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# Logistic regression for ordered dependant variable with more than 2 levels

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Logistic Regression for ordered dependant variable with more than 2 levels

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### Logistic regression for ordered dependant variable with more than 2 levels

1. 1. Multinomial Logistic Regression ModelsJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
2. 2.  Logistic regression CAN handle dependant variables with more than two categories  It is important to note whether the response variable is ordinal (consisting of ordered categories like young, middle-aged, old) or nominal (dependant is unordered like red, blue, black)  Some multinomial logistic models are appropriate only for ordered response  It is not mathematically necessary to consider the natural ordering when modeling ordinal response but,  Considering the natural ordering  Leads to a more parsimonious model  Increase power to detect relationships with other variablesJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
3. 3.  Applying logistic regression considering the natural order is done using a modeling technique called the “Proportional Odds Model”  Say the dependant variable Y has 4 states measuring the impact of radiation on the human body; fine, sick, serious,dead  Let p1=prob of fine, p2=prob of sick, p3=prob of serious, p4=prob of dead  Let us define a baseline category: fine, since this is the normal stage (we shall see why we need this later)January 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
4. 4.  What if we break up the modeling of the 4 level ordered dependant into 3 binary logistic situations: 1 – (fine,sick), 2 – (fine,serious),3 – (fine,dead)?  Then we would have 3 logit equations:  Log(p2/p1)=B11+B12X1+B13X2  Log(p3/p1)=B21+B22X1+B23X2  Log(p4/p1)=B31+B32X1+B33X2 X is the degree of radiation dummy with 3 levels so broken into 2 binary dummies  So, 9 parameters to be estmatedJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
5. 5.  Now consider an alternative model for the same situation  Cumulative logit model:  L1=log(p1/p2+p3+p4)  L2=log(p1+p2/p3+p4)  L3=log(p1+p2+p3/p4)  The obvious way to introduce covariates is  L1=B11+B12X1+B13X2  L2=B21+B22X1+B23X2  L3=B31+B32X1+B33X2January 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
6. 6.  Let us simplyfy the model by specifying that the slope parameters are identical over the logit equations. Then,  L1=A1+B1X1+B2X2  L2=A2+B1X1+B2X2  L3=A3+B1X1+B2X2  This is the proportional odds cumulative logit modelJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
7. 7.  Suppose that the categorical outcome is actually a categorized version of an unobservable (latent) continuous variable which has a logistic distribution  The continuous scale is divided into ﬁve regions by four cut-points c1, c2, c3, c4 which are determined by nature  If Z ≤ c1 we observe Y = 1; if c1 < Z ≤ c2 we observe Y = 2; and so on  Suppose that the Z is related to the X’s through a linear regression  Then, the coarsened categorical variable would be related Y will be related to the X’s by a proportional- odds cumulative logit modelJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
8. 8.  Let us go back to the model  L1=A1+B1X1+B2X2  L2=A2+B1X1+B2X2  L3=A3+B1X1+B2X2  Note that Lj is the log-odds of falling into or below category j versus falling above it  Aj is the log-odds of falling into or below category j when X1 = X2 = 0  B1 is the increase in log-odds of falling into or below any category associated with a one-unit increase in Xk, holding all the other X-variables constant.  Therefore, a positive slope indicates a tendency for the response level to decrease as the variable decreasesJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
9. 9.  Our example of 4 levels of impact of radiation corresponding to 3 levels of radiation proc logistic data=radiation_impact; freq count; class radiation / order=data param=ref ref=first; model sickness (order=data descending) = radiation / link=logit aggregate=(radiation) scale=none; run;January 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
10. 10.  Freq=count  This is important for specifying grouped data  Count is the variable that contains the frequency of occurrance of each observation  In its absence, each row would be considered a separate row of data  Class=radiation  Specifies that radiation is a classification variable to be used in the analysis  SAS would automatically generate n-1 binary dummies for n categories of radiation with param=ref optionJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
11. 11.  Order=data  Simply tells SAS to arrange the response categories in the order they occur in the input data 1,2,3,4  Param=ref  This implies that there is going to be dummy coding for the classification variable ‘radiation’listed in class  Ref=first  Designates the first ordered level, in this case ‘fine’ as the reference levelJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
12. 12.  Order=data descending  This tells SAS to reverse the order of the logits  So, instead of the cumulative logit model being  L1=log(p1/p2+p3+p4)  L2=log(p1+p2/p3+p4)  L3=log(p1+p2+p3/p4), it becomes  L1=log(p4/p1+p2+p3)  L2=log(p4+p3/p1+p2)  L3=log(p4+p3+p2/p1)  Now, a positive B1 indicates that a higher value of X1 leads to greater chance of radiation sicknessJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
13. 13.  Link=logit  fits the cumulative logit model when there are more than two response categories  Aggregate=radiation  Indicates that the goodness of fit statistics are to be calculated on the subpopulations of the variable: radiation  Scale=none  No correction is need for the dispersion parameter  To understand this, read up. This happens when the goodness of fit statistic exceeds its degrees of freedom and need to be corrected forJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
14. 14.  When we ﬁt this model, the first output we see: Score Test for the Proportional Odds Assumption Chi-Square DF Pr > ChiSq 17.2866 21 0.6936  Null hypothesis is that the current proportional-odds cumulative logit model is true  Seems like we fail to reject the null and so can proceed to the rest of the output under the current assumptionJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India
15. 15.  Ultimately we are interested in the predicted probabilities OUTPUT <OUT=SAS-data-set><options>  Predicted=  For a cumulative model, it is the predicted cumulative probability (that is, the probability that the response variable is less than or equal to the value of _LEVEL_);  PREDPROBS=I or C  Individual|I requests the predicted probability of each response level.  CUMULATIVE | C requests the cumulative predicted probability of each response levelJanuary 1, 2013 ©Arup Guha - Indian Institute of Foreign Trade - New Delhi, India