title: Multinomial regression using multinom function in R
body: <p>I was thinking about posting my question in Cross-Validated, but decided to come here. I am using the multinom() function from the nnet package to estimate the odds of becoming employed, unemployed, or out of labor force conditioned on age and education. I need some help with the interpretation.</p>
  
  <p>I have the following dataset of one dependent categorical variable employment status(EmpSt) and two independent categorical variables: age (Age) and education level (Education). </p>
  
  <pre><code>&gt;head(df)
                 EmpSt   Age                         Education
  1           Employed   61+   Less than a high school diploma
  2           Employed 50-60 High school graduates, no college
  3 Not in labor force 50-60   Less than a high school diploma
  4           Employed 30-39       Bachelor's degree or higher
  5           Employed 20-29  Some college or associate degree
  6           Employed 20-29  Some college or associate degree
  </code></pre>
  
  <p>Here is the summary with the levels:</p>
  
  <pre><code>&gt;summary(df)
                  EmpSt          Age                                    Education    
   Not in universe   :    0   16-19: 6530   Less than a high school diploma  :14686  
   Employed          :61478   20-29:16031   High school graduates, no college:30716  
   Unemployed        : 3940   30-39:16520   Some college or associate degree :28525  
   Not in labor force:38508   40-49:17403   Bachelor's degree or higher      :29999  
                              50-60:20779                                            
                              61+  :26663                                    
  </code></pre>
  
  <ul>
  <li>First,what is the estimation equation(model)</li>
  </ul>
  
  <p>I want to determine what is the estimation equation(model) for the call</p>
  
  <p><code>df$EmpSt&lt;-relevel(df$EmpSt,ref="Employed")
   multinom(EmpSt ~ Age + Education,data=df)</code> </p>
  
  <p>so I can write it down in my research paper. In my understanding the Employed is the base level and the logit model for this call is:</p>
  
  <p><img src="https://i.stack.imgur.com/bG0Dv.gif" alt="enter image description here">
  <img src="https://i.stack.imgur.com/UfP1N.gif" alt="enter image description here"></p>
  
  <p>where i and n are the categories of the variables age and education respectively (sorry for confusing notation). Please, correct me if my understanding of the logistic model produced by multinom() is incorrect. I am not going to include the summary of the test because it is a lot of output, so below I just include the the output for call <code>&gt;test</code>:</p>
  
  <pre><code>&gt; test
  Call:
  multinom(formula = EmpSt ~ Age + Education, data = ml)
  
  Coefficients:
                     (Intercept)   Age20-29   Age30-39   Age40-49   Age50-60     Age61+
  Unemployed           -1.334734 -0.3395987 -0.7104361 -0.8848517 -0.9358338 -0.9319822
  Not in labor force    1.180028 -1.2531405 -1.6711616 -1.6579095 -1.2579600  0.8197373
                     EducationHigh school graduates, no college EducationSome college or associate degree
  Unemployed                                         -0.4255369                                 -0.781474
  Not in labor force                                 -0.8125016                                 -1.004423
                     EducationBachelor's degree or higher
  Unemployed                                    -1.351119
  Not in labor force                            -1.580418
  
  Residual Deviance: 137662.6 
  AIC: 137698.6 
  </code></pre>
  
  <p>Given that my understanding of the logit model produced by the multinom() is correct the coefficients are the logged odds where the base level is Employed. To get the actual odds I antilog by the call <code>exp(coef(test))</code> which gives me the actual odds:</p>
  
  <pre><code>&gt; exp(coef(test))
                     (Intercept)  Age20-29  Age30-39  Age40-49  Age50-60    Age61+
  Unemployed           0.2632281 0.7120560 0.4914298 0.4127754 0.3922587 0.3937724
  Not in labor force   3.2544655 0.2856064 0.1880285 0.1905369 0.2842333 2.2699035
                     EducationHigh school graduates, no college EducationSome college or associate degree
  Unemployed                                          0.6534189                                 0.4577308
  Not in labor force                                  0.4437466                                 0.3662560
                     EducationBachelor's degree or higher
  Unemployed                                    0.2589504
  Not in labor force                            0.2058891
  </code></pre>
  
  <p>which brings me to my next question.</p>
  
  <ul>
  <li>Second, the probabilities</li>
  </ul>
  
  <p>I wonder if there is a way to get the actual probabilities of being unemployed vs employed based on the combination of age and education,e.g what is the probability of being unemployed if I am 22 and have a high school diploma. Sorry for the lengthy question. Thanks for your help. Let me know if additional clarification is needed.</p>