East Carolina University
Multinomial Logistic RegressionAs with binomial logistic regression, this technique is employed to predict a categorical variable from a collection of continuous and/or categorical predictors. Unlike with binomial logistic regression, there are more than two levels of the predicted categorical variable.In the summer of 2014 my colleagues and I received feedback on a manuscript we had submitted to a scholarly journal. The categorical variable being predicted was the status of engineering students here at ECU – they were classified as still being in the program, having left the program but in good status, or having left the program in poor status. One of my coauthors had used a discriminant function analysis, but one of the reviewers suggesting using a multinomial logistic regression instead, to avoid the restrictive assumptions associated with a discriminant function analysis. So, I taught myself how to do a multinomial logistic regression, with some help from a colleague in biostatistics. Since the data were in SPSS format, I employed SPSS.Below I present the multinomial logistic analysis recommended by one of our reviewers. Although I have done it in a sequential fashion, for pedagogical purposes, we reported a simultaneous analysis (all the variables thrown in at once, that is, the last step shown below). All of the predictor variables were continuous. To make it easier to compare predictors’ relative importance, I standardized them all to mean 0, standard deviation 1.MSAT is score on the math SAT. VSAT is score on the verbal SAT. HSGPA is high school GPA. ALEKS is score on a mathematics assessment test designed to test a college student’s readiness to take courses that require mastery of mathematics. LOC is locus of control, with high scores representing an external locus of control. The NEO predictors are scores on a Big Five personality test: Openness, Conscientiousness, Extroversion, Agreeableness, and Neuroticism.Descriptive StatisticsNMinimumMaximumMeanStd. DeviationMSAT256410780565.4762.174VSAT256350670492.9359.728HSGPA2562.224.003.1167.34986ALEKS256179753.7418.985LOC25603613.795.950NEOOpen256115026.835.663NEOC256144931.576.649NEOE256104630.685.946NEOA256124328.735.460NEON25665325.3111.284Valid N (listwise)256First I entered the Big Five predictors as a set. Analyze, Regression, Multinomial Logistic.Case Processing SummaryNMarginal PercentagegroupsPoor6826.6%Good8533.2%Stay10340.2%Valid256100.0%Missing0Total256Subpopulation256aa. The dependent variable has only one value observed in 256 (100.0%) subpopulations.Model Fitting InformationModelModel Fitting CriteriaLikelihood Ratio Tests-2 Log LikelihoodChi-SquaredfSig.Intercept Only555.273Final522.38132.89210.000Using these predictors significantly improved the model (compared to a model based only on the differences in group sample sizes).Pseudo R-SquareCox and Snell.121Nagelkerke.136McFadden.059This is an R-squared-like statistic, but cannot really be interpreted as a proportion of variance. I avoid it, but one of our reviewers wanted it.Likelihood Ratio TestsEffectModel Fitting CriteriaLikelihood Ratio Tests-2 Log Likelihood of Reduced ModelChi-SquaredfSig.Intercept533.14510.7642.005ZNEOOpen523.6561.2742.529ZNEOC537.58715.2062.000ZNEOE523.370.9892.610ZNEOA523.208.8262.662ZNEON527.8385.4572.065The chi-square statistic is the difference in -2 log-likelihoods between the final model and a reduced model. The reduced model is formed by omitting an effect from the final model. The null hypothesis is that all parameters of that effect are 0.Removing consciousness from the model would significantly lower fit between model and data. Neuroticism is nearly significant (but look below).Each predictor has k-1 B weights, each one comparing the reference group with one of the other groups. Here I designated the poor group as the reference group.Parameter EstimatesgroupsaBStd. ErrorWalddfSig.Exp(B)GoodIntercept.404.1844.8461.028ZNEOOpen-.135.187.5191.471.874ZNEOC.658.2139.5621.0021.932ZNEOE.078.200.1541.6951.081ZNEOA-.030.189.0251.873.970ZNEON.233.2141.1851.2761.262groupsaBStd. ErrorWalddfdf.Sig.Exp(B)StayIntercept.561.1799.7911.002ZNEOOpen-.208.1851.2701.260.812ZNEOC.741.21112.3721.0002.099ZNEOE-.092.196.2211.638.912ZNEOA.121.189.4101.5221.129ZNEON.467.2114.8931.0271.595For each one standard deviation increase in conscientiousness, the odds of being in the stay group rather than the poor group more than doubled.For each one standard deviation increase in conscientiousness. the odds of being in the good group rather than the poor group nearly doubled.For each one standard deviation increase in neuroticism the odds of being in the stay group rather than the poor group increased multiplicatively by 1.60.Locus of control was added in the next step. Its addition did not significantly improve the model.Model Fitting InformationModelModel Fitting CriteriaLikelihood Ratio Tests-2 Log LikelihoodChi-SquaredfSig.Intercept Only555.273Final520.18735.08612.000Pseudo R-SquareCox and Snell.128Nagelkerke.145McFadden.063Likelihood Ratio TestsEffectModel Fitting CriteriaLikelihood Ratio Tests-2 Log Likelihood Chi-SquaredfSig.Intercept531.08710.9012.004ZNEOOpen521.3621.1752.556ZNEOC536.24516.0582.000ZNEOE521.040.8532.653ZNEOA521.134.9472.623ZNEON524.5914.4052.111ZLOC522.3812.1942.334Parameter EstimatesgroupsaBStd. ErrorWalddfSig.Exp(B)GoodIntercept.403.1854.7591.029ZNEOOpen-.128.188.4591.498.880ZNEOC.706.21810.5281.0012.026ZNEOE.062.200.0971.7551.064ZNEOA-.065.193.1121.738.938ZNEON.091.236.1481.7001.095ZLOC.282.1982.0351.1541.326StayIntercept.567.1809.9461.002ZNEOOpen-.201.1861.1721.279.818ZNEOC.759.21412.6051.0002.136ZNEOE-.096.196.2401.624.909ZNEOA.105.192.2991.5851.111ZNEON.410.2303.1601.0751.506ZLOC.114.192.3551.5511.121a. The reference category is: Poor.On the third step, ALEKS was added to the model.Model Fitting InformationModelModel Fitting CriteriaLikelihood Ratio Tests-2 Log LikelihoodChi-SquaredfSig.Intercept Only555.273Final502.49552.77714.000Pseudo R-SquareCox and Snell.186Nagelkerke.210McFadden.095Likelihood Ratio TestsEffectModel Fitting CriteriaLikelihood Ratio Tests-2 Log Likelihood of Reduced ModelChi-SquaredfSig.Intercept514.75112.2552.002ZNEOOpen502.969.4732.789ZNEOC517.76015.2652.000ZNEOE503.311.8162.665ZNEOA503.7601.2652.531ZNEON505.6893.1932.203ZLOC504.8772.3822.304ZALEKS520.18717.6912.000Parameter EstimatesgroupsaBStd. ErrorWalddfSig.Exp(B)GoodIntercept.502.1976.5011.011ZNEOOpen-.104.191.2941.587.901ZNEOC.743.22211.1841.0012.103ZNEOE.081.203.1621.6871.085ZNEOA-.075.194.1501.698.928ZNEON.084.239.1221.7271.087ZLOC.290.1982.1361.1441.337ZALEKS.341.1873.3381.0681.406StayIntercept.630.19410.5361.001ZNEOOpen-.129.193.4511.502.879ZNEOC.740.22011.2711.0012.096ZNEOE-.076.202.1401.708.927ZNEOA.124.197.3951.5301.132ZNEON.362.2382.3141.1281.436ZLOC.107.198.2891.5911.112ZALEKS.733.18615.4991.0002.081Adding ALEKS significantly improved the model. Each increase of one standard deviation in ALEKS was associated with a more than doubling of the odds of being in the stay group rather than the poor group. The effect of ALEKS on the odds ratio for good versus poor fell just short of statistical significance.In Step 4 the SAT variables were added.Model Fitting InformationModelModel Fitting CriteriaLikelihood Ratio Tests-2 Log LikelihoodChi-SquaredfSig.Intercept Only555.273Final493.74861.52518.000The chi-square for this step is 502.495 – 493.748 = 8.747 on 18-14 = 4 degrees of freedom. That yields a p value of .068.Pseudo R-SquareNagelkerke.241Likelihood Ratio TestsEffectModel Fitting CriteriaLikelihood Ratio Tests-2 Log Likelihood of Reduced ModelChi-SquaredfSig.Intercept505.47411.7262.003ZNEOOpen494.425.6772.713ZNEOC509.56715.8192.000ZNEOE494.480.7322.693ZNEOA494.550.8022.670ZNEON496.5862.8382.242ZLOC496.6342.8862.236ZALEKS504.00610.2582.006ZMSAT500.9767.2282.027ZVSAT496.8243.0762.215Removing math SAT from the model would significantly reduce the fit of the model to the data, but the effects of math SAT on the two contrasts (stay versus good and stay versus poor) fall short of statistical significance. In another analysis I found that math SAT was significantly associated with the difference between the stay and the good groups, with the odds of being in the stay group rather than the good group increasing multiplicatively by 1.63 for each standard deviation increase in math SAT.Parameter EstimatesgroupsaBStd. ErrorWalddfSig.Exp(B)GoodIntercept.494.2016.0451.014ZNEOOpen-.124.194.4081.523.883ZNEOC.752.22511.1781.0012.121ZNEOE.058.205.0801.7771.060ZNEOA-.048.196.0601.807.953ZNEON.084.244.1181.7311.088ZLOC.327.2022.6201.1061.387ZALEKS.406.2053.9271.0481.500ZMSAT-.241.2131.2781.258.786ZVSAT.315.2062.3351.1261.370PoorIntercept.629.19710.1511.001ZNEOOpen-.157.195.6491.421.855ZNEOC.777.22511.9651.0012.176ZNEOE-.091.205.1991.655.913ZNEOA.112.198.3191.5721.119ZNEON.348.2402.1011.1471.416ZLOC.126.201.3891.5331.134ZALEKS.630.2029.7351.0021.878ZMSAT.266.2141.5451.2141.305ZVSAT.071.206.1201.7291.074In the last step, high school GPA was added to the model.Model Fitting InformationModelModel Fitting CriteriaLikelihood Ratio Tests-2 Log LikelihoodChi-SquaredfSig.Intercept Only555.273Final473.25382.02020.000Pseudo R-SquareCox and Snell.274Nagelkerke.310McFadden.148Likelihood Ratio TestsEffectModel Fitting CriteriaLikelihood Ratio Tests-2 Log LikelihoodChi-SquaredfSig.Intercept488.05314.8002.001ZNEOOpen473.641.3882.824ZNEOC488.93315.6802.000ZNEOE473.844.5912.744ZNEOA473.951.6982.705ZNEON475.2361.9832.371ZLOC475.3502.0962.351ZALEKS482.5469.2922.010ZMSAT480.0106.7572.034ZVSAT475.9472.6942.260ZHSGPA493.74820.4952.000Parameter EstimatesgroupsaBStd. ErrorWalddfSig.Exp(B)GoodIntercept.625.2148.5261.004ZNEOOpen-.102.202.2511.616.903ZNEOC.763.22811.1401.0012.144ZNEOE.118.215.3011.5831.125ZNEOA-.114.202.3191.573.892ZNEON.056.253.0491.8251.058ZLOC.276.2091.7541.1851.318ZALEKS.404.2083.7621.0521.498ZMSAT-.238.2221.1501.284.788ZVSAT.288.2151.7961.1801.334ZHSGPA.667.19711.4801.0011.949StayIntercept.734.21212.0341.001ZNEOOpen-.125.204.3731.541.883ZNEOC.807.23012.2981.0002.241ZNEOE-.008.216.0011.972.992ZNEOA.030.207.0221.8831.031ZNEON.289.2521.3141.2521.335ZLOC.087.211.1721.6781.091ZALEKS.619.2088.8331.0031.858ZMSAT.249.2261.2151.2701.283ZVSAT.049.217.0511.8201.051ZHSGPA.838.20217.1611.0002.312High School GPA, Conscientiousness, ALEKS, and high school GPA contributed significantly to the model.For each one standard deviation increase in high school GPA, the odds of being in the good group rather than the poor group nearly doubled, and the odds of being in the stay group rather than the poor group more than doubled.For each one standard deviation increase in conscientiousness, the odds of being in the stay group rather than the poor group more than doubled, and the same was true when comparing to the good group.For each one standard deviation increase in ALEKS, the odds of being in the stay group rather than the poor group were multiplied by 1.86. The effect of ALEKS on the contrast between the good group and the poor group fell just short of statistical significance.Although the removal of math SAT from the model would significantly reduce the fit of the model to the data, the effect of math SAT on the two focal contrasts fell short of statistical significance. Recall that math SAT did Given the final model, I thought it would be helpful to compare the group means on conscientiousness, ALEKS, math SAT, and high school GPA. I did so with REGWQ tests. When interpreting the results of these tests, it is important to remember that each tests the group differences on one continuous variable ignoring the other continuous variables. The corresponding effects in the logistic regression test the group differences after controlling for all of the other continuous variables.A Posteriori Pairwise Comparisons Between Group Means.VariableGroupConscientiousnessHS GPAALEKSMath SATPersisting33.23A3.21A59.82A583.30ALGS32.24A3.14A52.34B554.00BLPS28.21B2.94B46.28B552.79BNote: Within each column, means sharing a superscript are not significantly different from each other. N = 256.Karl L. Wuensch, September, 2019.Fair Use of this DocumentReturn to Wuensch’s Stats Lessons ................
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