Logistic Regression Models for Ordinal Response Variables

(2006) - Ann O'Connell

Journal:
Link:: http://methods.sagepub.com/book/logistic-regression-models-for-ordinal-response-variables
DOI:: 10.4135/9781412984812
Links::
Tags:: #paper #Methods #Logit #OrdinalLogit #ContinuationRatio
Cite Key:: [@OConnell2006]

Abstract

As we saw in Chapter 4, the cumulative odds model uses all the data available to assess the effect of independent variables on the log-odds of being at or beyond (or the reverse, at or below) a particular category. The odds are found by considering the probability of being at or beyond a category relative to the probability of being below that category. A restrictive assumption made in the CO analysis is that across all cumulative logit comparisons, the effect of any independent variable is similar; that is, the odds of being in higher categories relative to being in any category below it remains constant across the categories. However, these logit comparisons for the cumulative odds may not be theoretically appropriate in every research situation. If interest lies in determining the effects of independent variables on the event of being in a higher stage or category, then a comparison group that includes all people who failed to make it to a category may not lead us to the best conclusions or understanding of the data in terms of differences between people at a low stage versus all higher stages. Rather than grouping together all people who failed to make it to a category at any point, an alternative ordinal approach involves comparisons between respondents in any given category versus all those who achieved a higher category score. This approach forms the class of models known as continuation ratio (CR) models. The focus of a CR analysis is to understand the factors that distinguish between those persons who have reached a particular response level but do not move on from those persons who do advance to a higher level. Fox (1997) refers to this process as the analysis of a series of “nested dichotomies” (p. 472).

Notes

"A restrictive assumption made in the CO analysis is that across all cumulative logit comparisons, the effect of any independent variable is similar; that is, the odds of being in higher categories relative to being in any category below it remains constant across the categories" (O'Connell 2006:2)

"A continuation ratio is a conditional probability" (O'Connell 2006:2)

"The odds ratio, on the other hand, is a ratio of two odds, where the odds are a quotient of complementary probabilities, p/(1 − p)." (O'Connell 2006:3)

"The first approach, using the logit link, has been called the "logistic continuation ratio model" (Greenland, 1994, p. 1668). The model treats time to an event as a discrete quantity" (O'Connell 2006:3)

"The CR model using the logit link provides the odds for a child of being beyond a particular category, conditional on being at or beyond that category" (O'Connell 2006:3)

"The second approach considers time to the event as a continuous quantity" (O'Connell 2006:3)

"The probabilities of interest for the CR model are the probabilities of being beyond any category, given that a person has already attained at least that specific category, λj cat. j). These continuation ratios are conditional probabilities, rather than cumulative probabilities," (O'Connell 2006:5)

"when we start to consider the estimation of conditional probabilities, the models we derive and their interpretations explicitly depend on the manner in which the ordinal outcome is coded—either increasing or decreasing." (O'Connell 2006:6)

"CR models are not invariant to reversal of the outcome codes (Allison, 1999; Greenland, 1994)" (O'Connell 2006:6)

"models presented here use the forward approach, which corresponds naturally to the process of progression through the six hierarchically structured early reading skill categories" (O'Connell 2006:6)

"interest lies in modeling the probability that the ith child would advance beyond a particular category, given that he or she achieved mastery at least for that category" (O'Connell 2006:6)

"CR analysis, the first question that should be addressed is whether or not the model fits. There are two components to this: (1) overall model fit, which can be assessed by comparing the likelihood of the fitted model with the likelihood of the null, or intercept-only, model; and (2) investigating the assumption of parallel odds, or equal slopes in the logit model across the different responselevel comparisons being conducted" (O'Connell 2006:8)

"SAGE SAGE Research Methods 2006 SAGE Publications, Ltd. All Rights Reserved. Somers' D for these data was .691, indicating good correspondence between observed and predicted probabilities for being beyond a given category, although this rank-order correlation coefficient is for the Page 11" (O'Connell 2006:11)

"R2L restructured binary-outcome data and not the original ordinal responses. The likelihood ratio = .289, indicating moderate reduction in deviance for the CR model, relative to the null or intercept-only model" (O'Connell 2006:12)

"Advantages of the logit link include its simplicity of interpretation in terms of odds and odds ratios. Advantages of the clog-log link include its interpretation in terms of hazards and hazards ratios, and its direct connection to the proportional hazards model." (O'Connell 2006:16)

""time" is not a structural component of how the data were generated, so a stronger argument could be made in favor of the logit approach" (O'Connell 2006:16)

"Substantively, when the dependent responses represent an ordinal progression through sequential stages, a stronger understanding of factors affecting this progression can be obtained if variable effects on the conditional probabilities (i.e., conditional on reaching a particular stage or not), rather than cumulative probabilities, are estimated." (O'Connell 2006:18)