One-Sample Trinomial Test
Introduction
Would the majority of people in the population not see accounting as very scientific? The majority would be more than 50% of the people, so in other words, is the median (the score in the middle) in the population significantly different from 2.5 (since 2 = pretty scientific, and 3 = not scientific)?
Three tests could be used for this, the first does exactly what is described above, and is known as a sign-test, however another test is more frequently used and has a lot scarier name: one-sample Wilcoxon signed rank test (Wilcoxon, 1945). This second test uses rankings (it ranks the scores) and because of this might give a slightly different result. However the Wilcoxon signed rank test requires the data to be symmetrical, while the sign test does not. The third is a trinomial test, it is similar as a sign test but doesn't exclude the values tied with the hypothesized value.
The one-sample trinomial test uses the multinomial distribution, using the number of scores below, equal and above to the hypothesized median as counts, and the sample proportion of scores equal to the hypothesized median and dividing the rest over the two other options.
Performing the Test
with Excel
Excel file from video: TS - Trinomial (One-Sample) (E).xlsm
with stikpetE
without stikpetE
with Python
Notebook from video: TS - Trinomial (One-Sample) (P).ipynb
with stikpetP
without stikpetP
with SPSS (not possible?)
To my knowledge this test is not possible with SPSS
Formulas
The one-sample trinomial test can be performed using:
- Determine the number of scores above (\(n_{+}\)), below (\(n_{-}\)) and equal (\(n_{0}\)) to the hypothesized median
- Determine the sample probability for a difference of 0, i.e. \(p_0 = \frac{n_0}{n}\)
- Set the probability for the other to categories to half what remains, i.e. \(p_{+} = p_{-} = \frac{1 - p_0}{2}\)
- Determine the difference in number of positive and negative differences with the hypothesized median, i.e. \(n_d = \left|n_+ - n_-\right|\)
- determine the probability of such a difference, or even larger given the three probabilities, using the multinomial distribution, i.e. \(p = 2\times \sum_{i=n_d}^n \sum_{j=0}^{\lfloor \frac{n - i}{2} \rfloor} \text{mpmf}\left(\left(j, j+i, n - i\right), \left(p_{pos}, p_{neg}, p_0\right) \right)\)
WARNING: Do not use the multinomial cumulative density function (mcdf). The mcdf is for the probability of a distribution of counts or as extreme, not for a difference.
Since we use the multinomial distribution with three counts (and hence three probabilities) it becomes a trinomial distribution, which gives the name for this test.
This test can for example be found in Bian et al. (2009, p. 6)
Interpreting the Result
The assumption about the population for this test (the null hypothesis) is that the median is a specific value.
The test provides a p-value, which is the probability of a test statistic as from the sample, or even more extreme, if the assumption about the population would be true. If this p-value (significance) is below a pre-defined threshold (the significance level \(\alpha\) ), the assumption about the population is rejected. We then speak of a (statistically) significant result. The threshold is usually set at 0.05. Anything below is then considered low.
If the assumption is rejected, we conclude that the median in the population will be different than the one used in the test.
Note that if we do not reject the assumption, it does not mean we accept it, we simply state that there is insufficient evidence to reject it.
Writing the results
APA (2019) does not specify how to write the results of a one-sample trinomial test. It shows for most tests to report the test-statistic, the degrees of freedom (if applicable) and the p-value. However with an exact test as the one-sample sign test, there isn't a test-statistic. I would therefor suggest to list the number of scores above, equal and below the median and the p-value.
So for example:
A one-sample trinomial test, indicated that the median was significantly different from the neutral option, n+ = 13, n0 = 12, n- = 29, p = .016.
The p-value is shown with three decimal places, and no 0 before the decimal sign. If the p-value is below .0005, it can be reported as p < .001.
APA (2019, p. 88) states to also report an effect size measure.
Google adds