Class Notes: Week 6. Multinomial Outcomes
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1 Ronald Hek Class Notes: Week 6 1 Class Notes: Week 6 Multinomial Outomes For the next ouple of weeks or so, we will look at models where there are more than two ategories of outomes. Multinomial logisti regression provides a logial extension of the binary ase, sine it ompares two or more ategories to a referene ategory. Multinomial analyses produe one less set of outomes than the number of ategories (C 1). This an be hallenging to interpret when there are several outomes in the response variable. This approah is a good option when the outome is nominal, or when the model does not meet the riteria for an ordinal model. Multinomial models an be used with ordinal data but ordinal methods annot be used with nominal data. Provided the assumptions of ordered ategories an be supported (i.e., a parallel struture of preditors aross ategories of the outome), the advantage of an ordinal model is that it is generally easier to interpret beause the analysis produes only one set of estimates. Multinomial Outome Today, we ll onentrate on the multinomial logisti model. Coneptually, the fous of a multinomial model is to ompare the ategory of interest () to the referene ategory (C). It is important to take some are in thinking about the ategory that one wishes to use as the referene ategory. In this first model we will use the data desribed on page 263 in the text book. The data have to do with students post high shool plans. The outome is a three-level nominal variable with 0 = attend four year, 1 = attend teh/voational or two year institution, and 2 = no plan to obtain further eduation. The total number of outome ategories is often denoted as C and eah individual ategory an be indexed by, suh that we obtain the probability of being in the th ategory P(Y = ) is, where = 1,2,,C an be expressed suh that their sum is 1.0. The last ategory is treated as the referene by SPSS so there are C-1 equations to estimate. For individual I then we have the probability of being in one of the other ategories against the probability of being in the referene ategory C, where we an transform the outome sores using the generalized logit link funtion to obtain the log odds ( ): ln. C In general, then, we an obtain C-1 logits for the ategories in the outome. Let s first obtain a model for the odds of being in any of the three ategories of the postseondary plans outome.
2 Ronald Hek Class Notes: Week 6 2 Model 1: Interepts Only We an speify the first model with interept only as follows: ln = 0( ). C We an think of this as two separate log odds models: 0 0(0), 1 0(1) keeping in mind that the referene group is students who do not plan to attend shool further (oded 2). In this formulation, sine the last group (no shool) is the referene group, we will obtain log odds oeffiients for (0) attending a 4-year institution and (1) attending a tehnial, voational, or two-year institution. It is important to note that urrently in GENLIN, the multinomial model for nominal outomes, whih uses a generalized logit link funtion, is not available. Therefore, there are two hoies. One option is to run the multinomial logisti regression model in REGRESSION (Multinomial), but note that we annot run an interept-only model in that routine. Therefore, if we wish to run an initial model with only the interepts, in order to determine the probability of being in eah ategory, we have to use GENLINMIXED. This is found in ANALYZE (MIXED) and Generalized Linear Model. When we do that, we obtain the following information. Note also the model fit indies are not based on the kernel log likelihood so they will be different from what I have added below from the multinomial logisti regression output. Table 1: Preliminary Interept Model Estimate SE Sig OR Interept model (4 YEAR) Interept(Two Year) _ Note: Referene ategory is 2; -2LL (Interept only model) = Let s look more losely at this model. This suggests that graduating students say there are about 3.1 times more likely to enroll in a 4-year institution than to not pursue any further shool. With respet to the seond model, they report being about 2.4 times more likely to enroll in a voational/tehnial or ommunity ollege (2 year institution) than to not pursue any further shooling.
3 Ronald Hek Class Notes: Week 6 3 One again, we an make use of the odds ratios to help find the probability of being in eah ategory. We use a form of the following when there are three or more ategories: exp( 0 x ) 1 exp( x ) 0 In this ase, with just the interepts, the probability of being in eah of the three response ategories is as follows. The probably of going to a four-year institution P(Y = 1) versus no further shooling (where the odds ratios have been substituted into the formula above) is 3.102/( ) = 3.102/6.510 = The probably of going to a voational/tehnial or two-year institution P(Y = 2) is 2.408/( ) = 2.408/6.510 = The probability of being in the referene ategory P(Y = 3), sine the umulative probabilities must add up to 1.0, is then 1/( ) = 1/6.510 = These estimates losely math the ategories of the outome in the SPSS output for Model 1. These are attend a four-year institution (47.7%); attend a voational/tehnial or two-year institution (37.0%), and obtain no further eduation (15.4%). Model 2: Adding A Set of Preditors For our next model we will add three preditors students grade point average (gmgpa), their SES status (oded 1 = partiipant in free/redued lunh versus 0 = other) and gender (0 = female and 1 = male). The GPA variable has been entered on the grand mean of the sample. This an be aomplished by using Compute and subtrating GPA from the Mean of GPA for the sample). This adjusts the mean for the sample to be equal to 0. This failitates interpreting the interept as the log odds for the person who has a GPA at the sample average. We an speify the model as follows: ln = 0( ) 1 gmgpa 2 lowses 3 male. C Again, this will result in two sets of estimates: gmgpa lowses male 0 0(0) 1(0) 2(0) 3(0)
4 Ronald Hek Class Notes: Week 6 4 gmgpa lowses male 1 0(1) 1(1) 2(1) 3(1) This will result in eight parameters to estimate, for a resulting addition of 6 parameters relative to the interept-only model (Model 1). It should be noted that the odds ratios are not provided for the interepts in the multinomial logisti regression model (so we an alulate them by hand). For the first (4 year) model the interept odds ratio will be raised to the power of the log odds (exp ), or For the seond model the odds ratio will be estimated Table 2: Multinomial Model with Three Preditors Estimate SE Sig OR Interept model (4 YEAR) Gmgpa Lowses Male Interept(Two Year) Gmgpa Lowses Male _ Note: Referene ategory is 2; -2LL (Model with 3 preditors) = 3, ; Chi-square Test (6 df) = 3, We an see the models that GPA is a strong preditor of likelihood of obtaining further eduation. In the 4-year versus no further shooling, the odds of enrolling versus obtaining no further eduation are signifiantly inreased by a fator of about 7.6 for a 1-SD inrease in GPA. For pursuing voational/tehnial or two-year eduation versus no further eduation, the odds are inreased by a fator of about 3.5 for a 1-SD inrease in GPA. Note that for a 2-SD inrease in GPA, the odds of enrolling in a 4-year versus no further shooling are inreased by a fator of over 58 (7.62*7.62 = 58.06). This is the same as adding the two log odds together ( = 4.062) and then exponentiating the result ( = 58.09), with slight differene due to rounding. We an also note that gender and soioeonomi status are different in eah of the two settings. More speifially, for low SES students (oded 1), the odds of enrolling in a four-year institution versus no further shooling are signifiantly redued by a fator of (or 67.3%) ompared with students of average or higher SES. Gender is not signifiant in that model, however. For the voational/tehnial or two-year model, the odds of enrolling versus not obtaining further shooling are inreased for males by a fator of 1.18 (or 18%) ompared to females. In this model, however, student SES does not affet likelihood to enroll versus not obtain any further eduation after high shool.
5 Ronald Hek Class Notes: Week 6 5 Finally, we an see that the models with 3 preditors added results in a signifiantly better model fit than the interept only model (hi square, 6 df, = 3, , p <.001). We might further investigate whether there are possible interations by reating a set of interation terms and entering them in the model. We an then remove any interation terms that turn out to be not statistially signifiant, either though single parameter tests or likelihood ratio tests. Model 3: Adding a Set of Two-Way Interations The next model adds three interation terms. Note that the three-way interation was also added by was not signifiant, so it was removed. We an first ompare the fit of this model against the interept only model. We an see in Table 3 the final deviane (-2LL) for this model with six effets (and 12 degrees of freedom) is Table 3. Model Fitting Information Model Model Fitting Criteria Likelihood Ratio Tests -2 Log Likelihood Chi-Square Df Sig. Interept Only Final Next, in Table 4 the effets of the preditors are presented with the interations inluded. The table suggests that the lowses*male interation might be retained. The other two interations are not signifiant in either eduational setting (p >.10).
6 Ronald Hek Class Notes: Week 6 6 Table 4. Parameter Estimates plans a B Std. Error Wald df Sig. Exp(B) Interept Gmgpa Lowses Male gmgpa * lowses lowses * male gmgpa * male Interept Gmgpa Lowses Male gmgpa * lowses lowses * male gmgpa * male a. The referene ategory is: 2. Model 4: Retaining the LowSES*Male Interation First, in this final model we an examine the model fit riteria. First, we an see that the final hi-square for this model (with 8 degrees of freedom) is This final deviane (-2LL) is atually not muh different from the previous model (-2LL = ). We would likely favor this model sine it is more parsimonious, with 4 fewer parameters estimated. Table 5. Model Fitting Information Model Model Fitting Criteria Likelihood Ratio Tests -2 Log Likelihood Chi-Square Df Sig. Interept Only Final Notie also, that from Model 2 (with main effets only), the -2LL was 3, We an ompare these two models by examining the differene in deviane between them. The differene is distributed as hi-square, with 2 degrees of freedom (for the single interation term remaining in eah model). We an estimate the differene as This differene is formally
7 Ronald Hek Class Notes: Week 6 7 summarized in Table 6, where we an observe that the hi square oeffiient for the lowses*male effet is (with slight differene due to rounding), whih is signifiant at p =.09. Table 6. Likelihood Ratio Tests Effet Model Fitting Criteria Likelihood Ratio Tests -2 Log Likelihood of Redued Model Chi-Square df Sig. Interept Gmgpa Lowses Male lowses * male The hi-square statisti is the differene in -2 log-likelihoods between the final model and a redued model. The redued model is formed by omitting an effet from the final model. The null hypothesis is that all parameters of that effet are 0. The final parameter estimates are presented in Table 7. We an see that the interation is signifiant in the four-year versus no shooling model (p <.05). It is also signifiant at p <.062 in the seond model. We interpret an interation as the relationship of interest, for example, lowses and likelihood to enroll is different for males and females. For the four-year situation, we would say that the ombined effets of lowses are for females [ (0)], and for males they are stronger at [ (1) = ], whih through exponentiating will result in an odds ratio of This result an also be obtained by multiplying 0.449*0.720 = This suggests SES works differentially for males and females in terms of odds of enrolling in 4-year institutions. Table 7. Parameter Estimates plans a B Std. Error Wald df Sig. Exp(B) Interept gmgpa lowses male lowses * male Interept gmgpa lowses male lowses * male a. The referene ategory is: 2.
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