A 2005 study was conducted to evaluate the influence of fear avoidance beliefs (FAB), chronicity of low back pain (CHR), and severity of low back pain (SEV) on disability (DIS) in 209 patients from seven different regions in Spain who were treated for low back pain (LBP) in health-care centers belonging to the Spanish National Health Service. The sample was balanced for acute, subacute, and chronic LBP (i.e., the three categories of the CHR variable). Validated scales and questionnaires were used to assess DIS and FAB during the first visit (day 1) and also 14 days later (day 15). The analyses performed in the study were linear regression analyses using disability measurement (DIS) treated as a continuous outcome variable. However, in the questions to follow, we will treat the disability outcome (DIS) as either a polytomous or an ordinal outcome variable in a logistic regression model. The variables being considered are listed as follows:
DIS = disability level at day 15 (0 = none, 1 = mild, 2 = moderate, 3 = severe)
FAB = fear avoidance measure at day 1 (low = 0, high = 1)
CHR = chronicity of LBP at day 1(1= acute, 2 = subacute, 3 = chronic)
SEV = severity of LBP at day 1(1= low, 2 = medium, 3 = high)
SL = sick leave status at day 1 (0 = not on sick leave, 1 = on sick leave)
SEX = male (= 0) or female (=1)
AGE (continuous variable)
a. State the logit form of a no-interaction proportional odds logistic model for the relationship of disability outcome DIS to FAB, CHR, SEV, SL, SEX, and AGE. Make sure to treat the variables CHR and SEV as nominal (rather than ordinal) variables and to treat AGE as a continuous variable. In stating this model, let "acute" denote the referent category for CHR, and let "low" denote the referent category for SEV.
b. Based on the answer to part (a) above,
i. What is the formula for the odds for moderate or severe disability (DIS = 2 or 3) to none or mild disability (DIS = 0 or 1) for a subject with high fear avoidance behavior (FAB =1), chronic low back pain (CHR = 3), and severe low back pain (SEV = 3) who is on sick leave, is female, and is 40 years old?
ii. What is the formula for the odds ratio that compares the odds for moderate or severe disability (DIS = 2 or 3) to none or mild disability (DIS = 0 or 1) for a subject with high fear avoidance behavior (FAB = 1), chronic low back pain (CHR = 3), and high severity of low back pain (SEV = 3) to the corresponding odds for a subject with low fear avoidance behavior (FAB = 0), acute low back pain (CHR = 1), and low severity of low back pain (SEV = 1), controlling for SL, SEX, and AGE?
c. Consider the following 4 × 2 table that describes the crude relationship between fear avoidance behavior (FAB) and disability outcome (DIS):

i. If the proportional odds assumption is satisfied for these data, describe the 2 × 2sub-tables whose corresponding odds ratios are assumed to be equal (i.e., use the table above to assign letters to the appropriate cells and corresponding odds ratios in the sub-tables below):

ii. Give two reasons why one might want to consider using a polytomous logistic regression model instead of a proportional odds logistic regression model for analyzing these data.
Suppose that one decided to use polytomous logistic regression, instead of ordinal logistic regression, to analyze these data. Also, suppose that the variables CHR and SEV are treated as ordinal variables. Suppose also that the variables FAB, CHR, and SEV are considered as exposure (i.e., E) variables, with the variables SL, SEX, and AGE treated as control (i.e., V) variables.
Suppose further that this revised model is expanded to allow for two-way interactions (i.e., products of two variables) between FAB and each of the control variables SL, SEX, and AGE; two-way interactions between CHR and each of the control variables SL, SEX, and AGE; two-way interactions between SEV and each of the control variables SL, SEX, and AGE; and two-way interactions among FAB, CHR, and SEV. This "initial" model can thus be written as follows:
In [P(DIS = g|X)/P(DlS = 0|X)} = ag + β1g(FAB + β2gCHK) + β3g(SEV + γ1gSL)
+ γ2g(SEX + γ3gAGE) + δF1g(FAB × SL)
+ δF2g(FAB × SEX) + δF3g(FAB × AGE)
+ δC1g(CHR × SL) + δC2g(CHR × SEX)
+ δC3g(CHR × AGE) + δS1(SEV × SL)
+ δS2g(SEV × SEX) + δS3g(SEV × AGE)
+ δFC1g(FAB × CHR) + δFC2g/FAB × SEV)
+ δCS3g(CHR × SEV)
g = 1, 2, 3
d. Using the above polytomous logistic regression model, describe how one would simultaneously test for the significance of all EjVj product terms in this model. Make sure to state the null hypothesis in terms of model parameters, describe the formula for the test statistic, and give the distribution and degrees of freedom of the test statistic under the null hypothesis.
e. At the end of the interaction assessment stage, suppose that it was determined that the variables (CHR × SEV) and (CHR × SL) are the only product terms remaining in the model as significant interaction effects. For the reduced model obtained from the interaction results, give a formula for the odds ratio that compares the odds for severe disability (DIS = 3) to mild disability (DIS = 1) for a subject with high fear avoidance behavior (FAB = 1), chronic low back pain (CHR = 3), and high severity low back pain (SEV = 3) to the corresponding odds for a subject with low fear avoidance behavior (FAB = 0), acute low back pain (CHR = 1), and low severity low back pain (SEV = 1), controlling for SL, SEX, and AGE.