1.1 The running example

A large literature in social and political psychology argues that perceived competition and status threat shape attitudes toward immigrant groups. Within Ethnic Competition Theory (Scheepers et al., 2002), socioeconomic position — social class, education, and income — is positively associated with attitudes toward immigrants. Different theoretical accounts, however, predict different orderings of these three effects. We compare three theory-consistent variants plus a complement, using European Values Study (EVS) data from Germany.

We fit a linear regression of attitude on class, education, and income, with gender as a covariate, and test:

\[ \begin{aligned} \mathcal{H}_1 &:\ \beta_{\text{class}} > \beta_{\text{education}} > \beta_{\text{income}} > 0 \quad(\text{strict ordering: class matters most}),\\ \mathcal{H}_2 &:\ \beta_{\text{education}} > (\beta_{\text{class}}, \beta_{\text{income}}) > 0 \quad(\text{education largest; class and income unordered}),\\ \mathcal{H}_3 &:\ \beta_{\text{class}} = \beta_{\text{education}} = \beta_{\text{income}} > 0 \quad(\text{three equal positive effects}),\\ \mathcal{H}_4 &:\ \text{complement (none of } \mathcal{H}_1,\mathcal{H}_2,\mathcal{H}_3). \end{aligned} \]

A traditional point-null in which all three effects are zero is not of substantive interest here and is therefore excluded.

1.2 What you will be able to do

By the end you will be able to:

• fit a linear regression and read the exact coefficient names with get_estimates();
• encode competing ordering / equality expectations as a BFpack hypothesis string using >, =, (,), and ;;
• compute Bayes factors and posterior probabilities with BF() and interpret them; and
• see why the equivalent classical route — separate coefficient and contrast tests with a multiplicity correction — gives only fragmented, and sometimes intransitive, evidence.

3.1 Step 1 — Read the exact parameter names

This step is essential. The names used in a hypothesis string are exactly the names printed by get_estimates(). For this model the three predictors of interest appear simply as class, education, and income (the factor gender appears as gender1).

get_estimates(fit1)
#> Coefficient names include: education, income, gender1, class

3.2 Step 2 — Formulate the hypotheses

Each theory becomes one constrained hypothesis; hypotheses are separated by semicolons. (class, income) in \(\mathcal{H}_2\) means both are larger than 0 but unordered relative to each other. We do not write \(\mathcal{H}_4\) explicitly: complement = TRUE adds it automatically.

set.seed(1)   # the equality/order tests use sampling; fix the seed.

BF_App1 <- BF(fit1,
  hypothesis = "class > education > income > 0;
                education > (class, income) > 0;
                class = education = income > 0",
  complement = TRUE)

print(BF_App1)

By default the four hypotheses receive equal prior probabilities of \(1/4\). To weight them differently, pass prior.hyp.conf (one weight per hypothesis, in order, including the complement) — this shifts the posterior probabilities but leaves the Bayes factors, which measure the evidence in the data, unchanged.

3.3 Results

Table 1. Bayes factors (rows vs. columns) and posterior probabilities for the four hypotheses (equal prior probabilities, seed = 1).

All three theory-consistent hypotheses receive strong evidence over the complement, and \(\mathcal{H}_2\) is favored among \(\mathcal{H}_1\)–\(\mathcal{H}_3\) (though the evidence is mild, with Bayes factors around 3). A possible write-up:

“The model comparison favored \(\mathcal{H}_2\) with a posterior probability of .620, suggesting the largest positive effect for education, followed by class and income. Evidence distinguishing \(\mathcal{H}_2\) from \(\mathcal{H}_1\) and \(\mathcal{H}_3\) was mild (Bayes factors of approximately 3). All three theory-consistent hypotheses strongly outperformed the complement (Bayes factors of 49.5, 159, and 46.7), indicating that parameter configurations inconsistent with Ethnic Competition Theory are unlikely given the data.”

Note how a single comparison expresses the relative support for each whole theory — exactly what the substantive question asks, and exactly what a collection of separate tests cannot deliver.