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I am looking for a way to find the maximum in a quadratic regression. Specifically, I have two variables X and Y. Y is a discrete and commonly used scale representing the severity of a disease, ranging from 0 to 20 and X is a biological parameter. Since I expect the relationship between X and Y to take on an inverted u-shape, I want to find out at what level of disease severity X is greatest (e.g., let's say the maximum of X lies at Y = 13, then we can expect X to increase and then decrease before and after a disease severity of 13). I would also like to find out the predicted value of X and get a confidence interval for X. Since I have unbalanced multi-panel data, I am looking for an approach using the nlme or lme4 packages.

Any ideas?

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I found the "two-line test" (Simonsohn, 2016), which fits a segmented regression based on a breakpoint (i.e. the maximum in quadratic regression) identified with a particular algorithm. However, the procedure does not account for random variance.

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  • $\begingroup$ "the maximum of X lies at Y = 13, then we can expect X to increase and then decrease before and after a disease severity of 13" I think you've got X and Y the wrong way round. If not, then I don't understand your question. $\endgroup$
    – Nick Cox
    Commented Aug 9 at 15:57
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    $\begingroup$ A similar question Confidence interval for GLM or the maximum of a function? $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 10 at 1:22

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If you've fitted $$\hat y = b_0 + b_1 x + b_2 x^2$$ then the gradient of that quadratic curve vanishes when $$d\hat y / dx = b_1 + 2 b_2 x = 0$$ whence at that turning point $$x = -b_1 / 2 b_2,$$ which you can calculate simply from your parameter estimates.

Naturally you should always check that

a. the model is a good idea -- by plotting the data and the fitted curve together and judging the suitability of a quadratic

b. the turning point is indeed a minimum

and

c. the turning point occurs within the data range or at least is a plausible value of $x$.

It's unclear to me why or how having unbalanced multi-panel data complicates the question, unless

  1. Lack of balance and/or other heterogeneity makes a single model moot

  2. You really fitted a more complicated model, in which case there isn't likely to be a simple minimum but at best the minimum is a set of points on some more complicated prediction surface. In fact, your mention of those R packages is worrying insofar as it implies that you did that.

Detail: Statistically speaking $x$ is here a variable, and not a parameter, the last term being better reserved for the quantities $\beta_0, \beta_1, \beta_2$ of which $b_0, b_1, b_2$ are estimates.

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    $\begingroup$ To avoid the problems with estimating the standard error of $-\hat b_1/(2\hat b_2),$ I suggest analyzing the question of testing the contrast $2(x-x^*)\hat b_2 + \hat b_1$ where $x^*$ is the estimated peak location and $x$ is any other location. Intuitively, values of $x$ for which this test does not reject the null hypothesis are consistent with being a possible peak location, thereby producing fiducial limits for $x^*.$ This is sometimes known as inverse regression., q.v. $\endgroup$
    – whuber ♦
    Commented Aug 9 at 16:06
  • $\begingroup$ Thanks for your answer, @Nick Cox. Regarding the multi-panel nature, the dataset contains data from patients and some of them were tested over multiple years. I wanted to do something as an LMM to incorporate random effects (and possibly covariates). How could I statistically test the maximum of x? $\endgroup$
    – a.henrietty
    Commented Aug 9 at 16:27
  • $\begingroup$ also thanks for your comment, @whuber . $\endgroup$
    – a.henrietty
    Commented Aug 9 at 16:28
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    $\begingroup$ Your question about testing can't be answered well without an explicit mention of whatever model you fitted. Indeed what is X and what is Y still seems confused to me. $\endgroup$
    – Nick Cox
    Commented Aug 9 at 16:33
  • $\begingroup$ I am thinking of a model such as lmer(Y ~ X + I (X^2) + random(1 | patient)) , @NickCox. Y is expected to decrease linearly (disease severity will only get worse over time), while X, the biological variable, will increase as the disease worsens, but will decrease after a certain disease stage is passed. $\endgroup$
    – a.henrietty
    Commented Aug 9 at 16:47

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