The lambda should be in there somewhere. How to get it out of SPSS, I don't know.

If you can't get it out of SPSS, it might be easy enough to do the Box-Cox transformation in R, import the new variable in SPSS, and redo the analysis with the knowledge of the lambda used. The transformation may not be exactly the same between the packages, as I think the procedure --- in R at least --- just tries a bunch of different values for lambda to see which works the best.

Another idea: In theory, with what you have, even if you just examine one value, un-transformed and transformed, you should be able to solve for lambda. Maybe just try lambda values iteratively until you hone in on the right one.

Be aware that some people are really against the idea of back-transforming estimated parameters. It's common. But just be aware you might meet some criticism.

Hello Svetlana Bondar. How (and why) did you apply the Box-Cox transformation? I have never used it myself, but I see that relevant dialogs can be found under Transform > Prepare Data for Modeling. At the next level, the 3 options are:

• Automatic
• Interactive
• Backtransform Scores

I wonder if that 3rd option is what you are looking for? If that does not solve things for you, I suggest that you post to SPSSX-L where people like Jon Peck may be able to offer some help. I mention him because he is a very knowledgeable former IBM-SPSS employee, and because he has posted a relevant comment on this tutorial by Ruben Geert van den Berg:

https://www.spss-tutorials.com/normalizing-variable-transformations/comment-page-1/

Back to my parenthetical question above: Why did you feel it necessary to apply a Box-Cox transformation? If you are estimating some type of general linear model (e.g., OLS regression), then there is no requirement that the Y-variable be normally distributed. The sufficient normality condition is that the errors are (approximately) normally distributed; and the necessary normality condition is that the sampling distributions of the parameter estimates (i.e., the coefficients) are approximately normal. I am always curious, therefore, about why people transform to achieve (a better approximation) to normality. Thanks for clarifying.

Hello Svetlana Bondar If you're just looking to get and find lambdas, this is easy.

Based on the lambda calculation formula (I have given in the attached image) from W (transformed response data) and also from Y (actual response data, before transformation), take the natural logarithm.

Then calculate this formula.

Lambda = Ln (W) / Ln (Y)

Lambda is located there.

Bruce Weaver I did this to satisfy the assumption for multiple linear regression that the distribution of residuals should be normal (Oxford handbook of medical statistics, Peacock).

The residuals of the dependent variable in my case are not normally distributed – a bimodal distribution. I tried to apply log and square root transformation, but it did not help – the distribution did not become normal.

Then I read that I can try the Box-Cox transformation, but there is almost no information anywhere on how to do this in SPSS. I applied the method indicated here https://www.youtube.com/watch?v=lqEzupKtIDI - the distribution became normal, but it seems that this is not a Box-Cox transformation. And I don’t know how to do a back transformation from this case. Sal Mangiafico it turns out I can use this transformation and not do the back transformation?

I also applied your option (Transform > Prepare Data for Modeling > Automatic > Settings > Rescale Fields > Rescale a continuous target with Box-Cox transformation to reduse skew), but this did not make the distribution of residuals normal. So, the data obtained in this way did not coincide with the data from the YouTube video.

From what I read, it seemed to me that it necessary to perform a back transformation to correctly represent the data. But maybe that's not true.

You CAN not do a back transformation. The problem with this is that the parameter estimates (e.g. means) and e.g. confidence intervals are on some scale that no one would understand. Like if lambda were 0.2, and the original value were 10, then the transformed value is 3. Would any reader know what that 3 values means ?

I watched that video. I don't think that's Box-Cox transformation. I'm guessing it's inverse normal scores transformation. Because it involves taking ranks, I don't think you can back-transform it. The ranking step loses information.

Sal Mangiafico thank you a lot. Then I will have to use this type of transformation, because it is the only one that made the distribution normal.

What is the dependent variable, Svetlana Bondar? I ask because I wonder if some other type of model could be used--e.g., some kind of GzLM. Thanks for clarifying.

I can't comment on medical journal style guidelines.

Bruce Weaver the dependent variable is VCAM-1 (ng/mL), and the predictors are gender (I used dummy variables), age, and GFR (mL/min/1.73 m2).

Thanks for the additional info, Svetlana Bondar. Just to remind everyone, earlier, you wrote:

"The residuals of the dependent variable in my case are not normally distributed – a bimodal distribution."

I don't remember ever running into this problem myself, but a bit of digging around suggests that one possible cause of this is failure to include an important dichotomous explanatory variable. Do any possibilities come to mind?

Another suggestion I've seen that might be helpful is to use quantile regression rather than OLS regression, and to perhaps model the 25th and 75th percentiles instead of (or in addition to) the median.

Finally, I found this article, which seems to be addressing the same problem. Maybe you'll find some good suggestions in it.

https://www.redalyc.org/pdf/899/89949526005.pdf

HTH.

Thank you Bruce Weaver for clarification, I will try the option you suggested. I've checked the possible options for why this might be happening, but I'll give more thought to what might be driving the binominality of the distribution. Unfortunately, I can't download the pdf file from the link you provided, it doesn't open.

I agree with Bruce Weaver , a bimodal residual distribution is very unusual. Instead of blindly applying a (Box-Cox) transformation, you should think about the cause for the bimodality. I am not sure and someone may correct me here, but Box-Cox is not suitable for bimodal distributions to normalize them, is it?

Bruce's suggestion about a missing dichotomous predictor is very reasonable. Imagine you want to predict the height of people in a population. The distribution of height will be somewhat bimodal and skewed. If you predict height with age alone, th skew might diasappear, but the residuals will still be bimodal. If you add biological sex as predictor, the residuals will be fine, since the model implies the mean height of male and female, which in turn caused the bimodality.

Maybe this makes it more clear.

P.S.: the height example does not match reality, here the overall distribution would look unimodal, since the mean values of men and women are (relatively) close together.

I currently have an MS where clusters appear in the data, I cannot find what might be the cause. An option would be to cluster the residual groups and account for it with a 'random effect', as Bruce seems to hint. Then at least the bimodal might be averted. This can drastically change the outcome (e.g., https://en.wikipedia.org/wiki/Simpson%27s_paradox). However, this artificial clustering beats the purpose of inferential statistics, as we would like to learn something about the process that generated your data. Clustering it artificially solves the bimodal issue, but you have no idea what caused it (thus learn). Another, less useful option would be to discard the inferentials except the point estimate. The point estimate is ofcourse still biased, but since people do not interpret this in general (as significant or effect) it might still be considered "novel", it is not the worst decision imho. Moreover, its still useful reference work for priors, sample size estimation, storyline and comparison if the bias is not too severe. Not all data observe is useable for all inferential or causal inference questions, but all data is valuable in its own respect.

Rainer Duesing , I think you're right that Box-Cox wouldn't help make a bimodal distribution normal. But it appears that the transformation OP did was actually a form of inverse normal scores transformation (INT). (With the video, and my unfamiliarity with SPSS, I can't tell exactly what is being done in the video, but I'm pretty sure it's a form of INT.)

I think you are right, Sal Mangiafico. Here is documentation on the IDF.NORMAL function that is used in the video:

IDF.NORMAL. IDF.NORMAL(prob, mean, stddev). Numeric. Returns the value from the normal distribution, with specified mean and standard deviation, for which the cumulative probability is prob.

Source: https://www.ibm.com/docs/en/spss-statistics/29.0.0?topic=functions-inverse-distribution

Svetlana Bondar , hopefully you've found an acceptable approach to your problem. I have just a few potential suggestions.

1. Change the model to include the variable that is likely causing that bimodal distribution of residuals.

2. Use an alternate model or estimating procedure. Bruce Weaver mentioned quantile regression. For the model you have, I imagine there would also be non-parametric models that could be fit. Or other estimating procedures like by a standard linear regression model by permutation test. Quantile regression is probably a good option, and it might be worth exploring if the others are available and appropriate.

3. Keep the normal scores transformation you have, and use that for p-values in the model. If the transformed variable is standard normal (mean = 0, sd = 1), it might be fine to present everything according to the transformed variable. The results might be interpretable by readers. But also, you might present estimates for parameters on the original data. That is, you might use the p-value for Gender from the normal scores transformed model, but you present the means for each Gender just on the raw data.

Hello there, me either want to know how to back tranformation IDF and Inversion data tranformation