# Logit, Ordered Logit, and Multinomial Logit Models in Stata: A Hands-on Tutorial

An introductory guide to estimate logit, ordered logit, and multinomial logit models using Stata

## 1. Basic Concept

Logit Model

If the outcome or dependent variable is binary and in the form 0/1, then use logit or probit models.

Example 1:

Did you vote in the last election?

0 ‘No’

1 ‘Yes’

Example 2:

Do you prefer to use public transportation or to drive a car?

0 ‘Prefer to drive’

1 ‘Prefer public transport’

Note: Logit and probit models are basically the same; the difference is in the distribution:

• Logit – Cumulative standard logistic distribution (F)
• Probit – Cumulative standard normal distribution (Φ)

Both models provide similar results.

Ordered Logit Model

If the outcome or dependent variable is categorical but ordered (e.g., low to high), use ordered logit or ordered probit models.

Example 1:

Do you agree or disagree with the President?

1 ‘Disagree’

2 ‘Neutral’

3 ‘Agree’

Example 2:

1 ‘Low’

2 ‘Middle’

3 ‘High’

Multinomial Logit Model

If the outcome or dependent variable is categorical without any particular order, then use multinomial logit.

Example 1:

If elections were held today, for which party would you vote?

1 ‘Democrats’

2 ‘Independent’

3 ‘Republicans’

Example 2:

What do you like to do on the weekends?

1 ‘Rest’

2 ‘Go to movies’

3 ‘Exercise’

## 2.1. Estimating log-odds ratio

To get the data. Type:

use "https://dss.princeton.edu/training/logit.dta"

To run a logit model, type:

logit y_bin x1 x2 x3 i.opinion

Where y_bin is the dependent variable, and x1 x2 x3 i.opinion are independent variables. We use i. before opinion because it is a categorical variable.

Stata will give us the following output table.

###### ------------------------------------------------------------------------------        y_bin | Coefficient  Std. err.      z    P>|z|     [95% conf. interval] -------------+----------------------------------------------------------------           x1 |   1.133556   .9340937     1.21   0.225    -.6972338    2.964346           x2 |   .3021217   .3566358     0.85   0.397    -.3968716    1.001115           x3 |   .3976277   .4763941     0.83   0.404    -.5360876    1.331343              |      opinion |       Agree  |  -1.916357   .9046227    -2.12   0.034    -3.689385    -.143329       Disag  |   .3270627   .9984349     0.33   0.743    -1.629834    2.283959   Str disag  |   .6891686   1.249471     0.55   0.581     -1.75975    3.138087              |        _cons |   .8816118   .8587917     1.03   0.305     -.801589    2.564813 ------------------------------------------------------------------------------

Interpretation

• The P>|z| column shows the two-tailed p-values testing the null hypothesis that the coefficient equals zero (i.e., no significant effect). We usually reject the null hypothesis if the p-value <0.05. By this measure, none of the coefficients except for Agree significantly affect the log-odds ratio of the dependent variable. The coefficient for Agree is significant at the 5% (0.034<0.05) level.
• The z value also tests the null that the coefficient equals zero. For a 5% significance, the z-value should fall outside the ±1.96.
• The Coefficient column shows the coefficients in log-odds form. For example, when x1 increases by one unit, the expected change in the log odds is 1.133556 (an increase), holding all other variables in the model constant. However, this increase is not statistically significant because the p-value is not < 0.05.

Note: What you get from the Coefficient column is whether the effect of the predictors is positive or negative. See the following sub-sections (2.2., 2.3., and 2.4.) for an extended explanation of logit outcomes.

## 2.2. Estimating the odds ratio

The odds ratio allows an easier interpretation of the logit coefficients. They are the exponentiated value of the logit coefficients. We can get the odds ratio by using the following procedure.

To get the data. Type: (You don't need to get data again if you already estimated log-odds following the instruction in section 2.1.)

use "https://dss.princeton.edu/training/logit.dta", clear

To get odds ratio rather than logit coefficients, type:

logit y_bin x1 x2 x3 i.opinion, or

Note: We added or at the end of the command to get the odds ratio.

Stata will give us the following output table.

###### ------------------------------------------------------------------------------        y_bin | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------           x1 |   3.106685   2.901935     1.21   0.225     .4979609    19.38203           x2 |   1.352726   .4824304     0.85   0.397     .6724204    2.721314           x3 |    1.48829   .7090125     0.83   0.404     .5850326    3.786125              |      opinion |       Agree  |    .147142    .133108    -2.12   0.034     .0249874     .866469       Disag  |   1.386888   1.384718     0.33   0.743     .1959622    9.815465   Str disag  |   1.992059    2.48902     0.55   0.581     .1720878    23.05972              |        _cons |   2.414789     2.0738     1.03   0.305     .4486155    12.99822 ------------------------------------------------------------------------------ Note: _cons estimates baseline odds.

Interpretation:

• Odds Ratio: They represent the odds of Y=1 when X increases by 1 unit. These are the exp(logit coeff).

If the OR > 1 then the odds of Y=1 increases

If the OR < 1 then the odds of Y=1 decreases

• Let's interpret the odds ratio for x1 which is 3.106685.

It indicates, keeping all other variables constant, when x1 increases by one unit, the odds of Y = 1 increase by 211%. However, this increase is not statistically significant as the p-value is not < 0.05. We get this 211% as follows: (3.106685 - 1) * 100 = 210.67.

Or, we can also say the odds of Y =1 are 3.1 times higher when x1 increases by one unit, keeping all other predictors constant.

Note: To calculate the odds ratio by hand, you need to exponentiate the logit coefficient. The formula is:

odds ratio = exp(coef(logit))

In section 2.1., we got logit coefficient for x1 = 1.133556. Therefore, the odds ratio for x1 = exp(1.133556) = 3.106685

## 2.3. Estimating predicted probabilities after logit

First, we will estimate the probability that the outcome variable (Y) = 1, given that all predictors are set to their mean values.

To get the data. Type:

use "https://dss.princeton.edu/training/logit.dta", clear

To get the predicted probability, type:

###### margins, atmeans post

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .8575618   .0512873    16.72   0.000     .7570405    .9580832 ------------------------------------------------------------------------------

Interpretation:

The probability of y_bin = 1 is 85%, given that all predictors are set to their mean values.

Second, we will estimate the probability that the outcome variable (Y) = 1, setting a predictor to a specific value (x2 = 3)

To get the data. Type:

use "https://dss.princeton.edu/training/logit.dta", clear

To get the predicted probability, type:

###### margins, at(x2=3) atmeans post

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .9346922   .0732788    12.76   0.000     .7910683    1.078316 ------------------------------------------------------------------------------

Interpretation:

The probability of y_bin = 1 is 93%, given that x2 = 3 and the rest of the predictors are set to their mean values.

Third, we will estimate the probability that the outcome variable (Y) = 1, setting specific values for a couple of predictors  (i.e., x2 = 3 and x3 = 5)

To get the data. Type:

use "https://dss.princeton.edu/training/logit.dta", clear

To get the predicted probability, type:

###### margins, at(x2=3 x3=5) atmeans post

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .9872112   .0357288    27.63   0.000      .917184    1.057238 ------------------------------------------------------------------------------

Interpretation:

The probability of y_bin = 1 is 99% given that x2 = 3, x3 = 5, and the rest of the predictors are set to their mean values.

Fourth, we will estimate the probability that the outcome variable (Y) = 1, setting specific values for multiple predictors   (i.e., x2 = 3, x3 = 5, and opinion=(1 2))

To get the data. Type:

use "https://dss.princeton.edu/training/logit.dta", clear

To get the predicted probability, type:

###### margins, at(x2=3 x3=5 opinion=(1 2)) atmeans post

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------          _at |           1  |   .9891283   .0305393    32.39   0.000     .9292724    1.048984           2  |   .9304941   .1915434     4.86   0.000     .5550758    1.305912 ------------------------------------------------------------------------------

Interpretation:

1. The probability of y_bin = 1 is 98% given that x2 = 3, x3 = 5, the opinion is “strongly agree”, and the rest of the predictors are set to their mean values.

2. The probability of y_bin = 1 is 93% given that x2 = 3, x3 = 5, the opinion is “agree”,  and the rest of the predictors are set to their mean values.

Fifth, we will estimate the probability that the outcome variable (Y) = 1 for each category of the categorical variable  ( i.e., opinion)

To get the data. Type:

use "https://dss.princeton.edu/training/logit.dta", clear

To get the predicted probability, type:

###### margins opinion, atmeans post

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------      opinion |   Str agree  |   .8764826   .0739471    11.85   0.000      .731549    1.021416       Agree  |   .5107928   .1509988     3.38   0.001     .2148405    .8067451       Disag  |    .907761   .0673524    13.48   0.000     .7757527    1.039769   Str disag  |    .933931   .0644709    14.49   0.000     .8075704    1.060292 ------------------------------------------------------------------------------

Interpretation:

- Holding all variables at their mean values. The probability of y_bin = 1 is:

• 87% among those who “strongly agree”,

• 51% among those who “agree”,

• 91% among those who “disagree” and

• 93% among those who “strongly disagree”

- We can show the above predictions with the help of a graph. To do this, type:

marginsplot

The command will produce the graph below

## 2.4.Estimating marginal effects after logit

Marginal effects show the change in probability when the predictor or independent variable increases by one unit. For continuous variables, this represents the instantaneous change, given that the ‘unit’ may be very small. For binary variables, the change is from 0 to 1, so one ‘unit’ is as it is usually thought.

To calculate marginal effects after logit, type:

use "https://dss.princeton.edu/training/logit.dta", clear

quietly logit y_bin x1 x2 x3 i.opinion

margins, dydx(*) atmeans post

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------           x1 |   .1384634   .1093955     1.27   0.206    -.0759478    .3528746           x2 |    .036904   .0421082     0.88   0.381    -.0456266    .1194346           x3 |     .04857   .0548416     0.89   0.376    -.0589176    .1560577              |      opinion |       Agree  |  -.3656898   .1670551    -2.19   0.029    -.6931118   -.0382678       Disag  |   .0312784   .0945857     0.33   0.741    -.1541062    .2166629   Str disag  |   .0574484    .098205     0.58   0.559    -.1350299    .2499268 ------------------------------------------------------------------------------ Note: dy/dx for factor levels is the discrete change from the base level.

Interpretation of the numbers in dy/dx column :

x1 = .1384634  The change in probability for one instant change in x1 is 13 percentage points (pp). However, the change is not statistically significant because the p-value is not <0.05.

x2 = .036904 The change in probability for one instant change in x2 is 3 percentage points (pp). However, the change is not statistically significant because the p-value is not <0.05.

Agree = -.3656898 The change in probability when opinion goes from ‘strongly agree’ to ‘agree’ decreases by 36 percentage points or -0.36. This change is statistically significant because the p-value is 0.029 which is  <0.05.

Disag = .0312784 The change in probability when opinion goes from ‘strongly agree’ to ‘disagree’ increases by 3 percentage points or 0.03. However, the change is not statistically significant because the p-value is not <0.05.

Str Disag = .0574484 The change in probability when opinion goes from ‘strongly agree’ to ‘strongly disagree’ increases by 5 percentage points or 0.05. However, the change is not statistically significant because the p-value is not <0.05.

We can publish the above results in a Word document by using outreg2 command. To do this, type:

use "https://dss.princeton.edu/training/logit.dta", clear

quietly logit y_bin x1 x2 x3 i.opinion

margins, dydx(*) atmeans post

outreg2 using test.doc, word replace ctitle(Marginal effects)

Stata will give us the following outputs

. outreg2 using test.doc, word replace ctitle(Marginal effects)
test.doc
dir : seeout

- Windows users: click on test.doc to open the file in Word (you can replace this name with your own).

- Mac users: click on dir to go to the directory where test.doc is saved, and open it with Word (you can replace this name with your own)

The outputs in the Word document look as follows.

## 3. Estimating the Ordered Logit Model using Stata

As discussed earlier, we should use the ordered logit model when the dependent variable is categorical but ordered (e.g., low to high).

## 3.1. Estimating log-odds ratio

To get the data. Type:

use "https://dss.princeton.edu/training/ologit.dta"

Before running the regression, let's check the nature of the dependent variable first. Type:

tab opinion_or
tab opinion_or, nolab

Stat will give us the following tables.

###### Outcome |    variable |      Freq.     Percent        Cum. ------------+-----------------------------------           1 |         16       22.86       22.86           2 |         19       27.14       50.00           3 |         15       21.43       71.43           4 |         20       28.57      100.00 ------------+-----------------------------------       Total |         70      100.00

To run an ordered logit model, type:

ologit opinion_or x1 x2 x3 i.gender

Where opinion_or is the dependent variable, and x1 x2 x3 gender are independent variables. gender is a dummy variable defined by assigning 0 for female and 1 for male.

Stata will give us the following output table.

###### ------------------------------------------------------------------------------   opinion_or |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------           x1 |   -.946804   .5639295    -1.68   0.093    -2.052086    .1584776           x2 |  -.2233003   .2099466    -1.06   0.288    -.6347881    .1881875           x3 |  -.0579506   .1580635    -0.37   0.714    -.3677493    .2518481              |       gender |        male  |  -.5636039    .524222    -1.08   0.282     -1.59106    .4638524 -------------+----------------------------------------------------------------        /cut1 |  -2.401919   .6513011                     -3.678446   -1.125392        /cut2 |  -1.108298   .5878275                     -2.260419    .0438225        /cut3 |  -.1540898   .5775204                     -1.286009    .9778293 ------------------------------------------------------------------------------

Interpretation

• LR chi2(4)= 4.63 and Prob > chi2 = 0.3276. The likelihood ratio chi-square of 4.63 with a p-value of 0.33 indicates that our model as a whole is not statistically significant. To be statistically significant, we need a p-value  <0.05.
• x1 = -.946804 indicates that when x1 increases by one unit, the expected change in the log-odds of being in a higher level of opinion_or  is - 0.95 (a decrease), holding all other variables in the model constant. This decrease is not statistically significant at the 5% level because the p-value is not < 0.05 (however, the drop is statistically significant at the 10% level as the p-value is < 0.10).
• male = -.5636039 indicates that when gender increases by one unit (i.e., goes from 0 (female) to 1(male)), the expected change in the log-odds of being in a higher level of opinion_or  is - 0.56, holding all other variables in the model constant. However, this decrease is not statistically significant because the p-value is not < 0.05.

Note: The log-odds in the above output table mainly help us to understand the direction of the relationship between the dependent and independent variable; for more comprehensive interpretations of the ordered logit outcomes, see the following sub-sections (3.2, 3.3. and 3.4.).

## 3.2. Estimating the odds ratio

The odds ratio allows an easier interpretation of the ordered logit coefficients. They are the exponentiated value of the ordered logit coefficients. We can get the odds ratio by using the following procedure.

To get the data. Type: (You don't need to get data again if you already estimated log-odds following the instruction in section 3.1.)

use "https://dss.princeton.edu/training/ologit.dta", clear

To get odds ratio rather than ordered logit coefficients, type:

ologit opinion_or x1 x2 x3 i.gender, or

Where opinion_or is the dependent variable, and x1 x2 x3 gender are independent variables. gender is a dummy variable defined by assigning 0 for female and 1 for male.

Note: We added or at the end of the command to get the odds ratio.

Stata will give us the following output table.

###### ------------------------------------------------------------------------------   opinion_or | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------           x1 |    .387979   .2187928    -1.68   0.093     .1284667    1.171726           x2 |   .7998746    .167931    -1.06   0.288     .5300478     1.20706           x3 |   .9436966   .1491639    -0.37   0.714     .6922907    1.286401              |       gender |        male  |   .5691542   .2983632    -1.08   0.282     .2037095    1.590188 -------------+----------------------------------------------------------------        /cut1 |  -2.401919   .6513011                     -3.678446   -1.125392        /cut2 |  -1.108298   .5878275                     -2.260419    .0438225        /cut3 |  -.1540898   .5775204                     -1.286009    .9778293 ------------------------------------------------------------------------------ Note: Estimates are transformed only in the first equation.

Interpretation

• Odds Ratio: They represent the odds of Y=1 when X increases by 1 unit. These are the exp(logit coeff).

If the OR > 1 then the odds of Y=1 increases

If the OR < 1 then the odds of Y=1 decreases

• Let's interpret the odds ratio for x1 which is .387979.

It indicates, keeping all other variables constant, when x1 increases by one unit, the odds of moving to a higher category in the outcome variable decrease by 61%. However, this decrease is not statistically significant as the p-value is not < 0.05. We get this 61% as follows: (.387979 - 1) * 100 = - 61.2021.

Or, we can also say for a one-unit increase in x1, the odds of the high category of opinion_or versus the combined other categories of opinion_or are 0.39 times greater, given that all other variables in the model are constant.

Note: To calculate the odds ratio by hand, you need to exponentiate the ordered logit coefficient. The formula is:

###### odds ratio = exp(coef(ordered logit))

In section 3.1., we got logit coefficient for x1 = -.946804. Therefore, the odds ratio for x1 = exp(-.946804) = 0.387979.

## 3.3. Estimating predicted probabilities after ordered logit

As we discussed earlier, in an ordinal logit model, the outcome (dependent) variable has categories in a meaningful order. In our example, the variable opinion_or has four categories: 1, "Strongly disagree", 2, "Disagree", 3 "Agree",  and  4 "Strongly agree".

First, we will estimate the probability of the outcome variable for different values of opinion_or, given that all predictors are set to their mean values.

Case 1:  opinion_or = 1 ( "Strongly disagree")

Run the following Stata codes:

use "https://dss.princeton.edu/training/ologit.dta", clear
ologit opinion_or x1 x2 x3 i.gender
margins, predict(outcome(1)) atmeans post

Stata will give us the following outputs.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .2203782   .0501331     4.40   0.000     .1221192    .3186373 ------------------------------------------------------------------------------

Interpretation

_cons =.2203782 indicates that the probability of opinion_or = 1 ("Strongly disagree") given that all predictors are set to their mean values is 22%.

Case 2:  opinion_or = 2 ("Disagree")

we can estimate the probability of the outcome variable when opinion_or = 2, given that all predictors are set to their mean values.

Run the following Stata codes:

use "https://dss.princeton.edu/training/ologit.dta", clear
ologit opinion_or x1 x2 x3 i.gender
margins, predict(outcome(2)) atmeans post

Stata will give us the following outputs.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .2871605   .0563818     5.09   0.000     .1766542    .3976668 ------------------------------------------------------------------------------

Interpretation

_cons =.2871605 indicates that the probability of opinion_or = 2 ("Disagree"), given that all predictors are set to their mean values, is 28%.

Case 3:  opinion_or = 3 ( "Agree")

we can estimate the probability of the outcome variable when opinion_or = 3, given that all predictors are set to their mean values.

Run the following Stata codes:

use "https://dss.princeton.edu/training/ologit.dta", clear
ologit opinion_or x1 x2 x3 i.gender
margins, predict(outcome(3)) atmeans post

Stata will give us the following outputs.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .2204349   .0505508     4.36   0.000     .1213571    .3195126 ------------------------------------------------------------------------------

Interpretation

_cons =.2204349 indicates that the probability of opinion_or = 3 ("Agree"), given that all predictors are set to their mean values, is 22%.

Case 4:  opinion_or = 4 ( "Strongly agree")

we can estimate the probability of the outcome variable when opinion_or = 4, given that all predictors are set to their mean values.

Run the following Stata codes:

use "https://dss.princeton.edu/training/ologit.dta", clear
ologit opinion_or x1 x2 x3 i.gender
margins, predict(outcome(4)) atmeans post

Stata will give us the following outputs.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .2720264   .0540224     5.04   0.000     .1661444    .3779084 ------------------------------------------------------------------------------

Interpretation

_cons =.2720264 indicates that the probability of opinion_or = 3 ("Strongly agree"), given that all predictors are set to their mean values, is 27%.

Second, we will estimate the probability of the outcome variable for different values of opinion_orsetting predictors to specific values.

Case 1:  opinion_or = 1 ( "Strongly disagree"), x3=5, and the other variables are at their mean values.

Run the following Stata codes:

use "https://dss.princeton.edu/training/ologit.dta", clear
ologit opinion_or x1 x2 x3 i.gender
margins, predict(outcome(1)) at(x3=5) atmeans post

Stata will give us the following outputs.

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .2654444   .1388834     1.91   0.056    -.0067621    .5376509 ------------------------------------------------------------------------------

Interpretation

_cons =.2654444 indicates that the probability of opinion = 1, given x3=5, and the rest of the variables are at their mean values is 26.5%.

Case 2:  opinion_or = 2 ( "disagree"), x3=5, and the other variables are at their mean values.

Run the following Stata codes:

use "https://dss.princeton.edu/training/ologit.dta", clear
ologit opinion_or x1 x2 x3 i.gender
margins, predict(outcome(2)) at(x3=5) atmeans post

Stata will give us the following outputs

###### ------------------------------------------------------------------------------              |            Delta-method              |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+----------------------------------------------------------------        _cons |   .3030621   .0687894     4.41   0.000     .1682375    .4378868 ------------------------------------------------------------------------------

Interpretation

_cons =.3030621 indicates that the probability of opinion = 2, given x3=5, and the rest of the variables are at their mean values, is 30%.

Do Case 3 and Case 4 by yourself.

## 3.4.Estimating marginal effects after ordered logit

Marginal effects show the change in probability when the predictor or independent variable increases by one unit. For continuous variables, this represents the instantaneous change, given that the ‘unit’ may be very small. For binary variables, the change is from 0 to 1, so one ‘unit’ is as it is usually thought.

To calculate marginal effects after ordered logit with respect to the independent variable x1, type:

use "https://dss.princeton.edu/training/ologit.dta", clear

ologit opinion_or x1 x2 x3 i.gender

margins, dydx(x1)

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- x1           |     _predict |           1  |   .1620537   .0962109     1.68   0.092    -.0265162    .3506236           2  |   .0625812   .0402706     1.55   0.120    -.0163477    .1415101           3  |  -.0417126   .0283309    -1.47   0.141    -.0972401    .0138148           4  |  -.1829222   .1065133    -1.72   0.086    -.3916845      .02584 ------------------------------------------------------------------------------

Interpretation

The marginal effects indicate that for one instant change in x1, it is 16 percentage points more likely to strongly disagree, 6 percentage points more likely to disagree, 4 percentage points less likely to agree, and 18 percentage points less likely to strongly agree.

To calculate marginal effects after ordered logit with respect to the categorical independent variable gender, type:

margins, dydx(gender)

Stata will give us the following output table.

###### ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- 0.gender     |  (base outcome) -------------+---------------------------------------------------------------- 1.gender     |     _predict |           1  |   .0880757   .0750215     1.17   0.240    -.0589637    .2351151           2  |   .0466492   .0515201     0.91   0.365    -.0543283    .1476267           3  |  -.0179264   .0161309    -1.11   0.266    -.0495423    .0136896           4  |  -.1167985   .1133705    -1.03   0.303    -.3390007    .1054037 ------------------------------------------------------------------------------

Interpretation

The marginal effects indicate that, on average, males are 8.8 percentage points more likely than females to say strongly disagree, 4.6 percentage points more likely to say disagree, 1.8 percentage points less likely to say agree, and about 12 percentage points less likely to say strongly agree.

## 4. Estimating the Multinomial Logit Model using Stata

As discussed earlier, we should use the multinomial logit model when the dependent variable is categorical but without any particular order.

## 4.1. Estimating log-odds ratio

To get the data. Type:

use "https://dss.princeton.edu/training/mlogit.dta"

Before running the regression, let's check the nature of the dependent variable first. Type:

tab opinion_ml
tab opinion_ml, nolab

Stat will give us the following tables.

###### Outcome |    variable |      Freq.     Percent        Cum. ------------+-----------------------------------           1 |         36       51.43       51.43           2 |         15       21.43       72.86           3 |         19       27.14      100.00 ------------+-----------------------------------       Total |         70      100.00

To run a multinomial logit model, type:

mlogit opinion_ml x1 x2 x3 i.gender

Where opinion_ml is the dependent variable, and x1 x2 x3 gender are independent variables. gender is a dummy variable defined by assigning 0 for female and 1 for male.

Stata will give us the following output table.

###### ------------------------------------------------------------------------------   opinion_ml |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- Democrats    |  (base outcome) -------------+---------------------------------------------------------------- Independent  |           x1 |   .5438799   .9719664     0.56   0.576    -1.361139    2.448899           x2 |  -.2741146   .3610776    -0.76   0.448    -.9818137    .4335846           x3 |  -1.036186   .5886012    -1.76   0.078    -2.189823    .1174509              |       gender |        male  |  -2.255614   .8000494    -2.82   0.005    -3.823682   -.6875462        _cons |   .7328207   .8913779     0.82   0.411    -1.014248    2.479889 -------------+---------------------------------------------------------------- Republicans  |           x1 |  -.3814889   .7140076    -0.53   0.593    -1.780918     1.01794           x2 |  -.0402157   .2757943    -0.15   0.884    -.5807626    .5003313           x3 |   .2691887   .2230701     1.21   0.228    -.1680206    .7063981              |       gender |        male  |  -.0693857    .935593    -0.07   0.941    -1.903114    1.764343        _cons |  -.6036808   .9478304    -0.64   0.524    -2.461394    1.254033 ------------------------------------------------------------------------------

Interpretation

LR chi2(8)20.46 and Prob > chi2 = 0.0087The likelihood ratio chi-square of 20.46 with a p-value of 0.009 indicates that our model as a whole is statistically significant.

Notice that the above output table has two parts, and they are labeled with the categories of the outcome variable opinion_ml. When we explain the coefficients, we will explain them from the perspective of both categories. Remember that our benchmark category is  Democrats

• The .5438799 coefficient of x1 under Independent category suggests that for one unit increase in x1, the logit coefficient for Independent relative to Democrats will increase by that amount, 0.54. In other words, if  x1 increases by one unit, the chances for voting for an independent candidate are higher compared to voting for a Democrat candidate. Note that the coefficient is not statistically significant as the p-value is not < 0.05.
• The -.3814889 coefficient of x1 under Republicans category suggests that for one unit increase in x1, the logit coefficient for Republicans relative to Democrats will decrease by that amount, 0.38. In other words, if  x1 increases by one unit, the chances for voting for a Republican candidate are lower compared to voting for a Democrat candidate. Note that the coefficient is not statistically significant as the p-value is not < 0.05.
• The -2.255614 coefficient of male under Independent category suggests that when gender increases by one unit (i.e., goes from 0 (female) to 1(male)), the expected change in the log-odds of voting an independent candidate will decrease by 2.3 than voting a democrat, holding all other variables in the model constant. Note that this decrease is statistically significant because the p-value is < 0.05.

## 4.2. Estimating relative risk ratios

Relative risk ratios allow an easier interpretation of the multinomial logit coefficients. They are the exponentiated value of the logit coefficients.

To get the data. Type: (You don't need to get data again if you already estimated log-odds following the instruction in section 4.1.)

use "https://dss.princeton.edu/training/mlogit.dta", clear

To get odds ratio rather than multinomial logit coefficients, type:

mlogit opinion_ml x1 x2 x3 i.gender, rrr

Note: We added rrr at the end of the command to get the odds ratio.

Stata will give us the following output table.

###### ------------------------------------------------------------------------------   opinion_ml |        RRR   Std. Err.      z    P>|z|     [95% Conf. Interval] -------------+---------------------------------------------------------------- Democrats    |  (base outcome) -------------+---------------------------------------------------------------- Independent  |           x1 |   1.722678   1.674385     0.56   0.576     .2563685    11.57559           x2 |   .7602449   .2745074    -0.76   0.448      .374631    1.542778           x3 |   .3548052   .2088388    -1.76   0.078     .1119365    1.124626              |       gender |        male  |   .1048092   .0838525    -2.82   0.005     .0218472    .5028083        _cons |   2.080942   1.854906     0.82   0.411     .3626751    11.93994 -------------+---------------------------------------------------------------- Republicans  |           x1 |    .682844   .4875558    -0.53   0.593     .1684834    2.767488           x2 |   .9605822   .2649231    -0.15   0.884     .5594715    1.649268           x3 |   1.308902   .2919769     1.21   0.228     .8453365    2.026678              |       gender |        male  |   .9329667   .8728772    -0.07   0.941     .1491035    5.837735        _cons |   .5467953   .5182692    -0.64   0.524     .0853159    3.504446 ------------------------------------------------------------------------------ Note: _cons estimates baseline relative risk for each outcome.

Interpretation:

• Keeping all other variables constant, if your x1 increases by one unit, you are 1.72 times more likely to vote for an independent candidate as compared to the benchmark category Democrats (the risk or odds is (1.722678 - 1) * 100 = 72% higher). The coefficient, however, is not significant.

• Keeping all other variables constant, if your x1 increases by one unit, you are .68 times more likely to vote for a Republican candidate as compared to voting for a Democrat (the risk or odds is (.682844 - 1) * 100 = 32% lower). The coefficient, however, is not significant.

Note: To calculate the relative risk ratio by hand, you need to exponentiate the multinomial logit coefficient. The formula is:

relative risk ratio = exp(coef(logit))

In section 4.1., we got the multinomial logit coefficient for x1 = .5438799. Therefore, the relative risk ratio for x1 = exp(.5438799 ) = 1.7226

## References / Useful Resources

Adjusted Predictions & Marginal Effects for Multiple Outcome Models & Commands (including ologit, mlogit, oglm, & gologit2) / Richard Williams, University of Notre Dame, 2021. Available at https://www3.nd.edu/~rwilliam/stats3/Margins05.pdf

Applied Regression Analysis and Generalized Linear Models / John Fox, Sage, 2008.

Data analysis using regression and multilevel/hierarchical models / Andrew Gelman, Jennifer Hill. Cambridge ; New York : Cambridge University Press, 2007.

DSS Data Analysis Guides. Available at https://library.princeton.edu/dss/training

Econometric analysis / William H. Greene. 8th ed., Upper Saddle River, N.J. : Prentice Hall, 2018.

Introduction to econometrics / James H. Stock, Mark W. Watson. 4th ed., Boston: Pearson Addison Wesley, 2019.

UCLA, Stata Data Analysis Example. Multinomial Logistic Regression. Available at https://stats.oarc.ucla.edu/stata/dae/multinomiallogistic-regression/

UCLA, Stata Data Analysis Example. Ordered Logistic Regression. Available at https://stats.oarc.ucla.edu/stata/dae/ordered-logistic-regression/

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