In my opinion, you can run a binary logistic regression. For such a regression the IV can be either categorical/nominal or continuous.

In R: https://stats.idre.ucla.edu/r/dae/logit-regression/

SPSS: https://stats.idre.ucla.edu/spss/output/logistic-regression/

In my opinion, you can run a binary logistic regression. For such a regression the IV can be either categorical/nominal or continuous.

In R: https://stats.idre.ucla.edu/r/dae/logit-regression/

SPSS: https://stats.idre.ucla.edu/spss/output/logistic-regression/

plus to Jos' answer:

https://towardsdatascience.com/the-binomial-regression-model-everything-you-need-to-know-5216f1a483d3

Hello Anders Dahl. I agree that binary logistic regression would be a good model to use in your situation. However, simulation studies have suggested that in order to avoid over-fitting a logit model, you should have 10 to 20 "events" per explanatory variable. As I understand it, a categorical explanatory variable with k categories must be counted as k-1 (indicator) variables when applying that rule of thumb. What are the frequencies for your two outcome categories? The number of events is defined as the lower of those two frequencies.

Here are some references regarding the rule-of-thumb I mentioned.

• https://people.duke.edu/~mababyak/papers/babyakregression.pdf
• https://discourse.datamethods.org/t/author-checklist/3407

The first item is an article by Babyak (2004). Based on simulation studies that were available when he published, he recommended 10-15 EPV for binary logit models. The second item has textbook author Frank Harrell's "Author Checklist". He advises 20 EPV-do a Ctrl-F search on to find what he says.

HTH.

Thank you very much for the answers @Jos Feys, @Jochen Wilhelm and @Bruce Weaver.

The binary logistic regression will only work if I choose a reference level (variable) right?

The two linked examples from Jos (the SPSS one) and Jochen (the Titanic data) both use reference values of the nominal variable. That is the key part of my question - "is it possible to do this analysis without using a reference value of the nominal variable?".

If I do not use a reference level then what will the odds ratios be in relation to?

Thank you Bruce Weaver for mentioning the challenge with number of events.

I might very well have a problem here (as seen below I would just make it above the lower limit to K-1 = 9 multiplied by 10 = 90)

An example of my data (smoke yes / no)

Yellow: 41 yes and 25 no

Green: 41 yes and 9 no

Orange: 27 yes and 10 no

Blue: 18 yes and 14 no

Red: 15 yes and 12 no

Purple: 20 yes and 6 no

Black: 9 yes and 9 no

White: 13 yes and 4 no

Brown: 7 yes and 5 no

Pink: 7 yes and 1 no

This would leave me with 95 no`s and 198 yes in a total sample of 293 people.

Is a binary log regression really the best option here? (if yes can I perform it with 10 dummy variables and no reference?)

Or is data simply to scarce to perform statistics?

Thanks again for your time

Anders

Anders Dahl The reference variable is chosen for you in SPSS. In any other package I've used e,g. SAS, R, BMDP You Choose the reference variable by coding it as all zeros (i.e. it correspond to the constant term in the model). If you don't have it yet Try R. Here is a great intro for all you need to know about R (it's free) and here's a book to get you started, it has great code for logistic regression:

R for everyone: advanced analytics and graphics | Lander, Jared P | download (b-ok.cc) Best wishes, David Booth

The SPSS LOGISTIC REGRESSION command actually allows you to choose whichever category you wish as the reference category. Copy the syntax below, paste it into a new syntax editor window, and run all. Then study the results. HTH.

* --- Start of syntax --- .

NEW FILE.

DATASET CLOSE ALL.

DATA LIST LIST / Colour Yes n (3F5.0).

BEGIN DATA

1 1 41

1 0 25

2 1 41

2 0 9

3 1 27

3 0 10

4 1 18

4 0 14

5 1 15

5 0 12

6 1 20

6 0 6

7 1 9

7 0 9

8 1 13

8 0 4

9 1 7

9 0 5

10 1 7

10 0 1

END DATA.

VALUE LABELS colour

1 "Yellow"

2 "Green"

3 "Orange"

4 "Blue"

5 "Red"

6 "Purple"

7 "Black"

8 "White"

9 "Brown"

10 "Pink"

.

WEIGHT by n.

* When using the LOGISTIC REGRESSION command,

* use Indicator(#) to select the category number you want

* to be the reference category for odds ratios.

* Use 1st colour as the reference category.

LOGISTIC REGRESSION VARIABLES yes

/METHOD=ENTER colour

/CONTRAST (colour)=Indicator(1)

/PRINT=CI(95)

/CRITERIA=PIN(0.05) POUT(0.10) ITERATE(20) CUT(0.5).

* Use 2nd colour as the reference category.

LOGISTIC REGRESSION VARIABLES yes

/METHOD=ENTER colour

/CONTRAST (colour)=Indicator(2)

/PRINT=CI(95)

/CRITERIA=PIN(0.05) POUT(0.10) ITERATE(20) CUT(0.5).

* Use 10th colour as the reference category.

LOGISTIC REGRESSION VARIABLES yes

/METHOD=ENTER colour

/CONTRAST (colour)=Indicator(10)

/PRINT=CI(95)

/CRITERIA=PIN(0.05) POUT(0.10) ITERATE(20) CUT(0.5).

* Look at the Categorical Variables Codings table to verify

* which category is serving as the referent--it will be the row

* in that table with all codes = 0.

* You can also estimate the model via the GENLIN command.

* One advantage is that it can generate a table of esimated marginal means.

* If you choose the SCALE=TRANSFORMED option, the EM Means show

* the fitted log-odds.

* The following command will generate the same model as the last

* LOGISTIC REGRESSION command above, with Indicator(10).

* Generalized Linear Models.

GENLIN yes (REFERENCE=FIRST) BY colour (ORDER=ASCENDING)

/MODEL colour INTERCEPT=YES

DISTRIBUTION=BINOMIAL LINK=LOGIT

/CRITERIA METHOD=FISHER(1) SCALE=1 COVB=MODEL MAXITERATIONS=100 MAXSTEPHALVING=5

PCONVERGE=1E-006(ABSOLUTE) SINGULAR=1E-012 ANALYSISTYPE=3(WALD) CILEVEL=95 CITYPE=WALD

LIKELIHOOD=FULL

/EMMEANS TABLES=colour SCALE=TRANSFORMED

/MISSING CLASSMISSING=EXCLUDE

/PRINT CPS DESCRIPTIVES MODELINFO FIT SUMMARY SOLUTION (EXPONENTIATED).

* Compare the parameter estimates for the last two models.

* Notice that the EM Mean for category 10, the reference category,

* is equal to the Intercept for that model.

* --- End of syntax --- .

Anders Dahl , your colours example could be seen as resolutely nominal, is there no ordinality allowing you to group from a theoretical perspective?

I'll offer an alternative to Kelvyn Jones , Bruce Weaver , David Eugene Booth , Jos Feys , and Jochen Wilhelm . All of them seem pretty set on color being nominal, and that it is not possible (or practical) to think of metrics for them, so they are all put forward solutions counter to how artists, physicists, etc., think of color.

Suppose there is a relationship among the colors. Could represent these in polar coordinates, which would make the stats a little trickier, or as rgb format (or whatever). Would need to no more about what you are interested in evaluating and how large your data set is to see how practical this would be. It depends on your theories about why color liking relates to smoking. Worth adding that it is probably only worth checking hypotheses you have already written are worth checking. See Gelman's Garden of Forked Paths paper for using color to predict things.

http://www.stat.columbia.edu/~gelman/research/unpublished/p_hacking.pdf

I'd like to add that Fisher s exact text does not use a chi square so the two will give you different results

I agree with the logistic regression approach but I understand what you are looking for I would suggest computing the risk ratios or odds ratios for each color rather than focusing on testing, what you are really after is the relationship rather than a P-value

@anders: you also say that you would like an outcome such as

People with favorite color yellow smoke significantly more than others

Please remember that estimates are what you are really looking for, with a large enough sample size all pvalues from the tests will be arbitrarily small, some journals will also reject any publication that uses "significant" in the main text.

Daniel Wright AS an old time physical chemist and often a spectroscopy user and emeritus member of the American Chemical Society Color is not a continuous variable but rather the visible spectrum is bounded as is the infrared and the ultraviolet thus IMO can only be treated as a bounded variable . Now it is well known that different organisms have different wavelength sensitivities As do individual humans. Given this how do you decide appropriate bounds and find equipment to do the experiments. how would you experimentally determine what I was seeing when I say it looks green to me. Given the current state of the art I believe these can be measured in organisms such as cats because we have wavelength information on feline vision but my cat prefers to just say I see something in the UV. Is it worth it a t this stage to go to the wavelength level given cost, informed consent, vision problems that can be caused outside our usual ROYGBIV visible spectrum?. At this stage I think i would not go there. If the subject says my favorite color is green, I'd go with that for this study but probably not if the issue was development of say, eye tumors. with tongue firmly in cheek except for the last sentence I would stick with ROYGBIV for this study. Best wishes, David Booth

David Eugene Booth , I said it depends on what theories the questioner has which approaches are appropriate, but I guess you are arguing otherwise. I don't believe we have enough information to give an answer yet on this question. Since the question is about favorite color the research probably has more to do with social psychology than chemistry, but I'll take your word for it that those ten colors listed are the appropriate ones to discuss for any research question.

Your question related to the qualia of the green experience is relevant how the questionner measured favorite color (and the same issues apply for most survey questions).

As I said, I was just presenting an alternative. I don't have enough information to be so certain to say absolutely that one approach is appropriate.

I agree with Daniel, in principle. The variable to be operationalized in the original question certainly was the subjective quality of light perception. The problem here is that this may not be possible to code that as one single number; in the worst case the response variable would be an entire intensity spectrum over the wavelengths from about 400 to 800 nm. This seems to make it unattractive or next to impossible to handle this as a one-dimensional numerical quantity. Having said this, the categorization into practically useful (this is where phychology enters the topic) color classes is a reasonable way to go, especially because here a "reference value" for the color impression can be represented by a specific intensity spectrum that can be communicated independent of any subjective perception (to get a well-defined operationalization).

I cannot judge if the selection of colors is reasonable, sufficient of exhaustive. This has to be set by expert judgement of phychologists. And this may be subject to change as we learn more about how our brain works with the input signals.

Wherever physical characteristics of the radiation is relevant, it's usually possible, practical and useful to select a reference wavelength that actually reresents more an energy level than a "color" (we use color here as a synonym for the photon energy).

In attachment the R script for your example. In includes the chi-squared test plus post hoc tests, and the logistic regression.

The chi-squared test is not sign. p = .06; hence, of course, no sign. post hoc diffenerence.

The logistic regression reveals that the colour Green as a sign. coefficient.

Thanks a lot all for the really great answers - amazing to receive assistance in this way.

Thank you for the comprehensive answer Bruce Weaver and Jos Feys

I ran the syntax and it leads me back to my main concern that as far as I see it remains unanswered:

"Is it possible to do this WITHOUT defining a reference level (colour)?"

I understand that I can switch my reference however I want in the logistic regression but as I see my data I am unable to define a relevant reference value (despite the interesting colour discussion - thank you Daniel Wright David Eugene Booth and Jochen Wilhelm

To answer Kelvyn Jones I do not think that I can define an order in my variable - unfortunately. I made the colour and smoking (yes no) example to simplify the words - what I am in fact evaluating is blood infections with different bacterial species and various dichotomous outcomes (all yes/no). My numbers in the above colour example are representative of my data for one of the dichotomous outcomes (could be e.g. "abscess development" yes vs. no).

In other words colours represent different bacteria and smoking is an example of a dependent variable.

What I have been wondering is if the following approach is too biased:

Compare each level of the nominal variable against all the others - that would be "Yellow yes vs. yellow no" - then I would get an OR with 95% CI

Next "Green yes vs. green no" - etc. to get OR for each colour (bacterial species) for any given dependent variable.

In this way I do not need to define a reference but I am unsure as to whether this is too opportunistic a method to use?

Stefano Nembrini is the above the solution that you are suggesting?

"Computing OR for each level (variable)?

Good point with the sample size and P-value. In my case the data is so sparse that my problem will most likely be the wide 95%CIs and p-values well above 0.05.

Once again thank you all for taking your time to assist me.

Best regards

Anders

Anders Dahl You must have a reference variable because that specifies the intercept. However you can choose it to be whatever color you like. You just must have a base to compare the other regression coefficients with. Best, D. Booth

You can also specify a model without intercept. The coefficients in such a model will then represent expected values for each group. But usually you fit the model to get some kind of effects (e.g. the difference to a reference group; other contrasts can be defined via the coding system, see link).

https://stats.idre.ucla.edu/spss/faq/coding-systems-for-categorical-variables-in-regression-analysis-2/

https://stats.idre.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/

https://marissabarlaz.github.io/portfolio/contrastcoding/

Now that you explained better i would definitely compute risk ratios: odds ratios are used to estimate risk ratios in practice

But risk ratios are more straightforward

Also I would personally use maximum likelihood to estimated it or a bayesian or fiducial interval, these last two can be interpreted in terms of probability distributions, while confidence intervals cannot

Hi Anders Dahl. As Jochen Wilhelm said, yes, you can fit a model without an intercept, but it is very rarely done. Here's how to do it using the LOGISTIC REGRESSION command. Notice the addition of the /ORIGIN sub-command--that is what instructs SPSS to omit the intercept.

LOGISTIC REGRESSION VARIABLES Yes

/METHOD=ENTER Colour

/CONTRAST (Colour)=Indicator

/ORIGIN

/PRINT=CI(95)

/CRITERIA=PIN(0.05) POUT(0.10) ITERATE(20) CUT(0.5).

Thank you David Eugene Booth , Jochen Wilhelm and Bruce Weaver

Following your additional comments I think I have reached the conclusion that logistic regression cannot solve my problem as long as I am unable to choose a relevant reference level.

I ran the syntax without an intercept Bruce Weaver - but the results I got was then what I interpret as "odds" and not "odds ratios" since they are now not compared to a reference level. As a matter of fact entering the syntax without intercept I still only got exp(B) of the 9 first colours and these values were simple odds e.g. 1.4 for colour 9 brown (7 yes / 5 no) and 1.0 for colour 7 black (9 yes / 9 no) etc.

These odds doesn't really serve for much (just as several of you have mentioned).

My problem is that I withhold that in my case it does not make any sense to choose a fixed reference bacteria.

In the above process I got an idea (I have no idea if it is a statistically no go):

Use the whole population as a reference level in a logistic regression.

So include a base variable with 198 yes and 95 no (odds 2.08).

And then compare all the different colours/bacteria to this base level by conducting a log reg with all of the above variables (on top of the base variable).

My data would look like this on e.g. abscess Y/N:

All patients: 198 yes and 95 no (use as reference)

Bacteria 1: 41 yes and 25 no

Bacteria 2: 41 yes and 9 no

Bacteria 3: 27 yes and 10 no

Bacteria 4: 18 yes and 14 no

Bacteria 5: 15 yes and 12 no

Bacteria 6: 20 yes and 6 no

Bacteria 7: 9 yes and 9 no

Bacteria 8: 13 yes and 4 no

Bacteria 9: 7 yes and 5 no

Bacteria 10: 7 yes and 1 no

This would give me an OR with 95% CI for each bacteria (comparing to the whole cohort with "self-inclusion".

Would that be a way to go Bruce Weaver , David Eugene Booth , Jochen Wilhelm Kelvyn Jones ?

And Stefano Nembrini thanks for continuing the explanation of your suggestion with computing risk ratios on the direct chat.

Best regards

Anders

Anders Dahl, the EMMEANS sub-command for GENLIN allows various kinds of contrasts, including what the manual calls a Deviation contrast:

DEVIATION (value). Each level of the factor is compared to the grand mean. Deviation contrasts are not orthogonal.

Source: https://www.ibm.com/support/knowledgecenter/en/SSLVMB_26.0.0/statistics_reference_project_ddita/spss/advanced/syn_genlin_emmeans.html

Are you after something like that?

If your question is more about how to get SPSS to produce some output you envision, then maybe the SPSSX-L discussion forum would be a better place to post. The Nabble archive is a convenient way to interact with that forum:

http://spssx-discussion.1045642.n5.nabble.com/

Click near the top of the page for info about how to join the underlying mailing list and set your options.

Your n is small for treating "colors" as nominal, but my big question is what is the reference to bacteria? Is type of bacteria your variable?

Thanks once again Bruce Weaver - I will try to look into the deviation contrast.

I am not fixed on neither specific output or statistical program (prefer R or SPSS).

Daniel Wright I used the colour example with the intention to describe my nominal variable in easier words than "streptococcus anginosus" vs. "streptococcus gallolyticus" vs. "streptococcus agalactiae" etc. So yeah - the bacterial species is my nominal variable.

As you mention I have a problem with a total N of only around 300.

My hope was that I could lift my analysis above the simple description of frequencies in the different species.

In other words, right now I can describe that e.g. cardiac abscess occurs in:

15% of bacteria A infections

4% of bacteria B infections

25% of bacteria C infections

and 10, 11 and 13% of bacteria D,E,F respectively.

These numbers are fictive yet comparable to my actual data. I would like to be able to say (with a statistical test) if bacteria C cause more abscesses (and bacteria B less) than "the others" without defining a single bacteria as reference.

So far the only strait forward solution I see is to create RR or OR for each bacteria vs. all the others and then continue with the next bacteria compared to all the others. But here I am in doubt if this "allowed" statistically?

Thanks

Anders

The reference level is irrelevant - use the logistic regression to test the overall significance first, then consider pairwise comparisons if the overall test rejects.

BUT something doesn't seem right here. Are all ten colors (bacteria?) independent groups, or could more than one be present in each subject? If you are studying bacterial abscesses in patients, then it's almost certainly the later, and you want a logistics regression with dummy variables to indicate the presence of each bacteria type.

You may also have correlation between bacteria types, which will complicate the analysis. You can run a linear regression to generate the VIF statistic to look at this. VIF doesn't use the dependent variable, but you can you if to calculate VIF and ignore everything else about that regression.

I added the Row and Column percentages and the Masaic plot of the residuals.

I agree with Dan Eastwood . The presence of bacteria is unlikely to be exclusive, and there are are surely interactions. Since this is obviously an observational study (you did not perpuselully infect the people, did you?) you must control for confounders (sex, age, comorbidities, other possibly relevant variables).

Btw: it was a bad idea to use some made-up example ("colors") and not to clearly state the what the experimental design and the relevant variables are. Everything depends on this, and giving "wrong" information or leaving information aside will only produce a lot of fuss, largely waisting our times.

Thanks for your input Dan Eastwood and Jochen Wilhelm - your assumption is not correct - the type of bacteria is exclusive - if it wasn't I wouldn't have posted the question like I did.

Monomicrobial blood infections - no patient has more than one type of bacteria in the blood - therefore I thought the colour example would work fine (I get your point that it might end up being more confusing Jochen Wilhelm ).

I am not sure I understand your suggestion Dan Eastwood - do you suggest that for each dichotomous dependent variable I perform a log reg with a reference (e.g. bacteria type A) and then if the overall model is significant then I state that the values differ and if not I perform additional intergroup tests ad hoc?

Of course you are right about confounders Jochen Wilhelm - intention was to perform descriptive statistics describing the differences and adding a statistical comparison. Is your point that with my data it will have no real value to perform statistics and that I should instead stick to just describing the data?

Jos Feys and Stefano Nembrini thanks for your direct messages as well - I will see if I am able to understand the more complicated models suggested and respond.

Thanks for all the assistance

Yes, I would keep it descriptive. Any conclusion beyondyour particular sample can be very wrong, and there is no way to how wrong it can possibly or likely be.

I think it is important to do more than a descriptive analysis as the average for each ‘colour’ is based on a different number of observations, the trials as it were, so that there is potentially differential uncertainty. I am not trying to generalise to some hypothetical population, but to appreciate uncertainty in what data has been collected. Blimey, having written that, I see I am heading towards bayes!

So for a bit of fun in a wet lockdown, I tried to impose as little as possible on the analysis. I used a binomial model with observed proportion of ‘Yes’ as the dependent variable and the number of trials (Yes + No) as the denominator. I used a model without a constant and with a fixed effect for each and very colour, so each logit is separately estimated. I used MLwiN’s MCMC estimation with a Burn- in of 500 following quasi-likelihood estimation, and then a monitoring chain of 50,000. The chains looked like healthy white noise. I used MCMC to allow for potentially asymmetric credible intervals on the logit scale. I then transformed the estimates back to proportions for ease of interpretation and made the plot; see attached.

My reading of the graph shows that there is considerable uncertainty around the estimates; for me the evidence is not strong enough to tell the ‘colours’ apart in terms of the outcome. There would be more power if the colours could be arranged ordinally, but apparently no such luck. Anyway that would be cheating......