## Who is afraid of whom?

The liberal German weekly Zeit has commissioned a YouGov poll which demonstrates that Germans are more afraid of right-wing terrorists than of Islamist terrorists. The question read “What is, in your opinion, the biggest terrorist threat in Germany?” On offer were right-wingers (41 per cent), Islamists (36.6 per cent), left-wingers (5.6 per cent), other groups (3.8 per cent), or (my favourite) “no threat” (13 per cent). This is a pretty daft question anyway. Given the news coverage of the Neo-Nazi gang that has killed at least ten people more or less under the eyes of the authorities, and given that the authorities have so far managed to stop would-be terrorists in their tracks, the result is hardly surprising.

Nonetheless, the difference of just under five percentage points made the headlines, because there is a subtext for Zeit readers: Germans are worried about right-wing terrorism (a few weeks ago many people would have denied that there are right-wing terrorists operating in Germany), which must be a good thing, and they are less concerned about Islamist terrorists, which is possibly a progressive thing. Or something along those lines.

But is the five-point difference real?

YouGov has interviewed 1043 members of its online access panel. If we assume (and this is a heroic assumption) that these respondents can be treated like a simple random sample, what are the confidence intervals?

## Binomial Confidence Intervals

First, we could treat the two categories as if they were distributed as binomial and ask Stata for exact confidence intervals.

cii 1043 round(1043*.41) cii 1043 round(1043*.366)

The confidence intervals overlap, so we’re lead to think that the proportions in the population are not necessarily different. But the two categories are not independent, because the “not right-wingers” answers include the “Islamists” answers and vice versa, so the multinomial is a better choice.

## Multinomial Model

It is easy to re-create the univariate distribution of answers in Stata:

set obs 5 gen threat = _n lab def threat 1 "right-wingers" 2 "islamists" 3 "left-wingers" 4 "other" 5 "no threat" lab val threat threat gen number = round(1043* 0.41) in 1 replace number = round(1043* 0.366) in 2 replace number = round(1043* 0.056) in 3 replace number = round(1043* 0.038) in 4 replace number = round(1043* 0.13) in 5 expand number

Next, run an empty multinomial logit model

mlogit threat,base(5)

The parameters of the model reproduce the observed distribution exactly and are therefore not very interesting, but the estimates of their standard errors are available for testing hypotheses:

test [right_wingers]_cons = [islamists]_cons

At the conventional level of 0.05, we cannot reject the null hypothesis that both proportions are equal in the population, i.e. we cannot tell if Germans are really more worried about one of the two groups.

## Simulation

Just for the fun of it, we can carry out one additional test and ask a rather specific question: If both proportions are 0.388 in the population and the other three are identical to their values in the sample, what is the probability of observing a difference of at least 4.4 points in favour of right-wingers?

The idea is to sample repeatedly from a multinomial with known probabilities. This could be done more elegantly by defining a program and using Stata’s simulate command, but if your machine has enough memory, it is just as easy and possibly faster to use two loops to generate/analyse the required number of variables (one per simulation) and to fill them all in one go with three lines of mata code. Depending on the number of trials, you may have to adjust maxvars

local trials = 10000 foreach v of newlist s1-s`trials' { qui gen `v' = . } mata: probs =(.388,.388,.056,.038,.13) st_view(X.,.,"s1-s`trials'",) X[.,.] = rdiscrete(1043,`trials',probs) end local excess = 0 forvalues sample = 1/`trials' { qui tab s`sample' if s`sample' == 1 local rw = r(N) qui tab s`sample' if s`sample' == 2 local isl = r(N) if (`rw' / 1043 * 100) - (`isl' / 1043 * 100) >=4.4 local excess = `excess' +1 } display "Difference >=4.4 in `excess' of `trials' samples"

Seems the chance of a 4.4 point difference is between 5 and 6 per cent. This probability is somewhat smaller than the one from the multinomial model because the null hypothesis is more specific, but still not statistically significant. And the Zeit does not even have a proper random sample, so there is no scientific evidence for the claim that Germans are more afraid of right-wing extremists than of Islamists, what ever that would have been worth. Bummer.