# Understanding Maximum Likelihood Estimation (MLE) and its confidence intervals

I'm trying to figure out if I am actually understanding MLE correctly, or at least applying it correctly to my data. My data consists of several patients for which I have some data, which is used in the following equations.

My problem is to find the best fit parameters (three) of a model through MLE for my data. And from what I can read in literature, for my type of example, the log-likelihood function is this:

$$LLH (D_{50}, m, n) = \sum\limits_{y(i)=1}\log(NTCP(D_{50}, m, n)) + \sum\limits_{y(i)=0}\log(1- NTCP(D_{50}, m, n))$$

Where ´NTCPandt is given by:

$$NTCP = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{t} e^{-\frac{x^2}{2}} dx$$

$$t = \frac{gEUD - D_{50}}{m\times D_{50}}$$

$D_{50}$ can be values from 0 to in principle infinite. However, it is most likely that the optimal parameter would be in the range of 0-150. $m$ is probably in the range of 0-3, and $gEUD$ is calculated from the equation below.

$$gEUD = \left(\sum\limits_{i} v_i d_{i}^{\frac{1}{n}}\right)^{n}$$ As stated, I have several patients, and for each patient I have a two column list with values that is used for the $v_i$ and $d_i$ in the above equation to get the $gEUD$. Each patient is also associated with a value of 1 or 0, which tells me if this patient has had any toxicity from treatment (1 = yes, 0 = no). In turn, this is used in the $LLH$ function to divide patients into either one of the sums or the other.

So what I have done so far is to calculate the NTCP (as seen in the equation, which happens to be a sigmoid function given by scipy.stats.norm(0, 1).cdf(t) in python, where t has all the parameters) for all patients. I then create a grid of parameters (D50, m, and n), let's say a (100, 100, 100) grid, which means 1 million calculated NTCP values for each patient (different parameters lead to different NTCPs). So in my case, the NTCP can take values such as 1 or 0 (0% to 100% probability of getting toxicity), which means that in the cases where I have log(0) I will have infinity. So in my python code I check for 0's or 1's in all 1 million calculations for each patient, and combine these indexes (of 0's and 1's) and remove it from the parameter grid for all patients. So in turn I am losing some parameter sets since they won't be calculated otherwise. After that I take index (0, 0, 0) for all patients, and divide them accordingly to the LLH equation (y(i) = 1 if they have toxicity, or y(0) = 0 if they don't), and then get a LLH for that one parameter set. And this is obviously done for the entire grid for all patients, and I end up with approximately 1 million LLH values (minus the ones that has been removed due to the possible log(0)). I then find the maximum value, and that is supposed to be my MLE, right ?

I have no idea if there is an built-in python function/package that can do this, but nonetheless, I have written this my self. Obviously it takes some time calculating 1 million vales for x-number of patients, but I guess that is how it is.

My problem is, however, that I have data where the least square method has been used to find the best fitted parameters, and if I do this MLE method on the same data, I don't quite get the same result. Should that worry me, or...?

Also, for finding confidence intervals I have read that profile log likelihood should be the way to go, but also bootstrapping. What would be recommended in my case if I actually end up with a list of approximately 1 million LLH values ?

• I am confused: what exactly are the parameters that you want to maximize over? So far what you have written is a fixed expression with a bogous 't' in there. Could you provide a reference to the description of the model in the literature? Also: What exactly do you mean by 't contains all the parameters'? I am especially puzzled about the fact that you have a cdf in there and not a density (because if this indeed is log likelihood then exp(your expression) should be a density!). – Fabian Werner Aug 23 '18 at 9:51
• Sorry for the confusion. I have updated my OP with another figure of the equations used for calculating t in the error function. So basically D50 and m are just the values they are for each calculation, whereas n is used to calculate another thing (EUD) that is going to be different for each patient. So when putting all parameters (one set at a time) to use you can then calculate the NTCP (i.e. the expression with the bogus t`). This NTCP is then used in the LLH equation, again for all parameter sets in the grid, where I then take the log of each individual NTCP and sum through the patients – Denver Dang Aug 23 '18 at 10:53
• So summarized: $D_50, m, n$ are the parameters and you want to find $D_50, m, n$ such that the expression above becomes maximal, yes? Are $D_50, m, n$ supposed to come from a certain range or are these just real numbers? In the latter case you could try to do that in an analytical exact manner by exlicitly finding a point where the gradient becomes zero. That would be the optimal solution. It seems involved at first but you probably need to compute the gradient in any case for applying numeric methods (gradient ascend) if that first approach does not work. But I guess that m, n have to be... – Fabian Werner Aug 23 '18 at 11:41
• integers, is that right? – Fabian Werner Aug 23 '18 at 11:41
• Not integers no, unfortunately. The range of D50 is around 0-150 (in reality I would like in steps of 0.1), n can take values from 0 to 1, and m is also in the small range from 0-2 or 3. – Denver Dang Aug 23 '18 at 11:47

### Probit model

What you are describing

$\log \mathcal{L}(D_{50},m,n|y_i) = \sum\limits_{y(i)=1}\log(P(D_{50}, m, n)) + \sum\limits_{y(i)=0}\log(1- P(D_{50}, m, n))$

is a probit model.

Note that

$$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^t e^{-\frac{x^2}{2}}dx = \Phi(t)$$

is the cumulative distribution function of standard normal distribution.

Your loglikelihood model corresponds to a Bernouilli distribution for each individual $Y_i$:

$$P(Y_i=1|t_i) = 1 - P(Y_i=1|t_i) = \Phi(t_i)$$

### Faster estimation method by solving linear part seperately

Note that if you keep $n$ fixed then the parameter $t$ is basically linear:

$$t_i = \frac{x_i - D_{50}}{mD_{50}} = ax_i + b$$

where $x_i = gEUD_i(n)$, $a = \frac{1}{mD_{50}}$, $b = -\frac{1}{m}$.

Now you can put your model in a very standard (GLM) form for which many solution methods (and standard software) exists:

$$P(Y_i=1|t_i) = 1 - P(Y_i=1|t_i) = \Phi(ax_i + b)$$

and

$$f(y_i|x_i) = y_i \Phi(ax_i + b) + (1-y_i) (1-\Phi(ax_i + b))$$

then find which $n$, $a$ and $b$ maximize the likelihood function and you can compute your estimates for $m$ and $D_{50}$ from the estimates for $a$ and $b$.

This (only a grid search for $n$) is a faster solution method for the MLE than solving on a 100x100x100 grid.

• I then find the maximum value, and that is supposed to be my MLE, right ? Your grid search is valid (if you keep in mind that you check whether the maximum is not on the edge of the grid) but not very fast and scalable.

Possibly you could use a non-linear optimization method to solve the model without a grid search. I do not know about such methods that are specialized for probit models.

• I have data where the least square method has been used to find the best fitted parameters, and if I do this MLE method on the same data, I don't quite get the same result.

It is different to say exactly what is wrong when not knowing the details about the data and the least squares method. But it is very reasonable to expect that the MLE method which you use is gonna be different from least squares. With a probit model you assume that the data is distributed according to a Bernouilli distribution with a least squares model you assume that the data is distributed according to a Gaussian distribution, they are different things and particularly the Gaussian distribution (which does not model discrete values 0 or 1) is not very logical/correct to use and likely deviates from your results with a probit model.

• for finding confidence intervals ... profile log likelihood ... but also bootstrapping. What would be recommended

Both methods are not gonna give an exact solution and this will be a matter about how you prefer to choose between pro's and con's

The profile likelihood is approximately correct near the MLE and is gonna do better when the interval is not too far away from the center, as well when the interaction between the parameters is not to much. You can verify this on plots of the profiles.

The bootstrapping is neither correct but you can make it as close to the correct CI as you wish given enough computation and time (so the question here is how much time and computer power do you have)

• Thank you for that answer. I haven't thought about just reducing it to an 1D grid (n) instead of a 3D 100x100x100. I will try to see if I can incorporate that into my code, because it is indeed a time consuming process now, especially when doing it on a local computer. I do however have two questions if I may? 1) In my current grid search there are indeed probabilities that give 1 or 0 for some parameter sets. Is it correctly to assume, that in the case of the log() returning inf I should discard that parameter set all together, or...? 2) If bootstrapping and profile does not yield exact – Denver Dang Aug 23 '18 at 14:26
• solutions, is there another way of doing it, or is that just how it is ? That you get an interval that is justifiable, and you have to work with that. – Denver Dang Aug 23 '18 at 14:27
• I imagine that you get those 0's (bad predictions) because you are far away from the optimal MLE and the likelihood goes very much down to 0. You could verify this with a plot where you mark those occurrences of zero. The 1's may occur if your $y_i=1$ and $y_i=0$ cases are very much separated. Then your model will fit the parameters such that all the probabilities for those parameters go very close to 1. – Sextus Empiricus Aug 23 '18 at 14:43
• If you have some code where you use a custom made function for optimization then you could use functions that compute the logarithms of functions (your formula involves the error function) more efficiently. That is, directly rather than in two steps (1: first compute the function and 2: use that function output as input for the logarithm). In that case you will get less cases of 1's and 0's. – Sextus Empiricus Aug 23 '18 at 14:46
• I see, that makes sense. One last thing. For the solution to the linear regression you say that the x-values of the problem should be the $gEUD$ for each patients, with a fixed $n$. But what about the "y part". Have I stared myself blind on this, or...? Because $t_i$, which seems to be the y-values, well, that is calculated from the parameters I am trying to find, right ? I don't have values for that I can plot and optimize against the x-values (that's the way I'm seeing it now). So am I missing something here, or...? – Denver Dang Aug 23 '18 at 19:14