SPSS Data Analysis Examples: Multinomial Logistic Regression



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SPSS Data Analysis Examples
Multinomial Logistic Regression

Examples

Example 1. People's occupational choices might be influenced by their parents' occupations and their own education level. We can study the relationship of one's occupation choice with education level and father's occupation. The occupational choices will be the outcome variable which consists of categories of occupations.

Example 2. A biologist may be interested in food choices that alligators make. Adult alligators might have difference preference than young ones. The outcome variable here will be the types of food, and the predictor variables might be the length of the alligators and other environmental variables.

Example 3. Several brands of similar products are on the market, and you want to study brand choices based on gender and age. For example, a recent finding of a market research group claims that among digital camera choices, women prefer Kodak more than men and men prefer Canon more than women.

Description of the Data

For our data analysis example, we will expand our third example with a hypothetical data set. The data set contains information on 735 subjects who were asked their preference on three brands of some product (e.g., car or TV). Included in the data set are the information on subjects' gender and age. You can download the data here .

get file "D:\mlogit.sav".

list
/cases from 1 to 25.

    brand   female       age

        1        0        24
        1        0        26
        1        0        26
        1        1        27
        1        1        27
        3        1        27
        1        0        27
        1        0        27
        1        1        27
        1        0        27
        1        0        27
        1        1        27
        2        1        28
        3        1        28
        1        1        28
        1        0        28
        1        0        28
        2        1        28
        1        0        28
        1        0        28
        1        1        28
        1        1        28
        3        0        28
        1        1        28
        3        0        28

Number of cases read:  25    Number of cases listed:  25

The outcome variable is brand. The variable female is coded as 0 for male and 1 for female. Let's start with some descriptive statistics of the variables of our interest.

frequencies var = brand.

sort cases by brand.
temporary.
split file by brand.
descriptives var = age female.
split file off.

Some Strategies You Might Try

Multiple logistic regression analyses, one for each pair of outcomes: One problem with this approach is that each analysis is run on a different sample. The other problem is that without constraining the logistic models, we can end up with the probability of choosing all possible outcome categories greater than 1.
Collapsing number of categories to two and then doing a logistic regression: This approach suffers from loss of information and changes the original research questions to very different ones.
Ordinal logistic regression: If the outcome variable is truly ordered and if it also satisfies the assumption of proportional odds, then switching to ordinal logistic regression will make the model more parsimonious.
Linear regression: Sometimes the coding of a variable can be deceiving, making the outcome variable look like a continuous variable. This approach does not have any merit to it when the outcome variable is truly multinomial.
Multiple Discriminant Analysis: This is a multivariate technique of profiling, differentiation and classification of groups. The goals are different from multinomial logistic regression (MLR) in that MLR is more about description, inference and prediction.

Using the Multinomial Logit Model

Now we have warmed up to building our model. Our goal is to associate the brand choices with age and gender. We will assume a linear relationship between the transformed outcome variable and our predictor variables female and age. Since there are multiple categories, we will choose a base category as the comparison group. Here our choice is the first brand (brand=1).

nomreg brand (base = first) with female age
/print = lrt cps mfi parameter summary.

The table above, titled Parameter Estimates, has two parts, labeled with the categories of the outcome variable brand. They correspond to two equations:

log(P(brand=2)/P(brand=1)) = b_10 + b_11*female + b_12*age
log(P(brand=3)/P(brand=1)) = b_20 + b_21*female + b_22*age,

with b's being the raw regression coefficients from the output.

For example, we can say that for one unit change in the variable age, the log of the ratio of the two probabilities, P(brand=2)/P(brand=1), will be increased by 0.368, and the log of the ratio of the two probabilities P(brand=3)/P(brand=1) will be increased by 0.686. Therefore, we can say that, in general, the older a person is, the more he/she will prefer brand 2 or 3.

The ratio of the probability of choosing one outcome category over the probability of choosing the reference category is often referred as relative risk (and it is also sometimes referred as odds). So another way of interpreting the regression results is in terms of relative risk. We can say that for one unit change in the variable age, we expect the relative risk of choosing brand 2 over 1 to increase by exp(.3682) = 1.45. So we can say that the relative risk is higher for older people. For a dichotomous predictor variable such as female, we can say that the ratio of the relative risks of choosing brand 2 over 1 for female and male is exp(.5238). We can see the results displayed as relative risk ratios in the column labeled Exp(B) in the table above.

Sample Write-up of the Analysis

Below is one way of describing the results.

Both female and age are statistically significant across the two models. Females are more likely to prefer brands 2 or 3 compared to brand 1. Also, the older a person is, the more likely he/she is to prefer brands 2 or 3 to brand 1. Both of these findings are statistically significant.

Cautions, Flies in the Ointment

The Independence of Irrelevant Alternatives (IIA) assumption: Roughly, the IIA assumption means that adding or deleting alternative outcome categories does not affect the odds among the remaining outcomes.
Diagnostics and model fit: Unlike logistic regression where there are many statistics for performing model diagnostics, it is not as straightforward to do diagnostics with multinomial logistic regression models.
Pseudo-R-Squared: The R-squares offered in the output are basically the change in terms of log-likelihood from the intercept-only model to the current model. These do not convey the same information as the R-square for linear regression, even though it is still "the higher, the better".
Sample size: Multinomial regression uses a maximum likelihood estimation method. Therefore, it requires a large sample size. It also uses multiple equations. Therefore, it requires an even larger sample size than ordinal or binary logistic regression.
Perfect prediction: Perfect prediction means that only one value of a predictor variable is associated with only one value of the response variable. You can tell from the output of the regression coefficients that something is wrong. You can then do a two-way tabulation of the outcome variable with the problematic variable to confirm this and then rerun the model without the problematic variable.
Empty cells or small cells: You should check for empty or small cells by doing a crosstab between categorical predictors and the outcome variable. If a cell has very few cases (a small cell), the model may become unstable or it might not run at all.

SPSS Data Analysis Examples Multinomial Logistic Regression