I graphed a variable and it looked kind of bimodal but I'm not sure. Is there a more quantitative method of establishing this? Once again, I'm using Minitab.
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2$\begingroup$ Could you provide some context? What does the variable represent? Why are you interested in modality? What would be the consequences of making an incorrect decision about the number of modes? $\endgroup$– whuber ♦Apr 7, 2015 at 14:53
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$\begingroup$ The variable is the oxides of nitrogen emmission from car exhausts. It is continuous interval data and so I would have to split it into two separate groups (petrol vs diesel) before calculating any measures of location & spread. $\endgroup$– user72943Apr 7, 2015 at 14:57
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2$\begingroup$ So what you really want to ask is whether the data should be divided into two clusters. That has a different thrust than the question about modes (which are primarily descriptive and usually are meaningful only for discrete data). Asking about clustering can also attract people who are experts in that discipline. The context also suggests there will be other variables available, such as other exhaust components and maybe attributes of the cars themselves. These can be exploited to improve the clustering (often substantially). Consider editing your post accordingly. $\endgroup$– whuber ♦Apr 7, 2015 at 15:01
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$\begingroup$ I was told that if the distribution is unimodal then you use appropriate measures (mean and SD if symmetric and median and IQR if skewed or if it has outliers) whereas if it is multimodal then no measure is appropriate therefore it must be split into different groups and analysed separately. What I have said there is what applies to continuous interval data. The only method of grouping them is via petrol vs diesel. $\endgroup$– user72943Apr 7, 2015 at 15:08
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3$\begingroup$ Sometimes it's a good idea to challenge what you have been told. This is especially true in statistics, because each problem has unique characteristics that ought to be considered when selecting a statistical solution. In this case the sense of "appropriate" is vague: understanding it requires knowledge of the intended use of the descriptive statistics. In many cases they will do fine; in others they will be fine when supplemented by a few more; in yet other cases you might want to separate the data into two groups--with considerable added complications. $\endgroup$– whuber ♦Apr 7, 2015 at 15:11
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2 Answers
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To think about ways to infer whether your data is bimodal or unimodal you need to hypothesize on whether there is a good fundamental underlying reason that the thing creating your data is bimodal or not.
If we change your question slightly to say "given a measurement of nitrogen oxide emission, what is the probability the emission came from a petrol or diesel vehicle?". From this we can begin to estimate what the distribution of diesel emission looks like vs distribution of petrol and can do tests to see if these two distributions are statistically different.
Using standard Bayesian Inference and a Mixture Model, you can calculate these distributions and probabilities. Until you clarify your answer further, I will point you to this:
- General Wikipedia reference: http://en.wikipedia.org/wiki/Mixture_model
- python-based (and better theoretical explination) http://scikit-learn.org/stable/modules/mixture.html
If you can separate out the two types sources of diesel emission you can just run a the following tests to see if they are equal:
- Eyeball test (simple sanity check, not rigorous)
- Two-sample Kolmogorov–Smirnov http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
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$\begingroup$ This is for a very basic course, it need not be remotely that complicated. From what I've seen from a quick Google, there is a difference between the two (not surprising) and so it would be logical to split it but I'm not sure if that is correct nor how to actually do that in Minitab. If you can provide a way to quickly check and I find that it is bimodal then I will find out how to split them and see if that improves the situation. $\endgroup$ Apr 7, 2015 at 15:14
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$\begingroup$ Your first link discusses mixture designs, (experimental designs for investigating the effects of factors constrained to sum to a given amount), not mixture models. $\endgroup$ Apr 7, 2015 at 15:50
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$\begingroup$ Thank you for feedback. I have removed the incorrect link and added a more general reference. I also added a more simple test per @user72943 suggestion. $\endgroup$ Apr 8, 2015 at 14:13
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(1) get more data to make sure it doesnt smooth out
(2) look for humps
(3) google "mixture distribution"
(4) Use log normal plot of cumulative weight vs size in phi units.
Unimodal give one straight line plot, bimodal two different straight line plots and the inflection point is the change of modal distributions.
It's not uncommon for mixing of traction and saltation loads or traction and suspension loads mixing as velocity of transport decreases
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3$\begingroup$ At present, this is a bit sparse for an answer here. Can you flesh this out a little? $\endgroup$ Apr 8, 2015 at 20:28
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2$\begingroup$ Could you define a "phi unit"? $\endgroup$ Apr 10, 2015 at 11:00