Researchers interested in measuring people's underlying attitudes towards an object (e.g., abortion) often collect Likert data by administering a survey. Likert data consist of surveyees' responses to statements about the object, where responses fall into ordered categories running from `Strongly agree' to `Strongly disagree' or into a `Don't Know / Can't Choose' category. Two examples of Likert data are used for illustrative purposes. The first dataset was collected by the author from American and British graduate students at Oxford University and contains items measuring underlying abortion attitudes. The second dataset was taken from British and American responses to the 1995 National Identity Survey (NIS) and contains items measuring underlying national pride and immigration attitudes. A model for Likert data and underlying attitudes is introduced. This model is more principled than existing models. It treats people's underlying attitudes as latent variables, and it specifies a relationship between underlying attitudes and responses that is consistent with attitudinal research. Further, the formal probability model for responses allows people's interpretation of the response categories to differ. The model is fitted by maximising an appropriate likelihood. Variants of the model are used to analyse Likert data in three contexts; in each, the method using our model compares favourably to existing methods. First, the model is used to visualise the structure underlying the abortion attitude data. This method of visualization produces more sensible plots than analogous multivariate data visualization methods. Second, the model is used to select the statements whose responses (in the abortion attitude data) best reflect underlying abortion attitudes. Our method of statement selection more closely adheres to attitude researchers' stated aims than popular methods based on sample correlations. Third, the model is used to investigate how underlying national pride varies with nationality in the NIS data and also how underlying abortion attitude varies with gender, religious status, and nationality in the abortion attitude data. Unlike methods currently used by social scientists to model the relationship between attitudes and covariates, our method controls for the effects of differing response category interpretation. As a result, inferences about group differences in underlying attitudes are more robust to group differences in response category interpretation.