But Can we Fit Models to Them?Now that we some idea of the trends in admit rate based on GPA and MCAT, we would like to know if we can fit a (not too complex) mathematical model to these trends to predict admit rate for any combination of GPA and MCAT score with a decent degree of accuracy.

Before tackling the whole problem, let’s just see how good we can do at predicting those sigmoids from the last figure.

Here we took the three Admit Rate vs.

MCAT curves from the last section and chose (carefully) some sigmoids to best fit this data.

The sigmoid fits are shown with a red dotted line.

Recall that the three curves represent High, Medium, and Low GPA bands.

We see that at least for the high GPA band, our prediction is almost prefect.

As we get to lower GPA bands, the prediction is not as strong but still captures the dynamics of the data.

Using the fact that our model seems to predict well for higher values of GPA, we will limit our prediction to applicants whose GPA was 3.

1 and higher.

We also limit our prediction to applicants whose MCAT score was 492 or higher, to reduce noise from lower MCAT applicants.

A Full ModelAll the pieces are here.

But we need to put them together in order to create an all-encompassing model which, given your GPA and MCAT score, will give a (hopefully accurate) prediction of your chance of admission to medical school in the United States.

To recap, let’s list out what such a model needs to include:Given some fixed value of GPA, the model needs to be sigmoidal for changing values of MCAT score.

Given some fixed value of GPA, the properties of this sigmoid (slope, offset, etc.

) need to change as a function of GPA.

Given some fixed value of MCAT score, the model needs to be (roughly) linear for changing values of GPA.

The model will be limited to applicants with a GPA of 3.

1 or higher and with an MCAT score of 492 or higher.

Taking all these rules into account, and using mathematical fitting techniques to minimize our error, we arrive at a prediction function of our two interacting variables, GPA and MCAT.

Given our prediction function, we can now pick any set of GPA and MCAT from the valid range and plot the corresponding admit rate surface.

Here the x axis represents GPA, the y axis represents MCAT score, and the z axis represents admit rate.

It turns out that the Root Mean Squared Error (RMSE) from this model is about 3.

8% That is, on average we are 3.

8% away from the true value of admit rate across all GPA and MCAT values.

Not too bad!So How Does a Prospective Med School Student Use This?So we have a fairly accurate model to predict admit rate given just GPA and MCAT score.

How does a student who is trying to optimize her chances of medical school admission use this tool?Let’s say you are a student at some point in your academic track with a GPA of 3.

4 and an MCAT score of 512.

Your natural question should be “Should I dedicate my time to boosting my MCAT or my GPA?” After all, you still only get 24 hours in your day.

A natural way to answer this question would be to focus on the one which gives you the greatest admit rate boost.

If that seems like a strange notion, pretend you are a student in a more extreme situation.

You are a student who has a GPA of 3.

96 and an MCAT score of 485.

It really doesn’t make sense to spend all of your free time studying for exams and trying to boost your GPA when it is already so high.

It especially doesn’t make sense given that your MCAT score is below the median and might seriously impede your chances of admission if you don’t work on it.

So, the question is, how do we measure the admit rate boost from an increase in GPA vs an increase in MCAT score?We can do this pretty easily if we take a look at that surface above and treat it as a kind of ‘hill’.

That is, our goal is to climb higher and higher on that hill since that implies reaching higher and higher chances of admission.

Pretend you’re standing on that hill at some point and want to know the best way to gain some altitude.

You can see what happens to your admit rate by taking a few steps in the GPA direction and then what happens to it when you take a few steps in the MCAT direction.

We just need to be a bit careful here when we say few steps.

We note that GPA has a range from 0.

0 to 4.

0 (length 4) while MCAT has a range from 472 to 528 (length 56).

In lieu of any detailed information about the relative difficulty of GPA gain vs MCAT gain, we make the simplifying assumption of matching a 1 point gain in GPA with a 14 point gain in MCAT (since 56 divided by 4 is 14).

We show the table of predicted admit rate values for some intervals of MCAT and GPA below with relative GPA and MCAT step sizes determined by the scaling factor of roughly 14 discussed above.

Using this table, suppose you have a GPA of 3.

4 and an MCAT score of 512.

What is your best move?.Well, currently, your chance of admission is 47.

35%.

If you increase your GPA by a bit, your chance of admission jumps to 55.

6% while boosting your MCAT a bit boosts it only to 50.

25%.

Thus, you should focus now on your GPA.

Indeed this makes sense since your MCAT score is fairly above the median of 500, but your GPA could still use some work.

We can do this analysis for every cell in the table and reduce that decision to either a right arrow, indicating you should boost your GPA, or a down arrow, indicating you should boost your MCAT.

We get the result below.

Now suppose we are a student with a GPA of 3.

4 and an MCAT of 494.

We can carve out a path to success (higher and higher admit rates) by following the arrows starting at the initial state.

We can do this for any combination of GPA and MCAT score.

Just start at the initial state and follow the arrows to find an optimal path through the GPA — MCAT space.

To recap,We used raw admission statistics to capture trends between GPA, MCAT, and Admit RateWe built an accurate predictive model to extrapolate the admit rate for any combination of GPA and MCAT score (within a specified range)We used this predictive model to suggest a course of action for prospective medical students based on their current standings.

To finish, we show a much more fine grained table of values from the predictive model.

You might need to zoom in!.

Lastly, note that if you look up your GPA and MCAT and find the percentage to be not so high, you don’t necessarily need to stress!.Why?.Because first, this is a model which has some error (3.

8% RMSE).

Second, this doesn’t take into account other admission criteria such as extracurriculars, gender, race, etc.

, which have an impact as well.

So, use this as more of a benchmark than an end all prediction.

Best of luck!Dataset Reference[1] MCAT and GPA Grid for Applicants and Acceptees to U.

S.

Medical Schools (2017), Association of American Medical Colleges.