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I am in a weird position where I prespecified a plan to use linear regression to analyze my data, and stated I would use transformations to address any assumption violations. I'm pretty certain my data are not suitable for linear regression, nor will they be fixed with a transformation. I plan to use a more appropriate analysis, but I'd like to at least entertain the possibility that these data could be transformed OR that they are still acceptable for linear regression. Any ideas for transformations and/or opinions about whether or not it's acceptable to use linear regression with these data?

Edit: I know that survival analysis is more appropriate given that the data are right censored (the response variable is amount of time subjects waited before engaging in a certain behavior, and the experiment was ended if the subject waited 15 minutes). But what I'm asking is whether linear regression can be used on these data (ignoring the censoring issue)/if there is an appropriate transformation? Also, if censoring should not be ignored, why exactly (since it doesn't violate assumptions of linear regression)?

Edit2: Below are some diagnostic plots.

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  • $\begingroup$ Perhaps zero-inflated poisson regression? $\endgroup$
    – bdeonovic
    Commented May 13, 2016 at 1:51
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    $\begingroup$ There is no assumption for regression that suggests any of the original variables can't be bimodal. The assumptions don't relate at all to the marginal distribution of the DV. (Your plan relies on a mistaken idea. It would have been better to have asked some questions before you specified your plan.) ... What does your response variable consist of? Is it counts of something for example? Why is it bounded above by (what appears to be) a value near 16? $\endgroup$
    – Glen_b
    Commented May 13, 2016 at 2:14
  • $\begingroup$ The response variable is the amount of time participants waited before engaging in a specific action (or the max wait time, which was 15 minutes, at which point waiting was terminated). I believe survival analysis is most appropriate but what I'd like to know is whether it could be acceptable to use linear regression when the response variable is the way it is, or if there is a way to transform it so that it is more appropriate for linear regression. $\endgroup$
    – PanPsych
    Commented May 13, 2016 at 2:37
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    $\begingroup$ As @Glen_b rightly pointed, out, you shouldn't be focusing on the marginal distribution of the dependent variable as there are no assumptions made about that in linear regression. Instead, you should be examining the errors from your model. What do the residuals look like? They must be normally distributed in OLS -- not your dependent variable. Also, keep in mind that ordinary regression will not handle your censoring problem. That will likely need to be taken into account as you suspected. $\endgroup$
    – StatsStudent
    Commented May 13, 2016 at 4:35
  • $\begingroup$ @Glen_b perhaps the question is titled incorrectly, but the residuals clearly follow the same pattern as the dependent variable: they are bimodal. This does violate an assumption of regression. In this case, how does one handle residuals that are bimodal? I am stumped on the same issue. I have residuals that looks just like the residual histogram and Q-Q plot above. A better link function, maybe? A mixture model? $\endgroup$
    – Mark White
    Commented Nov 22, 2017 at 19:20

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I would argue that this is not a case of bimodal data, but rather right censoring. You're not interested in how long the experiment goes for (often terminated after 15 minutes), but rather time until action. Unfortunately, you don't also get to observe the time until event for every observation because of early termination. However, on these censored observations, you know that time until event is at least 15 minutes, which is somewhat informative.

This is well trodden territory in the field of survival analysis. Standard tools include Kaplan-Meier curves (for univariate fits) and Cox-PH (most common) or Accelerated Failure Models (probably an easier model to understand if you are unfamiliar with hazard rates, etc.)

EDIT: It was asked what, specifically, is wrong with using linear regression (without accounting for censoring) in this scenario. The answer is that your estimates will be biased (and in your case, this bias looks to be very serious). As an extreme example, suppose the population mean was greater than 15 (looking at your values, it looks like at least half of the recorded times are censoring 15, which implies that the mean is, in fact, likely to be greater than 15). Since all true response values greater than 15 enter your dataset as 15, you can't possibly estimate a mean of 15 (in fact the expected value of the mean will be much less than 15, even though that's the true mean value).

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  • $\begingroup$ I agree that survival analysis is the way to go, but am seeking more insight about why not addressing the censoring would make linear regression horribly inappropriate. The problem is that I preregistered the plan to run a lm and transform the dv if the data violated lm assumptions. Now I need to follow through or have a sound argument for abandoning the original plan. (Note, I will do the survival analysis either way, but it's a question of whether I report a lm at all.) Thanks! $\endgroup$
    – PanPsych
    Commented May 13, 2016 at 14:21
  • $\begingroup$ @user109053: updated the answer. $\endgroup$
    – Cliff AB
    Commented May 13, 2016 at 16:29
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Ignoring censoring in the data is a bad idea, especially because it's not a small amount of censoring. It looks like close to half the data is right- censored. To see why this might be bad, take a true linear regression $y_i=a+bx_i +e_i $ (assume $a,b>0$ for simplicity). Now suppose we trim all values $y_i $ above $15$ to $15$. What happens is for the large $y_i >15$ is that the corresponding large $x_i $ no longer sits on the straight line, and sits on a slope of roughly zero (not the "true slope" $b $). So when you fit the regression model using censored data the estimated slope will be smaller.

Having said all that, your set of predictors looks pretty limited - looks like there are only $3$ distinct combinations from the residual plot - so only $3$ fitted values are possible in any regression model. You could just run $3$ independent analysis within each group. Then you're just estimating univariate right censored data.

Hope this helps!

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  • $\begingroup$ Thanks! This is a very helpful explanation. Can you clarify what you mean in saying that running three independent analyses would estimate univariate right censored data? I don't follow how I would go about running this analysis. The three groups are three levels of my independent variable. $\endgroup$
    – PanPsych
    Commented May 13, 2016 at 14:26

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