【SPSS】Regression Analysis — Logistic Regression: Multinomial Logistic Regression

Steps

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1. Choose the analysis method: click Analyze -> Regression -> Multinomial Logistic... in the menu bar to open the multinomial logistic regression dialog box.

2. Set variables:

   • Dependent variable (Dependent): drag the multi-category dependent variable into this box.
   • Factors: drag categorical independent variables into this box.
   • Covariates: drag continuous independent variables into this box.

3. Set the reference category:

   • Click the Reference Category button and choose the baseline category.

Regression Equation

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In multinomial logistic regression analysis, log odds are used to represent the likelihood of each category.

Log Odds:

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$$ \ln \left( \frac{P_1}{P_3} \right) = B_0 + B_1 X_1 + B_2 X_2 + \cdots + B_n X_n $$

$$ \ln \left( \frac{P_2}{P_3} \right) = B_0 + B_1 X_1 + B_2 X_2 + \cdots + B_n X_n $$

Probability:

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$$ a = \left( \frac{P_1}{P_3} \right) = e^{B_0 + B_1 X_1 + B_2 X_2 + \cdots + B_n X_n} $$

$$ b = \left( \frac{P_2}{P_3} \right) = e^{B_0 + B_1 X_1 + B_2 X_2 + \cdots + B_n X_n} $$

$$ P_1 = \frac{a}{a + b + 1}, \quad P_2 = \frac{b}{a + b + 1}, \quad P_3 = 1 - P_1 - P_2 = \frac{1}{a + b + 1} $$

Derivation Process:

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Known conditions:

$$ a = \frac{P_1}{P_3}, \quad b = \frac{P_2}{P_3} $$

Step 1: Derive (P_1) and (P_2)

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From the given formulas:

$$ a = \frac{P_1}{P_3} \quad \Rightarrow \quad P_1 = a \cdot P_3 $$

$$ b = \frac{P_2}{P_3} \quad \Rightarrow \quad P_2 = b \cdot P_3 $$

Step 2: Use (P_1 + P_2 + P_3 = 1)

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According to the total-probability formula, we have:

$$ P_1 + P_2 + P_3 = 1 $$

Substitute (P_1) and (P_2) into the equation above:

$$ a \cdot P_3 + b \cdot P_3 + P_3 = 1 $$

Step 3: Extract the Common Factor (P_3)

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Extract (P_3):

$$ P_3 (a + b + 1) = 1 $$

Step 4: Solve for (P_3)

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Solve for (P_3):

$$ P_3 = \frac{1}{a + b + 1} $$

Step 5: Solve for (P_1) and (P_2)

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Substitute (P_3 = \frac{1}{a + b + 1}) into the previously derived (P_1) and (P_2):

$$ P_1 = a \cdot \frac{1}{a + b + 1} = \frac{a}{a + b + 1} $$

$$ P_2 = b \cdot \frac{1}{a + b + 1} = \frac{b}{a + b + 1} $$

Final Result:

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$$ P_1 = \frac{a}{a + b + 1}, \quad P_2 = \frac{b}{a + b + 1}, \quad P_3 = \frac{1}{a + b + 1} $$

Explanation

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Table:

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Model fitting information: if the significance value (p-value) is less than 0.05, the model is statistically significant and can effectively reflect the relationship between independent variables and the dependent variable.

Pseudo R-squared:

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Pseudo R-squared is used to evaluate model fit. It measures the match between model predictions and actual observed data. Common pseudo R-squared values include: 6. McFadden’s pseudo R-squared: suitable for discrete-choice models, widely used in economics and social-science research, and recommended as the first value to inspect. 7. Nagelkerke pseudo R-squared: an adjustment of McFadden’s pseudo R-squared that makes the model-fit range more intuitive. 8. Cox-Snell pseudo R-squared: another common pseudo R-squared estimation method.

The higher these pseudo R-squared values are, the better the model fit.

Parameter Estimates:

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The coefficients (B values) in the regression equation indicate the effect strength and direction of the independent variables on the dependent variable. Exp(B) represents the odds ratio of the corresponding variable for the dependent variable. These parameters make it easier to understand each variable’s contribution to the model’s prediction.

Summary:

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This article explains the steps of multinomial logistic regression, the derivation process of the regression equation, and how to interpret regression results. By understanding model fitting information, pseudo R-squared, and parameter estimates, readers can better understand the meaning and application of regression analysis.

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