1.485 | 0.337 | 19.479 | 0.000 | 4.417 |
PAS categories A, B or C vs. category D | 2.218 | 0.379 | 34.172 | 0.000 | 9.192 |
Group × consistency | −1.308 | 0.664 | 3.887 | 0.049 | 0.270 |
Group (experimental vs. control) | 0.248 | 0.426 | 0.339 | 0.561 | 1.282 |
Consistency (thin vs. mildly thick) | 0.025 | 0.419 | 0.003 | 0.953 | 1.025 |
Table 8
Post-hoc output for ordinal logistic regression analysis comparing the odds of different categorical PAS scores for the interaction of experimental group plus mildly thick liquids
PAS category A vs. categories B, C and D | 0.003 | 0.303 | 0.000 | 0.991 | 1.003 |
PAS categories A or B vs. categories C and D | 1.237 | 0.324 | 14.569 | 0.000 | 3.447 |
PAS categories A, B or C vs. category D | 1.970 | 0.266 | 28.915 | 0.000 | 7.173 |
Group × consistency | −1.308 | 0.664 | 3.887 | 0.049 | 0.780 |
Group (control vs. experimental) | −0.248 | 0.426 | 0.339 | 0.561 | 1.025 |
Consistency (mildly thick vs. thin) | 0.025 | 0.419 | 0.003 | 0.953 | 0.270 |